THREE DIMENSIONAL GEOMETRY
EXERCISE -1
1) The coordinates of a point which divides the line joining the points P(2,3,1) and Q(5,0,4) in the ratio 1:2 are
a) (7/3,1,5/3) b) (4,1,3) c) 3,2,2) d) (1,-1,1)
2) If a line OP through the origin O makes an angle α, 45° and 60° with x, y and z axis respectively then the direction cosines of OP are
a) 1/√2,1/2,1/2 b) 1/2,1/2,1/√2 c) 1/2,1/√2,1/2 d) none
3) The direction cosines of the line joining the points (1,2,-3) and (-2,3,1) are
a) -3,1,4 b) -1,4,-2 c) -3/√26, 1/√26,4/√26, d) -1/√30, 5/√30, -2/√30
4) An equation of z-axis is
a) z=0 , x=0 b) z=0 , y =0 c) x=0 , y=0 d) y =-k , x= k, (k≠0)
5) The ratio in which the yz-plane divides the segment joining the points (-2,4,7) and (3,-5,8) is
a) 2:3 b) 3:2 c) 4:5 d) -7:8
6) The coordinates of a point equidistant from the points (a,0,0),(0,a,0),(0,0,0) are
a) (a/3,a/3, a/3) b) (a/2, a/2, a/2) c) (a,a,a) d) (2a,2a, 2a)
7) If O is the origin and the line OP of length r makes an angle α with x-axis and lies in the xz-plane, then the coordinate of P are
a) (r cosα,0 r sinα) b) (0,0, r sinα) c) (0,0, r cosα) d) (r cosα,0,0)
8) If a line is equally inclined with the coordinates axes, then the angle of inclination is
a) cos⁻¹(1/2) b) cos⁻¹(1/√2) c) cos⁻¹(1/√3) d) cos⁻¹(√3/2)
9) A line makes an angle of 60° with each of x and y axis, the angle which it makes with z-axis is
a) 30° b) 45° c) 60° d) none
10) The projection of a line segment on x, y and z axis are respectively 3,4,12. The length of the line segment is
a) 5 b) 4√10 c) 3√17 d) 13
11) If P(x, y, z) is a point in the space at a distance r from the origin O, then the direction cosines of the lines OP are
a) r/x, r/y, r/z b) rx, ry, rz c) x/r, y/r, z/r d) none
12) Let N be the foot of the perpendicular of length p from the origin to a plane and l, m n be the direction cosines of IN, the equation of the plane is
a) px+ my+ nz= l
b) lx+ py+ nz= m
c) lx+ my+ pz= n
d) lx+ my+ nz= p
13) An equation of the plane passing through the point (1,2,3), the direction cosines of the normal to which are l, m,n is
a) lx+ my+ nz= l+ 2m + 3n
b) (x -1)/l + (y -2)/m + (z -3)/n =0
c) lx+ my+ nz= l
d) lx/1 + my/2 + nz/3 = 0
14) If a plane meets the coordinates axes in A, B, C such that the centroid of the triangle is the point (1, r, r²), the n equation of the plane is
a) x+ ry+ r²z= 3r²
b) r²x+ ry+ z= 3r²
c) x+ ry+ r²z= 3
d) r²x+ ry+ z= 3
15) Algebraic sum of the intercepts made by the plane x+ 3y - 4z + 6=0 on the axes is
a) -13/2 b) 19/2 c) -22/3 d) 26/3
16) An equation of the plane passing through the origin and containing the lines whose direction cosines are proportional to 1,-2,2 and 2,3,-1 is
a) x- 2y +2z =0 b) 2x + 3y -z =0 c) x + 5y - 3z =0 d) 4x- 5y - 7z =0
17) An equation of the plane passing through the point (1,-1,2) and parallel to the plane a) 3x + 4y - 5z +11 =0 b) 3x + 4y - 5z -11 =0 c) 6x + 8y - 10z -1 =0 d) 3x + 4y - 5z = 2
18) Vector equation of the line 6x -2= 3y +1= 2z -1 is
a) r= i - j + 3k + λ(i +2j + 3k)
b) r= i +2j + 3k + λ(i/3 - j/3 + k)
c) r= i/3 - j/3 + k + λ(i +2j + 3k)
d) r= -2i + j -2k + λ(6i +3j + 2k), λ being a parameter
19) Equation of a line through (2,-1,1) and parallel to the line whose equations are
(x -3)/2= (y+1)/7 = (z -2)/-3 are
a) (x -2)/3 = (y+1)/-1 = (z -1)/2
b) (x -2)/2= (y+1)/7 = (z -2)/-3
c) (x -2)/2= (y-7)/-1 = (z +3)/1
d) (x -3)/2= (y+1)/-1 = (z -2)/1
20) If M denotes the midpoint of the line joining A(4i +5 j -10k) and B(- i +2j + k), then equation of the plane through M and perpendicular to AB is
a) r=( -5i -3 j + 11k) + 135/2=0
b) r= 3i.2 + 7j/2 - 9k/2 + 135/2=0
c) r= 4i +5j -10k + 4=0
d) r= -i + 2j +k + 4=0
21) Equation of the plane through (3,4,-1) which is parallel to the plane r. (2i -3j +5k) + 7 =0 is
a) r. (2i -3j +5k) + 11=0
b) r. (3i -3j - k) + 11=0
c) r. (3i + 4j -k) + 7 =0
d) r. (2i -3j +5k) -7=0
22) The ratio in which the plane 2x -1=0 divides the line joining (-2,4,7) and (3,-5,8) is
a) 2:3 b) 4:5 c) 7:8 d) 1:1
23) A line passes through the points (6,-7,-1) and (2,-3,1). If the angle α which the line makes with the positive direction of x-axis is acute, the direction cosines of the line are
a) 2/3,-2/3,-1/3 b) 2/3, 2/3,-1/3 c) 2/3,-2/3, 1/3 d) 2/3, 2/3,1/3
24) The reflection of the point A(1,0,0) in the line (x -1)/2= (y+1)/-3 = (z +10)/8 is
a) (3,-4,-2) b) (5,-8,-4) c) (1,-1,-10) d) (2,-3,8)
25) If r.n= q is the equation of a plane normal to the vector n, the length of the Perpendicular from the origin on the plane is
a) q b) |n| c) q |n| d) q/|n|
26) An equation of the plane passing through the line of intersection of the planes x+ y+ z=6 and 2x+ 3y+ 4z= -5 and passing through (1,1,1) is
a) 2x+ 3y+ 4z= 9 b) x+ y+ z=3 c) x+ 2y+ 3z=6 d) 20x+ 23y+ 26z=69
27) If the foot of the perpendicular from the origin to a plane is (a,b,c), then equation of the plane is
a) x/a + y/b + z/c =1 b) ax+ by+ cz=1 c) ax+ by+ cz=a²+ b²+ c² d) ax+ by+ cz=0
28) If the equations of two lines l₁ and l₂ are given by R= a₁+ λb₁ and r= a₂+ μb₂ where λ , μ are parameters, the angle θ between them is given by
a) cosθ= (a₁.a₂)/|a₁||a₂|
b) cosθ= (b₁.b₂)/|b₁||b₂|
c) cosθ= (a₁.apb₂)/|a₁||b₂|
d) cosθ= (a₂.b₁)/|a₂||b₁|
29) Equation of the plane through three points A, B, C with position vectors -6i+ 3j + 2k, 3i -2j + 4k, 5i+ 7j + 3k is
a) r.(i- j + 7k)+23=0
b) r.(i+ j + 7k) -23=0
c) r.(i + j - 7k)+23=0
d) r.(i- j - 7k)-23=0
30) The lines whose vector equations are r= a+ tb, r= c+ t'd, r= c + t'd are coplanar if
a) (a - b). c x d=0
b) (a - c). b x d=0
c) (b - c). a x d=0
d) (b - d). a x c=0
31) The shortest distance between the skew lines l₁: r= a₁ + λb₁ and l₂: r= a₂+ μb₂ is
a) |(a₂ - a₁).b₁ x b₁b₂|/|b₁xb₂|
b) |(a₁ - b₁).a₂ x b₂|/|b₁ x b₂|
c) |(a₂ - b₂)a₁x b|/|b₁x b₂|
d) |(a₁ - b₂).b₁x a₂|/|b₁x a₂|
32) The length of the shortest distance between the lines r= 3i + 5j + 7k + λ (i - 2j + k) and r= -i - j - k + μ(7i - 6j + k) is
a) 83 b) √6 c) √3 d) 2√29
33) The angle between the lines whose direction cosines are given by the equations l²+ m²- n²=0, l+ m+ n=0 is
a) π/6 b) π/4 c) π/3 d) π/2
34) The volume of the tetrahedron included between the plane, 3x + 4y - 5z- 60=0 and the coordinate planes is
a) 60 b) 600 c) 720 d) none
35) The plane passing through the point (-2,-2,2) and containing the line joining the points (1,1,1) and (1,-1,2) makes intercepts on the coordinates axes, the sum whose lengths is
a) 3 b) 4 c) 6 d) 12
36) A line segment has length 63 and direction ratios are 3,-2,6. If the line makes an obtuse angle with x-axis, the components of the line vector are
a) 27,-18,54 b) -27, 18,54 c) -27, 18, -54 d) 27,-18, -54
37) Equation of a line passing through the point with position vector 2i - 3j + 4k and in the direction of the vector 3i + 4j - 5k is
a) 4x + 3y=17, 5y- 4z =1
b) 4x - 3y=17, 5y +4z =1
c) 4x + 5y=12, 3y + 4z =1
d) 4x + 3z =17, 5y + 4z =1
38) Equation of the plane passing through A (x₁, y₁, z₁) and containing the line (x - x₂)/d₁ = (y - y₂)/d₂ = (z - z₂)/d₃ is
a) |x+ x₁ y+ y₁ z+ z₁
x₂+ x₁ y₂+y₁ z₂+z₁ =0
d₁ d₂ d₃|
b) |x- x₂ y- y₂ z- z₂
x₁-x₂ y₁-y₂ z₁- z₂=0
d₁ d₂ d₃|
c) |x - d₁ y- d₂ z- d₃
x₁ y₁ z₁ =0
x₂ y₂ z₂
d) |x y z
x₁- x₂ y₁- y₂ z₁- z₂ =0
d₁ d₂. d₃|
39) The lines (x -2)/1= (y -3)/1= (z - 4)/-k and (x -1)/k = (y -4)/2 = (z - 5)/1 are coplanar if
a) k=0 b) k=-1 c) k=2 d) k=3
40) The lines r= (i- j ) + λ(2i+ k) and r= (2i- j ) + μ(i+ j - k)
a) intersect each other
b) do not intersect
c) intersect at r= 3i - j + k
d) are parallel
41) Equation of the line of shortest distance between the lines x/2= y/-3 = z/1 and (x-2)/3 = (y-1)/-5 = (z+2)/2 is
a) 3(x -21)= 3y + 92= 3z - 32
b) (x- 62/3)/1/3 = (y-31)/1/3 = (z+ 31/3)/1/3
c) (x-21)/1/3 = (y- 92/3)/1/3 = (z+ 32/3)/1/3
d) (x-2)/1/3 = (y+3)/1/3 = (z-1)/1/3
42) Equation of a plane bisecting the angle between the planes 2x - y + 2z +3=0 and 3x - 2y + 6z +8 =0 is
a) 5x - y - 4z - 45=0
b) 5x - y - 4z - 3=0
c) 23x + 13y + 32z -45=0
d) 23x - 13y + 32z +5=0
43) The lines whose vector equations are
r= 2i - 3j + 7k + λ (2i + pj + 5k)
r= i+ 2j + 3k + μ(3i - pj + pk) are perpendicular for all values of λ and μ if
a) p= -6 b) p= 0 c) p= 1 d) p= 6
44) An equation of the line passing through 3i - 5j + 7k and perpendicular to the plane 3x - 4y + 5z =8 is
a) (x-3)/3 = (y+5)/-4 = (z-7)/5
b) (x-3)/3 = (y+4)/-5 = (z-5)/7
c) r= 3i + 5j - 7k + λ (3i -4j - 5k)
d) r= 3i - 4j - 5k + μ(3i + 5j + 7k) ( λ , μ are parameter)
45) An plane x - 2y + 7z +21=0 contains the line
a) (x-1)/-3 = (y-3)/2 = (z+2)/1
b) (x+1)/-3 = (y+3)/2 = (z+2)/1
c) (x+1)/-3 = (y-3)/2 = (z+2)/1
d) x/1 = y/-2 = z/7
46) If the perpendicular distance of a point P other than the origin from the plane x+ y+ z = p is equal to the distance of the plane from the origin, then the coordinates of P are
a) (p,2p,0) b) (0,2p, -p) c) (2p, p, -p) d) (2p, -p, 2p)
47) The coordinates of a point on the line (x-1)/2 = (y +1)/-3 = z at a distance 4√14 from the point (1,-1,0) nearer the origin are
a) (9,-13,4) b) (8√14+1, -12√14-1, 4√14)
c) (-7,11,-4) d) (-8√14+1, 12√14-1, -4√14)
48) If p₁, p₂, p₃ denote the distance of the plane 2x - 3y + 4z +2=0 from the plane 2x - 3y + 4z +6 =0, 4x - 6y + 8z +3 =0 and 2x - 3y + 4z - 6 =0 respectively , then
a) p₁ + 8p₂ - p₃ = 0
b) p₃² = 16p₂²
c) 8p₂² = p₁²
d) p₁ + 2p₂ + 3p₃= √29
49) Equation of the plane containing the lines
r= i+ 2j - k + λ (i + 2j - k)
and i+ 2j - k + μ (i + j + 3k) is
a) r. (7i - 4j - k)=0
b) 7(x -1)- 4(y -1)- (z +3)=0
c) r. (i+ 2j - k)= 0
d) r. (i+ 2j + 3k)= 0
50) The foot of the perpendicular from the origin to the join of A(-9,4,5) and B(11,0,-1) divides AB in the ratio
a) 2:3 b) 3:2 c) 1:1 d) none
51) Cosine of the angle between the lines whose vector equations are r= r= 3i+ 2j - 4k + λ (i + 2j + 2k) and 4= 5i+ - 2k + μ (3i + 2j + 6k) ; λ, μ being parameter, is
a) -1/3√29 b) 3/7√29 c) 23/29 d) 19/21
52) The cartesian equation of the plane passing through the line of intersection of the planes r. (2i - 3j + 4k)=1 and r. (i - j)+ 4 =0 and perpendicular to the plane r. (2i - j + k)+ 8 = 0 is
a) 3x - 4y + 4z= 5 b) x - 2y + 4z= 3 c) 5x - 2y - 12z +47= 0 d) 2x +3y + 4= 0
53) The shortest distances between the lines
(x -3)/2= (y+15)/-7= (z -9)/5 and (x +1)/2= (y-1)/1 = (z -9)/-3 is
a) 4√5 b) 4√17 c) 4√3 d) 8√2
54) Equation of the line through the point (2,-1,1) and the intersection of the lines 2x - y -4=0= y -+2z, x +3z -4=0= 2x + 5z -8 is
a) x + y+ z -2=0, x+2y =0
b) x + y+ z= 2, x +2z -4=0
c) x + 2y+ z -1=0, x+2z =4
b) x + 2y+ z= 1, x +2y=0
55) The planes bx - ay = n, cy - bz = l, az - cx = m intersect in a line if
a) al - bm + cn = 1
b) al + bm + cn = 0
c) al - bm - cn = -1
d) al + bm + cn = 1
PAPER-1
1) A(1,-1,-3), B(2,1,-2) & C(-5,2,-6) are the position vectors of the vertices of a triangle ABC. The length of the bisectors of its internal angle at A is
a) √10/4 b) 3√10/4 c) √10 d) none
2) Let P is the P. V of the orthocentre and g is the p.v of the centroid of the triangle ABC where circumcentre is the origin. If p= K g, then K=
a) 3 b) 2 c) 1/3 d) 2/3
3) A vector 'a' has components 2p and 1 with respect to a rectangular cartesian atsr. The system is rotated through a certain angle about the origin in the counter clockwise sense. If with respect to the new system, a has components p+ 1 and 1 then,
a) p= 0 b) p=1 or p= -1/3 c) p= -1 or p= 1/3 d) p= 1 or p= -1
4) The number of vectors of unit length perpendicular to vectors a= (1,1,0) and b= (0,1,1) is:
a) 1 b) 2 c) 3 d) ∞
5) Four points A(1,-1,1); B(1,3,1); C(4,3,1) and D(4,-1,1) taken in order are the vertices of
(A) a parallelogram which is neither a rectangle nor a rhombus.
B) rhombus
c) an isosceles trapezium
d) a cyclic quadrilateral
6) Let α, β and γ be distinct real numbers. The points whose position vector's are αi + βj + γk; βi + γj + αk and γi +αj + βk
a) are collinear
b) form an equilateral r
c) form a scalene triangle
d) form a right angled triangle
7) If the vector a= 3i + j - 2k, b= - i + 3j + 4k and c= 4i - wj - 6k constitute the sides of a ∆ ABC, then the length of the median bisecting the vector c is
a) √2 b) √14 c) √74 d) √6
8) Let A(0,-1,1), B(0,0,1), C(1,0,1) are the vertices of a ∆ ABC. If R and r denotes the circumradius and in-radius of ∆ ABC, then r/R has value equal to
a) tan(3π/8) b) cot(3π/8) c) tan(π/12) d) cot(π/12)
9) A vector of magnitude 10 along the normal to the curve 3x²+ 8xy + 2y² -3= 0 at its point P(1,0) can be
a) 6i+ 8j b) -6i+ 8j c) 6i- 8j d) - 6i - 8j
10) Let OAB be a regular triangle with side length unity (O) being the origin). Also M, N are the points of tricsection of AB, M being closer to A and N closer to B. Position vectors of A, B, M and N are a, b, m and n respectively. Which of the following hold/s good?
a) m= xa + yb => 2/3 and y= 1/3
b) m= xa + yb => x= 5/6 and y= 1/6
c) m,p.n equals 13/18
d) m.n equals to 15/18
11) a, b, c are three non-zero vectors, no two of which are collinear and the vector a+ b is collinear with c, b+ c is collinear with a, then a+ b+ c is equal to
a) a b) b c) c d) none
12) Given the vector PQ= -6i - 4j and Q is the point (3,3), find the point P.
13) Find the unit vector (in xy plane) obtained by rotating j counter clockwise 3π/4 radian about the origin.
14) Show that the vector v= ai + bj is perpendicular to the line ax+ by = c.
15) If A(0,1, 0), B(0,0,0), C(1,0,1) are the vertices of a ∆ ABC. Match the entries of columns
Column I
A) Orthocentre of ∆ ABC
B) circumcentre of ∆ ABC
C) Area (∆ ABC)
D) Distance between orthocentre and centroid
E) Distance between orthocentre and circumcenter
F) Distance between circumcentre and centroid
G) Incentre of ∆ ABC
H) Centroid of ∆ ABC
Column II
P) √2/2
Q) √3/2
R) √3/3
S) √3/6
T) (0,0,0)
U) 1/2,1/2,1.2)
V) 1/3,1/3,1/3)
W) 1/(√1+√2+√3), √2/(√1+√2+√3), 1/(√1+√2+√3).
SCALAR OR DOT PRODUCT
EXERCISE - A
1) Find a.b when
i) a= 2i + 2j - k and b = 6i - 3j +2k
ii) a= (1,1,2) and b= (3,2,-1)
2) Find (a+ 3b).(2a - b), if a= i + j +2k and b = 3i + 2j - k.
3) Find the value of M so that the vector a= 2i + Mj +k and b = i - 2j +3k.
4) Find the value of p for which the vectors a= 3i + 2j +9k and b = i + pj + 3k are
i) parallel
ii) perpendicular
5) Find the angle between two vectors a and b with magnitude √3 and 2 respectively, and such that a.b=√6.
6) Find the angle between two vectors a and b having the same length √2 and their scalar product is -1.
7) Find the angle between the vectors 5i + 3j + 4k and 6i - 8j - k.
8) If i + j + k, 2i + 5j, 3i + 2j -3k and i -6j - k respectively are the position vector of points A, B, C and D, then find the angle between the straight lines AB and CD. Deduce that AB and CD are collinear.
9) Find the projection of the vector 7i + j - 4k on 2i + 6j + 3k.
10) let a and b be two vectors of the same magnitude such that the angle between them is 60° and a.b=8. Find |a| and |b|.
11) For any vector r, show that r= (r. i)i + (r.j)j + (r.k)k.
12) Let a= 4i + 5j -k, b = i - 4j + 5k and c = 3i + j - k. Find a vector d which is perpendicular to both a and b and satisfying d.c= 21.
13) Dot products of a vector with vector 3i - 5k, 2i + 7j and i + j + k are respectively -1,6 and 5. Find the vector.
14) Show that the projection vector of a on b (≠ 0) (component of a along b) is {(a.b)/|b|²}b.
15) Show that the projection vector of b on a≠ 0 is {(a.b)/|a|²}a.
16) For any two vectors a and b, prove that:
a) |a+ b|= |a|²+ |b|²+ 2 a.b
b) |a- b|= |a|²+ |b|²- 2 a.b
c) |a+ b|+ |a- b|²= 2(|a|²+ |b|²)
d) |a+ b|= |a - b| <=> a perpendicular to b.
e) |a+ b|= |a| + |b| <=> a is parallel to b.
f) |a+ b|² = |a|²+ |b|² <=> a, b are orthogonal.
17) If two a and b are such that |a|= 2, |b|=1 and a.b=1, find (3a - 5b). (2a+ 7b).
18) Find |x|, if for a unit vector a, (x - a). (x + a)= 15.
19) Find|a| and |b|, if (a - b). (a+ b)= 27 and |a|= 2 |b|.
20) If a= 5i - j - 3k and b = i +3j - 5k, then show that the Vectors a+ b and a - b are perpendicular.
21) If two vectors a and b are such that |a|= 3, |b|= 2 and a.b=6. Find |a+ b| and |a - b|.
22) Two vectors a and b, show that the vector |a| b + |b| a is orthogonal to the vector. | a| b - |b| a.
23) If a+ b + c=0, |a|= 3, |b|= 5 and |c|= 7, find the angle between a and b.
24) If a and b are unit vectors inclined at an angle θ, then show that sin(θ/2)= (1/2) |a - b|.
25) For any two vectors a and b, prove that
|a+ b|≤ |a|+ |b|.
26) (Cauchy-Schawarz inequality) for any two vectors a and b, show that
(a.b)²≤ |a|² |b|².
And hence show that
(a₁ b₁+ a₂b₂ + a₃b₃)² ≤ (a₁² + a₂² + a₃²)(b₁² + b₂² + b₃²).
27) If a, b are two vectors such that |a+ b|= |a|, then show that 2a+ b is perpendicular to b.
28) If a makes equal angles with i, j and k and has magnitude 3, then show that the angle between a and each of i, j and k is cos⁻¹(1/√3).
29) If a unit vector a makes angle π/4 with i, π/3 with j and an acute angle θ with k, then find the components of a and the angle θ.
30) Find the components of a unit which is perpendicular to the vectors i+ 2j - k and 3i - j +2k.
31) The scalar product of the vector a= i+ j + k with a unit vector along the sum of the vectors b= 2i+ 4j - 5k and c= Mi+ 2j + 3k is equal to 1. Find the value of M and hence find the unit vector b + c.
32) If a, b, c are three vectors such that a.b= a.c, then show that a=0 or, b= c or, a perpendicular to (b - c).
33) Show that the vectors 2i - j + k, i - 3j -5k, 3i - 4j - 4k form the sides of a right angled triangle.
34) Show that the points A, B, C with position vectors 2i - j + k, i - 3j - 5k and 3i - 4j - 4k respectively, are the vertices of a right angled triangle. Also, find the remaining angles of the triangle.
35) Show that the angle between two diagonals of a cube is cos⁻¹(1/3).
36) If with reference to a right handed system of mutually perpendicular unit vectors i, j, k, we have α = 3i - j, and β= 2i + j - 3k. Express β in the form β= β₁ + β₂, where β₁ is parallel to α and β₂ is perpendicular to α.
37) Find the values of x for which the angle between the vectors a= 2x²i + 4xj + k and b= 7i - 2j + xk is obtuse.
38) If l, m, n are scalar and a, b, c are vectors, show that | la + mb + nc|²= l² |a|²+ m²| |b|²+ n² |c|²+ 2{lm(a.b)+ mn(b.c)+ nl(c.a)}.
Also reduce that
|la + mb + nc|²= l² |a|²+ m²|b|²+ n² |c|² if a, b, c are mutually perpendicular vectors.
39) For any three vectors a, b, c, show that
|a+ b + c|²= |a|²+ |b|²+ |c|²+ 2(a.b + b.c + c.a).
40) If a, b, c are three mutually perpendicular vectors of equal magnitude, show that a+ b+ c is equally inclined with vectors a, b and c. Also, find the angle.
41) Let a, b, c be three vectors of magnitude 3, 4 and 5 respectively. If each one is perpendicular to the sum of the other two vectors, prove that |a+ b+ c|= 5√2./
42) If a, b, c are unit vectors such that a+ npb+ c =0. Find the value of a.b + b.c + c.a.
43) Three vectors a, b and c satisfy the condition a+ b+ c =0. Evaluate the quantity β= a.b + b.c + c.a, if |a|= 1, |b|= 4 and|c|= 2.
44) If a, b, c are mutually perpendicular unit vectors, find |2a + b + c|.
45) Find the values of c for which the vectors a= (c log₂x)i - 6j + 3k and b= (log₂x)i + 2j + (2c log₂x)k make an obtuse angle for any x ∈ (0,∞).
46) Find the values of 'a' for which the vector r= (a²- 4)i + 2j - (a²- 9)k makes acute angles with the coordinates axes.
47) If a, b, c are unit vectors, prove that|a - b|²+ |b - c|²+ |c - a|²≤ 9.
48) Let a, b, c be three vectors such that |a|= 1, |b|= 2 and |c|= 3. If the projection of b along a is equal to the projection of c along a and b, c are perpendicular to each other, find |3a - 2b + 2c|.
EXERCISE -B
1) If a, b, c are the lengths of the sides opposite respectively to the angles A, B and C of a triangle ABC, show that
a) cosA= (b²+ c²- a²)/2bc
b) cosB = (c²+ a²- b²)/2ac
c) cosC = (a²+ b²- c²)/2ab.
2) If a, b, c are the lengths of the sides opposite respectively to the angles A, B and C of a triangle ABC, show that
a) a= b cosC + c cosB
b) b= c cosA + a cosC
c) c= a cosB + b cosA
3) Prove using vectors: The median to the base of an isosceles triangle is perpendicular to the base.
4) Prove using vectors: if two medians of a triangle are equal, then it is isosceles.
5) Prove that the midpoint of the hypotenuse of a right angled triangle is equidistant from its vertices.
6) Prove that the altitude of a triangle are concurrent.
7) Prove that the perpendicular bisectors of the sides of a triangle are concurrent.
8) Prove using vectors: if the diagonals of a parallelogram are equal in length, then it is a rectangle.
9) Show that the diagonals of a rhombus bisect each other at right angles.
10) Using vector method, show that the angle in a semi-circle is a right angle.
11) Using vector ps: prove that
a) cos(A + B)= cosA cosB - sinA sinB.
b) cos(A - B)= cosA cosB + sinA sinB.
12) Prove that the cosine formula for triangles is equidistant to the definition of the scalar product.
VECTOR OR CROSS PRODUCT
PLANE
1. A first degree equation in x, y, z represents a plane
The most general equation of the first degree is
ax+ by + cz + d=0
Where a, b, c, d are real constants and a²+ b²+ c²≠ 0
Cor. 1
Equation of a plane passing through the point (x₁, y₁, z₁) is a a(x - x₁)+ b(y - y₁)+ c(z - z₁) =0.
Cor. 2
(a₁x+ b₁y + c₁z + d₁ = 0 and a₂x + b₂y + c₂z + d₂= 0
will represent the same plane, if and only if
a₁/a₂ = b₁/b₂ = c₁/c₂ = d₁/d₂ = λ (≠ 0 and real).
Cor. 3
Equation of the coordinates planes are x=0, y=0, z=0.
Cor. 4
Equation of the planes parallel to the coordinates planes are x= conste, y= constant, z= constant.
These planes are perpendicular to the axes.
Equation of the planes parallel to the x-, y- and z-ax hues are respectively of the form by+ cz + d=0, ax+ cz + d=0 and ax+ by + d=0. These planes are perpendicular to the yz-plane, zx- plane and xy-plane respectively.
Cor. 5
i) If d=0, then the plane passes through the origin.
ii) if a=0, then the plane is parallel to the x-axis .
iii) if a= b=0, then the plane is parallel to the xy-plane.
iv) if a= b = c = 9 but d is finite , then the plane is at an infinite distance; since, in that case, the x-axis (y= z = 0) meets the plane ax+ by + cz+ d=0 at y=0, z=0, ax+ d= 0.
The x-coordinate the point of intersection is given by (-d/a).
if now a be infinitely diminished while d remains finite , then the x-axis cuts the plane at an infinite distance. The same being true for other axes. Hence the plane is at an infinite distance from the origin.
Note:
The general equation of the plane ax+ by + cz+ d=0 can be put in the form (a/d)x + (b/d)y + (c/d)z = -1 having only 3 arbitrary constantr, (a/d, b/d, c/d). Hence a plane can be found to satisfy three conditions . On the other hand, if three conditions be given, then a plane can be found completely.
2. Equations Planes in different forms:
a) Normal form:
The equation of the plane: lx + my + nz= p.
If the normal makes an angles α, β, γ with the axes of x, y, z respectively, then the equation of the planes is
x cosα + y cos+ z cosγ= p.
Cor. 1
By Comparing the general equation the first degree ax+ by + cz+ d= 0 with lx + my+ nz= p, we see that the direction cosines of the normal to the plane given by the general equation ax+ by + cz+ d= 0 are proportional to a,b,c and hence are equal to
a/√(a²+ b²+ c²), b/√(a²+ b²+ c²), c/√(a²+ b²+ c²).
Also the length of the perpendicular from the origin to the plane is
s/√(a²+ b²+ c²),
The sign with the radical is taken to be positive or negative according as d is negative or positive, because p is taken to be positively by convention.
The general equation of the plane is reduced to the normal form by dividing it by (-√(a²+ b²+ c²), if d be positive or by √(a²+ b²+ c²), if d be negative.
Cor. 2
The equation a(x- x₁)+ b(y - y₁)+ c(z - z₁)= 0 representa plane through the point (x₁, y₁, z₁), where a, b, c are the direction ratios of the normal to the plane.
b) intercept form:
Let the equation of the plane be
Ax+ By+ Cz + D=0. (1)
let it make intercepts a,b,c on the coordinate Axes . then the points (a,0,0), (0,b,0), (0,0,c) lie on the plane and their coordinates must satisfy the equation. (1).
Therefore Aa + D= 0, Bb + D = 0, Cc + D= 0
Hence A= -D/a, B= - D/b, C= - D/c,
Putting these values in (1), we get
x/a + y/b + z/c = 1, (none of a,b,c is zero) as the equation of the plane.
3. Planes passing three given points
The general equation of a plane contains only 3 independant constants . Hence a plane can be made to pass through three non collinear points.
Let, ax+ by + cz+ d= 0 .....(1)
be the equation of the plane passing through the points (x₁, y₁, z₁), (x₂, y₂, z₂) and (x₃, y₃, z₃).
Since each point lies on the plane, its coordinates must satisfy the equation (1), that is
ax₁ + by₁ + cz₁ + d= 0. (2)
ax₂ + by₂+ cz₂ + d=0. (3)
ax₃+ by₃+ cz₃ + d=0. (4)
Eliminating a, b, c, d from these four eqs (1),(2),(3),(4) we get the required equation of the plane as
x y z 1 = 0
x₁ y₁ z₁ 1
y₂ y₂ z₂ 1
x₃ y₃ z₃ 1
Cor.
The condition that the four points (x₁, y₁, z₁), i= 1,2,3,4, are coplanar is
x₁ y₁ z₁ 1=0
x₂ y₂ z₂ 1
x₃ y₃ z₃ 1
x₄ y₄ z₄ 1
This condition is necessary as well as sufficient.
4. Angle between two planes.
Let two planes be a₁x + b₁y + c₁z + d₁=0 and a₂x + b₂y + c₂z + d₂ =0.
Let θ be the angle between the two planes. Then this must be the angle between their normals.
The direction ratios of the normals of the two planes are (a₁, b₁, c₁) and (a₂, b₂, c₂). Hence, if θ be the angles between them, then
Cosθ = (a₁a₂ + b₁b₂ + c₁c₂)/(√(a₁²+ b₁²+ c₁²). √(a₂²+ b₂²+ c₂²)).
Cor .1
If the two planes be perpendicular to each other, so are their normals whose direction cosines are proportional to (a₁, b₁, c₁) and (a₂, b, c₂). Hence the condition perpendicularity of the two planes is
a₁a₂ + b₁b₂ + c₁c₂ = 0
Cor. 2
if the two planes be parar, so are their normals. Hence the condition of parallelism of the two planes is
a₁/a₂ = b₁/b₂ = c₁/c₂.
From this, we can easily conclude that the equations of two parallel planes differ only by a constant.
Thus any plane parallel to the plane ax+ by+ cz + d=0 is ax+ by+ cz + k =0, where k is constant.
Exercise - A
1) Find the equations of the three planes through the points (3,1,1),(1,-2,3) parallel to the coordinates axes. 2y+3z= 5, x+ z=4, 3x -2y=7
2) Find the intercepts made on the coordinates axes by the plane x+ 2y - 2z = 6. Find also the direction cosines of the normal to the plane. (6,3,-3), x/3+2y/3-2z/3= 2, (1/3,2/3,-2/3)
3) Find the equation of the plane passing through the three points (2,2,-1),(3,4,2),(7,0,6). 5x+ 2y - 3z -17=0
4) Show that the point O(-1/2,2,0) is the circumcentre of the triangle formed by the points P(1,1,0), Q(1,2,1) and R(-2,2,-1).
5) Find the equation of the plane through the point (x₁, y₁, z₁) parallel to the plane ax+ by + cz=0. a(x - x₁)+ b(y - y₁)+ c(z - z₁) =0
6) Find the equation of the plane which passes through the point (2,1,-1) and is orthogonal to each of the planes x- y+z=1 and 3x + 4y-2z=0. 2x - 5y - 7z - 6 =0.
7) Find the equation of the plane passing through the points (1,1,2) and (2,4,3) and perpendicular to the plane x- 3y + 7z +5=0. 4x - y - z-1=0
8) If P be the point (2,3,-1), then find the equation of the plane through P at right angles to the straight line OP, where O is the origin. 2x + 3y - z= 14.
Exercise - B
1)a) Find the equation of the plane through the points (2,3,-4),(1,-1,3) and parallel to the x-axis. 7y + 4z- 5=0
b) Show that the intercepts made on the axes by the plane 4x +3y - 2z+ 12 =0 are (-3),(-4),(6).
c) Find the points where the plane ax+ by + cz+ d=0 (a,b,c≠0) meets the coordinates. (-d/a,0,0), (0,-d/b,0),+0,0,-d/c)
d) Does the point (4,-6,0) lies on the plane which intersects the positive x, y, z-axis at a distance 2,3,5 units respectively. No
2) a) A plane makes equal nonzero intercepts on the axes measured from the origin and passes through the point (1,2,3). Show that its equation is x + y + z - 6=0.
b) A plane passes through the point (-2,3,1) and its intercepts measured from the origin on the x and y axis are 4 and 3 respectively. Find the equation of the plane. 3x + 4y + 6z-12 =0
c) If the sum of the reciprocals of intercepts on the coordinates axes of a plane be constant, then show that the plane always passes through a fixed point.
3) a) The foot of the perpendicular from the origin to the plane is (3,2,-1). Show that the equation of the plane is 3x +2y - z-14 =0
b) If P be the point (-3,1,1) and Q be the point (3,4,2), then the equation of the plane through R at right angles to PQ, where R lies on PQ and PR= (1/3) PQ. 3x +4y + 6z-12 =0
c) Show that the plane bisecting the straight line joining the points (-1,2,3) and (3,-5,6) at right angles is 4x - 7y + 3z- 28=0
d) Find the equation of the plane through the point (2,-3,1) and which is perpendicular to the straight line joining the point (2,-3,1) and which is perpendicular to the straight line joining the two points (3,4,-1) and (2,-1,5). x +5y - 6z + 19 =0
4) Show that the equation of the plane through the point P(a,b,c) and perpendicular to the straight line OP, where O is the origin, is ax+ by + cz= a²+ b²+ c².
5) Reduce the equation of the plane 6x - 3y +2z -14 =0 to the normal form and show that the distance of the origin from the plane is 2 units and the direction cosines of the perpendicular from the origin to the plane are 6/7, (-3/7), 2/7.
6) a) Show that the equation of the plane through the points
i) (2,3,-3),(1,1,-2) and (-1,1,4) is 3x - y + z =0
ii) (2,2,2(3,1,1) and (6,-4,-6) is x +2y - z- 4 =0
iii) (3,3,1),(-3,2,-1) and (8,6,3) is 4x +2y - 13z-5 =0
b) Show that the points (3,9,4), (4,5,1),(-4,4,4) and (0,-1,-1) are coplanar.
c) Show that the points (2,1,-2) lies on the plane passing through the points (1,0,0),(0,1,0),(0,0,1).
7) a) Show that the equation of the plane through the point
i) (1,2,3) and parallel to the plane 3x +4y -5 z =0 is 3x +4y - 5z + 4 =0.
ii) (0,4,-3) and parallel to the plane 3x - 4y + 7z + 3 =0 is 3x - 4y + 7z+ 37 =0
iii) (2,-3,5) and parallel to the yz plane is x= 2.
b) Show that the equation of the plane parallel to the plane 2x +4y + 5z- 6 =0 and the sum of whose intercepts on the coordinates axes is 19, is 2x +4y + 5z- 20=0
8) Determine the values of K and L for which the two planes Kx +y - 2z + 4 =0 and 6x - My - 4z- 9 =0 are parallel. 3,-2
9) Determine the value of h for which the planes 3x - 2y + hz-1 =0 and x +hy + 5z +2 =0 may be perpendicular to each other. -1
10) Show that the angle between the planes
i) x - y +2z- 9=0 and 2x + y + z- 7 =0 is π/3.
ii) 2x - y +2z- 3 =0 and 3x +6 y +2z-4 =0 is cos⁻¹(4/21).
b) Find the angles that the plane x +8y - 6z + 16=0 makes with the cordinates planes. cos⁻¹(1/√101),cos⁻¹(3/√101), cos⁻¹(6/√101)
11) Show that the equations of the planes passing through the points (0,4,-3), (6,-4,3) and cutting off intercepts from the axes whose sum is zero, are 2x -3y - 6z - 6=0 and 6x +3y - 2z - 18=0.
12) Show that the equation of the plane through the points
i) (1,2,3), (3,2,-1) and
11) Show that the equations of the planes passing through the points (0, 4,-3), (6,-4,3) and cutting off intercepts from the axes whose sum is zero, are 2x -3y - 6z - 6=0 and 6x +3y - 2z - 18 =0.
12) Show that the equation of the plane through the points
i) (1,2,3), (3,2,-1) and perpendicular to the plane 3x +2y +6z - 4 =0 is 2x -6y + z +7=0,
ii) (0,2,3), (5,-1,4) and perpendicular to the plane x +2y +3=0 is 2x - y - 13z +4=0.
iii) (2,1,1), (3,2,2) and perpendicular to the plane x +2y - 5z - 3=0 is 7x -6y - z - 7=0.
13) Show that the equation of the plane passing through the point (-1,3,2) and perpendicular to the planes x +2y +2z - 5 =0 and 3x + 3y +2z - 8=0 is 2x -4y +3z - 8=0.
14) Find the equation of the plane which passes through the point (2,1,4) and is perpendicular to each of the planes 9x -7y + 6z +48=0 and x + y - z =0. x +15y +16z - 81=0
15) Show that the planes x +2y + 2z =0 and 2x +y -2z=0 are orthogonal. Find another plane through the origin which is perpendicular to each of the above planes. 2x - 2y + z= 0
16) prove that the equation of the plane through the points
i) (1,-2,4) and (3,-4,5) and parallel to the x-axis is y + 2z - 6=0
ii) (3,-1,2) and (2,1,-4) and perpendicular to the xz-plane is 6x -z -16=0 .
iii) (2,-3,4) and (-1,5,2) and perpendicular to the xy-plane is 8x +3y -7=0.
17) Perpendiculars OL, PM, PN are drawn from the point P(a,b,c) to the coordinates plane. Show that the equation of the plane LMN is x/a + y/b + z/c = 2.
18) A plane cuts the axes in A, B, C and the centroud of the triangle ABC is (a,b,c). Show that the equation of the plane is x/a + y/b + z/c = 3.
19) A variable plane which is at a constant distance 3p from the origin O cuts the axes in A, B, C. Show that
i) the locus of the centroid of the triangle ABC is
x⁻² + y⁻² + z⁻² = p⁻² and that of the tetrahedron OABC is 9(x⁻² + y⁻² + z⁻²) = 16p⁻².
ii) the locus of the point of intersection of the planes through A, B, C drawn parallel to the coordinate planes is 8(x⁻² + y⁻² + z⁻² = p⁻².
20) A variable plane passes through the point (f,g,h) and meets the axes in A, B, C. If the planes through A, B, C and parallel to the axes meet at P, then show that the locus of P is fx⁻¹ + gy⁻¹ + hz⁻¹ = 1.
21) A variable plane has intercepts on the co-ordinate axes, the sum of whose squares is a constant k². Show that the locus of the foot of the perpendicular from the origin to the plane is (x²+ y²+ z²)²(x⁻² + y⁻² + z⁻²) = k².
22) A point P moves on the plane x/a + y/b + z/c = 1, which is fixed and the plane through P perpendicular to OP meets the axes in A, B, C. if the planes through A, B, C parallel to the co-ordinate planes meet in a point Q. then show that the locus of Q is
1/x² + 1/y² + 1/z²= 1/ax + 1/by + 1/cz.
(the equation of the plane perpendicular to OP where P is the point
PAPER-3
1) If a+ b+ c= 0, |a|= 3, |b|= 5, |c|= 7, then the angle between a and b is
a) π/6 b) 2π/3 c) 5π/3 d) π/3
2) A line passes through the point A(i + 2j + 3k) and is parallel to the vector V(i + j + k). The shortest distance from the origin, of the line is
a) √2 b) √4 c) √5 d) √6
3) Let a, b, x be vectors of length 3, 4, 5 respectively. Let a be perpendicular to b+ c, b to c+ a and c to a+ b. Then |a+ b+ c| is
a) 2√5 b) 2√2 c) 10√5 d) 5√2
4) Given a parallelogram ABCD . If | AB|= a, | AD|= b & | AC|= c, then DB, AB has the value
a) (3a²+ b²- c²)/2
b) (a²+ 3b²- c²)/2
c) (a²- b²+ 3c²)/2 d) none
5) The set of values of x for which the angle between the vectors a= xi - 3j - k and b= 2xi + xj - k acute and the angles between the vector b and the axis of ordinates is obtuse, is
a) 1< x < 2 b) x> 2 c) x <1 d) x < 0
6) If a vector 'a' of magnitude 50 is collinear with vector b= 6i - 8j + 15k/2 and makes an acute angle with positive z-axis then :
a) a= 4b b) a= - 4b c) b= 4a d) none
7) A, B, C, D are 4 points in a plane with p.v's a,b,c,d respectively such that (a - d). (b - c)= (b - d). (c - a)= 0. Then for the triangle ABC , D is its
a) incentre b) circumcenter c) orthocentre d) centroid
8) a and b are unit vectors inclined to each other at an α, α∈(0,π) and |a+ b|< 1. Then α∈
a) (π/3,2π/3) b) (2π/3,π) c) (0,π/3) d) (π/4,3π/4)
9) Image of the point P with the position vector 7i - j +2k in the line whose vector equation is, r= (9i + 5j + 5k) + K(i + 3j + 5k) has the position vector
a) (- 9,5,2) b) (9,5,-2) c) (9,-5,-2) d) none
10) Let a, b, c are three unit vectors such that a+ b+ c is also a unit vector. If pairwise angles between a, b, c are θ₁, θ₂, and θ₃ respectively then cosθ₁ + cosθ₂+ cosθ₃ equals
a) 3 b) -3 c) 1 d) -1
11) A tangent is drawn to the curve y= 8/x² at a point A(x₁, y₁), where x₁= 2. The tangent cuts the x-axis at point B. Then the scalar product of the vectors AB and OB is
a) 2 b) -3 c) 6 d) -6
12) L₁ and L₂ are two lines whose vector equations are
L₁: r= λ[cosθ + √3)i + (√2 sinθ)j + (cosθ - √3)k]
L₂: r= (ai + bj + ck), where μ, λ are scalar and α is the acute angle between L₁, L₂.
If the angle α is independent of θ then the value of α is
a) π/6 b) π/4 c) π/3 d) π/2
1d 2a 3d 4a 5d 6b 7c 8b 9b 10d 11a 12a
Paper-2
1) If the three points with position vectors (1,a,b); (a,2,b) and (a, b, 3) are collinear in space, then the value of a+ b is
a) 3 b) 4 c) 5 d) none
2) Consider the following 3 lines in space
L₁: r= 3i - j + 2k +λ(2i + 4j - k)
L₂: r= i + j - 3k + μ(4i + 2j + 4k)
L₃: r= 3i + 2j - 2k + t(2i + j + 2k)
Then which one of the following part/s are in the same plane.
a) only L₁L₂ b) only L₂L₃ c) only L₃L₁ d) L₁L₂ & L₂L₃
3) The cute angle between the medians drawn from the acute angles of an isosceles right angle triangle is
a) cos⁻¹(2/3) b) cos⁻¹(3/4) c) cos⁻¹(4/5) d) none
4) The vectors 3i - 2j +k, i - 3j + 5k & 2i + j - 4k form the sides of a triangle. Then triangle is
a) an acute angled triangle
b) an obtuse angled triangle
c) an equilateral triangle
d) a right angled triangle
5) if the vectors 3p+ q; 5p - 3q and 2p+ q, 4p - 2q are pairs of mutually perpendicular vector then sin(p q) is
a) √55/4 b) √55/8 c) 3/16 d) √247/16
6) Consider the points A, B and C with position vectors (-2i + 3j +5k), (i + 2j +3k) and 7i -k respectively.
Statement 1: The vector sum AB+ BC+ CA=0
because
Statement 2: A, B and C form the vertices of a triangle.
i) Statement 1 is true, Statement 2 is true and Statement 2 is correct explanation for statement 1.
ii) Statement 1 is true, Statement 2 is true and Statement 2 is NOT the correct explanation for statement 1
iii) Statement 1 is true, Statement 2 is false
iv) Statement 1 is false, Statement 2 is true
7) The set of the values of c for which the angle between the vector cxi - 6j +3k & xi - 2j + 2cxk + is acute for x ∈R is
a) (0,4/3) b) [0,4/3] c) (11/9,4/3) d) [0,4/3)
8) Let u= i + j , v= i - j and w= i +2j + 3k . If n is a unit vector such that u. n =0 and v.n=0, then |w.n| is equal to
a) 1 b) 2 c) 3 d) 0
9) If the vector 6i -3j -6k is decomposed into vectors parallel and perpendicular to the vector i +j + k then the vectors are :
a) -(i +j + k) & 7i - 2j - 5k
b) -2(i +j + k) & 8i - j - 4k
c) 2(i +j + k) & 4i - 5j - 8k d) none
10) Let r= a+ λl and r= b + μm be two lines in space where a= 5i +j + 2k, b= -i + 7j + 8k , l= -4i +j - k) & m= 2i - 5j - 7k then the p.v of a point which lies on both of these lines, is
a) i + 2j + k b) 2i +j +k c) i +j + 2k d) non existent as the lines are skew
11) Let A(1,2,3), B(0,0,1), C(-1,1,1) are the vertices of a ∆ ABC
i) The equation of internal angle bisector through A to side BC is
a) r= i + 2j + 3k + μ(3i +2j +3k)
b) r= i + 2j + 3k + μ(3i +4j +3k)
c) r= i + 2j + 3k + μ(3i +3j +2k)
d) r= i + 2j + 3k + μ(3i +3j +4k)
ii) The equation of median through C to side AB is
a) r= -i + j + k + p(3i -2k)
b) r= -i + j + k + p(3i +2k)
c) r= -i + j + k + p(-3i +2k)
d) r= - i + j + k + p(3i + 2j)
iii) The area (∆ ABC ) is equal to
a) 9/2 b) √17/2 c) 17/2 d) 7/2
12) In ∆ ABC, a point P is choosen on side AB so that AP: PB= 1:4 and a point Q is choosen on the side BC so that CQ: QB = 1: 3. Segment CP and AQ intersect at M. If the ratio MC/PC is expressed as a rational numbers in the lowest term as a/b , find (a+ b).
PAPER-1
1) A(1,-1,-3), B(2,1,-2) & C(-5,2,-6) are the position vectors of the vertices of a triangle ABC. The length of the bisectors of its internal angle at A is
a) √10/4 b) 3√10/4 c) √10 d) none
2) Let P is the P. V of the orthocentre and g is the p.v of the centroid of the triangle ABC where circumcentre is the origin. If p= K g, then K=
a) 3 b) 2 c) 1/3 d) 2/3
3) A vector 'a' has components 2p and 1 with respect to a rectangular cartesian atsr. The system is rotated through a certain angle about the origin in the counter clockwise sense. If with respect to the new system, a has components p+ 1 and 1 then,
a) p= 0 b) p=1 or p= -1/3 c) p= -1 or p= 1/3 d) p= 1 or p= -1
4) The number of vectors of unit length perpendicular to vectors a= (1,1,0) and b= (0,1,1) is:
a) 1 b) 2 c) 3 d) ∞
5) Four points A(1,-1,1); B(1,3,1); C(4,3,1) and D(4,-1,1) taken in order are the vertices of
(A) a parallelogram which is neither a rectangle nor a rhombus.
B) rhombus
c) an isosceles trapezium
d) a cyclic quadrilateral
6) Let α, β and γ be distinct real numbers. The points whose position vector's are αi + βj + γk; βi + γj + αk and γi +αj + βk
a) are collinear
b) form an equilateral r
c) form a scalene triangle
d) form a right angled triangle
7) If the vector a= 3i + j - 2k, b= - i + 3j + 4k and c= 4i - wj - 6k constitute the sides of a ∆ ABC, then the length of the median bisecting the vector c is
a) √2 b) √14 c) √74 d) √6
8) Let A(0,-1,1), B(0,0,1), C(1,0,1) are the vertices of a ∆ ABC. If R and r denotes the circumradius and in-radius of ∆ ABC, then r/R has value equal to
a) tan(3π/8) b) cot(3π/8) c) tan(π/12) d) cot(π/12)
9) A vector of magnitude 10 along the normal to the curve 3x²+ 8xy + 2y² -3= 0 at its point P(1,0) can be
a) 6i+ 8j b) -6i+ 8j c) 6i- 8j d) - 6i - 8j
10) Let OAB be a regular triangle with side length unity (O) being the origin). Also M, N are the points of tricsection of AB, M being closer to A and N closer to B. Position vectors of A, B, M and N are a, b, m and n respectively. Which of the following hold/s good?
a) m= xa + yb => 2/3 and y= 1/3
b) m= xa + yb => x= 5/6 and y= 1/6
c) m,p.n equals 13/18
d) m.n equals to 15/18
11) a, b, c are three non-zero vectors, no two of which are collinear and the vector a+ b is collinear with c, b+ c is collinear with a, then a+ b+ c is equal to
a) a b) b c) c d) none
12) Given the vector PQ= -6i - 4j and Q is the point (3,3), find the point P.
13) Find the unit vector (in xy plane) obtained by rotating j counter clockwise 3π/4 radian about the origin.
14) Show that the vector v= ai + bj is perpendicular to the line ax+ by = c.
15) If A(0,1, 0), B(0,0,0), C(1,0,1) are the vertices of a ∆ ABC. Match the entries of columns
Column I
A) Orthocentre of ∆ ABC
B) circumcentre of ∆ ABC
C) Area (∆ ABC)
D) Distance between orthocentre and centroid
E) Distance between orthocentre and circumcenter
F) Distance between circumcentre and centroid
G) Incentre of ∆ ABC
H) Centroid of ∆ ABC
Column II
P) √2/2
Q) √3/2
R) √3/3
S) √3/6
T) (0,0,0)
U) 1/2,1/2,1.2)
V) 1/3,1/3,1/3)
W) 1/(√1+√2+√3), √2/(√1+√2+√3), 1/(√1+√2+√3).
VECTOR
COORDINATES
EXERCISE - A
1) Find the ratio in which the line segment joining the points (2,- 3,5) and (7,1,3) is divided by the xy plane. 5:3
2) Find the centroid of the triangle whose vertices are the points A(x₁, y₁, z₁) , B(x₂,y₂,z₂), C(x₃, y₃, z₃).
3) Find the projection of the line segment joining the points (3,3,5) and (5, 4,3) on the straight line joining the points (2,-1,4) and (0,1,5). - 4/3
4) Show that the triangle formed by the points (2,3,1), (-2, 2,0) and (0, 1,-1) is right angled . Find also the other angles. cos⁻¹(1/√3) and cos⁻¹(2/√6)
5) a) Show that the pair of straight line whose direction cosines are given by 3lm - 4ln+ mn=0 and l+ 2m + 3n=0 are at right angles.
b) Show that if the straight lines whose direction cosines are given by al+ bm+ cn =0, fmn+ gnl+ hlm=0 be parallel, then one of the relations √(af)+ √(bg) ± √(ch)=0 is true.
Prove further that if the lines be at right angles, then
f/a + g/b + h/c= 0.
6) Find the area of the triangle whose vertices are the points (2,-5,3),(1,-7,4) and (2,-3,5). √11
7) Find the foot of the perpendicular drawn from the point P(1,8,4) on the straight line joining the points A(0,-11,4) and B (2,-3,1). (4,5,-2)
8) Straight lines OP and OQ are drawn from O with direction numbers (1,-2,-1) and (3,- 2,3). Find the direction cosines of the normal to the plane POQ. 4/√29,3/√29,-2/√29,
EXERCISE -B
1)a) Show that the distance between the points
i) (-3,1,-2) and (-2,-5,1) is √46 units .
ii) (1,2,3) and (3, 4,5) is √12 units .
b) Find the perpendicular distances of the point (x,y,z) from the co-ordinate axes .
c) Find the distance of the point (4,3,5) from the y-axis and ZOX plane .
2) a) Show that if the distance between the points (5,-1,7) and (a,5,1) be 9 units , then a must be 2 or 8.
b) Find the coordinates of the point which is equidistant from the four points (0, 0,0), (a,0,0), (0,b, 0) and (0,0,c).
c) Find the locus of the point which is equidistant from the two points (2,0,4) and (0,2,3).
d) Find the equation of the locus of a point the sum of the squares of whose distances from to given points (-3,2,0) and (3,-2,0) is a constant equal to 28.
e) Find the point on the x-axis whose distance from the point (2 ,-2, 4) is 6 units.
f) Show that the points A (2,5,-4), B(1,4,-3) and C(4,7,-6) are collinear .
3) Show that the triangle formed by the points
i) (-1,-3,4),(-2,1,-4) and (3, - 11,5) is isosceles.
ii) (3,5,0),(2,3,-3),(6,1,-3) is right angled.
iii) (a,b,c),(b,c,a),(c,a,b) is equilateral
iv) (0,7,10),(-1,6,6) and (-4,9,6) is isosceles right angled.
4) a) Show that the four points A(7,6,3), B(4,10,1), C(-2,6,2) and D(1,2,4) form the vertices of a rectangle.
b) Show that the four Points (1,1,1),(-2,4,1),(-1, 5,5) and (2,2,5) are the vertices of a square.
c) Show that the points (5,-1,1),(7,-4,7),( 1,-6,10) and (-1,-3,4) are the vertices of a rhombus .
5) Find the points that divide the join the points (2,-3, 1) and (3, 4,- 5) internal and externally in the ratio 3:2.
6) a) Find the ratio in which the straight line joining the points (2,4,5) and (3,5,-4) is divided by the yz-plane.
b) Show that the yz-plane divides the straight line joining the points (3, 5,- 7) and (-2, 1, 8) in the ratio 3:2 at the point (0,13/5,2).
c) Show that the ratios in which the co-ordinate planes divide the join of the points (-2, 4, 7) and (3,- 5, 8) are respectively 2:3, 4 :5 and (-7) :8.
7) Find the length of the medians of the triangle whose vertices are (1,2,4),(3,-2,0) and (-1,4,2).
8) Show that the co-ordinates of the centroid of the triangle whose vertices are
i) (1,2,3),(2,3,4) and (6,-2,5) is (3,1,4).
ii) ( 2,-3,4),(5, 2,8) and (2,-2,-3) is (3,-1,3).
9) a) Find the direction cosines of the straight line joining the points
i) (5,-3,8),(6,-1,6).
ii) ( 4,3,-5),(- 2,1,8).
b) Show that the straight line joining the points (3, 4 ,-1) and (4, 2, 2) is parallel to the straight line joining the points (2,1,-5) and (4, -3,1).
10)a) The projections of a line segment on the axes are 2, 3,6. Show that its length is 7 and its direction cosines are (2/7, 3/7, 6/7).
b) The projection of a line segments on the axes are 3, 4,12. Find the length and the direction cosines of the line.
11) a) Find the direction numbers of a straight lines which makes equal angles with the axes.
What are the direction cosines ?
How many such straight lines are there ?
b) A directed straight line makes the angle 60°,45° with the axes of x and y respectively. What angle does it make with the z-axis .
12) If α, β, γ be the direction angles of a straight line, then show that sin²α+ sin²β+ sin²γ= 2.
13) a) Find the angle between the straight lines which direction ratios are
i) +5 ,-12, 13) and (-3,4,5)
ii) (1,1,2) and (√3-1, - √3-1,4).
b) calculating the angles of the triangle, show that the points (4,5,0),(2,6,2) and (2,3,-1) are the vertices of an isosceles triangle.
14) If P and Q be the two points (2,3,5) and (1,1,-1) respectively and O be the origin, show that the straight line OP is perpendicular to the straight line OQ.
15) Show that the direction cosines l, m, n of two straight lines connected by the relations l+ m+ n=0 and mn - 2nl - 2lm =0 are given by
(l:m:n)= (1:1:-2) and (l:m:n)= (1:-2:1).
16) Show that the direction cosines l, m, n of two straight lines which are connected by the relations l- 5m + 3n=0 and 7l²+ 5m²- 3n²=0 are (1/√6) (-1,1,2) and (1/√14) (1,2,3).
17) a) Find the angle between the two straight lines whose direction cosines are given by
i) l+ m+ n=0, l²+ m²- n²=0.
ii) 2l+ 2m- n=0, mn+ nl + lm =0.
b) Show that the angle between the straight lines whose direction cosines are given by l+ m+ n=0, fmn+ +gnl+ hlm=0 is π/3, if 1/f + 1/g + 1/h=0.
c) Show that the straight lines whose direction cosines are given by a²l+ b²m + c²n =0, mn + nl+ ln=0 are parallel, if a+ b+ c=0.
d) Show that the straight lines whose direction cosines are given by al+ bm+ cn=0 and ul²+ vm²+ wn²=0 are perpendicular, if a²(v + w)+ b²(w+ u)+ c²(u+ v)=0 and parallel, if a²/u + b²/v + c²/w =0.
18) Find the area of the triangle formed by the points
i) (3,0,1),(4,-1,-1),(3,-2,-2).
ii) (1,2,3),(-2,1,4),( 3,4,-2).
19) a) P, Q, R, S are the points (6,3,2), (5,1,4),(3,-4,7) and (0,2,5) respectively. Find the projection of PQ on RS.
b) The coordinates of the points P, Q, R, S are (3,4,5),(4,6,3),(-1,2,4) and (1,0,5) respectively.
Show that the projection of
i) PQ on RS is (-4/3)
ii) RS on PQ is (-4/3).
20) Prove , by finding the necessary direction cosines , that the points (1,2, 3), (4, 0,4), (-2,4, 2) and (7,- 2, 5) are collinear .
21) Find the point in which the perpendicular from the origin on the straight line joining the points A (-9,4,5) and B(11,0,-1) meets it.
22) The parallelogram ABCD has one vertex A at the origin and the vertices B and C are (3,-4,4) and (7,1,4) respectively. Find the point D.
23) a) prove that the three straight lines drawn from O with direction numbers 2, 1,5; 2, -1,1; 6, -4 ,1 are coplanar.
b) If the three concurrent straight lines whose direction ratios are (l₁, m₁, n₁), (l₂, m₂, n₂),(l₃, m₃, n₃) be coplanar, then prove that
|l₁ m₁ n₁
l₂ m₂ n₂ = 0
l₃ m₃ n₃
24) prove that the angle between the two diagonals of a cube is cos⁻¹(1/3).
25) a line makes an angle α, β, γ, δ with the four diagonals of a cube; show that cos²α+ cos²β+ cos²γ+ cos² δ= 4/3.
26) if the edges of a rectangular parallelopiped be a, b, c, then show that the angles between the four diagonals are given by
cos⁻¹{(a²±b²±c²)/(a²+ b²+ c²)}.
27) A variable line in two adjascent positions has direction cosines l, m, n and (l+ δl), (m+ δm), (n + δn). Show that the small angle δθ between the two positions is given by
δθ²= δl²+ δm²+ δn².
28) If (l₁,m₁,n₁), (l₂, m₂, n₂), and (l₃, m₃, n₃) be the direction cosines of three mutually parapendicular lines, then show that the line whose direction ratios are (l₁+ l₂+ l₃), (m₁+ m₂+ m₃) and (n₁+ n₂+ n₃) makes equal angles with them.
29) If l₁, m₁, n₁ and l₂, m₂, n₂ be the direction numbers of two intersecting straight lines, then show that the line through their intersection with direction numbers
(l₁+ λl₂),(m₁+ λm₂), (n₁+ λn₂) is coplanar with them for all values of λ.
30) a) The direction cosines of two concurrent lines are l₁, m₁, n₁ and l₂, m₂, n₂. Show that direction cosines of the lines bisecting the angles between them are proportional to (l₁± l₂), (m₁ ± m₂), (n₁ ± n₂).
b) If (l₁, m₁, n₁) ar(l₂, m₂, n₂) be the direction cosines of two perpendicular straight lines , then show that the direction cosines of the straight line perpendicular to both of them are
±(m₁n₂ - m₂n₁), ±(n₁l₂ - n₂l₁) and ±(l₁m₂ - l₂m₁).
31) If θ be the angle between two Straight lines whose direction cosines are l₁, m₁, n₁ and l₂, m₂, n₂, then show that the direction cosines of their angle bisectors are
(l₁+ l₁).(2 cos(θ/2)), (m₁+ m₂)(2cos(θ/2)), (n₁ + n₂)/(2 cos(θ/2) and (l₁- l₂)/(2sin(θ/2), (m₁ - m₂)/(2sin(θ/2)), (n₁ - n₂)/(2sin(θ/2))
32) a) if two pairs of opposite of a tetrahedron be at right angles , then show that the third pair is also right angles.
b) If a pair of opposite edges of a tetrahedron be perpendicular, then prove that the distance between the middle points of the other two pairs of opposite edges are equal.
c) prove that the three straight lines joining the middle points of the opposite sides of a tetrahedrons are concurrent.
1) b) √(y²+ z²), √(z²+ x²), √(x²+ y²)
c) √41, 3
2) b) (a/2, b/2, c/3) c) 4x - 4y + 2z -7=0. d) x²+ y²+ z² = 1 e) (6,0,0),(-2,0,0)
5) (13/5,6/5,-13/5);(5,18,-17
6) a) (-2):3
7) √10 units, √43 units, 5 units
9) a) i) (±1/3,±2/3,±2/3) ii) (±6/7,±2/7,±3/7)
10) b) 13, (3/13,4/13,12/13
11) a) (1,1,1); (±1/√3, ±1/√3, ±1/√3); two b) 60°
13) a) i) cos⁻¹(1/65) ii) π/3
17) a) i) 60° ii) 90°
18) i) √14/2 square units ii) √1218/2 square units
19) a) 13/7
21) (1,2,2) 22) (4,5,0)
MISCELLANEOUS
1) Represent graphically
a) a displacement of 40 km, 30° west of South.
b) 60 km, 40° east of north.
c) 50 km south-east.
2) Classify the following measures as scalar and vectors
a) 10 kg
b) 10 meters north-west.
c) 10 Newton
d) 30 km/hr.
e) 50 m/sec towards north
f) 10⁻¹⁹ coloumb.
3) In figure which of the vectors are:
a) collinear b) Equal c) Co-initial
4) in figure (a square), identify the following vectors:
a) Co-initial b) Equal c) Collinear but not equal.
5) If a, b, c be the vector represented by the sides of a triangle, taken in order, then prove that a+ b + c=0.
6) If P₁, P₂, P₃, P₄ are points in a plane or space and O is the origin of vectors , show that P₄ coincides with O iff OP₁+ P₁P₂ + P₂P₃+ P₃P₄ =0.
7) If PO+ OQ= QO+ OR, show that the points P,Q,R are collinear.
8) If a, b are any two vectors, then give the geometrical interpretation of the relation
|a+ b|= |a - b|.
9) if a and b are the vectors to determined by the two adjacent sides of a regular hexagon, what are the vectors determined by the other sides taken in order ?
10) if the sum of two unit vectors is a unit vector, prove that the magnitude of the difference is √3.
11) If a and b represent two adjacent sides AB and BC respectively of a parallelogram ABCD , then show that its diagonals AC and DB are equal to a+ b and a- b respectively.
12) Vectors drawn from the origin O to the points A, B and C are respectively a, b, and 4a- 3b. Find AC and BC.
13) If a, b, c and d are distinct non zero vectors represent by directed line segment from the origin to the points A, B, C and D respectively, and if b - a = c - d, then prove that ABCD is a parallelogram.
14) A, B, P, Q and R are 5 points in a plane. Show that the sum of the vectors AP, AQ, AR, PB, QB and RB is 3AB.
15) Let O be the centre of a regular hexagon ABCDEF. Find the sum of the vectors OA, OB, OC, OD, OE, OF. 0
16) for any two vectors a and b, prove that
a) |a+ b|≤ |a|+ |b|
b) |a - b|≤ |a|+ |b|
c) |a b|≥ |a|+ |b|
17) If c= 3a + 4b and 2c= a - 3b, show that
i) c and a have the same direction and |c|> |a|
ii) c and b have opposite direction and |c|> |b|.
18) The position vectors of points A, B, C, D are a, b,, 2a+ 3b and A- 2b respectively. show that DB= 3b - a and AC= a+ 3b.
19) Let ABCD be a parallelogram. If a, b, c be the position vectors A, B, C respectively with reference to the origin O, find the position vector of D with reference to O.
20) Find the position vectors of the points which divide the join of the points 2a - 3b and 3a - 2b internally and externally in the ratio 2:3. -5b
21) If a and b are position vectors of points A and B respectively, then find the position vector of points of tricsection of AB. (a+ 2b)/3
22) Find the position vector of a point R which divides the line segment joining P and Q whose position vectors are 2a+ b and a - 3b, externally in the ratio 1:2. Also, show that P is the midpoint of the line segment RQ. 3a+ 5b
23) If a and b are the position vectors of A and B respectively, find the position vector of a point C on BA produced such that BC= 1.5 BA.
24) Four points A, B, C, D with position vectors a, b, c, d respectively are such that 3a - b+ 2c - 4d =0. Show that the four points are coplanar. Also, find the position vector of the point of intersection of line segment AC and BD. (3a + 2c)/5 or (b + 4d)/5
25) Let a, b, c be the position vectors of three distinct points A, B, C. If there exist scalars x, y, z(not all zero) such that xa+ yb+ zc= 0 and x+ y+ z=0, then show that A, B, C lie on a line.
26) If D is the midpoint of the side BC of a triangle ABC, show that AB + AC = 2AD.
27) Points L, M, N divide the sides BC, CA, AB of ∆ ABC in the ratio 1:4,3:2,3:7 respectively. Show that AL+ BM+ CN is a vector parallel to CK, where K divides AB in the ratio 1:3.
28) Prove using vectors: Medians of a triangle are concurrent.
29) If G is the centroid of a triangle ABC, show that GA+ GB+ GC=0
30) If D and E are the midpoints of a side AB and AC of a triangle ABC respectively, show that BE+ DC= (3/2) DC.
31) If ABC and A'B'C' are two triangles and G, G' be their centroids, show that AA'+ BB' + CC' = 3GG'.
32) Show that the line segment joining the midpoints of two sides of a triangle is parallel to the third side and equal to half of it.
33) Show that the sum of the vectors directed from the vertices to the midpoints of opposite sides of a triangle is zero.
34) Show using vector: the diagonals of a quadrilateral bisect each other iff it is a parallelogram.
35) Using vector method, show that the line segments joining the midpoints of the adjacent sides of a quadrilateral taken in order form a parallelogram.
36) Prove that the segment joining the middle points of two non parallel sides of a trapezium is parallel to the parallel sides and half of their sum.
37) Prove by vector method that the line segment joining the midpoints of the diagonals of a trapezium is parallel to the parallel sides and equal to half of their difference.
38) If ABCD is quadrilateral and E and F are the midpoints of AC and BD respectively, prove that AB+ AD+ CB+ CD= 4EF.
39) Show that the line joining one vertex of a parallelogram to the midpoint of an opposite side trisects the diagonal and is trisected thereat.
40) ABCD is a parallelogram. E, F are midpoints of BC, CD respectively. AE, AF meet the diagonal BD at points Q and P respectively. Show that points P and Q trisect DB.
41) ABCD is a parallelogram. If L and M are the midpoints of BC and DC respectively, then express AL and AM in terms of AB and AD. Also, prove that AL+ AM = (3/2) AC.
42) If P and Q are the midpoints of the sides AB and CD of a parallelogram ABCD, prove that DP and BQ cut the diagonal AC in its points of tricsection which are also the points of tricsection of DP and BQ respectively.
43) The midpoints of two opposite sides of a quadrilateral and the midpoints of the diagonals are the vertices of a parallelogram. Prove using vectors.
44) If O is the circumcentre and O' the orthocentre of a triangle ABC, prove that
i) SA+ SB + SC = 3SG, where S is any point in the plane of triangle ABC whose centroid is at G.
ii) OA+ OB+ OC= OO'
iii) O'A+ O'B + O'C = 2O'O.
iv) AO'+ O'B + O'C = AP, where AP is the diameter of the circumcircle.
45) The lines joining the vertices of a tetrahedron to the centroids of opposite faces are concurrent.
46) Find the values of x and y so that the vectors 2i + 3j and xi + yj are equal.
47) Let O be the origin and let P(-4,3) be a point in the xy-plane. Express OP in terms of vectors i and j. Also, find |OP|.
48) Let a= i+ 2j and b= 2i+ j. Is |a|= |b| ? Are the vectors a and b equal ?
49) If the position vector 'a' of a point (12, n) is such that |a|= 13, Find the value of n.
50) If A=(0,1), B=(1,0), C=(1,2), D=(2,1), prove that AB= CD.
51) Find the coordinates of the tip of the position vector which is equivalent to AB, where the coordinates of A and B are (3,1) and (5,0) respectively.
52) If a is a position vector whose tip is (1,-3). Find the coordinates of the point B such that AB= a. If A has coordinates (-1,5).
53) ABCD is a parallelogram. If the coordinates of A, B, C are (2,3),(1,4) and (0,-2) respectively, find the coordinates of D.
54) Find a unit vector parallel to the vector -3i+ 4j.
55) Find a vector of magnitude 5 units which is parallel to the vector 2i - j.
56) Find the components along the coordinates axes of the position vector of each of the following points:
a) P(5,4)
b) Q(-4,3)
c) R(5,-7)
d) S(-4,-5).
57) Find the scalar and vector components of the vector with initial point A(2,1) and terminal point B(-5,7).
58) Write all the unit vectors in XY-plane.
59) Write down a unit in XY-plane, making an angle of 30° with the positive direction of x-axis.
60) A girl walks 4km towards west, then she walks 3km in a direction 30° east of North and stops. Determine the girl's displacement from her initial point of departure.
61) Find the values of x, y and z so that the vectors a= xi + 2j + zk and b= 2i + yj + k are equal.
62) Find the sum of vectors a= i - 2j + k, b= -2i + 4j + 5k and c= i - 6j - 7k.
63) Find the magnitude of the vector a= 3i - 2j + 6k.
64) Find the distance between the points A(2,3,1) and B (-1,2,-3), using vector method.
65) If a= 3i - 2j + k and b = 2i - 4j - 3k, find |a - 2b|.
66) If A, B, C have position vectors (2,0,0),(0,1,0),(0,0,2). Show that ∆ ABC is isosceles.
67) Show that the points A, B and C with position vectors a= 3i - 4j - 4k, b = 2i - j + k and c = i - 3j - 5k respectively, form the vertices of a right angled triangle.
68) Find the unit vector in the direction of 3i - 6j + 2k.
69) Find the unit vector in the direction of a+ b, if a= 2i - j + 2k and b = - i + j - k.
70) Find the unit vector in the direction of PQ, where P and Q are the points (1,2,3) and (4,5,6) respectively.
71) Find a vector of magnitude 11 in the direction opposite to that of PQ, where P and Q are the points (1,3,2) and (-1,0,8) respectively.
72) if a= i + 2j + 3k and b = 2i + 4j - 5k represent two adjacent sides of a parallelogram, find unit vectors parallel to the diagonals of a parallelogram.
73) Three vectors of magnitude a, 2a, 3a meet in a point and their directions are along the diagonals of the adjacent faces of a cube. Determine their resultant.
74) If a and b are non-collinear vectors such that x₁a + y₁b= x₂a + y₂b, then show that x₁ = x₂ and y₁= y₂.
75) If a and b are non-collinear vectors, find the value of x for which vectors α= (x -2)a + b and β= (3+ 2x)a - 2b are collinear.
76) If a and b are non-collinear vectors, find the value of x for which the vectors α= (2x +1)a - b and β= (x-2)a +b are collinear.
77) If a, b are the position vectors of the points (1,-1),(-2,m(, find the value of m for which a and b are collinear.
78) Let u = i + 2j, v = -2i + j and w = 4i + 3j. Find scalars x and y such that w= xu + yv.
79) Show that the vectors 2i - 3j + 4k and -4i + 6j -8k are collinear.
80) If a, b, c are three non-null vectors such that any two of them are non-collinear. If a+ b is collinear with c and b+ c is collinear with a, then find a+ b + c.
81) Let a, b, c be three non-zero vectors such that any two of them are non-collinear. If a+ 2b is collinear with c and b+ 3c is collinear with a, then prove that a+ 2b + 6c =0.
82) If a and b are non-collinear vectors and vectors α= (x + 4y)a + (2x + y +1)b and β= (-2x+ y +2)a + +2x - 3y -1)b are connected by the relation 3α= 2β, find x, y.
83) Show that the points with position vectors a- 2b + 3c, -2a + 3b - c and 4a - 7b + 7c are collinear.
84) Show that the points with position vectors a- 2b + 3c, -2a + 3b + 2c and -8a + 13b are collinear whatever be a, b, c.
85) Show that the points A, B, C with position vectors 2a + 3b + 5c, a + 2b + 3c and 7a- c respectively, are collinear.
86) Using vectors, show that the points A(-2,1), B(-5,-1) and C(1,3) are collinear.
87) If the points with position vectors 60i + 3j, 40i - 8j and ai - 52j are collinear, find the value of a.
88) Show that the three points A(-2,3,5), B(1,2,3) and C(7,0,-1) are collinear.
89) The position vectors of the points P, Q, R are i + 2j+ 3k, -2i + 3j+ 5k and 7i - k respectively. Prove that P, Q and R are collinear points.
90) If the position vectors of the points A, B, C, D are 2i + 4k, 5i + 3√3j+ 4k, -2√3j+ k and 2i + k respr, prove that CD is parallel to AB and CD= (2/3) AB.
91) Show that the points A(6,-7,0), B(16,-19,-4), C(0,3,-6) and D(2,-5,10) are such that AB and CD intersect at the point P(1,-1,2).
92) If the points (-1,-1,2), (2,m,5) and (3,11,6) are collinear, find the value of m.
93) If a, b are two non-collinear vectors, show that the points having position vectors l₁a+ m₁b, l₂a+ m₂b and l₃a+ m₃b are collinear, if
1 1 1
l₁ l₂ l₃ = 0
m₁ m₂ m₃
94) Show that the vectors a- 2b + 3c, a- 3b + 5c, and -2a+ 3b - 4c are coplanar, where a, b, c are non-coplanar.
95) Show that the vectors 2a- b + 3c, a + b - 2c, and a+ b - 3c are non-coplanar vectors.
96) Prove that four points 2a + 3b - c, a- 2b + 3c, 3a+ 4b - 2c and a- 6b + 6c are coplanar.
97) If a, b, c are non-coplanar vectors such that x₁a + y₁b + z₁c= x₂a+ y₂b + z₂c, then show that x₁ = x₂, y₁ = y₂ and z₁ = z₂.
98) A vector OP is inclined OX at 45° and OY at 60°. Find the angle at which OP is inclined to OZ.
99) If a vector makes an angle α, β, γ with OX, OY, OZ respectively, show that sin²α+ sin² β+ sin²γ = 2.
100) Find the direction cosines of a vector r which is equally inclined with OX, OY, OZ. If |r| is given, find the total number of such vectors.
101) A vector r is inclined at equal anglea to OX, OY, OZ. if the magnitude of r is 6 units, find r.
102) A vector r has length 21 and its direction ratios are proportional to 2, -3,6. Find the direction cosines and components of r, given that r makes an acute angle with x-axis.
103) Find the angles at which the vector 2i - j + 2k is inclined to each of the coordinates axes.
104) The projection of a vector on the coordinate Axes are 6, -3, 2. Find its length and direction cosines.
105) Find the direction cosines of the vector joining the points A(1,2,-3) and B (-1,-2,1) directed from A to B.
106) Find the position vector of a point A in space such wOA is inclined at 60° to OX and at 45° to OY and |OA|= 10 units.
STRAIGHT LINE IN SPACE
EXERCISE- A
1) Find, in the symmetrical form the equation of the straight line x - 2y + 3z= 4, 2x - 3y + 4z= 5 and find its direction cosines. (x +2)/1= (y -3)/2= z/1, ±1/√6, ±2/√6,±1/√6,
2) Find the point where the straight line joining the points (2,-3,1) and (3,-4,-5) cuts the plane 3x +y + z= 8. (1,-2,7)
3) Find the image of the point (-3,8,4) in the plane 6x - 3y -2z= -1. (9,2,0)
4) Find the equation of the straight line passing through the point (-1,1,-3) and perpendicular to the straight line (x - 3)/-2= (y + 1)/3= (z-2)/-4. (x +1)/48= (y -1)/44= (z+3)/9
5) Find the equation of the plane through the straight line 2x - y + 3z+2 = 0= 3x + 2y - z+3 parallel to the coordinate Axes. 7y -11z= 0
6) Find the equation of the plane through the point (α, β, γ) and through the straight line (x - x')/l = (y -y')/m = (z - z')/n.
7) If the axes be rectangular, then find the equations of the straight line through the point (α, β, γ) at right angles to the straight lines
x/l₁= y/m₁ = z/n₁ and x/l₂= y/m₂= z/n₂. (x- α)/(m₁n₂- m₂n₁) = (y - β)/(n₁l₂ - n₂l₁)= (z - γ)/(l₁m₂ - l₂m₁)
8) Find the equation of the lines of greatest slope and least slope on the plane 3x - 4y + 5z- 5 = 0 drawn through the point (1,2,2) given that the plane 4x - 5y + 6z -6 = 0 is horizontal. (x - 1)/- 7 = (y -2)/1 = (z-2)/5, (x -1)/1= (y -1)/2= (z-2)/1
EXERCISE - B
1) Show that the direction cosines of the straight line x + y - z+ 1 = 0, 4x + y - 2z+2 = 0 are (±1/√14, ±2/√14, ±3/√14)
2) Show that the equation of the straight line through the point (3,4,5) which is equally inclined to the axes are x- 3= y-4= z-5.
3) a) Find the equation of the straight lines passing through the points
i) (3,-9,4) and (-9,5,-4)
ii) (-7,5,3) and (2,6,8)
b) Show that the equation of the median AD of the triangle whose vertices are A(3,4,8), B(1,-6,2) , C(1,4,-2) are (x - 3)/2 = (y -4)/5 = (z-8)/8.
4) a) Show that the three points (-1,5,3),(5,1,5) and (8,-1,6) are collinear.
b) show that the straight line through the points (a,b,c) and (a', b', c') passes through the origin, if aa' + bb'+ cc' = pp' where p and P' are the distance of the points from the origin.
5) Put the equations of the straight lines in symmetrical form as given by
a) x + 5y - z -7 = 0, 2x -5y +3z+ 1 = 0
b) x + y + z +1 = 0, 4x +y -2z+ 2 = 0
6) Find the image of the point
a) (1,-2,3) in the plane 2x -3y +2z +3 = 0.
b) (1,3,4) in the plane 2x - y + z+ 3 = 0
c) Find the image of the straight line (x -1)/3= (y - 3)/5= (z -4)/2 in the plane 2x - y + z+ 3 = 0.
7)a) Find the equations of the straight line through the points (α, β, γ) which is
i) parallel to the z-axis
ii) Perpendicular to the z-axis
b) Find the equations of the straight line passing through the point (1,2,3) and parallel to the straight line
i) x/2= y/4= z/3
ii) x - y +2z -5 = 0= 3x + y + z -6 .
c) Find the equations of the straight line through the point (1,2,3) and parallel to the straight line joining the points (-4,7,2) and (5,-3,-2).
d) Find the equations of the straight line through the point (8,9, -10) and perpendicular to each of the straight lines
(x -2)/3= (y -3)/2= (z +4)/4 and (x +1)/5 = (y -2)/ -6 = (z +3)/3
8) Show that the equations of the straight line passing through the point (1,-2,3) and perpendicular to the plane 2x+ y+ 3z= 4 is (x -1)/2 = (y +2)/1 = (z -3)/3.
9) Show that the straight line (x -1)/-1 = (y +4)/3 = (z +5)/2 meets the plane 2x - 3y + 4z = 0 at the point (4/3,-5,-17/3).
10) a) Show that the distance of the point of intersection of the straight line (x -2)/3= (y +1)/4 = (z -2)/2 and the plane x+ y+ z= 12 from the point (-1,5,10) is 2√19
b) Find the coordinates of the point in which the straight line (x -1)/2 = (y +1)/ - 1 = z/3 intersects the plane 3x + 2y - z= 5.
c) Find the point where the straight line through the points (5,-2,3) and (3,0,1) pierces the xy-plane.
d) Find the coordinates of the point where the straight line x+ 3y - z= 6, y- z= 4 meets the plane 2x+ 2y + z= 0
e) Find the points where the straight line (x -a)/l = (y -b)/m = (z - c)/n meets the coordinates planes.
f) lA straight line is drawn through the points (-6,6,-5) and (12,-6,1). Find the points in which it meets the coordinates planes.
11) a) Show that the distance of the point (3,-4,5) from the plane 2x + 5y - 6z=0 measured along the straight line whose direction ratios are (2,-1,-2) is 12 units.
b) Find the distance of the point (1,-2,3) from the plane x - y + z= 5 measured parallel to the straight line x/2= y/3= z/6.
c) Show that the distance of the point (3,8,2) from the straight line (x -1)/2 = (y -3)/4 = (z -2)/3 measured parallel to the plane 3x + 2y - 2z +15=0 is 7 units.
12) a) Show that the foot of the Perpendicular from the point (-1,3,2) to the plane x+ 2y + 2z -3=0 is (-5/3,5/3,2/3).
b) Show that the equations of the projection of the straight line (x -1)/2 = (y +1)/-1 = (z -3)/4 on the plane x+ 2y+ z= 6 are and (x -3)/4 = (y +2)/ -7 = (z -7)/10.
13) a) Are the two straight lines (x -2)/3= (y -3)/2= (z +4)/4 and (x +1)/5 = (y -2)/ -6 = (z +3)/2 Perpendicular to each other?
b) Find whether the following straight lines are mutually perpendicular
(x -1)/3= (y +2)/2= (z -6)/5 and 2x + y - 3z-2=0= 3x + 2y + 5z +7.
c) Show that the straight lines 3x + 2y + z -5=0=x + y - 2z -3 and 2x - y - z =0= 7x + 10y - 8z are at right angles.
d) Show that the straight lines x= ay+ b, z= cy+ d and x'= a'y+ b', z= c'y + d' are right angles, if aa'+ cc'+1=0.
14) a) Show that the angle between the two straight lines x + 5y - z =0= 2x - 5y + 3z and x +y +z =0= 4x + y - 2z is cos⁻¹⁽1/2√21)
b) Show that the angle between the straight line (x -4)/7 = (y -1)/4 = (z +3)/4 and the plane x - 2y -2z -8 =0 is sin⁻¹⁽1/3)
15) a) Show that the straight lines 3x + 2y - 3z +5=0= x - 2y +z -3 and 11x- 4y - 3z +13 =0= 9x + 2y - 6z +3 are parallel .
b) Prove that the straight line 2x + 3y - z +3=0= 3x - 2y +2 z -6 is not parallel to the z-axis.
16) a) Show that the straight line x- 1= y - 2= (1/2) (z -3) lies on the plane 2x + 4y - 3z = 1.
b) Find the values of b and c for which the straight line (x -1)/2 = (y -2)/-1 = (z +3)/3 lies on the plane 9x + by + cz - 30 =0.
17) a) Show that the line x - y - z + 3 =0= 3x + 3y - z -15 is normal to the plane 2x- y + 3z + 4 =0.
18) A straight line is given by x= t-2, y= 3 - 4t, z= 5t +6 is parallel to the plane x - y - z = 1.
19) a) Show that the straight line 2x + 2y - z - 6 =0= 2x + 3y - z -8 is parallel to a cordinate plane and find the equation of the plane normal to this straight line and passing through the point where this straight line meets the plane x= 0.
b) Show that the equation of the plane through the origin and containing two straight lines whose direction ratios are (1,0,2) and (-1,5,0) is 10x + 2y - 5z =0.
20) a) Show that the equation of the plane containing the straight line x + y + z -1=0= 2x + 3y +4z -5. and perpendicular to the plane x - y +z =0 is and x- z +2 =0
b) Find the equation of the plane which is perpendicular to the plane x + 2y - z +1=0 and which contains the line intersection of the planes x + 2y + 3z -4 =0 and 2x + y +z +2=0
21) a) Show that the equations of the straight line through the point (3,1,-6) and parallel to each of the planes and x + y + 2z -4 =0 and 2x - 3 y + z +5= 0 are (x -3)/7= (y - 1)/3 = (z +6)/-5. are
b) Obtain the equations of the straight line passing through the point (2,3,5) and parallel to the intersection of the planes x + 2y - 1=0 and 2y + 3z -5 =0.
c) Find the equation of the straight line through the point (1,2,4) and perpendicular to the straight line 3x + 2y - z -4=0= x - 2y - 2z -5.
22) a) Show that the plane containing the straight line (x -1)/3= (y +6)/4= (z +1)/2 and parallel to the straight line
(x - 2)/2= (y - 1)/3 = (z +4)/5 is 26x -11y - 17z - 109 =0.
b) Show that the equation of the plane through the point (2,3,3) and parallel to the straight lines x - 1= 2y - 5= 2z and 3x - 4y - 11= 3z -4 is x - 4y + 2z + 4=0.
23) a) Show that the plane passing through the point (-2,-2,2) and containing the straight line joining the two points (1,-1,2) and (1,1,1) is x - 3y - 6z +8=0.
b) Show that the equation of the plane through the point (0,7,-7) and containing the straight line (x + 1)/3= (y - 3)/-2 = (z +2)/-1 is x + y + z =0
c) Prove that the plane through the point (α, β, γ) and the straight line x= pyn+ q = rz + s is given by determinant
x py+ q rz+ s
α pβ+ q rγ+ s = 0
1 1 1
24) a) Find the equation of the Perpendicular from the point (5,9,3) to the straight line (x -1)/2= (y - 2)/3= (z -3)/4. Also find the foot of the Perpendicular.
b) Find the equations of the Perpendicular from the point (1,6,3) to the straight line x + y - z +1 =0= 2x - 7y +4z - 1. Also find the foot of the Perpendicular.
25) a) Find the plane through the point (3,-2,1) and perpendicular to the straight line 3x - 5y - 2z +6= 0= 4x + y +3z -7.
b) Find the equation of the plane through the point (1,2,3) and perpendicular to the straight line
i) (x -2)/2= (y +1)/5= (z -1)/3
ii) x +2y +z =0= 2x - y +z -1.
iii) x + 2y +3z -2=0, 3x + 2y +4z =0.
c) Find the equation of the plane which passes through the z-axis and is perpendicular to the straight line
(x -1)/Cosθ = (y+2)/sinθ = (z -3)/0.
26) a) Obtain the equation of the plane which passes through the intersection of the two planes x+ y - z - 1=0 and 2x + 3y + z - 6= 0 and which is parallel to the straight line x/2= y/3= z/6.
b) Find the equation of the plane through the straight line x + y - 2z + 4=0= x - y + 2z +1 and parallel to the straight line (x + 2)/2= (y - 2)/2= (z -1)/-1 .
27) a) Show that the equation of the plane through the straight line 3x - 4y + 5z - 10=0= 2x + 2y - 3z -4 and parallel to the straight line x= 2y= 3z is x - 20y + 20z - 14=0.
b) Show that the equation of the plane containing the straight line y/b - z/c= 1, x= 0 and parallel to the straight line x/a + z/c = 1, y= 0 is x/a - y/b + z/c +1=0.
28) a) Show that the equation of the plane through the points (2,-1,0),(3,-4,5) and parallel to the straight line 2x = 3y= 4z is 29x - 27y - 22z - 85=0.
b) Show that the equation of the plane through the points (1,0,-1), (3,2,2) and parallel to the straight line (x - 1)/1= (y - 1)/2= (z -2)/3 is 3y - 2z= 2.
29) a) Show that the equation of the plane which contains the straight line x = (y - 3)/2= (z -5)/3 and perpendicular to the plane 2x +7y - 3z -1= 0 is 9x - 3y - z +14=0.
b) Show that the equation of the plane through the straight line (x - 1)/2= (y +2)/-3= z/5 Perpendicular to the plane x - y + z +2 =0 is 2x + 3y + z +4=0.
c) Show that the equation of the plane through the line x/l = y/m = z/n and perpendicular to the plane containing the lines x/m= y/n = z/l and x/n = y/l = z/m is (m - n)x + (n - l)y + (l - m)z= 0.
30) Find the equation of the plane which contains the parallel straight lines x -4= (-1/4) (y - 3)= (1/5) (z -2) and x -3= (-1/4) (y+2)= z/5.
31) Find the equatios of the three planes through the straight line (x - 2)/3= (y - 1)/2= (z -4)/4 parallel to the coordinate Axes.
32) Assuming the plane 4x -3y +7z -4=0 to be horizontal, show that the equations of the line of greatest slope through the point (2,1,1) in the plane 2x + y - 5z =0 are (x - 2)/3= (y - 1)/-1= (z -1)/1.
33) If the plane 3x + 4y +5z =0 be horizontal then show that the equations of the lines of greatest and least slopes on the plane x + 2y + 3z -4=0 through the point (2,-2,2) are respectively
(x - 2)/4= (y + 2)/1= (z -2)/-2 and (x - 2)/1= (y +2)/-2= (z -2)/1.
34) A, A', B, B', C, C' are points on the axes. Show that the lines of intersection of the plans A'BC, AB'C', B'CA, BC'A', C'AB, CA'B' are coplanar.
35) The plane x/a+ y/b +z/c = 1 meets the axes in A, B, C. Show that the planes through the axes and the internal bisectors of angles ABC pass through the straight line
x/a√(b²+ c²)= y/b√(c²+ a²) = z/apc√(a²+ b²).
1) a) (x +9)/6 = (y -5)/-7 = (z +4)/4
b) (x +7)/9 = (y -5)/1 = (z -8)/5
5) a) (x - 2)/2 = (y -1)/-1= z/-3
b) (x + 1/3)/1= (y +2/3)/-2= z/1
6) a) i) (-3,4,-1) ii) (-3,5,2)
b) (x +3)/1 = (y -5)/6 = (z -2)/1
7) a) i) (x - α)/0 = (y -β)/0 = (z -γ)/1
ii) (x - α)/l= (y- β)/m = (z - γ)/0
b) i) (x - 1)/2 = (y - 2)/4 = (z -3)/3
ii) (x - 1)/-3 = (y - 2)/5 = (z -3)/4
c) (x - 1)/-9 = (y - 2)/10 = (z -3)/4
d) (x - 8)/2 = (y -9)/1 = (z +10)/10-2
10) b) (9,-5,12) c) (2,1,0) d) (2,0,-4) e) (0, b- am/l, c - an/l), (a - bl/m, 0, c- bn/m), (A- cl/n, b - cm/n, 0)
f) (9,-4,0),(3,0,-2),(0,2,-3)
11) b) 7/5 units
13) a) no b) yes
16) b) b= 3, c= -5
19) x+ 2z +4=0
20) b) x - 4y - 7z + 16=0
21) b) x -2= -2(y -3)= 3(z -5)
c) (x - 1)/4p-202 = (y - 2)/460 = (z -4)/439
24) a) (x - 5)/4 = (y -9)/2= (z -3)/-2, (3,5,7)
b) (x - 1)/0 = (y -6)/3 = (z -3)/-2 , (1,3,5)
25) a) 13x + 17y - 23z + 18=0
b) i) 2x - 5y +3z - 1=0 ii) 3x +y -5z + 10=0 iii) 2x + 5y -4z =0
c) xcosθ + y sinθ= ó
26) a) 21x + 22y - 18z - 25=0 b) 8x - 4y + 8z - 1=0
30) 11x - y - 3z - 35 =0
31) 2y - z +2 =0 , 4x - 3z - +4=0 , 2x - 3y - 1=0
EXERCISE - C
1) Prove that the straight lines (x - 1)/2= ( y - 2)/3= (z - 3)/4 and 4x - 3y +1 =0 = 5x - 3z +2 =0 are coplanar.
Find also the equation of the plane. 4x - 3y +1 - (1/2) (5x - 3z +2) =0 .
2) Show that the straight lines (x - 1)/2= ( y +1)/-3= (z +10)/8 and (x - 4)/1= (y +3)/-4 =(z +1)/7 intersect and find the plane through thm. Also find thir point of intersection. (5,-7,6)
3) Find the distance of the point (4,1,1) from the straight line x + y +z -4 =0 and x - 2y - z -4 =0.
Find the equation of the Perpendicular. Also find the foot of the Perpendicular. √(27/14), (x -4)/1= (y -1)/16= (z -1)/11, (55/14,-1/7,3/14)
4) Find the condition that the straight lines x/α = y/β= z/γ, x/aα= y/bβ = z/cγ and x/l = y/m= z/n will lie on a plane. (l/α) (b - c) + (m/β) (c - a)+ (n/γ) (a- b)= 0
5) Find the shortest distance between the straight lines
(x - 3)/-3 = ( y -8)/1 = (z -3)/-1 and (x +3)/3 = (y +7)/-2 =(z -6)/-4
And the equations of the line of shortest distance. 3√30, (x - 3)/6 = ( y -8)/15 = (z -3)/-3.
6) Show that the equation to the plane containing the straight line y/b + z/c= 1, x=0 and parallel to the straight line x/a - z/c =1, y=0 is x/a - y/b - z/c +1=0 and if 2d be the shortest distance between the lines, then show that 1/d²= 1/a² + 1/b²+ 1/c².
EXERCISE -D
1) Show that the straight lines
i) (x - 5)/4 = (y -7)/4 =(z +3)/-5 and (x - 8)/7 = (y -4)/1 =(z -5)/3.
ii) (x - 3)/3 = (y -2)/-4 =(z +1)/1 and x +2y +3z =0= 2x +4y +3z +3
iii) x +2y -5z +9=0= 3x -y +2z -5 and 2x +3y -z -3=0= 4x -5y + z +3 are coplanar.
2) Determine the value of k so that the straight lines
(x - 1)/2 = (y -4)/1 =(z -5)/2 and (x - 2)/-1 = (y -8)/k =(z -11)/4.
may intersect.
Also find their point of intersection.
3) Show that the two straight lines given by
x - 2 =0= 2y - 6 = 3z and 4x - 11 = 4y -13= 3z are coplanar and the equation of the plane containing them is 2x - 6y + 3z + 14 =0
4) (x - 5)/4 = (y -7)/4 =(z +3)/-5 and (x - 8)/7 = (y -4)/1 =(z -5)/3.
Also find the equation of the plane on which they lie.
5) a) Show that the straight lines (x - 1)/2 = (y +1)/-3 =(z +10)/8 and (x - 4)/1 = (y +3)/-4 =(z +1)/7 intersect at the point (5,-7,6) and the equationz of the bisectors of the angles between the two lines are (x - 5)/(2/√77± 1/√66) = (y +7)/(-3/√77 ± 4/√66) =(z -6)/(8/√77± 7/√66).
b) Show that the bisectors of the angles between the two straight lines (x - 3)/2 = (y +4)/-1 =(z -5)/-2 and (x - 3)/4 = (y +4)/-12 =(z -5)/3 are (x - 3)/14 = (y +4)/23 =(z -5)/-35 and (x - 3)/-38 = (y +4)/4 =(z -5)/17.
6) Show that the distance of the point
i) 2,3,1) from the straight line y+ z-1 = 0= 2x - 3y - 2z +4 is 3 units.
ii) (4,-5,3) from the straight line (x - 5)/3 = (y +2)/-4 =(z -6)/5 is √457/5 units.
7) Show that the distance between the two parallel straight lines (x - 1)/3 = (y -2)/4 =(z +3)/-4 and (x - 2)/-3 = (y -2)/-3 =(z -8)/4 is 9 units
8) Show that the distance of the point (3,2,1) from the line of intersection of the planes
x+ y + z -4=0 and x - 2y - z-4=0 is √6 units
9) Show that the Perpendicular distance from the point (-1,3,9) to the straight line (x - 13)/5 = (y +8)/-8 =(z - 31)/1 is 21 units and the foot of the Perpendicular is (3,8,28).
10) a) Find the locus of the point P, if its distance from the straight line (x - 1)/2 = (y -2)/- 4=(z -3)/2 be always 5 units
b) Find the locus of the point whose distance from the xy-plane is equal to one fourth of the square of its distance from the y-axis.
11) Find the coordinates of the point on join of (-3,7,-13) and (-6,1,-10) which is nearest to the intersection of the planes 3x - y - 3z +32=0 and 3x + 2y - 15z =8.
12) Examine whether the following lines are skew:
a) (x - 2)/1 = (y -1)/-2 =(z -6)/1 and (x + 3)/7 = (y +3)/-6 =(z +3)/1.
13) a) Show that the length of the shortest distance between the straight lines (x - 3)/2 = (y + 15)/-7 =(z -9)/5 and (x +1)/2 = (y -1)/1 =(z -9)/-3 is 4√3 units and the equations of the line of the shortest distance are x= y= z.
b) Find the shortest distance between the two skew lines x+ 2y + 2z -13=0= x+ 4y - 2z -5 and 2x- 2y + z -1 =0= 2x- 4y - z -9
14) Obtain the shortest distance between the straight line given by ax+ by + cz + d =0= a'x+ b'y + c'z + d' and the axis of z.
15) Find the shortest distance between the straight lines (x - 1)/2 = (y -2)/3 =(z -3)/4 and (x - 5)/4 = (y -4)/4 =(z -5)/5.
Hence show that the two straight lines are coplanar.
16) a) Find the length of the shortest distance between the lines (x - 3)/1 = (y -5)/-2 =(z -7)/1 and (x + 1)/7 = (y +1)/-6 =(z +1)/1
Find also its equation and the point where it intersects the lines.
b) Find the length of the shortest distance between the lines (x - 3)/2 = (y -5)/2 =(z +1)/1 and y -z -1=0= x -2. Find also the points where the line of shortest distance meets the given lines.
17) Show that the straight lines px/l= qy/m = rz/n, x=/l= y/m= z/n and x/pl= y/qm = z/rn are coplanar, if p= q or Q= r or r= p.
18) Show that the straight lines x= nz + a, y= mz + b and x = z+1, y = z + 2 will be coplanar, if (a- 1)(m -1)= (b -2)(n -1).
19) Show that the straight line x= a y + b = cz + d and x = αy + β= γz + δ are coplanar, if
(γ- c)(aβ - bα)- (α - a)(δ- dγ)= p
20) Show that the straight lines (x - a₁)/a₂ = (y - b₁)/b₂ = (z - c₁)/c₂ and (x - a₂)/a₁ = (y - b₂)/b₁ = (z - c₂)/c₁, where a₁, b₁, c₁ are not proportional to a₂, b₂, c₂ intersect and the point of intersection is
(a₁ + a₂), b₁+ b₂, c₁ + c₂).
21) Show that the straight lines
(x - a+ d)/(α-δ) = (y - a)/α = (z - a- d)/(α+ δ) and
(x - b+ c)/(β - γ) = (y - b)/β= (z - b - c)/(β+γ) are coplanar and they lie on the plane x+ z - 2y= 0.
22) Two straight lines
(x - α₁)/l₁ = (y - β₁)/m₁ = (z - γ₁)/n₁ and
(x - α₂)/l₂ = (y - β₂)/m₂ = (z - γ₂)/n₂ are cut by a third line whose direction cosines are λ, μ, ν . Show that d, the length intercepted on the third line, is given by
d| l₁ m₁ n₁ = α₁ - α₂ β₁- β₂ γ₁- γ₂
l₂ m₂ n₂ l₁ m₁ n₁
λ μ ν. l₂ m₂ n₂
23) A square ABCD of diagonal 2a is folded along the diagonal AC so that the planes DAC, BAC are at right angles. Show that the shortest distance between AB and DC is 2a/√3.
24) Show that the shortest distance between any two opposite edges of the tetrahedron formed by the planes y+ z=0, z+ x=0, x+ y=0, x+ y+ z= a is 2a/√6 and that the three lines of shortest distances interesect at the point x= y = z = - a.
25) Prove that the shortest distances between the diagonals of a rectangular parallelopiped and the edges not meeting it, are
bc/√(b²+ c²), ca/√(c²+ a²), ab/√(a²+ b²), where a, b, c are the lengths of the edges.
2) 3; (3,5,7) 4) (1,3,2); 17x - 47y - 24z + 172=0. 10) a) 5x²+ 2y²+ 5z²+ 4yz - 2zx + 4xy - 12x - 24y - 36z = 66 b) x²+ z²= 4z 11) (-7,-1,-9) 12) skew lines 13) b) 7 units
14) (c'd- cd')/√{(bc'- b'c)²+ (ca' - c'a)²} units 16) a) 2√29 units; (x-3)/2= (y-5)/3= (z - 7)/4; (3,5,7), (-1,-1,-1)
b) 3 units; (1,3,-2), (2,1,0).
EXERCISE - E
1) Find the equation of the straight line drawn from the origin to meet the straight lines 3x + 2y + 4z - 5 =0= 2x - 3y + 4z +1 and 2x - 4y + z + 6 =0= 3x - 4y + z -3. 13x - 13y + ²4z =0= 8x - 12y + 3z or x/249= y/153 = z/-52
2) Find the equation of the straight line that intersects the straight lines x + y + z - 4 =0 = 2x - y - z -3, x - y + z - 3 =0= x + y + z -3= 0= x + 4y - z +1 and passes through the point (1,0,1). x - 2y - 2z +1 =0= 2x + 3y -2.
3) a) Show that the planes 2x - 3y +6z -1=0, 3x +6y -z -2=0 and x + y + z -1 =0 interesect at a point. Also find the point of intersection. (-1,1,1)
b) Show that the planes 2x - 3y - z - 3=0, x + 2y + 3z -2=0 and -x + 2y + z - 2 =0 intersect in a straight line.
4) Show that the planes 2x - 5y + z - 1 =0, x + y + 4z = 2 and x + 3y + 6z = 3 form a triangular prism.
5) Find the surface generated by the straight line which intersects the lines y=z=0 and x + 3z - 5 = a = y + z and is parallel to the plane x+ y=0. (y+ z)(x+ y)= 2a(x + z).
6) Prove that the locus of the straight lines which intersects the three lines y - z - 5 =0, x= 0, z - x=q, y=0; x - y =1, z=0 is x² + y² + z² - 2yz - 2zx - 2xy= 1.
7) Find the surface generated by the straight line which intersects the lines x + y= 0, z=0; x- y= z, x+ y = 2a and the parabola x²= 2az, y=0. x²- y²= 2az.
8) A straight line moves so as to intersect the straight line x= y, z=0 and the circles y²+ z²= r², x=0; z²+ x²= r², y=0. Show that the equation to the locus of the moving line is (x + y)² {(x- y)² + z² = r²(x - y)².
EXERCISE - F
1) Find the equation of the straight line passing through the point (1,1,1) and intersecting the straight lines x + y + z - 1 =0= 2x - y - z -2 and x - y - z - 3 =0= 2x + 4y - z -4.
2) Find the equations of the straight line drawn from the origin to intersect the straight lines 3x + 2y + 4z - 3 =0= 2x - 3y + 4z +1 and 2x - 4y + z - 5 =0= 3x - 4y + z - 1.
3) Find the equation of the straight line drawn through the origin which will intersect both the straight lines (x -1)/1= (y + 3)/4= (z - 5)/3 and (x - 4)/1= (y +3)/1= (z -14)/2.
4) Find the equations of the straight line which passes through the point (3,-2,-4) is parallel to the plane 3x - 2y - 3z - 7 =0 and intersects the straight line (x -2)/3= ( y + 4)/-2= (z - 1)/2 .
5) Find the equation of the straight line which can be drawn from the point (2,-1,-3) to intersect the straight line (x - 1)/2= (y - 2)/3= (z -3)/4 and (x - 4)/4= y/5= (z + 3)/3.
6) A straight line with directio ratios 2,7,-5 is drawn to intersect the straight lines (x -5)/3= (y - 7)/-1= (z +2)/1 and (x + 3)/-3= (y - 3)/2= (z -6)/4
Find the coordinates of the points of intersection and the length intercepted on it.
7) Find the equations of the two straight lines through the origin which intersect the straight line (x -3)/2= y -3= z at an angle of 60°.
8) Show that the following planes intersects at a point:
a) 2x + 3y - z -2=0, 3x + 3y + z - 4 =0, x - y + 2z - 5= 0.
b) 2x - y + z - 4 =0, 0= 5x + 7y + 2z -2 and 3x + 4y - 2z + 3 =0.
Show that in the second case the planes intersects at the point (1,-1,1).
9) Show that the following planes intersect in a straight line:
a) 4x + 3y + 2z +7 =0, 0= 2x + y - 4z +1, and x - 7z - 2.
b) 2x + y + z +4=0, y - z +4=0, 3x + 2y + z +8 =0.
10) Show that the following planes form a triangular prism:
a) x - 2y + z -3=0, x + y - 2z -3 =0, 0= x - z -1
b) 2x - 4y + 2z - 5 =0, 5x - y - z =8, x + y - z - 7.
11) a) Find the valus of k, for which the planes kx + 3y +2z -4=0, 3x + ky + z - 1 =0.
i) intersect at a point
ii) interesect in a straight line
iii) form a triangular prism.
b) Determine the values of a and b, for which the planes 2x - y + 3z -1=0, x + 2y - z + b =0, x + 2y - 6z +10=0
i) have a common point
ii) intersect in a line
iii) intersect in three distinct parallel lines.
12) Show that, if the planes x = cy + bz, y= az+ c and z= bx + ay pass through one straight line, then a²+ b²+ c²+ 2abc= 1 and the common line intersection is
x/√(1- a²)= y/√(1- b²)= z/√(1- c²).
13) Show that the planes cy- bz = l, az- cx = m and bx - ay= n will intersect in a line, if al+ bm + cn=0 and in that case the direction ratios of the straight line a,b,c.
14) Find the condition that the plans lx+ my+ n- z =0, mx + ny + lz = 0 and nx + ly + mz =0 will pass through a straight line.
15) Prove that the plane x + ay + (b + c)z +d=0, x + by + (c + a)z +d =0 and x + cy + (a+ b)z +d= 0 have a common line of intersection.
16) A line of constant length has its extremities on two fixed straight lines. Show that the locus of its midpoint is an ellipse.
17) A line is drawn to meet the straight lines y= tanα , z= c and y= - tan α, z= - c, so that the length intercepted on it is constant.
Show that its equation may be put in the form
(x - k sin θ cotα)/k cosθ = (y - k cosθ tanα)/k sinθ = z/c, where k is a constant and θ is a parameter.
18) A variable straight line always interesects the lines x= k, y= 0; y= k, z=0; z= k, x=0. Show that the equation to its locus is xy+ yz+ zx = k(x+ y + z - k)=0.
19) A variable straight line intersects the straight lines x= b, y= c ; y= c, z= -a; z= a, x=b .
Show that the equation of the locus of the straight line is axy+ byz+ czx+ abc=0.
20) Show that the surface generated by the straight line which intersects the straight lines y= mx, z= c and y= - mx, z= - c and x-axis is mzx= cy.
21) Show that all lines which cut the z-axis and the lines (x +3)/2= (y-6)/3= (z -3)/-2; x/2= (y - 6)/2= z/-1 lie on the surface 7(x - y +6)(x+ z)= (3x -2y + 21)(x + 2z).
22) Show that the locus of the variable line which intersects the three lines
y= mx, z= c; y= - mx, z= - c; y= z, mx= - c is the surface y²- m²x²= z² - c²
23) Prove that the line of shortest distance between the z-axis and the variable line
x/a+ z/c= K (1+ y/b), x/a - z/c = (1/k)(1- y/b), (where k varies) generate the surface abz(x²+ y²)= (a²- b²)cxy.
24) a) Show that the locus of a point which is equidistant from two given straight lines y= mx, z= c and y= -mx, z= - c is mxy+ c(1+ m²)z= 0
b) Show that the surface generated by a straight line which meets two straight lines y= mx, z= c and y= -mx, z= - c at the same angle is mcx = yz.
c) A variable line intersects the lines y= 0, z= c ; x= 0, z= - c and is parallel to the plane lx+ m + nz= p. Show that the surface generated by its lx(z- c)+ my(z+ c)+ n(z² - c²)= 0.
25) A and B are two variable points on two non intersecting straight lines and AB is of constant length 2k. Show that the surface generated by AB is (mzx - cy)²+ m²(yz - mcx)²= m²(k²/c² -1)(z²- c²)², where the equations of the two non intersecting straight lines are y= mx, z= c and y= -mx, z= - c .
26) a) Through two non intersecting straight lines y= x tan k, , z= c and y= - x tan k, z= - c two planes are drawn at right angles to each other.
Show that the locus of their line of intersection is x² sin² k - y² cos² k= (z²- c²) cos2k.
b) A straight line moves so as to meet the straight lines y= mx, z= c and y= -mx, z= - c in A and B and intersects the curve yz = k², x=0. Show that the locus of the middle point of AB is (m²x²- y²)mcx = k²y².
27) Show that the surface generated by a straight line which intersects the straight lines y= 0, z= c; x= 0, z= - c and the curve z=0, xy+ c²= 0 is z² - c²= xy.
28) Show that the locus of a line which meets the lines y= mx, z= c and y= -mx, z= - c which intersects
i) the circle x²+ y²= a², z=0 is
c²m²(cy- mxz)²+ c²(yz - cmx)²= a²m²(z²- c²)².
ii) The ellipse x= 0, y²/a²+ z²/b²= 1 is
(y²- m²x²)²/a²+ (yz - cmx)²/b²= (cy - mzx)²/c².
iii) the Hyperbola xy= c², z=0 is
(cmx - yz)(mzx - cy)+m(c²- z²)²=0.
29) A straight line is parallel to the yz-plane and interacts the curves x²+ y²= a², z=0 and x²= az , y=0. Show that
x⁴y²= (a²- x²)(x²- az)².
30) Show that the locus of the straight line which moves parallel to the xz-plane and meets the curves
xy= c², z=0; y²= 4cz, x=0 is
(c²- xy)(y²- 4cz)= 4cxyz.
31) a) A variable line intersects OX and the curve x= y, y²= cz and is parallel to the plane YOZ. Show that it generates the surface xy= cz.
b) A variable plane through the x-axis and a variable plane through the y-axis are inclined at a constant angle. Show that their line of intersection generates the surface z²(x²+ y²+ z²)= x²y² tan²α.
32) A straight line is parallel to the plane y+ z= 0 and interesects the circles x²+ y²= a², z= 0 and x²+ z²= a², y= 0. Show that it generates the surface x²+ (y+z)²= a².
MISCELLANEOUS -1
1) Find the vector equation of a line which passes through the point with position vector 2i - j + 4k and is in the direction of i+ j - 2k. Also, reduce it to cartesian form. r= (2i - j + 4k)+ λ(i + j - 2k), (x-2)/1= (y+1)/1= (z -4)/-2
2) Find the vector equation of the line through A(3,4,-7) and B (1,-1,6). Find also, its cartesian equations. r= (3i + 4j -7k)+ λ(-2i -5 j +13k), (x-3)/-2 = (y-4)/-5 = (z +7)/13
3) The point A(4,5,10), B(2,3,4) and C(1,2,-1) are three vertices of a parallelogram ABCD. Find vector and cartesian equations for the sides AB and BC and find the coordinates of D. r= (4i + 5j + 10k)+ λ(i + j +3k) where λ= -2k (x-4)/1= (y-5)/1= (z -10)/3, r= (2i +3j + 4k)+ v(i + j +5k), (x-2)/1= (y-3)/1= (z -4)/5, (3,4,5)
4) Find the vector equation of a line passing through a point with position vector 2i - j + k, and parallel to the line joining the points - i + 4j + k and i+ 2j + 2k. Also, find the cartesian equivalent of this equation. r= (2i - j + k)+ λ(2i - 2j + k), (x-2)/2 = (y+1)/-2 = (z -1)/1.
5) Find the cartesian equation of a line passing through the points A(2,-1,3) and B (4,2,1). Also, reduce it to vector form. (x-2)/2 = (y+1)/3 = (z -3)/-2, r= (2i - j + 3k)+ λ(2i + 3j - 2k)
6) The cartesian equations of a line are 6x -2= 3y +1= 2z -2. Find its direction ratios and also find vector equation of the line. 1,2,3, r= (i/3 - j/3 + k)+ λ(i + 2j +3k),
7) Find the direction cosines of the line (x-2)/2 = (2y-5)/-3 , z= -1. Also , find the vector equation of the line. 4/5,-3/5,0, r= (2i +5j/2 - k)+ λ(2i -3j/2+ 0k)
8) Show that the points whose position vector are 5i + 5k), 2i + j + 3k) and -4i+ 3j - k are collinear .
9) If the points A(-1,3,2), B(-4,2,-2) and C(5, 5, k) are collinear, find the value of k. 10
10) Find the point on the line (x+2)/3 = (y+1)/2 = (z -3)/2 at a distance of 3√2 from the point (1,2,3). (-2,-1,3) And (56/17,43/17,111/17)
11) Find the angle between the lines r= (3i + 2j - 4k)+ λ(i + 2j + 2k) and r= (5j - 2k)+ K(3i + 2j + 6k). cos⁻¹(19/21)
12) Find the angle between the lines (x-2)/3 = (y+1)/-2, z = 2 and (x- 1)/1= (2y+3)/3 = (z +5)/2. π/2
13) Show that the line x= at+ b, z= cy+ d and x= a'y + b', z= c'y + d' are perpendicular if aa' + cc' +1=0.
14) Find the angle between two lines whose direction ratios are proportional to 1,1,2 and (√3-1),(-√3-1), 4. π/3
15) Find the equation of a line passing through a point (2,-1,3) and parallel to the line r= (i + j)+ λ(2i + j - 2k). r= (2i - j + 3k)+ K(2i + j - 2k)
16) Find the equation of a line passing through (1,-1,0) and parallel to the line (x-2)/3 = (2y+1)/2 = (5-z)/1. (x-1)/3 = (y+1)/1= (z - 0)/-1
17) Find the cartesian equation of the line passing through the point (-1,3,-2) and perpendicular to the lines x/1= y/2 = z/3 and (x + 2)/-3 = (y-1)/2 = (z +1)/5. (x+ 1)/2 = (y+3)/-7 = (z +2)/4
18) A line passes through (2,-1,3) and is perpendicular to the line r= (i + j -k)+ λ(2i -2 j + k), and r= (2i - j - 3k)+ K(i + 2j + 2k), obtain its equation. r= (2i - j + 3k)+ M(2i + j - 2k), where M= -3L
19) Find the value of K so that the lines l₁: (1-x)/3 = (7y-14)/2K= (z -3)/2 and l₂: (7- 7x)/3K = (y-5)/1= (6-z)/5 are at right angle. Also find the equation of a line passing through the point (3,2,-4) and parallel to line l₁. 70/11, (x-3)/-3 = (y-2)/20/11= (z +4)/2
20) Show that the line (x-1)/2= (y-2)/3 = (z -3)/4 and (x-4)/5 = (y-1)/2 = z intersect. Find their point of intersection. (-1,-1,-1)
21) Show that the lines (x-1)/3 = (y+1)/2 = (z -1)/5 and (x+ 2)/4 = (y-1)/3 = (z + 1)/-2 do not intersect.
22) Show that the lines r= (i + j - k)+ λ(3i - j) and r= (4i - k)+ K(2i + 3k) intersect . Find their point of intersection. (4,0,-1)
23) Find the equations of the two lines through the origin which intersect the line (x- 3)/2 = (y -3)/1= z/1 at angle of π/3 each. x/1= y/2 = z/-1 and x/-1= y/1 = z/-2.
24) AB= 3i - j + k) and CD= -3i + 2j +4k) are two vectors . The position vectors of the points A and C are 6i + 7j + 4k and -9j + 2k respectively . Find the position vector of a point P on the line AB and a point Q on the line CD such that PQ is perpendicular to AB and CD both. 3i +8 j + 3k) , -3i -7j +6k)
25) Find the foot of the Perpendicular from the point (0,2,3) on the line (x+3)/5 = (y-1)/2 = (z + 4)/3. Also, find the length of the Perpendicular. (2,3,-1), √21
26) Find the length of the Perpendicular from the point (1,2,3) to the line (x-6)/3 = (y-7)/2 = (z -7)/-2. 7 units
27) Find the foot of the Perpendicular drawn from the point 2i - j + 5k to the line r= (11i - 2j - 8k) + λ(10i -4j - 11k), also , find the length of the Perpendicular. 14
28) Find the image of the point (1,6,3) in the line x/1= (y-1)/2 = (z -2)/3. Also, write the equation of the line joining the given point and its image and find the length of the segment joining the given point and its image. (1,0,7), (x-1)/0 = (y-6)/-6 = (z -3)/4, 2√13
29) Show that the distance d from point P to the line l having equation r= a + λb is given by d= |b x PQ|/|b|, where Q is any point on the line l.
30) Find the distance from the point P(3,-8,1) to the line (x-3)/3 = (y+ 7)/-1 = (z +2)/5 by using the formula derived in question 29. √(94/35)
31) Vertices B are C of ∆ ABC lie along the line (x+2)/2 = (y-1)/1 = (z -0)/4, find the area of triangle given that A has coordinates (1,-1,2) and line segment BC has length 5. √(1774/28)
32) Find the shortest distance between the lines r= (4i - j) + λ(i + 2j - 3k) and r= (i - j + 2k) + μ(2i + 4j - 5k).
33) Find the shortest distance between the lines (x-1)/2 = (y-2)/3 = (z -3)/4 and (x-2)/3 = (y-4)/4 = (z -5)/5. 1/√6
34) By computing the shortest distance determine whether the following pairs of lines intersect or not:
a) r= (i - j) + λ(2i + k) ; and r= (2i - j) + μ(i - j - k).
b) (x -1)/2 = (y+1)/3 = z; (x +1)/5 = (y-2)/1 = z= 2.
35) Find the shortest distance between the lines whose vector equations are
r= (i + 2j-3k) + λ(2i + 3j + 4k) ; and r= (2i + 4j+ 5k ) + μ(4i + 6j + 8k).
TEST PAPER -1
PLANES
THE PLANE
1) The sum of the intercepts made by the plane ax+ by+ cd= d on the three axes of reference is
a) a+ b+ c b) 1/a+ 1/b+ 1/c c) d(1/a+ 1/b+ 1/c) d) (1/d) √(a²+ b²+ c²)
2) If the sum of of the reciprocals of the intercepts made by the plane ax+ by + cz= 1 on the three axes is 1 then the plane always passes through the point.
a) (2,-1,0) b) (1,1,1) c) (-1,-1,-1) d) 1/2,-1,1/2)
3) The direction cosines of the Perpendicular from the origin to the plane 3x - y + 4z= 5 are
a) 4,-1,3 b) 3,-1,4 c) 3/√26,-1/√26,4/√26 d) 4/√26,-1/√26,3/√26
4) The length of the Perpendicular from the origin to the plane 2x + 3y + Kz = 1 (K> 0) is 1/5. Then K is
a) 2√3 b) 3√2 c) 0 d) 1
5) The direction cosines of the normal to the plane 5(x -2) = 3(y -z) are
a) 5,-3,3 b) 5/√33, -3/√33, 3/√43 c) 1/2,-3/10, 3/10 d) 1,-3/5,3/5
6) A plane passing through the line joining the points A(1,-3,5) and B(4,1,-1) is turned about AB till it passes through the origin. The equation of the plane in the new position is
a) 3x + 4y - 6z= 0 b) 2x - 21y +13z= 0 c) 2x - 21y - 13z= 0 d) none
7) The equation of a line passing through the point (-1,0,3) and perpendicular to the plane 4x + 3y - 5z= 12 are
a) (x-1)/4 = y/3 = (z+3)/-5
b) 5(3x -1) - 20(y - 1)= -4(z -4)
c) (x + 1)/-5 = y/3 = (z -3)/4 d) none
8) The equation of the plane passing through the line
(x -1)/2 = (y+1)/-1= z/3 and parallel to the direction whose direction numbers 3,4,2 is
a) 14x -5y - 11z= 19
b) 3x + 4y + 2z + 1= 0
c) 2x - y +3z= 3 d) none
9) If the line (x -1)/1 = (y+1)/-2 = (z+1)/K lies in the plane 3x - 2y +5z= 0 then K is
a) 1 b) -7/5 c) 5/7 d) no possible value
10) The equation of the line of intersection of the plane x+ y+ z= 2 and 3x - y + 2z= 5 in symmetric form are
a) (x - 7/4)/4 = (y-1/4)/-1= z/-3
b) x/3 = (y+1/3)/1= (z-7/3)/-4
c) x/1 = (3y+1)/1= (3z-7)/-4 d) none
11) The direction cosines of a line parallel to the plane 3x + 4y + z= 0 and x - 2y - 3z = 5 are
a) (-1,1,-1) b) (-1/√3, -1/√3, 1/√3) c) (-1/√3, 1/√3, -1/√3) d) none
12) If (3, K, L) is a point on the line 2x + y + z -3= 0= x - 2y + z -1= 0 then K, L is
a) -8/3,-1/3 b) -1/3,-8/3 b) -1,-5 c) -5, -1
13) The equations of the perpendicular from the point (-2,4,1) to the plane 7x -2y + 3z -1= 0 are
a) (x - 5)/7 = (y-2)/-2= (z-4)/3
b) (x - 2)/7 = (y+4)/-2= (z+1)/3
c) (x +2)/1 = (y-4)/4= (z- 1)/-2 d) none
14) P is a point on the y- z plane, making equal angles with y-axis and z-xis and at a distance 2 from the origin M is the foot of the perpendicular from P to the plane 3x + y - √2 z = 2√2. The coordinates of M are
a) (1,5/3, √2/3) b) (1,-3,-2) c) (1/√2,5/3√2, 1/3) d) none
15) The distance of the point (2,0,3) from the plane 5x - 12y = 0 is
a) 10/13 b) 46/13 c) 36/13 d) none
16) The image of the point P(α, β,γ) by the plane lx + my + nz=0 is the point Q(α', β',γ'). then
a) α²+ β²+ γ²= l²+ m²+ n²
b) α²+ β²+ γ²= α'²+ β'²+ γ'²
c) αα'+ ββ'+ γγ'= 0
d) l(α-α')+ m(β-β') + n(γ-γ') = 0
17) The image of the point (2,-1,1) by the plane 3x + 4y - 5z =0 is
a) (-2,1,-1) b) (2/3,-1/4,-1/5) c) (59/25,-13/25,2/5) d) none
18) if the image of the point (1,1,1) by a plane (3,-1,5) then the equation of the plane is
a) x - y + 2z -8= 0 b) x - y + 2z -16= 0 c) x - y + 2z - 14= 0 d) none
19) The angle between the line x= y= z and the plane 4x - 3y + 5z - 2 = 0 is
a) cos⁻¹(√6/5) b) sin⁻¹√(6/5) c) π/2 d) sin⁻¹(1/√6)
20) The equation of the plane passing through the origin and containing the line of intersection of the plane 5x + y -3z - 2= 0 and x + y + 3z - 1= 0 is
a) 2x + y - 1= 0 b) x - y - 3z= 0 c) 4x - y - 6z= 0 d) 7x + 5y + 3z = 0
21) What is the equation of the plane passing through the line of intersection of the planes x - y + 3z - 4= 0 and 2x + y + 3z - 5= 0 and parallel to the plane x + y + z - 1= 0 ?
a) x + y + z - 2 = 0 b) x + y + z + 2 = 0 c) 2x= y+ z d) no plane exists
22) What is the equation of the plane passing through the line 3x + y - 5z =2 = x -2y + 3z and perpendicular to the plane x - y + z - 3 = 0
a) 2x +3y +z = 2 b) 3x +2y - z = 2 c) 7(x - z) = 0 d) no plane exists
23) The equation of the plane passing through the line x +y - 2 = 0 = x - y -2z and at a distance 1 from the point (0,1,1) is
a) 2x + y - z - 3= √3(2- x - y)
b) x - y -2z - 2 + √3(x + y - 2) = 0
c) x + y - 2 = √3(x - y - 2z) d) none
24) The angle between the planes x + y + z = 0 and 3x - 4y + 5z = 0 is
a) cis⁻¹(1/5 √(2/3)) b) π/2 c) π/3 d) cos⁻¹(2/4 √(2/3))
25) The variable plane (2K +1)x + (3- K)y + z - 4 = 0 always passes through the line
a) x/0= y/0= (z+4)/1 b) x/1= y/2= z/-3 c) x/1= y/2 = (z -4)/-7 d) none
26) The distance between the plane 4x - 5y + 3z - 5=0 and 4x - 5y + 3z +2 = 0 is
a) 7/2√5 b) 7 c) 7/5√2 d) 3
27) The distance between the planes x +2y - 3z -4 = 0 and 2x +4y -6z - 1= 0 along the line x/1= y/-3 = z/2 is
a) 19/22 c) 3/22 c) 5 d) none
28) The shortest distance between the lines x - y = 0 = 2x + z and x + y - 2 = 0 = 3x - y + z - 1 is
a) 1/√3 b) 1/2√3 c) 1/2 d) 1
29) Which of the following planes is equally inclined to the planes 4x +3y -5z = 0 and 5x +3y -5z = 0 and 5x - 12y + 13z = 0 ?
a) 11x - 3y = 0 b) 3x +11y= 0 c) 3x +11y -65z = 0 d) none
30) The equation of the plane bisecting the angle between the planes 3x +4y - 4=0 and 6x - 2y + 3z +5 = 0 that contains the origin, is
a) 9x - 38y + 15z +43 = 0
b) 51x +18y + 15z - 13 = 0
c) 9x +2y + 3z +1 = 0 d) none
31) The equation of the plane bisecting the acute angle between the planes x - y + z - 1 = 0 and x + y + z - 2 = 0 is
a) x + z - 3/2 = 0 2y= 1 c) x - y - z - 3 = 0 d) none
32) The direction cosines of the projection of the line x/2= ( y-1)/1= (z +1)/-1 = 0 on the plane 2x + y -3z - 5 = 0 are
a) 2,-1,1 b) 2/7,-1/7,1/7 c) -2/√6, 1/√6, -1/√6 d) 2/√6,-1/√6,1/√6
33) Two systems of rectangular axes have the same origin. If a plane cuts them at distances a, b, c are A', b', c' from the origin then
a) a⁻² + b⁻² - c⁻² + a'⁻² + b'⁻² - c'⁻²=0
b) a⁻² - b⁻² - c⁻² + a'⁻² - b'⁻² - c'⁻²=0
c) a⁻² + b⁻² + c⁻² - a'⁻² + b'⁻² - c'⁻²=0 d) none
34) The lines x= ay +b, z= cy+ d and x= a'y +b', z= c'y+ d'
will be perpendicular and only if
a) aa' + bb' + cc'=0
b) (a+a')(b + b') + c+ c'=0
c) aa' + cc' +1=0
d) aa' + bb' + cc'+ 1=0
35) Which of the following planes intersects the plane x - y + 2z - 3 = 0 and 4x + 3y - z - 1 = 0 along the same line?
a) 11x + 10y -4z = 0
b) 7x + 7y -4z = 0
c) 5x + 2y + z - 2 = 0 d) none
36) The line (x -1/0)/2= y.-1= (z + 2)/2 cuts the plane x + y + z - 1 = 0 at P. If the foot of the perpendicular from P to a plane be (3,-4,1) then the equation of the plane
a) 3x -2y - z = 0 b) 2x -y + 2z - 12 = 0 c) 2x -10y + 5z - 51= 0 d) none
37) A variable plane at a distance of 1 unit from the origin cuts the coordinates axes at A, B and C. If the centroid D(x, y, z) satisfies the relation x⁻² + y⁻² + z⁻²= k then the value of k is
a) 3 b) 1 c) 1/3 d) 9
38) A plane through the line (x -1)/1= +y+1)/-2 = z/1 has the equation
a) x + y +z = 0
b) 3x + 2y - z = 0
c) 4x + y -2z - 3= 0
d) 3x + 2y + z= 0
39) The equation of a plane is 2x - y - 3z = 5 and A(1,1,1), B(2,1,-3) C(1,-2,-2) and D(-3,1,2) are four points. Which of the following line segments are intersected by the plane?
a) AD b) AB c) AC d) BC
1c 2b 3c 4a 5b 6c 7b 8a 9b 10b 11c 12b 13a 14d 15a 16b 17c 18a 19b 20b 21a 22c 23a 24d 25c 26c 27d 28b 29a 30b 31a 32d 33c 34c 35a 36c 37d 38ac 39bc
Exercise - A
1) Find the equations of the three planes through the points (3,1,1),(1,-2,3) parallel to the coordinates axes. 2y+3z= 5, x+ z=4, 3x -2y=7
2) Find the intercepts made on the coordinates axes by the plane x+ 2y - 2z = 6. Find also the direction cosines of the normal to the plane. (6,3,-3), x/3+2y/3-2z/3= 2, (1/3,2/3,-2/3)
3) Find the equation of the plane passing through the three points (2,2,-1),(3,4,2),(7,0,6). 5x+ 2y - 3z -17=0
4) Show that the point O(-1/2,2,0) is the circumcentre of the triangle formed by the points P(1,1,0), Q(1,2,1) and R(-2,2,-1).
5) Find the equation of the plane through the point (x₁, y₁, z₁) parallel to the plane ax+ by + cz=0. a(x - x₁)+ b(y - y₁)+ c(z - z₁) =0
6) Find the equation of the plane which passes through the point (2,1,-1) and is orthogonal to each of the planes x- y+z=1 and 3x + 4y-2z=0. 2x - 5y - 7z - 6 =0.
7) Find the equation of the plane passing through the points (1,1,2) and (2,4,3) and perpendicular to the plane x- 3y + 7z +5=0. 4x - y - z-1=0
8) If P be the point (2,3,-1), then find the equation of the plane through P at right angles to the straight line OP, where O is the origin. 2x + 3y - z= 14.
Exercise - B
1)a) Find the equation of the plane through the points (2,3,-4),(1,-1,3) and parallel to the x-axis. 7y + 4z- 5=0
b) Show that the intercepts made on the axes by the plane 4x +3y - 2z+ 12 =0 are (-3),(-4),(6).
c) Find the points where the plane ax+ by + cz+ d=0 (a,b,c≠0) meets the coordinates. (-d/a,0,0), (0,-d/b,0),+0,0,-d/c)
d) Does the point (4,-6,0) lies on the plane which intersects the positive x, y, z-axis at a distance 2,3,5 units respectively. No
2) a) A plane makes equal nonzero intercepts on the axes measured from the origin and passes through the point (1,2,3). Show that its equation is x + y + z - 6=0.
b) A plane passes through the point (-2,3,1) and its intercepts measured from the origin on the x and y axis are 4 and 3 respectively. Find the equation of the plane. 3x + 4y + 6z-12 =0
c) If the sum of the reciprocals of intercepts on the coordinates axes of a plane be constant, then show that the plane always passes through a fixed point.
3) a) The foot of the perpendicular from the origin to the plane is (3,2,-1). Show that the equation of the plane is 3x +2y - z-14 =0
b) If P be the point (-3,1,1) and Q be the point (3,4,2), then the equation of the plane through R at right angles to PQ, where R lies on PQ and PR= (1/3) PQ. 3x +4y + 6z-12 =0
c) Show that the plane bisecting the straight line joining the points (-1,2,3) and (3,-5,6) at right angles is 4x - 7y + 3z- 28=0
d) Find the equation of the plane through the point (2,-3,1) and which is perpendicular to the straight line joining the point (2,-3,1) and which is perpendicular to the straight line joining the two points (3,4,-1) and (2,-1,5). x +5y - 6z + 19 =0
4) Show that the equation of the plane through the point P(a,b,c) and perpendicular to the straight line OP, where O is the origin, is ax+ by + cz= a²+ b²+ c².
5) Reduce the equation of the plane 6x - 3y +2z -14 =0 to the normal form and show that the distance of the origin from the plane is 2 units and the direction cosines of the perpendicular from the origin to the plane are 6/7, (-3/7), 2/7.
6) a) Show that the equation of the plane through the points
i) (2,3,-3),(1,1,-2) and (-1,1,4) is 3x - y + z =0
ii) (2,2,2(3,1,1) and (6,-4,-6) is x +2y - z- 4 =0
iii) (3,3,1),(-3,2,-1) and (8,6,3) is 4x +2y - 13z-5 =0
b) Show that the points (3,9,4), (4,5,1),(-4,4,4) and (0,-1,-1) are coplanar.
c) Show that the points (2,1,-2) lies on the plane passing through the points (1,0,0),(0,1,0),(0,0,1).
7) a) Show that the equation of the plane through the point
i) (1,2,3) and parallel to the plane 3x +4y -5 z =0 is 3x +4y - 5z + 4 =0.
ii) (0,4,-3) and parallel to the plane 3x - 4y + 7z + 3 =0 is 3x - 4y + 7z+ 37 =0
iii) (2,-3,5) and parallel to the yz plane is x= 2.
b) Show that the equation of the plane parallel to the plane 2x +4y + 5z- 6 =0 and the sum of whose intercepts on the coordinates axes is 19, is 2x +4y + 5z- 20=0
8) Determine the values of K and L for which the two planes Kx +y - 2z + 4 =0 and 6x - My - 4z- 9 =0 are parallel. 3,-2
9) Determine the value of h for which the planes 3x - 2y + hz-1 =0 and x +hy + 5z +2 =0 may be perpendicular to each other. -1
10) Show that the angle between the planes
i) x - y +2z- 9=0 and 2x + y + z- 7 =0 is π/3.
ii) 2x - y +2z- 3 =0 and 3x +6 y +2z-4 =0 is cos⁻¹(4/21).
b) Find the angles that the plane x +8y - 6z + 16=0 makes with the cordinates planes. cos⁻¹(1/√101),cos⁻¹(3/√101), cos⁻¹(6/√101)
11) Show that the equations of the planes passing through the points (0,4,-3), (6,-4,3) and cutting off intercepts from the axes whose sum is zero, are 2x -3y - 6z - 6=0 and 6x +3y - 2z - 18=0.
12) Show that the equation of the plane through the points
i) (1,2,3), (3,2,-1) and
11) Show that the equations of the planes passing through the points (0, 4,-3), (6,-4,3) and cutting off intercepts from the axes whose sum is zero, are 2x -3y - 6z - 6=0 and 6x +3y - 2z - 18 =0.
12) Show that the equation of the plane through the points
i) (1,2,3), (3,2,-1) and perpendicular to the plane 3x +2y +6z - 4 =0 is 2x -6y + z +7=0,
ii) (0,2,3), (5,-1,4) and perpendicular to the plane x +2y +3=0 is 2x - y - 13z +4=0.
iii) (2,1,1), (3,2,2) and perpendicular to the plane x +2y - 5z - 3=0 is 7x -6y - z - 7=0.
13) Show that the equation of the plane passing through the point (-1,3,2) and perpendicular to the planes x +2y +2z - 5 =0 and 3x + 3y +2z - 8=0 is 2x -4y +3z - 8=0.
14) Find the equation of the plane which passes through the point (2,1,4) and is perpendicular to each of the planes 9x -7y + 6z +48=0 and x + y - z =0. x +15y +16z - 81=0
15) Show that the planes x +2y + 2z =0 and 2x +y -2z=0 are orthogonal. Find another plane through the origin which is perpendicular to each of the above planes. 2x - 2y + z= 0
16) prove that the equation of the plane through the points
i) (1,-2,4) and (3,-4,5) and parallel to the x-axis is y + 2z - 6=0
ii) (3,-1,2) and (2,1,-4) and perpendicular to the xz-plane is 6x -z -16=0 .
iii) (2,-3,4) and (-1,5,2) and perpendicular to the xy-plane is 8x +3y -7=0.
17) Perpendiculars OL, PM, PN are drawn from the point P(a,b,c) to the coordinates plane. Show that the equation of the plane LMN is x/a + y/b + z/c = 2.
18) A plane cuts the axes in A, B, C and the centroud of the triangle ABC is (a,b,c). Show that the equation of the plane is x/a + y/b + z/c = 3.
19) A variable plane which is at a constant distance 3p from the origin O cuts the axes in A, B, C. Show that
i) the locus of the centroid of the triangle ABC is
x⁻² + y⁻² + z⁻² = p⁻² and that of the tetrahedron OABC is 9(x⁻² + y⁻² + z⁻²) = 16p⁻².
ii) the locus of the point of intersection of the planes through A, B, C drawn parallel to the coordinate planes is 8(x⁻² + y⁻² + z⁻² = p⁻².
20) A variable plane passes through the point (f,g,h) and meets the axes in A, B, C. If the planes through A, B, C and parallel to the axes meet at P, then show that the locus of P is fx⁻¹ + gy⁻¹ + hz⁻¹ = 1.
21) A variable plane has intercepts on the co-ordinate axes, the sum of whose squares is a constant k². Show that the locus of the foot of the perpendicular from the origin to the plane is (x²+ y²+ z²)²(x⁻² + y⁻² + z⁻²) = k².
22) A point P moves on the plane x/a + y/b + z/c = 1, which is fixed and the plane through P perpendicular to OP meets the axes in A, B, C. if the planes through A, B, C parallel to the co-ordinate planes meet in a point Q. then show that the locus of Q is
1/x² + 1/y² + 1/z²= 1/ax + 1/by + 1/cz.
(the equation of the plane perpendicular to OP where P is the point (α, β, γ) is α(x - α)+ β(y - β) + γ(z - γ)=0. Also αa⁻¹+ βb⁻¹+ γc⁻¹= 1.
if this plane meet pa the axes in A, B, C then the equation of the plane through A parallel to yz-plane is xα = α²+ β²+ γ² and similarly for others. Find the locus of P.)
23) A makes intercepts OA= a, OB= b and OC= c on the axes. Show that the area of the triangle ABC is
(1/2) √(b²c²+ c²a²+ a²b²).
24) Through the point P(α, β, γ) a plane is drawn at right angles to OP to meet the rectangular axes in A, B, C. Show that the area of the triangle ABC r⁵/2αβγ, where OP= r.
25) A plane triangle of sides of the lengthta, b, c is placed so that the mid points of the sides are on the axes. Show that the lengths α, β, γ of intercepts on the axes are given by 8α²= b²+ c²- a², 8β²= c²+ a²- b², 8γ²= a²+ b²- c² and that the coordinates of the vertices are (-α, β, γ), (α, -β, γ) and (α, β, -γ).
Exercise -C
1) a) Find the distance of the point (1,2,-3) from the plane 5x - 3y + z+5=0. 1/√35
b) Prove that the plane 3x - 4y -2z+5=0 cuts the line segment bounded by the points (3,-2, 1) and (-2,5,2).
2) Find the distance between the two parallel planes 2x + 5y + 4z -12=0 and 2x +5y + 4z+6 =0. 6√5/5
3) Find the coordinates of the point the situated on the z-axis equidistant from the point (1,-2,0) and the plane 3x - 2y + 6z - 9 =0. (0,0,-2) & (0,0,-82/13)
4) Find the equation of the plane parallel to the plane 2x - 2y - z- 3=0 and situated at a distance of 7 units from it. 2x - 2y - z -24=0 & 2x - 2y - z +18 =0.
5) Determine if the origin lies inside the acute or the obtuse angle formed by the two planes x - 2y +3z - 5=0 and 2x - y - z +3 =0. Between
6) Find the equation of the planes bisecting the angles between the planes 2x +y - 2z - 4=0 and 2x - 3y + 6z + 2=0. Discuss their positions. 10x - y +2z - 11=0 & 4x +8y - 16z -17=0.
7) Find the equations of the plane passing through the intersection of the planes 2x +y +2z -9=0 and 4x - 5y - 4z - 1=0 and the point (3,2,-1). 11x - 5y - z -24=0
8) Prove that the equation 2x² - 6y² - 12z²+ 18yz + 2zx + xy =0
represent a pair of planes. Find the angle between them .
Exercise -D
1) Show that the points (1,-1,3) and (3,3,3) are equidistant from the plane 5x + 2y - 7z -+9 =0 and on opposite sides of it.
2) a) Find the distance of the point (1,2,0) from the plane 4x +3y +12z +16 =0.
b) Find the distance of the point
i) (2,-1,-1) from the plane 16x - 12y +15z -4=0.
ii) (3,-6,7) from the plane 4x - 3z - 1=0.
c) Prove that, if a plane has intercepts l, m, n on the axes and be at a distance p from the origin, then 1/l²+ 1/m²+ 1/n²= 1/p².
3) Find the distance of the point (-1,1,-2) from the plane passing through the point (1,-1,1),(-2,1,3) and (4,-5,-2).
4)a) Show that the origin and the point (2,-1,1) lies on the same side of the plane 2x +7y +3z +1=0 but on opposite sides of the plane x +5y +12z - 1=0.
b) Show that the plane 5x - 2y + z - 1=0 does not cut the line segment joining (1,4,-3) and (2, 5,0).
5) Show that the distance between the parallel planes
i) 2x - 2y + z +6 =0 and 4x - 4y + 2z +7 =0 is 5/6 unit
ii) 2x - 2y + 6z -14 =0 and 4x - 6y + 12z +21 =0 is 7/2 units .
Find the equation of the plane parallel to and midway between the two parallel planes of (i).
6) Find the planes which are parallel to the plane 3x - 2y + 6z + 8 =0 and at a distance of 2 units from it.
7) a) Find the points on the y-axis situated at a distance of 4 units from the x + 2y -2z -2 =0.
b) Find the plane on the z-axis which are is equidistant from the two planes
3x - 6y + 2z +1 =0 and 2x - y + 2z -3 =0.
8) Determine whether the origin lies inside the acute or obtuse angle formed by the planes .
x - 2y + 3z -5 =0 and 2x - y -z +3 =0.
9) Show that the point (3,2,-1) lies inside the acute angle formed by the planes
5x - y + z +3=0 and 4x - 3y + 2z +5 =0.
10) a) Find the equation of the plane bisecting the angle between the planes 3x - 6y + 2z + 5 =0 and 4x - 12y + 3z -3 =0 which contains the origin.
b) Find the equation of the plane bisecting the obtuse the dihedral angle between the planes
x + 2y + 2z -19 =0 and 4x - 3y + 12z +3 =0.
11) a) Show that the plane 14x - 8y + 13 =0 bisects the obtuse angle between the planes , 3x +4y -4z +1 =0 and 5x +12y - 13z =0.
b) Show that the bisector of the acute angle between the planes 2x - y + 2z +3 =0 and 3x - 2y + 6z + 8 =0 is 22x - 13y + 32z + 45 =0.
12) a) Find the locus of a point the sum of squares of whose distances from the planes x +y + z =0, x + y - z =0 and x - 2y + z =0 is 9.
b) Find the locus of a point the sum of square of whose distances from the planes x + y + z =0, x - y =0 and x + y - 2z =0 is 4.
13) a) show that equation of the plane passing through the point (-1,0,1) and the line of intersection of the planes 4x - 3y + 1=0 and y - 4z +13 =0 is 3x - 2y - z + 4=0.
b) Show that the equation the plane through the intersection of the planes x - 2y + 3z + 4 =0 and 2x - 3y + 4z -7 =0 and the point (1,-1,1) is 9x - 13y + 17z - 39 =0.
14) a) Show that the equation of the plane through the intersection of the planes x +2y + 3z -4 =0 and 2x + y - z +5 =0 and which is perpendicular to the plane 5x + 3y + 6z +8 =0 is 51x +15y -50z + 173 =0.
b) Show that the equation of the plane passing through the line of intersection of the Planes x +y + z -6 =0 and 2x + 3y + 4z +5 =0 and perpendicular to the plane 4x +5y - 3z -8 =0 is x +7y + 13z + 96=0.
15) Show that the equation of the planes through the intersection of the planes x + 3y + 6 =0 and x - y - 4z =0 whose perpendicular distance from the origin is unity, are
2x +y -2z +3 =0 and x - 2y - 2z - 3 =0.
16) Show that the equation of the plane through the intersection on the Planes ax + by + cz +d =0 and a'x + b'y + c'z + d' =0 and perpendicular to the xyplane is
(ac' - a'c)x + (bc' - b'c)y + (dc'- d'c) =0.
17) Find the equation of the plane through the intersection of the Planes 2x +2y -3z -2 =0 and x - 5y - 3z -1 =0 and cutting a triangle of area 2 units in the xyplane by its intersection with the xy-plane and the xaxis and y axis.
18) a) The plane ax+ by=0 is rotated about its line intersection with the Plane z=0 through an anglesp α. Show that the equation of the plane in the new position is
ax + by ± z √(a²+ b²) tan.
( the plane ax+ by + Kz= 0 makes an angle α with the plane the plane ax+ by=0)
b) The plane x - 2y + 3z -4= 0 is rotated through a right angle about its line of intersection with the plane 2x +3y -4z - 5= 0. Show that the equation of the plane in its new position is 22x +5y -4z -67= 0.
19) Show that the equation 2x² - y² + 3z² -xy+ 7xz + 2yz= 0 represents two planes and the angle between the two planes tan⁻¹(5√2)/4.
20) Show that the equation a/(y - z)+ b/(z - x)+ c/(x - y)=0 represents a pair of planes.
2) a) 2 units b)i) 1 unit ii) 2 units 3) 4 units 5) 8x - 8y + 4z +19= 0 6) 3x - 2y + 6z - 6= 0, 3x - 2y + 6z +22 = 0
7) a) (0,7,0), and (0,-5,0)
b) (0,0,3) and (0,0,9/10)
8) inside
10) a) 66x - 162y + 47z + 44= 0
b) x + 35y - 10z - 256= 0
12) a) 5x²+ 8y²+ 5z²+ 4xy -4yz + 2zx - 54= 0
b) x² + y² + z² -4= 0
17) 16x - 4y - 29z -16= 0, 16x +4y -27z - 16= 0
Miscellaneous - 1
1) Find the equation of the plane through the points A(2,2,-1), B(3,4,2) and C(7,0,6). 5x + 2y - 3z= 17
2) Find the equation of the plane passing through P(1,1,0), Q(1,2,1) and R(-2,2,-1). 2x + 3y - 3z= 5
3) If from a point P (a,b,c) perpendiculars PA and PB are drawn to yz and zx-plane, find the equation of the plane OAB. x/a + y/b - z/c = 0.
4) Show that the four points (0,-1,-1),(-4,4,4),(4,5,1) and (3,9,4) are coplanar.
5) Write the equation of a plane whose intercepts on the coordinates axes are -4,2 and 3 respectively. -3x + 6y +4z= 12
6) Reduce the equation of the plane 2x + 3y - 4z= 12 to intercept form and find its intercepts on the coordinate Axes. x/6+ y/4+ z/-3= 1, (6,4,-3)
7) A plane meets the coordinates axes in A, B, C such that the centroid of triangle ABC is the point (p,q,r). Show that the equation of the plane is x/p + y/q + z/r= 3.
8) A variable plane moves in such a way that the sum of the reciprocals of its intercepts on the three coordinate Axes is constant. Show that the plane passes through a fixed point.
9) Find the vector equation of a plane passing through a point having position vector 2i + 3j - 4k and perpendicular to the vector 2i - j + 2k. Also, reduce it to cartesian form. r. (2i- j+ 2k)= -7, 2x - y + 2z= -7
10) Find the equation in cartesian form of the plane passing through the point (3,-3,1) and normal to the line joining the points (3,4,-1) and (2,-1,5). r. (-i - 5j + 6k)= 18, x+ 5y - 6z +18=0
11) The foot of perpendicular drawn from the origin to the plane is (4,-2,-5). Find the equation of the plane. r. (4i- 2j - 5k)= 45, 4x - 2y - 5z= 45
12) If the line drawn from the point (-2,-1,-3) meets a plane at right angle at the point (1,-3,3), find the equation of the plane. r. (3i- 2j + 6k)= 27, 3x - 2y +6z= 27
13) Find the equation of the plane which bisects the line segment joining the points A (2,3,4) and B (4,5,8) at right angles. r. (i + j + 2k)= 19, x +y +2z= 19
14) Find the vector equation of the plane whose cartesian form of equation is 3x - 4y + 2z= 5. r. (3i- 4j + 2k)= 5
15) Find a normal vector to the plane 2x - y +2z= 5. Also, find a unit vector normal to the plane. n= 2i - j + 2k, (1/3) (2i - j + 2k)
16) Find the equation of the plane passing through the point (1,-1,2) having 2,3,2 as direction ratios of normal to the plane. r. (2i + 3j + 2k)= 3, 2x +3y +2z= 3
17) Let n be a vector of magnitude 2√3 such that it makes equal acute angles with the coordinates axes. Find the vector and cartesian form of the equation of a plane passing through (1,-1,2) and normal to n. r. (i + j + k)= 2, x + y + z= 2
18) Find the angle between the normals to the planes 2x - y + z = 6 and x+ y + 2z= 7. π/3
19) Show that the normals to the planes r. (i- j +k)= 3, r. (3i+ 2j - k)= -5 are perpendicular to each other.
20) Find the angles at which the normal vector to the plane 4x + 8y + z= 5 is inclined to the coordinates axes. cos⁻¹(4/9), cos⁻¹(8/9), cos(1/9).
21) A vector n of magnitude 8 units is inclined to x-axis at 45°, y-axis at 60° an acute angle with z-axis. If a plane passes through a point (√2,-1,1) and is normal to n, find its equation in vector form. r. (√2i+ j + k)= 2
22) Find the vector equation of a plane at a distance of 5 units from the origin and has i as the unit vector normal to it. r. i= 5.
23) Find the vector equation of a plane which is at a distance of 8 units from the origin and which is normal to the vector 2i+ j + 2k. r. (2i + j + 2k)= 24
24) Reduce the equation r. (3i- 4j + 12k)= 5 to the normal form and hence find the length of perpendicular from the origin to the plane. r. (3i/13 - 4j/13 +12k/13)= 5/13, 5/13
25) Reduce the equation of the plane x - 2y - 2z = 12 to normal form and hence find the length of the perpendicular from the origin to the plane. Also, find the direction cosines of the normal to the plane. x/3- 2y/3- 2z/3= 4, 4, 1/2,-2/3,-2/3.
26) Find the vector equation of a plane which is at a distance of 6 units from the origin and has 2,-1,2 as the direction ratios of a normal to it. Also, find the coordinates of the foot of the normal drawn from the origin. r. (2i/3 - j/3 +2k/3)= 6, 4i - 2j +4z, 4,-2,4
27) Find the coordinates of the foot of the perpendicular drawn from the origin to the plane 2x - 3y + 4z - 6=0. (12/29,-18/29,24/29)
28) Find the direction cosines of perpendicular from the origin to the plane r. (2i- 3j - 6k)+5 = 0. -2/7,3/7,6/7
29) Find the equation of the plane passing through the point (-1,2,1) and perpendicular to the line joining the points (-3,1,2) and (2,3,4). Find also the perpendicular distance of the origin from this plane. r. (5i + 2j +2k)= 1, 1/√33
30) Find the vector equation of the plane passing through the points A(2,2,-1), B(3,4,2) and C(7,0,6). Also, find the cartesian equation of the plane. r. (5i + 2j - 3k)= 17, 5x +2y - 3z= 17
31) Find the vector equation of the plane passing through the points having position vectors i+ j - 2k), 2i- j+ k and i+ 2j + k. r. (9i + 3j - k)= 14
32) If from a point P (a,b,c) perpendicular PA and PB are drawn to YZ and zx-plane, find the vector equation of the plane OAB. r. (bci + caj - abk)= 0
33) Find the angle between the planes r. (2i- j +k)= 6, r. (i+ j + 2k)= 5. π/3
34) Find the angle between the planes x +y +2z= 9 and 2x - y + z= 15. π/3
35) Show that the planes 2x + 6y + 6z= 7 and 3x + 4y - 5z= 8 are at right angles.
36) If the planes r. (2i- j - λk)= 5, and r. (3i+ 2j +2k)= 4 are perpendicular . Find the value of λ. -2
37) Find the equation of the plane passing through the point (1,1,-1) and perpendicular to the planes x + 2y +3z -7= 0 and 2x - 3y + 4z= 0. 17x + 2y - 7z= 26
38) Find the equation of the plane through the q(2,1,-1) and (-1,3,4) and perpendicular to the plane x - 2y + 4z= 10. Also, show that the plane thus obtained contains the line r= - i+ 3j + 4k + λ(3u - 2j - 5k). 18x +17y +4z= 49
39) Find the equation of the plane through the points (3,4,2) and (7,0,6) and is perpendicular to the plane 2x - 5y= 15. 5x + 2y - 3z -17=0
40) Find the vector equation of the following plane in scalar product form: r= (i - j)+ λ(1+ j + k)+ μ(i - 2j + 3k). r. (5i- 2j - 3k)= 7
41) Find the cartesian form of the equation r= (s- 2t)i+ (3- t)j +(2s + t)k. r. (2i- 5j - k)= -15, 2x - 5y - z= -15
42) Find the vector equation of the plane r = (1+ s - t)i + (2- s)j +(3- 2s+ 2t)k in non parametric form r. (2i + 0j +k)= 5
43) Find the equation of a plane through the intersection of the plane r. (i + 3j - k)= 5 and r. (2i- j +k)= 3 and passing through the point (2,1,-2). r. (3i + 2j + 0k)= 8
44) Find the equation of the plane containing the line of intersection of the plane x+ y + z -6=0 and 2x + 3y + 4z +5=0 and passing through the point (1,1,1). 20x + 23y +26z= 69
45) Find the direction ratios of the normal to the plane passing through the point (2,1,3) and the line of intersection of the planes x+ 2y + z= 3 and 2x - y - z= 5. 13,6,1
46) Find the equation of the plane which is perpendicular to the plane 5x+ 3y + 6z + 8= 0 and which contains the line of intersection of the planes x+ 2y + 3z - 4= 0 and 2x+ y - z +5 = 0. 51x+ 15y - 50z +173 = 0
47) Find the equation of the plane through the line of intersection of r. (2i -3j + 4k)= 1 and r. (i - j) + 4= 0 and perpendicular to r. (2i - j + k)= -8. r. (-5i + 2j + 12k)= 47.
48) Find the equation of the plane passing through the intersection of the planes 2x - 3y + z -4=0 and x - y + z +1=0 and perpendicular to the plane x+ 2y - 3z +6=0. x- 5y - 3z - 23=0
49) Find the cartesian as well as vector equations of the planes through the intersection of the planes r. (2i + 6j) +12= 0 and r. (3i - j + 4k)= 0 which are at a unit distance from the origin. r. (2i + j + 2k)+3= 0 and r. (-i + 2j -2k)+3= 0
50) The plane lx + my= 0 is rotated through an angle α about its line of intersection with the plane z=0. Show that the equation of the plane in its new position is lx + my± √(l²+ m² tanα)z= 0.
51) The plane x- 2y +3z=0 is rotated through a right angle about the line of intersection with the plane 2x+ 3y - 4z -5 =0, find the equation of the plane in its new position. 21x+ 5y - 4z -35 =0.
52) Find the equation of the point 2i + j - k from the plane r. (i - 2j + 4k) -9 = 0. 13/√21
53) Find the distance of the point (2,1,0) from the plane 2x + y + 2z + 5= 0. 10/3
54) Show that if a plane has the intercepts a, b, c and is at a distance of p units from the origin, then 1/a²+ 1/b²+ 1/c²= 1/p².
55) Show that the points i - j + 3k and 3(i + j + k) are equidistant from the plane r. (5i +2j + k)+ 9= 0 and lie on opposite sides of it.
56) Find the equations of the plane parallel to the plane x - 2y + 2z -3=0 which is at a unit distance from the point (1,2,3). x - 2y + 2z=0 and x - 2y + 2z -6=0
57) If the points (1,1,λ) and (-3,0,1) be equidistant from the plane r. (3i+ 4j - 12k)+13=0, find λ. 1 or 7/3
58) Find the distance between the point P(6,5,9) and the plane determined by the points A(3,-1,2) , B(5,2,4) and C(-1,-1,6). 6/√34
59) Find the equation of a plane passing through the point P(6,5,9) and parallel to the plane determined by the points A(3,-1,2), B(5,2,4) and C(-1,-1,6). Also find the distance of this plane from the point A. r . (3i - 4j + 3k) - 25 = 0, 6/√34
60) Two system of rectangular axes have the same origin. If a plane cuts them at distance a, b, c and A', b', c' respectively, show that 1/a²+ 1/b²+1/c²= 1/a'²+ 1/b'²+ 1/c'².
61) A variable plane which remains at a constant distance 3p from the origin cut the coordinate Axes at A, B, C. Show that the locus of the centroid of triangle ABC is 1/x²+ 1/y²+ 1/z²= 1/p².
62) A variable plane is at a constant distance p from the origin and meets the coordinates axes in A, B, C. Show that the locus of the centroid of the tetrahedron OABC is 1/x²+ 1/y²+ 1/z²= 16/p²
63) If a variable plane at a constant distance p from the origin meets the coordinate Axes in points A, B And C respectively. Through thess points, planes are drawn parallel to the coordinate planes. Show that the locus of the point of intersection is 1/x²+ 1/y²+ 1/z²= 1/p².
64)
65)
66) Find the distance between the two parallel planes x + y - z +4=0 and x +y -z +5=0. 1/√3
67) Find the distance between the parallel planes 2x - y + 2z +3 =0 and 4x - 2y + 4z +5 =0. 1/6
68) Find the distance between the parallel planes r . (2i - 3j + 6k) - 5 = 0 and r . (6i - 9j + 18k) +20 = 0. 5/3
69) Reduce in symmetrical form, the equation of the line x - y + 2z -5 =0 and 3x +y + z -6 =0. (4x - 11)/-3= (4y + 9)/5= (z -0)/1.
70) Reduce in symmetrical form, the equation of the line x = ay +b, z = cy+ d. (x - b)/a= (y -0)/1= (z -d)/c.
71) Find the angle between the lines x - 2y + z =0= x + 2y -2z and x+ 2y + z=0 and 3x + 9y + 5z. cos⁻¹(8/√406)
72) Find the angle between the lines r= (i+ 2j- k)+ λ(i - j + k) and the plane r. (2i - j + k)=4. sin⁻¹(2√2/3)
73) Find the angle between the lines (x +1)/3= (y -1)/2= (z -2)/4 and the plane 2x + y - 3z +4=0. Sin⁻¹(-4/√406)
74) If the line r= (i- 2j+ k)+ λ(2i +j + 2k) is parallel to the plane r. (3i - 2j + mk)=14, Find the value of m. -2
75) Show that the line whose vector equation is r= (2i- 2j+ 3k)+ λ(i - j + 4k) is parallel to the plane whose vector equation is r. (i + 5j + k)=5. Also, find the distance between them. 10/√27
76) Find the vector equation of the line passing through the point with position vector 2i - 3j- 5k) and perpendicular to the plane r. (6i - 3j + 5k) +2= 0. r= (2i- 3m- 5k)+ λ(6i - 3j + 5k).
77) Find the equation of the line passing through the point (3,0,1) and parallel to the planes x+ 2y= 0 and 3y+ z= 0. (x-3)/-2= (y-0)/1= (z -1)/3 or r= (3i+ k)+ λ(-2i + j + 3k)
78) Find the equation of the plane passing through the line of intersection of the planes 2x-0+ y- z=3, 5x- 3y + 4z+9=0 and parallel to the line (x -1)/2= (y-3)/4= (z -5)/5. 7x + 9y - 10z -27=0.
79) Find the equation of the plane passing through the intersection of the planes r. (i + j + k)=1 and r. (2i +3j - k) +4=0 is. r. (-j/2+ 3k/2)-3=0 or r. (- j + 3k)=6.
80) Find the equation of the plane through the w(1,0,1),(3,2,2) and parallel to the line (x-1)/1= (y-1)/-2= +z -2)/3. 4x - y - 2z-6=0
81) State when the line r= a+ λb is parallel to the plane r.n= d. Show that the line r= (i+ j)+ λ(2i + j + 4k) is parallel to the plane r. (-2i + k)=5. Also, find distance between the line and plane. 7/√5
82) Find the equation of the plane passing through the intersection of the planes 4x - y + z=10 and x+ y - z= 4 and parallel to the line with direction ratios proportional to 2,1,1. Find also the perpendicular distance of point (1,1,1) from this plane. 5y- 5z -6=0, 3√2/5
83) Find the equation of the plane passing through the point A(1,2,1) and perpendicular to the line joining the points P(1,4,2) and Q(2,3,5). Also, find the distance of this plane from the line (x+3)/2= (y-5)/-1= (z -7)/-1. x- y + 3z= 2, √11
84) Find the equation for the line passes through the point P(2,3,1) and is parallel to the line of intersection of the plane x- 2y + z= 0. (x -2)/1= (y -3)/1= (z -1)/1
85) Find the plane passing through (4,-1,2) and parallel to the lines (x +2)/3 = (y -2)/-1= (z -1)/2 and (x -2)/1= (y -3)/2 = (z -4)/3. x + y - z -1=0
86) Find the coordinates of the point where the line through the points A (3,4,1) and B (5,1,6) crosses the xy-plane. (13/5,23/5,0)
87) Find the distance between the point with position vector - i- 5j - 10k and the point of intersection of the line (x -2)/3 = (y +1)/4 = (z -2)/12 with the plane x - y +z= 5. 13
88) Find the distance of the point (1,-2,3) from the plane x - y + z= 5 measured parallel to the line whose direction cosines are proportional to 2,3, -6. 7/6
89) Show that the lines r= (i+ j - k)+ λ(3i - j) and r= (4i- k)+ μ(2i +3k) are coplanar . Also, find the plane containing these two lines. r. (3i + 9j - 2k)= 14
90) Show that the lines (x +1)/3 = (y +3)/5 = (z +5)/7 and (x -2)/1 = (y -4)/4 = (z -6)/7 are coplanar. Also, find the plane containing these two lines. x - 2y+ z= 0
91) Show that (x -a+ d)/(α- δ) = (y -a)/α = (z -a- d)/(α +δ) and (x -b+ c)/(β-γ) = (y -b)/β = (z -b - c)/(β+γ) are coplanar.
92) Find the vector equation of the plane that contains the lines r= (i + j)+ λ(i + 2j - k) are r= (i+ j)+ μ(-i +j - 2k). Also, find the length of the Perpendiculars drawn from the point (2,1,4) to the plane thus obtained. r. (-i+ j+ k)= 0, √3
93) Find the equation of the plane passing through the point (0,7,-7) and containing the line (x +1)/-3 = (y -3)/2 = (z +2)/1 . x + y+ z=0
94) Show that the plane whose vector equation is r. (i+ 2j- k)= 3 contains the line whose vector equation is r= (i+ j)+ λ(2i + j+4k).
95) Find the vector and cartesian equation of the plane containing the two lines r= (2i+ j - 3k)+ λ(i +2j+5k) and r= (3i + 3j +2k)+ μ(3i -2j +5k). r. (10i+ 5j - 4k)= 37, 10x + 5y - 4z = 37.
96) If 4x + 4y - λz= 0 is the equation of the plane through the origin that contains the line (x +1)/2 = (y +1)/3 = z/4, find the value of
and r= (4i- k)+ μ(2i +3k) are λ. 5
97) If this line (x -1)/2 = (y +1)/3 =( z-1)/4, and (x -3)/1 = (y - k)/2 = z/1 interesect, then find the value of k. And hence find the equation of the plane containing these lines. 9/2, 5x - 2y - z= 6.
98) Find the shortest distance between the skew lines M= (x -1)/2 = (y +1)/1 = (z-2)/4, N= (x +2)/4 = (y -0)/-3 = (z+1)/1. 5/√465
99) Find the distance the line r= (-i+ 3k)+ λ(i -2j) and the line passing through (0,-1,2) and (1,-2,3). 2/√6
100) Find the shortest distance between the lines (x -2)/2 = (y +1)/3 = (z-0)/4 and 2x +3y - 5z- 6=0= 3x - 2y - z +3. 97/13√7
101) Find the image of the point (3,-2,1) in the plane 3x - y + 4z= 2. (0,-1,-3)
102) Find the length and the foot of the perpendicular from the point (7,14,5) to the plane 2x + 4y - z = 2. Also, find the image of the point P in the plane. (1,2,8), 3√21, (-5,-10,11)
103) Find the image of the point having position vector i + 3j + k in the plane r. + (2i - j +k)+3=0 . -3i+ 5j +2k.
MAXIMUM AND MINIMUM
Sap-1
1) For the function f(x)= x + 1/x
a) x= 1 is a point of maximum
b) x= -1 is a point of minimum
c) maximum value > minimum value
d) maximum value <minimum value
2) Let f(x)= x³+ 3x²- 9x +2, then, f(x) has
a) a maximum at x= 1
b) a minimum at x= 1
c) neither maximum nor minimum at x= 3 d) none
3) The minimum value of f(x)= x⁴ - x² - 2x +6 is
a) 6 b) 4 c) 8 d) none
4) The number which exceeds its square by the greatest possible quantity is
a) 1/2 b) 1/4 c) 3/4 d) none
5) Let f(x)= = (x - a)² + (x - b)²+ (x - c)². Then f(x) has a minimum at x=
a) (a+ b+ c)/3 b) 3√(abc) c) 3/(1/a + 1/b + 1/c) d) none
6) The sum of two non zero numbers is 8, the minimum value of the sum of their reciprocals is
a) 1/4 b) 1/2 c) 1/8 d) none
7) If x lies in the interval [0,1], then the least value of x²+ x +1 is
a) 3 b) 3/4 c) 1 d) none
8) the least value of the function f(x)= x³ - 18x²+ 96x in the interval [0,9] is
a) 126 b) 135 c) 160 d) 0
9) The maximum value of f(x)= x/(4- x + x²) on [-1,1] is
a) -1/4 b) -1/3 c) 1/6 d) 1/5
10) The point on the curve y²= 4x which is nearest to the point (2,1) is
a) (1,2√2) b) (1,2) c) (1,-2) d) (-2,1)
11) If x+ y=8, then the maximum value of xy is
a) 8 b) 16 c) 20 d) 24
12) the least and greatest value of f(x)= x³- 6x²+ 9x in [0,6] are
a) 3,4 b) 0,6 c) 0,3 d) 3,6
13) The minimum value r(x²+ 250/x) is
a) 75 b) 50 c) 25 d) 55
14) If f(x)= x+ 1/x, x> 0, then its greatest value is
a) -2 b) 0 c) 3 d) none
15) If f(x)= 1/(4x²+ 2x +1), then its maximum value is
a) 4/3 b) 2/3 c) 1 d) 3/4
16) Let x, y be two variables and x> 0, xy = 1, then minimum value of x+ y is
a) 1 b) 2 c) 5/2 d) 10/3
17) The function f(x)= 2x³- 15x²+ 36x +4 is maximum at x=
a) 3 b) 0 c) 4 d) 2
18) The maximum value of f(x)= x/(4+ x + x²) on [-1,1] is
a) -1/4 b) -1/3 c) 1/6 d) 1/5
19) Let f(x)= 2x³- 3x²- 12x +5 on [-2,4]. The relative maximum occurs at x=
a) -2 b) -1 c) 2 d) 4
20) The minimum value of x logx is equals to
a) e b) 1/e c) -1/e d) 2e f) - e
21) The minimum value of the function f(x)= 2x³- 21x²+ 36x - 20 is
a) -128 b) -126 c) - 120 d) none
22) If x is real, the minimum value of x⅖- 8x +17 is
a) -1 b) 0 c) 1 d) 2
23) The maximum value of (1/x)ˣ is
a) e b) eᵉ c) e¹⁾ᵉ d) (1/e¹⁾ᵉ)
24) The function f(x) = xˣ has a stationary point at
a) x= e b) x= 1/e c) x= 1 d) x=√e
25) maximum slope of the curve y= - x³+ 3x²+ 9x -27 is
a) 0 b) 12 c) 16 d) 32
26) The function f(x) = 2x³- 3x² -12x +4 has
a) two points of local maximum
b) 2 points of local minimum
c) 1 maximum and 1 minimum
d) no maximum no minimum
Sap-2
1) The positive real number x when added to its reciprocal gives the minimum value of the sum when , x= ____
2) The real number which must exceeds is cube is ____:_
3) The function f(x) = ax + b/x, a, b, x> 0 takes on the least value at x equal to ___'
4) If y = a logx + bx²+ x has its extreme values at x= 1 and x = 2, then (a,b)= ____
5) The maximum value of f(x) = xe⁻ˣ is ______
6) If the function f(x) = x⁴- 62x²+ ax +9 attains a local maximum at x= 1, then a = ____
7) if the sum of two non zero numbers is 4, then the minimum value of the sum of their reciprocals is____'
8) If x and y are two real numbers that x> 0 and xy= 1. then the minimum value of x+ y is ___
9) The number that exceeds its square by the greatest amount is____
10) If m and M respectively denote the minimum and the maximum value of f(x) = (x -1)²+ 3 in the interval [-3,1] , then the ordered pair (m, M)= ____
11) The minimum value of f(x) = x²+ 250/x is _____
12) The maximum slope of the curve y= x³+ 3x²+ 9x -27= _____
13) The function f(x) = x/2+ 2/x has a local minimum at x= ____
14) The least value of the function f(x) = ax + b/x (a> 0, b >0, x > 0) is ____
Sap-3
1) Write necessary condition for a point x= c to be extreme point of the function f(x) .
2) Write sufficient conditions for a point x= c to be a point of local maximum.
3) If f(x) attains a local minimum at x=c, then write the values of f'(c) and f"(c).
4) Write the minimum value of f(x) = x+ 1/x, x> 0.
5) Write the maximum value of f(x) = x+ 1/x, x< 0.
6) write the point where f(x) = x logx attains minimum value.
7) Find the least value of f(x) = ax + b/x, where a>0, b>0 and x > 0.
8) Write the minimum value of f(x) = x ˣ.
9) Write the maximum value of f(x) = x¹⁾ˣ.
10) Write the maximum value of f(x) = (logx)/x, if it exist.
MATRIX
1) if A= 2 -1
-1 2 and I is the unit matrix of order 2, then A² is equal to
a) 4A - 3I b) 3A - 4I c) A - I d) A+ I
2) The multiplicative inverse of matrix
2 1
7 4 is
a) 4 -1 b) 4 -1 c) 4 -7 d) -4 -1
-7 -2 -7 2 7 2 7 -2
3) Assuming that the sums and products given below are defined, which of the following is not a true for matrices?
a) AB= AC doesn't imply B= C
b) A+ B = B+ A
c) (AB)'= B'A'
d) AB= 0 implies A= O or B= O
4) If A= 1 0 2 & Adj A= 5 a -2
-1 1 -2 1 1 0
0 2 1 -2 -2 b
Then the values of a and b are
a) -4,1 b) -4, -1 c) 4, 1 d) 4, -1
5) If A= -1 0
0 2 then the value of A³- A² is equal to
a) I b) A c) 2A d) 2I
6) If A= - x - y
z t then the transpose of adj A is
a) t z b) t y c) t -z d) none
-y - x -z -x y -x
7) If A=3 5 & B= 1 17
2 0 0 -10 then|AB| is equal to
a) 80 b) 100 c) -110 d) 92
8) If A= 5. -2
3 1 find the inverse of A
9) If A is singular matrix of order n then A. (Adj A) is equal to
a) a null matrix
b) a row matrix
c) a column matrix
d) none
10) If A= 3 -5
-4 2 then find the value of A²- 5A is equal to
a) I b) 14I c) O d) none
11) If A= 1 2 & B= 1 2
2 3 2 1
3 4
Then
a) both AB and BA exist
b) neither AB nor BA exist
c) AB exists but BA does not exist
d) AB does not exist but BA exist
12) If A= 2 -1 & B= 1 0
0 1 -1 -1 then+A+ B)² is not equal to
a) A²+ AB+ BA+ B²
b) A²+ AB+ BA+ B²I
c) A²I + AB+ BA+ B²
d) A²+ 2AB+ B²
13) If A be an n × n matrix and k any scalar then det. kA is equal to
a) k detA b) nᵏdetA c) kⁿ detA d) kn detA
14) If A= 1 2
3 -5 find inverse of A
15) If A= -1 2 & B= 5
2 -1 7 and AX= B, then X is equal to
a) 19 17 b) 19/3 c)19/3 17/3 d) 19
17/3 17
16) If A≠ O and B≠ O are two 2 x 2 matrices such that AB= O, then which of the following is correct?
a) detA= 0 or detB= 0
b) detA= 0 and detB= 0
c) detA= 0 = detB≠ 0 d) none
17) If A is a square matrix 3x3 and k is a scalar, then adj(kA) is equal to which of the following?
a) k adj A B) k² adj A c) k³ adj A d) 2k adj A
18) If A= 0 1 2
1 2 3
3 1 1 and its inverse B= [bᵢⱼ] , then the element b₂₃ of matrix B is
a) -1 b) 1 c) -2 d) 2
19) If A=a₁₁ a₁₂ a₁₃ & B=1 2 3
a₂₁ a₂₂ a₂₃ 2. 3. 4
a₃₁ a₃₂ a₃₃ 3 4 5
C= -1 -2 & D= -4 -5 -6
3 0 0 0 1
0 -4
With the relation A= BCD, then the value of a₂₂ is
a) 40 b) -40 c) -20 d) 20
20) If A= 1 2 & B= 3 8
3 4 7 2 with the relation 2X+ A= B, then the matrix X is equal to
a) 2 6 b) 1 -3 c) 1 3 d) 2 -6
4 -2 2 -1 2 -1 4 -2
21) If A= a 2
2 a and |A³|= 125, then the value of a is
a) ±2 b) ±2 c) ±5 d) 0
22) If A= |aᵢⱼ| and Aᵢⱼ denotes the Cofactor of aᵢⱼ, then which of the following is not equal to zero ?
a) a₃₁A₁₁ + a₃₂A₁₂+ a₃₃A₁₃
b) a₁₁A₃₁+ a₁₂A₃₂+ a₁₃A₃₃
c) a₂₁A₂₁+ a₂₂A₂₂+ a₂₃A₂₃
d) a₃₁A₂₁+ a₃₂A₂₂+ a₃₃A₂₃
23) A= 1 0 0
a 1 0
b c 1 find inverse of A
24) The minors of (-4) and 9 and the Cofactor of (-4) and 9 in matrix
-1 -2 3
-4 -5 -6
-7 8 9 are respectively
a) 42,3; -42,3 b) -42,-3; 42,-3 c) 42,3; -42,-3 d) 42,3; 42,3
25) If A= 0 3 & kA= 0 4a
4 5 3b 60
Then the values of k, a and b are respectively
a) 12,9,16 b) 9,12,16 c) 12,9,12 d) 16,12,9
26) B= 1 3 & C= 1 1
0 1 0 -1
If the matrix A satisfies the equation BA= C, then which one of the following represents A?
a) 1 4 b) 1 4 c) 1 -4 d) 1 -2
-1 0 0 -1 1 0 0 -1
27) For any matrix A, if A⁻¹ exists then which of the following is not true?
a) (A⁻¹)⁻¹= A
b) (Aᵀ)⁻¹= (A⁻¹)ᵀ
c) (A²)⁻¹ = (A⁻¹)²
d) |A⁻¹|= |A|⁻¹
28) If A and B are two square matrices of the same order, then (A - B)² is equal to
a) A²- 2AB+ B²
b) A²- AB - BA + B²
c) A²- 2BA+ B²
d) A²+ 2AB+ B²
29) For how many values of x in the closed interval [-4,-1], the matrix
3 -1+ x 2
3 -1 x+2
x+3 -1 2
is singular
a) 0 b) 1 c) 2 d) 3
30) A= 7 1 2 & B= 3 & C= 4
9 2 1 4 2
5
find the relation AB+ 2C
a) 43 b) 43 c) 45 d) 44
44 45 44 45
31) If A= 3 4
5 7
Then the value of A(adj A) is equal to
a) I b) |A| c) |A| I d) none
32) If x+ y 2x + z = 4 7
x -y 2z+ w 0 10 then the values of x, y, z and w are
a) 2,3,1,2 b) 2,2,3,4 c) 3,3,0,1 d) 2,2,4,3
33) The matrix
2 k. -4
-1 3 4
1 -2 -3 is non-singular if
a) k≠ 2 b) k≠ 3 c) k≠ -3 d) k≠ -2
34) If matrix A= 3 2
4 5 and
AC= 19 24
37 46
Then the matrix C is equal to
a) 3 4 b) 3 5 c) 5 4 d) 3 2
5 6 4 6 2 6 6 4
35) Let A= [5] be a matrix of order 1x1. Then adj A is equal to
a) [1] b) [5] c) [0] d) 1
5
36) If A²- A + I= 0 then the inverse of matrix A is
a) A- I b) A+ I c) A d) I - A
37) Let A and B are two square matrices such that AB= A and BA= B. Then A² is equal to
a) O b) I c) A d) B
38) If A= 4 2
-1 1 then the value of (A - 2I)(A- 3I) is
a) A b) I c) O d) 4I
39) If A= 1 -1 & B= a 1
2 -1 b -1 and (A+ B)²= A²+ B², then the values of a and b are
a) a=4, b= 1 b) a=1, b= 4 c) a=9, b= 4 d) a=2, b= 4
40) Let A= a 0 & B= 1 0
1 1 5 1 if A²= B, then the value of a is
41) The matrix 0 7 4
-7 0 -5
-4 5 0 is
a) symmetric b) skew symmetric c) nonsingular d) orthogonal
42) Let A= 1 -1 1 & B= 4 2 2
2 1 -3 -5 0 a
1 1 1 1 -2 3
If B is the inverse of the matrix A, then the value of a is
a) 2 b) 1 c) -2 d) 5
43) If A= 0 0 -1
0 -1 0
-1 0 0
Then the only correct statement about the matrix A is
a) inverse of A does not exist
b) A= (-1) I
c) A is a zero matrix
d) A²= I
MATRIX & DETERMINANT
Section I (Single Correct Answer Type)
1) r x n(n+1)/2
If Dᵣ= 2r -1 y n²
3r-2 z n(3n -1)/2
Then ⁿᵣ₌₁∑ Dᵣ=
a) 0 b) -1 c) 1 d) n
2) The system of linear equations x+ y+ z=6; x+ 2y+ 3z= 10, x+ 2y+ az=b has no solution when
a) a=2, b≠ 3 b) b =2, a= 3 c) a=3, b≠ 10 d) b=3, a≠ 10
3) If A= 1 2
3 4 then A⁴- 5A³ - A² - 4A - 2I=
a) 0 b) A c) I d) 2I
4) If α, β, γ are the real roots of a cubical polynomial satisfying
α² β² γ²
(α+1)² (β+1)² (γ+1)² = 0
(α-1)² (β-1)² (γ-1)²
Then
a) all roots must be equal
b) atleast two roots are equal
c) atleast one root is nonzero d) none
5) Which of the following statements is/are true?
a) Every square matrix can be expressed as the sum of a symmetric and skew symmetric matrices in unique way.
b) For a square matrix 'A' , |adj A|= |A|ⁿ⁻¹ where 'n' is the order of 'A'.
c) If 'A' is a non singular square matrix then |A⁻¹|= 1/|A|
d) Two matrices 'A' and 'B' are said to be conformable for multiplication in the same order iff the number of rows of 'A' is equal to number of columns of B
6) If for a square matrix A=[a
ᵢⱼ], aᵢⱼ = i² - j² i, j= 1,2,....,n is of even order then
a) A is a skew symmetric
b) |A| is not a perfect square
c) A is symmetric and |A|= 0
d) A is neither symmetric nor skew symmetric
7) Let S={(a, b): A³= A, where
A= 1/2 1/2
a b } then n(S)=
a) 1 b) 2 c) 3 d) 0
8) If A and P are different matrices of order n satisfying A³ = P³ and A²P= P²A (where |A|≠ |P|) then |A² + P²| is equal to
a) n b) 0 c) | A| |P| d) |A+ P|
9) If A= 1 1 1
1 ω² ω
1 ω ω²
Where ω is a complex cube root of unity then adjA equals
a) (ω² - ω)A' b) (ω- ω²)A' c) - A' d) A'
10) Let A and B be square matrices of same order satisfying AB= A and BA= B, then A²B² equals (O being zero matrices of the same order as B)
a) A b) B c) I d) O
11) The product of all the values of t, for which the system of the equation (a - t)x + by + cz = 0, bx + (c - t)y + az = 0, cx + ay + (b - t)z = 0 has non-trivial solution, is
a) a -c -b b) a b c
-c b -a b c a
-b -a c c a b
c) a c b d) none
b a c
c b a
12) The number of 2x2 matrices A, that are there with the elements as real numbers satisfying A+ Aᵀ = I and AAᵀ = I is
a) zero b) one c) two d) infinite
13) If A, B, C, D be real matrices (not necessarily square) such that Aᵀ = BCD, Bᵀ = CDA, Cᵀ = DAB, Dᵀ = ABC for the matrix, S= ABCD consider the statements
I: S³ = S
II: S² = S⁴
a) II is true but not I
b) I is true but not II
c) both I and II are true
d) both I and II are false
14) Consider the set A if all determinants of order 3 with entries 0 and 1 only. Let B be the subset of A consisting of all determinants with value 1. Let C be the subset of A consisting of all determinants with value -1. Then
a) C is empty
b) B has as many elements as C
c) A= B U C
d) B has twice as many elements as C
Then λ
a) 0 b) 1 c) -1 d) 2
16) 1 x x+1
2x x(x-1) (x+1)x
3x(x -1) x(x-1)(x-2) x(x+1)(x-1)
Then f(100)=
a) 0 b) 100 c) 1 d) -100
17) The value of a for which the system of equations (a+1)³x + (a+2)³y= (a+3)³,(a+1)x + (a+2)y= a+3, x+ y=1 is consistent , is
a) -2 b) -1 c) 1 d) 2
18) Let A, B are square matrices of same order satisfying AB= A and BA= B then (A²⁰¹⁰ + B²⁰¹⁰)²⁰¹¹ equals
a) A+ B b) 2010(A+ B) c)2011(A+ B) d) 2²⁰¹¹(A+ B)
19 The system of homogenous equations tx + (t+1)y + (t -1)z= 0, (t+1)x + ty + (t +2)z= 0, (t-1)x + (t+2)y + tz= 0 has a non-trivial solution for
a) exactly three real values of t
b) exactly two real values of t
c) exactly one real value of t
d) infinite number of values of t.
20) If A is a square matrix of order 3 such that |A|= 5 then |adj (4A)|=
a)
5³.4² b) 5².4³ c) 5². 16³ d) 5³.16²
21) If p, q and r in AP then the value of determinant
a²+2ⁿ⁺¹+2p b²+2ⁿ⁺²+3q c²+ p
2ⁿ+ p 2ⁿ⁺¹+ q 2q
a²+2ⁿ+p b²+ 2ⁿ⁺¹+ 2q c²- r
a) 1 b) 0 c) a²b²c² - 2ⁿ d) (a²+ b²+ c²) - 2ⁿq
SECTION II (More Than one correct answer)
22) Which of the following statements is always true?
a) A(Adj(A))= (Adj(A))A
b) Adjoint of a diagonal matrix is a diagonal matrix
c) Adjoint of a symmetric matrix is a symmetric matrix
d) Adjoint of a unit matrix is a unit matrix.
23) If the determinant
3 3x 3x² + 2a²
3x 3x²+ 2a² 3x³ + 6a²x
3x²+2a² 3x³+ 6a²x 3x⁴+12a²x²+ 2a⁴
Then
a) f'(x)=0 b) f(x) is independent
c) ¹₀∫f(x) dx = 4a⁶
d) y² = f(x) represents a pair of lines
24) If B= 1 3 4 & C= 3 -1 5
3 -1 5 1 3 4
-2 4 -3 4 -8 6 with the relation AB= C then find matrix A
a) 1 0 0 b) 0 1 0
0 1 0 1 0 0
0 0 -2 0 0 1
c) 1 0 0 d) 0 1 0
1 0 0 1 0 0
0 0 -2 0 0 -2
25) If the determinant
x²- 5x +3 2x -5 3
3x²+ x + 4 6x+1 9
7x²-6x+9 14x -6 21
= ax³+ bx²+ cx + d
Then which of the following are correct?
a) a=0 b) b=0 c) c =0 d) d=0
26) If determinant
a b c
b c a
c a b
Which of the following are correct?
a) bc - a² ca- b² ab - c²
ca - b² ab - c² bc - a² =∆²
ab -c² bc - a² ca- b²
b)
a² c² 2ac - b²
2ab - c² b² a²
b² 2bc - a² c²
c) ∆= 0=> either a= b= c or a+ b+ c=0
d) ∆= 3abc - a³ - b³ - c³
27) If f(x) and g(x) are functions such that f(x + y)= f(x). g(y)+ g(x). f(y), then the determinant
f(α) g(α) f(α+θ)
f(β) g(β) f(β+θ)
f(γ) g(γ) f(γ+θ)
is independent of
a)α b) β c) γ d) θ
28) If AB= A and BA= B, then
a) B= I b) A= I c) A² = A d) B²= B
29) If the equations x+ ay - z= 0, 2x - y + az = 0, ax + y + 2z= 0 have non-trivial solutions, then a=
a) 2 b) -2 c) 1+√3 d) 1-√3
30) If A² + A + I=0, then
a) A is a nonsingular
b) A≠ 0 c) A is singular d) A⁻¹= - (A + I)
31) If the elements of a 2x2 matrix A are real positive and distinct such that det(A+ Aᵀ)ᵀ= 0 then
a) det A> 0 b) det A≥ 0 c) det(A- Aᵀ)> 0 d) det(AAᵀ)> 0
32) let the matrices
A= -3 -7 -5 B= a
2 4 3 b
1 1 2 1
And AB is a scalar multiple of B, then
a) 4a+ 7b +5=0 b) a+ b + 2=0 c) b - a=0 d) a+ 3b =0
33) If A is an odd order sequence matrix such that A²+ A + 2I= 0 then which of the following is/are true?
a) A is non singular
b) A is symmetric
c) A cannot be skew symmetric
d) A⁻¹ = (-1/2) (A+ I)
34) A∈Mₙ(c) (where M is a matrix of order n x n) is nonsingular matrix such that 3ABA⁻¹ + A = 2A⁻¹BA thn
a) A and B both are identity matrices
b) |A+ B|= 0
c) |ABA⁻¹ - AB⁻¹A|= 0
d) A+ B is not a singular matrix
35) A and B are square matrices with the same order AB = BA and A and B are skew hermitian.
a) A+ B is skew hermitian
b) AB is skew hermitian
c) The diagonal elements of A and B are all zero.
d) AB is hermitian
36) Let a, b, c be real numbers such that a²+ b²+ c² = 1 then determinant
ax - by - c bx + ay cx+ a
bx+ ay -ax+ by- c cy+ b = 0
cx+ a cy+ b - ax- by + c
is a
a) straight line
b) point circle
c) parabola
d) ellipse
37) If M=[A: A is 3x3 matrix whose entries are -1 and 1] thn
a) det A lies between -6 and 6
b) det A ∈ {-4,0,4}
c) Number of elements of M= 2⁹
d) number of elements of M= 3⁹
38) Let the determinant
1+ x x x²
x 1+ x x²
x² x 1+x
= ax⁵ + bx ⁴+ cx³ + dx² + ex + f then
a) f=1 b) e= 3 c) a+ c= -1 d) b+ d= 1
SECTION III
Comprehensive Type (39-41)
A square matrix A is said to be orthogonal if AᵀA= AAᵀ= I with this information answer the following
39) P₂ₓ₂ orthogonal matrix,
A= 29 -28
30 -29
If Q= PᵀAP. Then PQ²⁰¹²Pᵀ=
a) 2012 A B) I c) A d) A²⁰¹¹
40) P₃ₓ₃ orthogonal matrix,α, β, γ are the angles made by a straight line with OX, OY, OZ.
sin²α sinα sinβ sinα sinγ
sinα sinβ sin²β sinβ sinγ
sinα sinγ sinβ sinγ sin²γ
and Q= PᵀAP. If PQ⁶Pᵀ = 2ᵏA, then k=
a) 5 b) 7 c) 6 d) 0
41) 0 2b c
a b -c
a -b c
is orthogonal matrix then 36|abc|=
a) 4 b) 6 c) 9 d) 1
Paragraph for question (42 to 44)
If A and B are square matrices then
42) If AB+ BA= O then which of the following option is equivalent to A³ - B³ ?
a) (A- B)(A² + AB + B²)
b) (A- B)(A² - AB - B²)
c) (A+ B)(A² - AB - B²)
d) (A+ B)(A² + AB - B²)
43) If A, B are nonsingular matrices such that, B≠ I, A⁶ = I and AB² = BA. Then the least value of k for Bᵏ = I is
a) 7 b) 15 c) 63 d) 127
44) If |A- B|≠ 0, A⁴ = B⁴, C³A = C³B, B³A = A³B then |A³ + B³ + C³|=
a) 0 b) 1 c) 3|A³ d) 6
Paragraph for question (44-47)
If A= 1 0 0
2 1 0
3 2 1 U₁, U₂, U₃ are column matrices satisfying
AU₁= 1 , AU₂= 2 AU₃= 2
0 3 3
0 0 1
and 3 x 3 matrix whose columns are U₁, U₂, U₃.
45) The value of det U is
a) 3 b) -3 c) 3/2 d) 2
46) The sum of the elements of U⁻¹ is
a) -1 b) 0 c) 1 d) 3
47)[3 2 0] U [3
2
0
=
a) 5 b) 5/2 c) 4 d) 3/2
Paragraph for question (48-50)
A square matrix A is said to be unitary matrix iff AA*= I = A*A where A* is the conjugate transpose of A. Given that
A= k[1 1+ i
1- i -1]
Where k ∈ R
48) If A is an unitary matrix, then k is equal to
a) 1/√3 b) -1/√3 c) both a and b d) either a or b.
49) If Q= ABA* where
B= cosα sinα
- sinα cosα
Then the value of X= A*Q²⁰A is
a) cosα sinα
-sinα cosα
b) cos20α sin20α
-sin20α cos20α
c) cos20α -sin20α
sin20α cos20α
d) cos20α - sin20α
-sin20α cos20α
50) The value of|adj (adj X)| is
a) 1 b) 0 c) -1 d) 2
SECTION IV
51) Given system of equations x+ y + 2z= a, -2x - z= b, x+ 3y + 5z= c, (a,b,c) is ordered triplet then
Column I
A) (0,0,0)
B) (-1,1,1)
C) (1,1,4)
D) (2,1,1)
Column II
p) no solution
q) infinite solution
r) non-trivial solution
s) trivial solution
52) Suppose matrix
A= 7 a b 1
c α β d
3 e f 10
All the unknown numbers are distinct integers from set {2,4,5,6,8,9} such that sum of the entries in 1ˢᵗ row, 3ʳᵈ row, 1ˢᵗ column and 4ᵗʰ column are equal to k then
Column I
A) k=
B) a+ b+ c=
C) 2ef
D) c + d
Column II
p) 20
q) 17
r) 19
s) 16
53) If matrix
A= 1 2 2
2 1 -2
2 -2 1
And 3B= A then match the column
Column I
A) adj A is
B) B is
C) Bᵀ is
D) B⁻¹
Column II
p) Orthogonal matrix
q) Involuntary matrix
r) Cofactor matrix of A
s) -3A
54) If p(θ)= -√2 sinθ cosθ
1 cosθ sinθ
-1 sinθ -cosθ
q(θ)= sin2θ -1 1
Cos2θ 4 -3
2 7 -5
r(θ)= cosθ sinθ cosθ
-sinθ cosθ sinθ
-cosθ -sinθ cosθ
s(θ)= sec²θ 1 1
cos²θ cos²θ cosec²θ
1 cos²θ cot²θ
Then match
Column I (Function)
A) p(θ)
B) q(θ)
c) r(θ)
d) s(θ)
Column II (Range)
p) [0,1]
q) [0,2√2]
r) [-2,2]
s) [-√5 -2, √5 -2]
55) Column I
A) If ω is a non real cube root of unity, then a root of the following equation
|x +1 ω ω²
ω x+ω² 1 = 0
ω² 1 x+ ω|
B) If i²= -1, ω is a nonreal cube root of unity and if a= i, b= ω and c= ω², then the value of determinant
a a+ b a+b+ c
3a 4a+3b 5a+ 4b + 3c
6a 9a+ 6b 11a+9b+6c
C) If ω is a nonreal cube root of unity, the value of determinant
1 1 1
1 ω² ω
1 ω ω²
is equal to
D) Let the determinant
|x 2 x
x² x 6
x x 6
= Ax⁴+ Bx³+ Cx²+ Dx + E then the value of A+ B+ C+ D+ E=
Column II
p) 0
q) 2
r) i
s) 3√3 i
SECTION V
56) If x - 2y + z= -4; 2x - y + 2z= -2, x + y + λz= 4; are inconsistent then λ=
57) If determinant
1 n n
∆ₖ= 2k n²+n-1 n²+n
2k-1 n² n²+n+1
and ⁿₖ₌₁ₖ∑∆ₖ = 56, then n is equal to
58) If determinant
sinx cisx cosx
cosx sinx cosx = 0
cosx cosx sinx
(π/4≤ x≤π/4) Then tanx=
59) If x, y, z are nonzero real numbers and determinant
1+ x 1 1
1+ y 1+2y 1 = 0
1+z 1+z 1+3z
Then -(1/x + 1/y + 1/z)=
60) If 12 α²
-16 -12
is a nilpotent matrix of index 2, α > 0 then α=
61) 1+ sin²θ cos²θ 4sin4θ
sin²θ 1+cos²θ 4 sin4θ
sin²θ cos²θ 1+4sin4θ
Then number of solutions of f(θ)= 0 in [0,π/2] is
62) Rank of
2 4 7
17 38 57
33 70 113 is
63) If the determinant f(x)=
cos(x + α) cos(x+β) cos(x+γ)
sin(x+α) sin(x+β) sin(x+γ)
sin(β-γ) sin(γ-β) sin(α-γ)
and f(2)= 1/4 then [¹⁵ᵣ₌₁ ∑ f(x)ᵣ] is (where [ ] denotes greatest integer function)
64) If the system of equations 3x - 2y + z=0, λx - 14y + 15z = 0, x + 2y - 3z = 0 has a nonzero solution, then λ=
Sap-1
1) If two distinct chords of a parabola y²= 4ax passing through the point (a,2a) are bisected by the line x+ y =1, then the length of the latus rectum can be
a) 2 b) 7 c) 4 d) 5
2) If α, β are the roots of x² + px +1=0 and γ, δ are the roots of x¹+ qx +1=0, then (α-γ)(β+ δ)(α+δ)(β -α) is divisible by
a) α+β+γ+δ b) α- β+ γ- δ c) α+ β+ γ- δ d) α- β - γ - δ
3) If log₀.₂ {(x+1)/x ≥ 1, then x ∈
a) [-5/4,0) b) [-5/4,-1] c) [-5/4,-1( d) (-5/4,-1)
4) The coefficient of x⁵⁰ in the series
S= ¹⁰¹ᵣ₌₁∑ rxʳ⁻¹ (1+ x)¹⁰¹⁻ʳ is
a) (100 b) (101 c) (102 d) (103
50) 50) 50) 50)
5) The number of 3 digit numbers which contain not more than two different digits is
a) 216 b) 252 c) 264 d) 288
6) If +2,4) is a point interior to the circle x²+ y²- 6x - 10y + λ= 0 and the circle does not cut the axes at any point, then λ lies in the interval
a) (25,32) b) (9,32) c)(32, ∞) d) (4,25)
7) OABC is a regular tetrahedron of unit edge. It's volume is
a) 1/√3 b) 1/√6 c) 1/3√2 d) 1/6√2
8) The value of ⁶ᵣ₌₁∑ (sin(2πr/7) - i cos(2πr/7)) is
a) 0 b) 1 c) -1 d) i
9) A straight line with negative slope passes through the point P(8,2) and ends the coordinate Axes at the points A and B. The minimum value of OA+ OB is
a) 12 bc) 16 c) 18 d) 24
10) If f(a - x)= f(x) and g(x)+ g(a- x)= 2,
I₁ = ∫ᵃ₀ f(x) dx, I₂= ᵃ₀∫ f(x) g(x) dx, then I₂/I₁=
a) 1 b) 2 c) a d) a/2
11) The vector a is equally to the plane determined by 3i and 4j and to the plane determined by 3i and 4j at angle θ. Then θ∈
a) [0,π/2] b) [π/2,π] c) [π/3, 2π/3] d) [0,π/4], U [3π/4,π]
12) The greatest slope of a tangent to the curve, y= x/(1+ x²) is
a) 1/√2 b) -√2 c) 1 d) -1
13) lim ₓ→₀ (ⁿᵣ₌₁∑ ᵣcosec²x)sin²x.
a) 0 b) ∞ c) n d) 1/n
14) If z+ 1/z = 1 and a= z²⁰⁰⁵ + 1/z²⁰⁰⁵ and b is the last digit of the number ₂2ⁿ -1 , when the integer n> 1, then a² + b²=
a) 22 b) 24 c) 26 d) 27
15) Let matrix
A= 1. 4
3 2
If θ is the angle between the two nonzero column vectors X such that AX=λX for some scalar λ, then tanθ =
a) 2 b) 3 c) 5 d)
16) A group of 10 boys are randomly divided into two groups containing 5 boys each. The probability that the two tallest boys are in different teams is
a) 5/9 b) 4/9 c) 4/11 d) 5/11
17) Three numbers a,b,c are between 2 and 18 such that their sum is 25. 2, a, b, are in AP and b, c, 18 are in GP, then abc is
a) 360 b) 420 c) 480 d) 540
18) With 17 consonants and 5 vowels, the number of words of 4 letters, that can be formed having 2 different vowels in the middle 1 consonant (repeated or different) at each end, is
a) 3680 b) 4540 c) 5780 d) 6210
19) If f(x - y)= f(x)f(y) - f(a - x) f(a+ y), and x, y and f(0)= 1. Then f(2a - x)=
a) f(x) b) - f(x) c) f(- x) d) f(a) + f(a - x)
20) The number of discontinuous function y(x) on [-2,2] satisfying x² + y² = 4 is
a) 0 b) 1 c) 2 d) >2
21) If I₁= ¹₀∫ ₂x² dx, I₂= ¹₀∫ ₂x³ dx, I₃ = ²₁∫₂x² dx, I₄= ²₁∫ ₂x³ dx, then
a) I₁> I₂ b) I₁< I₂ c) I₃< I₄ d) I₃= I₄
22) The area between the curve y= 2x⁴ - x², the x-axis and the ordinates of the two minimum points on the curve is
a) 7/12 b) 7/30 c) 7/60 d) 7/120
23) If y- cosx dy/dx = y²(1- sinx)cosx, y(0)= 1, then y(π/3)=
a) 1 b) 2 c) 1/2 d) √3
24) In a ∆ ABC , if a,b,c are in AP and the largest angle exceeds the smallest by 90°, then cosB=
a) 1/2 b) 2/3 c) 3/4 d) 3/5
25) If |a|=|b|= 1, |c|= 3, b.c= -3/4, and a x c = 2(a x b), then |c - 2b|=
a) 1 b) 2 c) 3 d) 4
26) The locus of P(x,y) such that
√(x² + y² - 6x +9) + √(x² + y² + 8y + 16)= 5 is
a) hyperbola b) circle c) finite line segment d) infinite ray
27) If a line makes angles α, β, γ, δ with the four diagonals of a cube, then
cos²α + cos²β+ cos²γ+ cos²δ=
a) 1 b) 2 c) 2/3 d) 4/3
28) The normal at t on the parabola y²= 4ax subtends a right angle at the vertex. The value of t² is
a) 1 b) 2 c) 3 d) 4
29) Let a, b, c be distinct real numbers. If a, b, c are in GP and a+ b + c= bx, then x∈
a) (0, ∞) b) (-∞,0) c) (-1,3) d) R -(-1,3)
30) The coefficient of x⁶ in the expansion of (1+ x + x²)⁶ is
a) 131 b) 141 c) 151 d) 167
31) If f(x)= ˣ₋₁∫ |z| dz, and x≥ 0 f'(x) equas
a) x b) (1+ x²)/2 c) (1- x²)/2 d) - x
32) If y= cos⁻¹{2x/(1- x²)}, then dy/dx equal
a) 2/(1+ x²) ∀ |x|> 1
b) 2/(1+ x²) -1< x<1
c) - 2/(1+ x²) ∀ - ∞< x ∞
d) none
33) lim ₓ→₀ f(x)/x= 2, where
f(x)= minimum {sin √[m]x, |x|} and [ .] Represent greatest integer function, then
a) m ∈ {4} b) m ∈[4,5] c) m ∈ [4,5) d) m= {5}
34) If (h,k) be the point on the axis of the curve 2(x -1)²+ 2(y -1)² = (x + y +2)² from where three distinct normal can be drawn, then h satisfies the conditio
a) h< 4 b) h >6 c) h < 8 d) h > 2
35) If f(x)= ∫ sin⁻¹√z dz at (sin²x, 0) + ∫ cos⁻¹√z dz at (cos²x, 0), then f(x) is a/an
a) constant function
b) odd function
c) Even function d) none
INCREASING AND DECREASING FUNCTION
SAP-1
1) Let h(x)= f(x) - (f(x))²+ (f(x))³ for every real number x. Then
a) h is increasing whenever f is increasing.
b) h is increasing whenever f is decreasing.
c) h is decreasing whenever f is decreasing
d) nothing can be said in general
2) If f(x)= x/sinx and g(x)= x/tanx, where 0< x<1, then in this interval
a) both f(x) and g(x) are increasing functions
b) both f(x) and g(x) are decreasing functions
c) both f(x) is increasing functions
d) g(x) are increasing functions
3) The interval to which b may belong so that the function
f(x)= {1- √(21- 4b - b²)/(b+1)}x³+ 5x + √6 is increasing at every point of its domain is
a) (-7,-1) b) (-6,-2) c) (2,2.5) d) (2,3)
4) The interval of increase of the function f(x)= x - eˣ+ tan(2π/7) is
a) (0,∞) b) (-∞,0) c) (1,∞) d) (-∞ ,-1)
5) The function f(x)= cot⁻¹x+ x increases in the interval
a) (1,∞) b) (-1,∞) c) (-∞,∞) d) (0,∞)
6) The function f(x)= xˣ decreases on the interval
a) (0,e) b) (0,1) c) (0,1/e) d) none
7) The set of all x for which log(1+ x)≤ x is
a) (0,∞) b) (-1,∞) c) (-1,0) d) none
8) The function f(x)= x/logx increases on the interval
a) (0,∞) b) (0, e) c) (e,∞) d) none
9) The function f(x)= 2 log(x -2) - x²+ 4x +1 increases on the interval
a) (1,2) b) (2,3) c) (5/2,3) d) (2,4)
10) The set of all x for which 1+ logx < x is
a) (1,∞) b) (0,1) c) (0,∞) d) none
11) For x> 1, y= logx satisfying the inequality
a) x- 1> y b) x²-1 >y c) y > x -1 d) (x-1)/x <y
12) If the function f(x)= 2x²- kx + 5 is increasing on [1,2], then k lies in the interval
a) (-∞,4) b) (4,∞) c) (-∞,8) d) (8, ∞)
13) Let f(x)=x³+ ax²+ bx + 5 sin²x be an increasing function on the set R. Then a and b satisfy
a) a²- 3b - 15> 0
b) a²- 3b + 15> 0
c) a²- 3b + 15< 0
d) a>0 and b > 0
14) The function f(x)=log(x³+ √(x⁶+1)) is of the following types
a) even b) odd c) increasing d) decreasing
15) If a,b,c be real, then determinant
x+ a² ab ac
f(x)= ab x+ b² bc
ac bc x+ c²
is decreasing on
a) ((-2/3)(a²+ b²+ c²),0)
b) (0, (2/3)(a²+ b²+ c²))
c) ((a²+ b²+ c²)/3,0) d) none
1ac 2c 3abcd 4bd 5c 6c 7b 8c 9bc 10 abc 11abc 12a 13c 14bc 15a
Sap-2
1) if f and g are two increasing functions such that fog is defined, then
a) fog is an increasing function
b) fog is a decreasing function
c) fog is a neither increasing nor decreasing.
d) none
2) if f and g are two decreasing functions such that fog exists , then fog
a) is an increasing function
b) is a decreasing function
c) is neither increasing nor decreasing
d) none
3) If f is an increasing function and g is a decreasing function on an interval I such that fog exists , then
a) fog is an increasing function on I.
b) fog is decreasing function on I
c) fog is neither increasing nor decreasing on I
d) none
4) Let y= x² e⁻ˣ, then the interval in which y increases with respect to x is
a) (-∞, ∞) b) (-2,0) c) (2,∞) d) (0,2)
5) The interval in which the function f(x)= x e²⁻ˣ increases is
a) (-∞,0) b) (2, ∞) c) (0,2) d) none
6) The function f(x)= cos(π/x) is increasing in the interval
a) (2n +1, 2n), n ∈N
b) (1/(2n +1, 2n)), n ∈N
c) (1/(2n +2), 1/(2n+1)), n ∈N
d) none
7) The value of b for which the function f(x)= sinx - bx + c is
a) b<1 b) b ≥1 c) b > 1 d) b ≤1
8) The set of all real values of a for which the function
f(x)= [{√(a+4)/(1- a)} -1 ]x⁵- 3x + log5 decreases for all real x is
a) (-∞, ∞) b) [-4, 3 - √21/4] U [1, ∞)
c) (-3, 5 - √27/2) U (2, ∞)
d) [1, ∞)
9) The set of values of a for which the function f(x)=x²+ ax +1 is an increasing function on [1,2] is
a) (-2,∞) b) [-4,∞) c) [-∞,-2) d) (-∞,2)
10) On which of the following intervals is the function x¹⁰⁰+ sinx -1 decreasing?
a) (0,π/2) b) (0,1) c) π/2,π) d) none
11) which of the following function are decreasing on (0,π/2)?
a) cosx b) cos2x c) cos3x d) tanx
12) If f'(x)= g(x)(x - a)² where g(a)≠ 0 and g is continuous at x= a, the pb
a) f is increasing in the nbd of a if g(a)> 0
b) f is increases in the nbd of a if g(a)< 0
c) f is decreases in the nbd of a if g(a)> 0
d) f is decreasing in the nbd of a if g(a)< 0.
13) If f(x)= 2x + cot⁻¹+ log{√(1+ x² -X}, then f(x)
a) increase in [0, ∞)
b) decreases in [0, ∞)
c) neither increases nor decreases in (0,∞)
d)?increases in (-∞, ∞).
14) The function f(x)= log(1+ x) - 2x/(2+ x) is increasing on
a) (0,∞) b) (-∞,0) c) (-∞,∞) d) none
15) On which of the following intervals is the function f(x)= 2x²- log|x|, x ≠ 0 increasing?
a) (1/2, ∞)
b) (- ∞,-1/2) U(1/2, ∞)
c) (-∞, -1/2) U (0,1/2)
d) (-1/2,0) U (1/2,∞)
16) if the function f(x)= (K sinx + 2 cosx)/(sinx + cosx) is increasing for all values of x, then
a) K< 1 b) K > 1 c) K <2 d) K> 2
17) If f(x)= (a sinx + b cosx)/(c sinx + d cosx) i s decreasing for all x, then
a) ad - bc > 0
b) ad - bc < 0
c) ab - cd > 0
d) ab - cd < 0
18) If f(x)= Kx³- 9x² + 9x + 3 is increasing on R, then
a) K< 3 b) K > 3 c) K ≤3 d) none
19) The values of a for which the function (a+ 2)x³ - 3ax²+ 9ax -1 decreases monotonically throughout for all real x are
a) a< -2 b) a > -2 c) -3<a <0 d) -∞< a ≤ -3
20) The function y= x³- 3x²+ 6x -17
a) increases everywhere
b) decreases everywhere
c) increases for positive x and decreases for negative x
d) increases for negative x and decreases for positive x
21) The interval in which is the function x³ increases less rapidly than 6x²+ 15x +5 is
a) (-∞, -1) b) (-5,1) c) (-1,5) d) (5, ∞)
22) The value of a for which the function f(x)= sinx - cosx - ax + b decreases for all real values of x, is given by
a) a≥√2 b) a ≥1 c) a<√2 d) a< 1
23) The function y= x - cot⁻¹x - log{x + √(x²+1)} is increasing on
a) (- ∞,0) b) (0,-∞,) c) (0,∞) d) (-∞,∞)
24) The he function f(x)= |x -1|/x² is monotonically decreasing on
a) (2, ∞) b) (0,1) c) (0,1) U (2,∞) d) (-∞, ∞)
25) The value of a in order that f(x)= √3 sinx - cosx - 2ax + b decreases for all real values of x, is given by
a) a<1 b) a ≥1 c) a≥√2 d) a < √2
26) The function f(x)= x³- 3x is
a) increasing on (-∞,-1] U [1, ∞) and decreasing on (-1,1)
b) decreasing on (-∞,-1] U [1,∞) and increasing on (-1,1)
c) increasing on +0,∞) and decreasing on (-∞,0)
d) decreasing on (0,∞) and increasing on (-∞,0)
27) A condition for a function y= f(x) to have an inverse is that it should be
a) defined for all x
b) continuous everywhere
c) strictly monoton and continuous in the domain
d) an even function
28) Let g(x)= f(x)+ f(1- x) and f"(x)< 0, 0≤ x ≤ 1, then
a) g(x) increases on [1/2,1]
b) g(x) decreases on [1/2,1]
c) g(x) decrease in [0,1/2]
d) g(x) increases on [0,1/2]
29) The function f(x)= ln(π+x)/ln(e + x) is
a) increasing [0, ∞)
b) decreasing on [0, ∞)
c) increasing on [0,π/e) and decreasing on [π/e, ∞)
d) decreasing on [0,π/e) and increasing on [π/e, ∞)
30) The function f define by f(x)= (x +2)e⁻ˣ is
a) decreasing for all x
b) decreasing in (-∞, -1) and increasing in (-1, ∞)
c) increasing for all x
d) decreasing in (-1, ∞) and increasing in (-∞,-1)
31) y= [x(x -3)]² increases for all values of x lying in the interval
a) 0<x<3/2 b) 0<x <∞ c) - ∞<x <0 d) 1<x < 3
32) If a< 0, the function f(x)= eᵃˣ + e⁻ᵃˣ is a monotonically decreasing function for values of x given by
a) x>0 b) x <0 c) x >1 d) x < 1
33) The function f(x)= tanx - x
a) always increase
b) always decreases
c) never decreases
d) sometimes increases and some times decreases
34) The function f(x)= logx/x is increasing in the interval
a) (1,2e) b) (0,e) c) (2,2e) d) (1/e, 2e)
35) If the function f(x)= cos |x| - 2ax + b increases along the entire number scale, the range of values of a is given by
a) a≤b b) a= b/2 c) a≤ -1/2 d) a ≥ -3/2
36) If f(x)= kx - sinx is monotonically increasing, then
a) k> 1 b) k > -1 c) k < 1 d) k < -1
37) The function f(x)= x √(ax - x²), a> 0
a) increases on the interval (0,3a/4)
b) decreases on the interval (3a/4, a)
c) decreases on the interval (0,3a/4)
d) increases on the interval (3a/4, a)
38) The function f(x)= sin⁴x + cos⁴x increases if
a) 0< x <π/8
b) π/4 <x <3π/8
c) 3π/8 <x <5π/8
d) 5π/8 <x <3π/4
1a 2a 3b 4d 5a 6d 7c 8b 9a 10d 11ab 12ad 13ad 14a 15d 16d 17b 18b 19d 20a 21c 22a 23d 24c 25b 26a 27c 28b d 29b 30d 31a 32b 33a 34b 35c 36a 37a 38b
Sap-3
1) If the function x³ - 6x² -36x +7 is increasing with x, then x lies outside the range ___.
2) The value of b for which the function f(x) = sinx - bx + C increases on R are ____.
3) The interval of monotonically of the function f(x)= log |x| are given by ___ decreases, _____ increases.
4) The function y= x²- log |x| is monotonically increasing for the value of x ≠0 satisfying the inqualities _____ and monotonically decreasing for values of x satisfying the inequalities _____.
5) The larger of log(1+ x) and (tan⁻¹x)/(1+ x) is _____.
6) The larger of x- x³/6 and tan⁻¹x for 0< x≤ 1, is____
7) The function f(x)= (2x²-1)/x⁴ decreases in the interval _____.
8) The larger of sinx + tanx and 2x is _____(0< x<π/2)
1) (-2,6) 2) (-∞,-1) 3) (-∞,0) and (0,∞) 4) x> 1/2, -1/2< x < 0 and 0 <x < 1/2, x < -1/2 5) log(1+ x) 6) x - x³/6 7) (-1,0) U (1, ∞) 8) sinx + tanx
SAP- 4
1) The function f(x)= x²e⁻ˣ is monotonic increasing when
a) x∈ R -[0,2]
b) 0<x <2
c) 2< x <∞ d) x< 0
2) function f(x)= cosx - 2 λx is monotonic decreasing when
a) λ> 1/2 b) λ< 1/2 c) λ< 2 d) λ > 2
3) In the interval (1,2) function
f(x)= 2|x -1|+ 3|x -2| is
a) monotonically increasing
b) monotonically decreasing
c) not monotonic d) constant
4) function f(x)= x³ - 27x +5 is monotonically increasing when
a) x<-3 b) |x|> 3 c) x≤-3 d) |x|≥ 3
5) function f(x(= 2x³- 9x²+ 12x +29 is monotonically decreasing when
a) x< 2 b) x >2 c) x > 3 d) 1< x < 2
6) if the function f(x)= kx³- 9x²+ 9x +3 is mono technical increasing in every interval , thempn
a) k<3 b) k ≤3 c) k>3 d) k ≥ 3
7) f(x)= 2x - tan⁻¹x - log(x + √(x²+1)) is monotonically increasing when
a) x> 0 b) x <0 c) x ∈ R d) x ∈ R - {0}
8) Function f(x)= |x| - |x -1| is monotonically increasing when
a) x< 0 b) x >1 c) x<1 d) 0<x < 1
9) Every invertible function is
a) monotonic function
b) constant function
c) identity function
d) not necessarily monotonic function
10) In the interval (1,2) , functions f(x)= 2|x -1| +3|x -2| is
a) increasing b) decreasing c) constant d) none
11) if the function f(x)= cos|x| - 2ax + b increasing along the entire number scale, then
a) a= b b) a = b/2 c) a ≤ -1/2 d) a > -3/2
12) f(x)= 2x - tan⁻¹x - log(x + √(x²+1)) is monotonically increase when
a) x< 0 b) x >0 c) x∈ R₀ d) x ∈ R
13) The function f(x)= (λ sinx + 2 cosx)/(sinx + cosx) is increasing, if
a)λ<1 b) λ>1 c) λ<2 d) λ> 2
14) function f(x)= aˣ is increasing on R, if
a) a>0 b) a <0 c) 0< a <1 d) a > 1
15) function f(x)= logₑx is increasing on R, if
a) 0< a <1 b) a>1 c) a <1 d) a > 0
16) Let φ(x)= f'(x)+ f'(2a - x) and f"(x)> 0 for all x∈ [0,a]. Then φ(x)
a) increases on [0,a] b) decreases on [0,a] c) increases on [a,2a] d) decreases on [a, 2a]
17) If the function f(x)= x²- kx +5 is increasing on [2,4], then
a) k∈ (2, ∞) b)k∈ (-∞,2) c) k∈ (4, ∞) d) k∈ (-∞,4)
18) The function f(x)= -X/2 + sinx defined on [-π/3, π/3] is
a) increasing b) decreasing c) constant d) none
19) If the function f(x)= x³- 9kx²+ 27x +30 is increasing on R, then
a) -1≤ k < 1 b) k<-1 or k>1 c) 0<k < 1 d) -1< k < 0
20) The function f(x)= x⁹+ 3x⁷+ 64 is increasing on
a) R b) (-∞,0) c) (0,∞) d) R₀
1b 2a 3b 4d 5d 6c 7c 8d 9a 10b 11c 12d 13d 14d 15b 16bc 17d 18a 19a 20a
Tangent and Normal
SAP-1
1)a) Find the equation of the tangent to given curve at point (x,y)
a) xᵐ/aᵐ + yᵐ/bᵐ = 1.
b) i)Deduce the equation of tangents to
a) x²/a² + y²/b² = 1 ellipse
ii) (x/a)²⁾³+ (y/b)²⁾³ = 1. Hypo-cycloid Astroid
iii) x²⁾³ + y²⁾³ = a²⁾³.
iv) (x/a)ⁿ/⁽ⁿ⁻¹⁾ + (y/b)ⁿ/⁽ⁿ⁻¹⁾= 1
c) Does the straight line x/a + y/b =2 touch the curve (x/a)² + (y/b)² ? If it touches then determine the coordinates of the point of contact.
2a) x= a sin³θ, y= a cos³θ
Or
x²⁾³ + y²⁾³ = a²⁾³, Tangent and Normal.
b) Normal to x²⁾³ + y²⁾³ = a²⁾³ in the form y cos θ - x sinθ = a cos2θ.
Where θ is the angle which the normal makes with the axis of x.
3a) Show that θ is the angle which the perpendicular from the origin on the tangent makes with the x-axis for the curve whose parametric equations are x= a sin³θ, y= a cos³θ.
b) If p₁ and p₂ be the lengths of perpendiculars from the origin on the tangent and normal to the curve x²⁾³ + y²⁾³ = a²⁾³ respectively, show that 4p₁²+ p₂² = a².
c) If the tangent at any point of the curve x²⁾³ + y²⁾³ = a²⁾³ meets the axes of coordinates in A and B then locus of mid-point of AB is a circle.
4a) The equation of the normal to the curve x³+ y³= 6xy at the point (3,3).
b) The equation of the normal to the curve 2y = 3 - x² at the point (1,1).
5a) Normal to the parabola y²= 4ax in the form y= mx - 2am - am³ where m is the slope of the normal.
b) Three normals are drawn from the point (c,0) to the curve y²= x. Show that c must be greater than 1/2. One normal is always the x-axis. Find c for which the other two normals are perpendicular to each other.
c) Normal to the curve x²= 4y which passes through the point (1,2).
6a) Tangent to the parabola y²= 4ax in the form y= mx + a/m where m is the slope of the tangent.
Prove also that two perpendicular tangents to a parabola meet in the directrix.
b) The equation of the tangent at the point P(t), where t is any parameter, to the parabola y²= 4ax is
i) yt= x + at² ii) y= xt + at² iii) y= tx iv) y= tx + a/t
7) The parametric equation of a curve are x= a cos2t, y= 2√2 a sin t. Find the equation of tangent in the form
mx - y = a(m + 1/m)
Where m is the slope of the tangent. Hence show that two perpendicular tangents meet on the line x= 2a.
8) Tangent and Normal to
a) y²(a + x)= x²(3a - x)
b) y²(2a + x)= x³ at the point where x= a.
9) Normal to y= x³- 3x which is parallel to 2x + 18y = 9
10) Tangent to 3x²+ y²+ x + 2y = 0 which are perpendicular to the line 4x - 2y = 1.
11) a) Tangent to parabola y¹= 4x + 5 which is parallel to y= 2x +7.
b) Determine the constant c such that the straight line joining the points (0,3) and (5,-2) is tangent to the curve y= c/(x +1).
12) a) The straight line x+ y= a will be a tangent to the ellipse x²/9+ y²/16 = 1 if a=
a) 8 b) ±5 c) ±10 d) ±6
b) On the ellipse 4x²+ 9y² = 1, the points at which the tangents are parallel to the line 8x = 9y are
a) (2/5,1/5) b) (-2/5,1/5) c) (-2/5,-1/5) d) (2/5,-1/5)
13) a) Determine the equation of the tangent at the vertex of the parabola (x+4)² = -4( y -2).
b) if the normal to the curve y= f(x) at the point (3,4) makes an angle 3π/4 with the positive x-axis , then f'(3)=
i) -1 ii) -3/4 iii) 4/3 iv) 1
14) a) Tangent and normal to the curve x= 2at²/(1+ t²), y= 2at³/(1+ t²) at the point for which t= 1/2.
b) Normal at any point θ to the curve
x= a cosθ + aθ sinθ
y= a sinθ - aθ cosθ
Also show that it is at a constant distance from the origin.
15) a) The rectangular coordinates of a point on the curve are given by
x= 3 cosθ - cos³θ, y = 3 sinθ - sin³θ.
Find the equation of the normal at any point on the curve and show that at the point P where θ=π/4, the normal passes through the origin.
b) The parametrics equations of given curve are
x= a(2 cos t + cos2t), y= a(2sin t - sin2t).
Prove that the equations of tangent and normal at any point t are x sin(t/2) + y cos(t/2) = a sin(3t/2) and x cos(t/2) - y sin(t/2) = 3a cos(3t/2) respectively . Also establish the S. T= ycot(t/2) and S. N= y tan(t/2).
c) Prove that the portion of the normal to the curve
x= 2a sin t + a sin t cos²t
y= - a cos³t
which is intercepted between the coordinate Axes is of constant length.
16) a) Tangent to y= (x³-1)(x -2) at the points where the curve cuts the x-axis.
b) Find the equation of the tangent to the curve y= x²+1 at the point (1,2).
c) Find the slopes of the tangents of the curve
y= (x +1)(x -3) at the points where it cuts the x-axis .
17) a) tangents and normal to the curve
y(x -2)(x -3) - x +7=0
at any point where it cuts the axis of x.
b) Normal to the curve y²= x³ at the point whose abscissa x is 8.
c) Normal to the curve y= x³- 2x²+4 at the point where x= 2.
18) Find the points in the curve 9y²= x³ where normal to the curve makes equal intercepts with the axes.
19) a) Find the point on the curve
y= x⁴- 6x³+ 13x²- 10x +5
Where the tangent is parallel to the line y= 2x. Show that two of these points have the same tangent.
b) Find the points in the curve
x²+ y²- 2x -3=0 the tangent at which are parallel to x-axis.
c) Find the points on the curve y= x³, the tangents at which are inclined at an angle of 60° to x-axis .
20)a) Normals at the points on the curve
y= x/(1- x²) where the tangent makes an angle of π/4 with x-axis.
b) At which Points on the curve
y= x/(1- x²), the tangent makes an angle 45°with x-axis? Is there any point on the curve the tangent at which is parallel to y-axis.
c) The point/s on the curve y³+ 3x²= 12y where the tangent is vertical, is/are
a) (±4/√3,-2) b) (±√(11/3, 1) c) (0,0) (±4/√3,2)
21) a) The curve y - eˣʸ + x = 0 has a vertical tangent at the point
i) (1,1) ii) no point iiu) (0,1) iv) (0,1)
b) A and B are points (-2,0) and (1,3) on the curve y= 4 - x². If the tangent at P on the curve be parallel to chord AB then coordinates of point P are (-1/2, 15/4).
22) a) Prove that all points on the curve y²= 4a[x + a sin(x/a)] at which the tangent is parallel to the x-axis lie on the parabola y²= 4ax.
b) Tangents are drawn from the origin to the curve y= sinx. Prove that their points of contact lie on x²y²= x²- y².
23) Show that the line x/a + y/b = 1 touchase the curve be y= b e⁻ˣ/ᵃ at the point where the curve crosses the y-axis .
b) The line x/a + y/b = 1 touches the curve y= b e⁻ˣ/ᵃ at the point
i) (a, b/a) ii) (-a, b/a) iii) (a,a/b) iv) none
24) a) Show that the curve (x/a)ⁿ + (y/b)ⁿ = 2 touches the straight line x/a + y/b = 2 at that the point (a,b) whatever the value of n maybe.
b) if the line x/a + y/b = 1 be a tangent to hyperbola xy = c² then show that ab= 4c².
25) a) Show that tangents to the folium of descates x³+ y³= 3axy at the point where it meets the parabola y²= ax are parallel to the axis of y. Also determine the point on the curve at which the tangent is the parallel to x-axis .
b) Find the equation of the normal to the curve x³+ y³= 8xy at the point other than origin where it meets the curve y²= 4x.
26) a) Find the equations of the tangents drawn to the curve y²- 2x³ - 4y + 8= 0 from the point (1,2).
b) Find the equations of tangents to the curve y= x⁴ which are drawn from the point (2, 0).
MATRICES
1) If A= 1 -2 4 & B= 0 -2 4
2 3 2 1 3 2
3 1 5 -1 1 5 then A+ B is
a) 1 -2 4 b) 1 -2 8
3 3 2 3 3 4
2 1 5 2 1 10
c) 1 -4 8
3 6 4
2 2 10 d) none
2) If A²= 8A + kI where
A= 1 0
-1 7 then k is
a) 7 b) -7 c) 1 d) -1
3) The matrix
λ 7 -2
4 1 3
2 -1 2 is a singular matrix if λ is
a) 2/5 b) 5/2 c) -5 d) none
4) If A= a b
c d then A² is
a) a² b² b) a²+ bc ab+ bd
c² d² ac+ dc bc + d²
c) non-existent d) none
5) If A= α 0 & B= 1 0
1 1 5 1
Such that A² = B then α is
a) 1 b) -1 c) 4 d) none
6) If A= 2 -3 & B= 1 5 μ & C= 2 4 1
1 λ 0 2 -3 1 -1 13 with the relation AB= C then find λ and μ
a) λ= 3, μ= 4
b) λ= 3p4, μ= -3
c) no real value of λ, μ are possible d) none
7) A= coa²θ cosθ sinθ
cosθ sinθ sin²θ
B= cos²φ cosφ sinφ
cosφ sinφ sin²φ
With the relation AB= 0 then|θ - φ| is
a) 0 b) π/2 c) π/4 d) π
8) If A= 0 -4 1
2 λ -3
1 2 -1 then A⁻¹ exist (i.e., A is invertible) if
a) λ ≠ 4 b) λ ≠ 8 c) λ = 4 d) none
9) If A= 1 0 2
0 1 -1
1 2 1
Find the reciprocal of A
a) -3 -4 2
-1 1 -1
1 2 -1
b) 3 4 -2
1 -1 1
-1 -2 1
c) -3 -1 1
-4 1 2
2 -1 -1
d) none
10) If A= 1 -1 1
1 2 0
1 3 0 then the value of|adj A| is equal to
a) 5 b) 0 c) 1 d) none
11) If A= sinα - cosα 0
cosα sinα 0
0 0 1 then A⁻¹ is equal to
a) Aᵀ b) A c) adj A d) none
12) If A= 4 -1 -4
3 0 -4
3 -1 -3 then A² is equal to
a) A B) I c) Aᵀ d) none
13) If f(x)= cosx - sinx 0
sinx cosx 0
0 0 1 then f(x + y) is equal to
a) f(x) + f(y) b) f(x) - f(y) c) f(x) . f(y) d) none
14) If A= 1 ω ω² & B= ω ω² 1 & C= 1
ω ω² 1 ω² 1 ω ω
ω² 1 ω² ω ω² 1 ω² where ω is the complex cube root of 1 then (A + B)C is equal to
a) 0 b) 1 0 0 c) 1 d) 1
0 0 1 0 0 1
0 0 0. 1 1 1
15) If A= 0 c -b & B= a² ab ac
-c 0 a ba b² bc
b -a 0 ca cb c² then AB is equal to
a) 0 b) I c) 2I d) none
16) If the matrices such that AB= C
B= 1 -2 & C= 6 0
1 4 0 6 then Find the matrix A
a) 2 4 b) -1 1 c) 4 2 d) none
1 -1 4 2 -1 1
17) The rank of the matrix
-5 3 2
3 2 -5
4 -1 -3 is
a) 3 b) 2 c) 1 d) none
18) The rank of the matrix
1 2 3
λ 2 4
2. -3 1 is 3 if
a) λ≠ 18/11 b) λ= 18/11 c) λ= - 18/11 d) none
19) The rank of the matrix
4 1 0 0
3 0 1 0
5 0 0 1 is
a) 4 b) 3 c) 2 d)
20) The system of equations x+ y+ z=2, 2x- y+ 3z=5; x- 2y - z= -1 written in Matrix form is
a)
21) If A= 1 x 1 & B= 1 3 2 & C= 1
2 5 1. 2
x
With relation ABC= 0 then x is
a) 2 b) -2 c) 1 d) 4 d) none
22) If A= x+ y y & B= 2 & C= 3
2x x- y -1 2 with the relation AB= C then x. y is equal to
a) -5 b) 5 c) 4 d) 6
****
23) 1 -2 3
2 -1 4
3 4 1 is a
a) rectangular matrix
b) singular matrix
c) square matrix
d) non-singular matrix
24) If A= 3 1 & B= 5 4 6
-1 2 4 1 2
0 6 -5 -1 1 then
a) A+ B exists
b) AB exists
c) BA exists d) none
25) If A= 1 1 1
1 1 1
1 1 1
a) A³= 9A b) A³= 27A c) A+ A= A² d) A⁻¹ does not exist
PROGRESSION, RELATED INEQUALITIES AND SERIES
1) If a₁, a₂, a₃, are in AP then aₚ, aq, aᵣ are in AP if p.q,r are in
a) AP b) GP c) HP d) none
2) Let tᵣ denote the rth term of an AP. If tₘ = 1/n and tₙ = 1/m then tₘₙ equals
a) 1/mn b) 1/m + 1/n c) 1 d) 0
3) If p,q,r,s ∈ N and they are four consecutive terms of an AP then the pth, qth, rth, sth terms of a GP are in
a) AP b) GP c) HP d) none
4) If in a progression a₁, a₂, a₃, .....etc, (aᵣ - aᵣ₊₁) bears a constant ratio with aᵣ. aᵣ₊₁ then the terms of the progression are in
a) AP b) GP c) HP d) none
5) If a₂a₃/a₁a₄ = (a₂+ a₃)/(a₁+ a₄) = 3(a₂ - a₃)/(a₁ - a₄) then a₁, a₂, a₃, a₄ are in
a) AP b) GP c) HP d) none
6) Let x,y,z be three positive prime numbers. The progression in which √x, √y, √z can be three terms (not necessarily) consecutive is
a) AP b) GP c) HP d) none
7) Let f(x)= 2x +1. Then the number of real values of x for which the three unequal numbers f(x), f(2x), f(4x) are in GP is
a) 1 b) 2 c) 0 d) none
8) If a> 0, r∈ N and a₁, a₂, a₃, ....a₂ₙ are in AP then
(a₁+ a₂ₙ)/(√a₁+ √a₂) + (a₂ + a₂ₙ₋₁)/(√a₂ + √a₃) + (a₃ + a₂ₙ₋₂)/(√a₃ + √a₄) + ...+ (aₙ + aₙ₊₁)/(√aₙ + aₙ₊₁) is equal to
a) n -1 b) n(a₁ + a₂ₙ)/(√a₁ + √aₙ₊₁) c) (n -1)/(√a₁ + √aₙ₊₁) d) none
9) If a₁, a₂, a₃, ....a₂ₙ₊₁ are in AP then
(a₂ₙ₊₁ - a₁)/(a₂ₙ₊₁ + a₁) + (a₂ₙ - a₂)/(a₂ₙ + a₂) + .... + (aₙ₊₂ - aₙ)/(aₙ₊₂ + aₙ) is equal to
a) n(n +1)/2 . (a₂ - a₁)/aₙ₊₁
v) n(n +1)/2
c) (n +1)(a₂ - a₁) d) none
10) a₁, a₂, a₃, .... be in AP and aₚ, aq, aᵣ be in GP. Then aq : aₚ is equal to
a) (r - p)/(q - p) b) (q - p)/(r - q) c) (r - q)/(q - p) d) none
11) If a,b,c are in GP then a+ b, 2b, b + c are in
a) AP b) GP c) HP d) none
12) If a,b,c,d are non-zero real numbers such that
(a² + b² + c²)(b²+ c²+ d²) ≤ (ab+ bc+ cd)² then a,b,c,d are in
a) AP b) GP c) HP d) none
13) If 4a²+ 9b² + 16c²= 2(3ab + 6bc + 4ca) , where a,b,c are non-zero numbers, then a,b,c are in
a) AP b) GP c) HP d) none
14) If a,b,c are in AP then a/bc, 1/c, 2/b are in
a) AP b) GP c) HP d) none
15) If in an AP, t₁ = log₁₀a, tₙ₊₁ = log₁₀b and t₂ₙ₊₁ = log₁₀c then a,b,c are in
a) AP b) GP c) HP d) none
16) If n!, 3 x n! and (n+1)! are in GP then n!, 5x n! and (n +1)! are in
a) AP b) GP c) HP d) none
17) In an AP, the pth term is q and the (p + q)th term is 0. Then the qth term is
a) -p b) p c) p+ q d) p - q
18) In a sequence of (4n +1) terms the first (2n +1) terms are in AP whose common difference is 2, and the last (2n +1) terms are in GP whose common ratio is 0.5. if the middle terms of the AP and GP are equal then the middle term of the sequence is
a) (n. 2ⁿ⁺¹)/(2ⁿ -1) b) (n. 2ⁿ⁺¹)/(2²ⁿ -1) c) n. 2ⁿ) d) none
19) If x² + 9y² + 25z² = xyz (15/x + 5/y + 3/z) then x,y,z are in
a) AP b) GP c) HP d) none
20) If a,b,c,d and p are distinct real numbers such that
(a²+ b²+ c²)p² - 2(ab+ bc+ cd)(b²+ c² + d²)≤ 0 then a,b,c,d are in
a) AP b) GP c) HP d) none
21) The largest term common to the sequence 1,11,21,31,....to 100 terms and 31,36,41,46,....to 100 terms is
a) 381 b) 471 c) 281 d) none
22) The interior angles of a convex polygon are in GP, the common difference being 5°. If the smallest angle is 2π/3 then the number of sides is
a) 9 b) 16 c) 7 d) none
23) The minimum number of terms of 1+ 3+5+7+.... that add up to a number exceeding 1357 is
a) 15 b) 37 c) 35 d) 17
24) In the value of 100! the number of zeros at the end is
a) 11 b) 22 c) 23 d) 24
25) The sum of all the proper divisors the 9900 is
a) 33851 b) 23952 c) 23951 d) none
26) The sum of all odd proper divisors of 360 is
a) 77 b) 78 c) 81 d) non
27) In the sequence 1,2,2,3,3,3,4,4,4,4,..., where n consecutive terms have the value n, the 150th term is
a) 17 b) 16 c) 18 d) none
28) In the sequence 1,2,2,4,4,4,4,8, 8, 8, 8, 8, 8, 8, 8,.... where n consecutive terms have the value n, the 1025th term is
a) 2⁹ b) 2¹⁰ c) 2¹¹ d) 2⁸
29) Let [tₙ] be a sequence of integers in GP in which t₄ : t₆ = 1: 4 and t₂ + t₅ = 216. Then t₁ is
a) 12 b) 14 c) 16 d) none
30) If log(5c/a), log(3b/5c) and log(a/3b) are in AP, where a,b,c are in GP, then a,b,c are the lengths of sides of
a) an isosceles triangle
b) an equilateral triangle
c) a scalene triangle d) none
31) Let S be the sum, P be the product and R be the sum of the reciprocals of n terms of a GP. Then P²Rⁿ : Sⁿ is equal to
a) 1:1 b) (common ratio)ⁿ : 1 c) (first term)² : (common ratio)ⁿ d) none
32) If the pth, qth and rth terms of an AP are in GP then the common ratio of the GP is
a) (p+ q)/(r+ q) b) (r - q)/(q - p) c) (p - r)/(p - q) d) none
33) The number of terms common between the series 1+2+4+8+....to 100 terms and 1+4+7+10....to 100 terms is
a) 6 b) 4 c) 5 d) none
34) The 10th common term between the series 3+7+11+... and 1+6+11+...is
a) 191 b) 193 c) 211 d) none
35) Three consecutive terms of a progression are 30,24,20. The next term of the progression is
a) 18 b) 120/7 c) 16 d) none
36) If the numbers are in GP then the numbers obtained by adding the middle number to each of the three numbers are in
a) AP b) GP c) HP d) none
37) If a₁, a₂, a₃ are in AP, a₂, a₃, a₄ are in GP and a₃, a₄, a₅ are in HP then a₁, a₃, a₅ are in
a) AP b) GP c) HP d) none
38) If a,b,c,d are four numbers such that the first three are in AP while the last three are in HP then
a) bc= ad b) ac= bd c) ab= cd d) none
39) If the first two terms of an HP be 2/5 and 12/23 then the largest positive term of the progression is the
a) 6th term b) 7th term c) 5th term d) 8th term
40) If x, 2y, 3z are in AP, where the distinct numbers, x,y,z are in GP then the common ratio of the GP is
a) 3 b) 1/3 c) 2 d) 1/2
41) If x>1, y> 1, z> 1 are three numbers in GP then
1/(1+ lnx) , 1/(1+ ln y), 1/(1+ logz) are in
a) AP b) HP c) GP d) none
42) If a, a₁, a₂, a₃,....a₂ₙ₋₁, b are in AP a, b₁, b₂, b₃,....b₂ₙ₋₁, b are in GP and a, c₁, c₂, c₃,,.....c₂ₙ₋₁, b are in HP, where a,b are positive, then the equation aₙx² - bₙx + cₙ = 0 has its roots.
a) real and unequal
b) real and equal
c) imaginary d) none
43) If a, x, b are in AP, a, y, b are in GP and a,z, b in HP such that x = 9z and a> 0, b >0 then
a) |y|= 3z b) 3|y|= x c) 2y = x + z d) none
44) If there numbers are in HP then the numbers obtained by subtracting half of the number from each of them are in
a) AP b) GP c) HP d) none
45) a,b,c,d , e are five numbers in which the first three are in AP and the last three are in HP. If the three numbers in the middle are in GP then the numbers in the odd places are in
a) AP b) GP c) HP d) none
46) Let a₁, a₂, a₃,....a₁₀ be in AP and h₁, h₂, h₃,....h₁₀ be in HP. If a₁ = h₁ =2 and a₁₀ = h₁₀ = 3 then a₄h₇ is
a) 2 b) 3 c) 5 d) 6
47) If in an AP, Sₙ = p. n² and Sₘ = p. m², where Sᵣ denotes the sum of r terms of the AP, then Sₚ is equal to
a) p³/2 b) mnp c) p² d) (m + n)p²
48) If Sᵣ denotes the sum of the first r terms of an AP then (S₃ᵣ - Sᵣ₋₁)/(S₂ᵣ - S₂ᵣ₋₁) is equal to
a) 2r -1 b) 2r+ 1 c) 4r + 1 d) 2r +3
49) Sᵣ denotes the sum of the first r terms of a GP. Then Sₙ, S₂ₙ - Sₙ, S₃ₙ - S₂ₙ are in
a) AP b) GP c) HP d) none
50) If (1- p)(1+ 3x + 9x²+ 27x³+ 81x⁴+ 243x⁵)= 1- p⁶, p ≠ 1 then the value of p/x is
a) 1/3 b) 3 c) 1/2 d) 2
51) If the sum of the series 1+ 2/x + 4/x² + 8/x³ + .....to ∞ is a finite number then
a) x< 2 b) x > 1/2 c) x > -2 d) x< -2 or x > 2
52) Let Sₙ denotes the sum of the first n terms of an AP. If S₂ₙ= 3Sₙ then S₃ₙ : Sₙ is equal to
a) 4 b) 6 c) 8 d) 10
53) In a GP of even number of terms, the sum of all terms is 5 times the sum of the odd terms. The common ratio of the GP is
a) -4/5 b) 1/5 c) 4 d) none
54) In an AP, Sₚ = q, Sq = p and Sᵣ denotes the sum of the first r terms. Then Sₚ₊q is equal to
a) 0 b) -(p+ q) c) p+ q d) pq
55) The co-efficient of x¹⁵ in the product
(1- x)(1- 2x)(1- 2²x)(1- 2³x)...(1- 2¹⁵x) is equal to
a) 2¹⁰⁵- 2¹²¹ b) 2¹²¹ - 2¹⁰⁵ c) 2¹²⁰ - 2¹⁰⁴ d) none
56) The co-efficient of x⁴⁹ in the product (x -1)(x -3)...(x - 99) is
a) -99² b) 1 c) -2500 d) none
57) If a,b,c are in AP then a+ 1/bc, b+ 1/ca, c+ 1/ab are in
a) AP b) GP c) HP d) none
58) The AM of two given positive numbers is 2. If the larger number is increased by 1, the GM of the numbers becomes equal to the AM of the given numbers. Then HM of the given numbers is
a) 3/2 b) 2/3 c) 1/2 d) none
59) Let a,b be two positive numbers, where a> b and 4x GM = 5x HM for the numbers. Then a is
a) 4b b) b/4 c) 2b d) b
60) If a, a₁, a₂, a₃,......a₂ₙ, b are in AP and a, g₁, g₂, g₃, .....g₂ₙ, b are in GP and h is the HM of a and b then
(a₁+ a₂ₙ)/(g₁g₂ₙ) + (a₂ + a₂ₙ₋₁)/(g₂g₂ₙ₋₁) + ....+ (aₙ + aₙ₊₁)/(gₙgₙ₊₁) is equal to
a) 2n/b b) 2nh c) nh d) n/h
61) Let a₁ =0 and a₁, a₂, a₃,....aₙ be real numbers such that |aᵢ|= |aᵢ₋₁ +1| for all i then the AM of the numbers a₁, a₂, a₃,.....aₙ has the value A where
a) A< -1/2 b) A< -1 c) A≥ -1/2 d) A= -1/2
62) Let there be a GP whose first term is a and the common ratio is r. If A and H are the arithmetic mean and the harmonic mean respectively for the first n terms of the GP, A. H is equal to
a) a²rⁿ⁻¹ b) arⁿ c) a²rⁿ d) none
63) If the first and the (2n -1)th terms of an AP, a GP and an HP are equal and their nth terms are a, b and c respectively then
a) a= b = c b) a≥ b ≥ c c) a+ c = b d) ac - b²= 0
64) (aⁿ + bⁿ)/(aⁿ⁻¹ + bⁿ⁻¹) is the HM between a and b if n is
a) 0 b) 1/2 c) -1/2 d) 1
65) If the harmonic mean between P and Q be H then H(1/P + 1/Q) is equal to
a) 2 b) PQ/(P + Q) c) (P+ Q)/PQ d) 1/2
66) Let x be the AM and y,z be two GMs between two positive numbers. Then (y³ + z³)/xyz is equal to
a) 1 b) 2 c) 1/2 d) none
67) If HM : GM= 4: 5 for two positive numbers then the ratio of the numbers is
a) 4:1 b) 3:2 c) 3:4 d) 2:3
68) In a GP of alternatively positive and negative terms, any term is the AM of the next two terms. Then the common ratio is
a) -1 b) -3 c) -2 d) -1/2
69) If a,b, c are in AP and p, P' are the AM and GM respectively between a and b, while q, q' are the AM and GM respectively between b and c, then
a) p²+ q²= p'²+ q'² b) pq= p'q' c) p²- q²= p'²- q'² b) none
70) If -π/2< θ< π/2 then the minimum value of cos³θ + sec³θ is
a) 1 b) 2 c) 0 d) none
71) If a> 1, b > 1 then the minimum value of logᵥa+ logₐv is
a) 0 b) 1 c) 2 d) none
72) The minimum value of 4ˣ + 4¹⁻ˣ, x ∈ R, is
a) 2 b) 4 c) 1 d) none
73) If x= log₅3+ log₇5 + log₉7 then
a) x≥ 3/2 b) x ≥ 1/³√2 c) x ≥ 3/³√2 d) none
74) If aₙ > 1 for all n ∈N then
logₐ₂a₁ + logₐ₃a₂ +.....+ Logₐₙaₙ₋₁ + log₁aₙ has the minimum value
a) 1 b) 2 c) 0 d) none
75) The product of n positive numbers is 1. Their sum is
a) a positive integer
b) divisible by n
c) equal to n + 1/n
d) greater than or equal to n
76) If x,y,z are three real numbers of the sum sign then the value of x/y + y/z + z/x lies in the interval
a) [2, +∞) b) [3, +∞) c) (3, +∞) d) (-∞,3)
77) The least value of 2 log₁₀₀a - logₐ0.0001, a> 1 is
a) 2 b) 3 c) 4 d) none
78) If 0< x<π/2 then the minimum value of (sin x + cos x + cosec 2x)³ is
a) 27 b) 13.5 c) 6.75 d) none
79) If x,y,z are positive then the minimum value of
xˡᵒᵍʸ ⁻ˡᵒᵍᶻ + yˡᵒᵍᶻ⁻ˡᵒᵍˣ + zˡᵒᵍˣ⁻ˡᵒᵍʸ is
a) 3 b) 1 c) 9 d) 16
80) a,b,c are three positive numbers and abc² has the greatest value 1/64. Then
a) a=b =1/2, c= 1/4
b) a=b =1/4, c= 1/2
c) a=b = c= 1/3 d) none
81) if a> 0, b >0, c > 0 and the minimum value of
a(b² + c²) + b(c² + a²) + c(a² + b²) is λabc then λ is
a) 2 b) 1 c) 6 d) 3
82) The value of ¹⁰ₙ₌₁ ∑ ⁿ₀∫ x dx is
a) an even integer
b) an odd integer
c) a rational number
d) an irrational number
83) The sum of (0.2 + 0.004+ 0.00006+ 0.0000008+... to ∞) is
a) 200/891 b) 2000/9801 c) 1000/9801 d) none
84) If (2n + r)r, n ∈ N, r ∈ N is expressed as the sum of k consecutive odd natural numbers then k is equal to
a) r b) n c) r+1 d) n+1
85) ⁿᵣ₌₁ ∑r² - ⁿₘ₌₁ ∑ ᵐᵣ₌₁∑ r is equal to
a) 0 b) (1/2) (ⁿᵣ₌₁ ∑ r²+ ⁿᵣ₌₁ ∑ r)
c) (ⁿᵣ₌₁ ∑ r² - ⁿᵣ₌₁ ∑ r) d) none
86) If (1+ x)(1+ x²)(1+ x⁴)....(1+ x¹²⁸)= ⁿᵣ₌₀ ∑ xʳ then n is
a) 225 b) 127 c) 63 d) none
87) The value of ᵐₙ₌₁∑ log(a²ⁿ⁻¹/bᵐ⁻¹) (a≠ 0, 1; b ≠ 0, 1) is
a) m log(a²ᵐ/bᵐ⁻¹)
b) log(a²ᵐ/bᵐ⁻¹)
c) (m/2) log(a²ᵐ/b²ᵐ⁻²)
d) (m/2) log(a²ᵐ/bᵐ ⁺¹)
88) The sum of the product of the ten numbers ±1, ±2, ±3, ±4, ±5 taking two at a time is
a) 165 b) -55 c) 55 d) none
89) The sum of the series 1/log₂4 + 1/log₄4 + 1/log₈4 +....+ 1/log₂ⁿ 4 is
a) n(n +1)/2 b) n(n +1)(2n +1)/2 c) 1/n(n +1) d) n(n +1)/4
90) If ⁿₙ₌₁∑n, √10/3. ⁿₙ₌₁∑n², ⁿₙ₌₁∑n³ are in GP the value of n is
a) 2 b) 3 c) 4 d) none
91) The value of ⁿᵣ₌₁ ∑ {(2r -1)a + 1/bʳ} is equal to
a) an²+ (bⁿ⁻¹ -1)/(bⁿ⁻¹(b -1))
b) an² + (bⁿ -1)/(bⁿ(b -1))
c) an³ + (bⁿ⁻¹ -1)/(bⁿ(b -1)) d) none
92) If aₙ = ⁿₙ₌₁∑ (1+ 2+ 2²+.... To n terms)/2ⁿ then sₙ is equal to
a) 2ⁿ - (n +1) b) 1 - 1/2ⁿ c) n - 1 + 1/2ⁿ d) 2ⁿ -1
93) Let Sₙ denote the sum of the cubes of the first n natural numbers and sₙ denote the sum of the first n natural numbers. Then ⁿᵣ₌₁ ∑ Sᵣ/sᵣ is equal to
a) n(n+1)(n+2)/6
b) n(n+1)/2
c) (n²+ 3n+2)/6 d) none
94) It is known that ∞ᵣ₌₁ ∑ 1/(2r -1)² = π²/8. Then ∞ᵣ₌₁ ∑ 1/r² is equal to
a) π²/24 b) π²/3 c) π²/6 d) none
95) It is given that 1/1⁴ + 1/2⁴ + 1/3⁴+......to ∞= π⁴/90. Then 1/1⁴ + 1/3⁴ + 1/5⁴+......to ∞ is equal to
a) π⁴/96 b) π⁴/45 c) 88π⁴/90 d) none
96) If in a series tₙ ²⁰ₙ₌₁∑tₙ is equal to
a) (20! -1)/20! b) (21! -1)/21! c) 1/2(n -1)! d) none
97) If tₙ denotes the nth term of the series 2+3+6+11+18+.... then t₅₀ is
a) 49² -1 b) 49² c) 50²- 1 d) 50²+2
98) 2¹⁾⁴. 4¹⁾⁸. 8¹⁾¹⁶ ....is equal to
a) 1 b) 2 c) 3/2 d) none
99) The sum of n terms of the series
1²+ 2.2² + 3² + 2.4²+ 5² + 2.6²+... is n(n+1)²/2 when n is even. When n is odd, the sum is
a) n²(n +1)/2 b) n(n²-1)/2 c) 2(n +1)²(2n +1) d) none
100) If n is an odd integer greater than or equal to 1 then the value of n³ - (n -1)³ + (n -2)³ - .....+ (-1)ⁿ⁻¹. 1³ is
a) (n+1)²(2n -1)/4
b) (n -1)²(2n -1)/4
c) (n +1)²(2n +1)/4 d) none
101) Observe that
1³= 1, 2³= 3+5, 3³= 7+9+11, 4³= 13+15+17+19.
Then n³ as a similar series is
a) [2{n(n -1)/2 +1} - 1] + [2{n(n +1)/2 +1} + 1]+ .... [2{n(n +1)/2 +1} +2n -3]
b) (n²+ n +1)+ (n²- n +3)+ (n²- n +5)+ ....+ (n²+ 3n -1)
c) (n²- n +1)+ (n²- n +3)+ (n²- n +5)+ ....+ (n²+ n -1) d) none
102) Let tᵣ= 2ʳ⁾² + 2⁻ʳ⁾². Then ¹⁰ᵣ₌₁∑t²ᵣ is equal to
a) (2²¹ -1)/2¹⁰ + 20
b) (2²¹ - 1)/2¹⁰ + 19
c) (2²¹ -1)/2¹⁰ - 1 d) none
103) Let Sₖ = lim ₙ→∞ ⁿᵢ₌₀∑ 1/(k +1)ⁱ. Then ⁿₖ₌₁∑ k Sₖ equals
a) n(n +1)/2
b) n(n -1)/2
c) n(n +2)/2 d) n(n +3)/2
104) Let tₙ = n. (n!). Then ¹⁵ₙ₌₁∑tₙ is equal to
a) 15! -1 b) 15! + 1 c) 16! -1 d) none
105) The sum of
3/(1.2). 1/2 + 4/(2.3) . (1/2)² + 5/(3.4) . (1/2)³+ ....to n terms is equal to
a) 1- 1/(n+1)2ⁿ
b) 1- 1/(n. 2ⁿ⁻¹)
c) 1+ 1/(n +1)2ⁿ d) none
106) Let f(n)= [1/2 + n/100] where [x] denotes the integral part of x. Then the value of ¹⁰⁰ₙ₌₁∑ f(n) is
a) 50 b) 51 c) 1 d) none
107) Aᵣ ; r= 1,2,3,....n are n points on the parabola y²= 4x in the first quadrant. If Aᵣ = (xᵣ, yᵣ), where x₁, x₂, x₃, ....xₙ are in GP and x₁= 1, x₂= 2, then yₙ is equal to
a) (-2)⁽ⁿ⁺¹⁾/² b) 2ⁿ⁺¹ c) (√2)ⁿ⁺¹ d) (2)ⁿ⁾²
108) In the given square, a diagonal is drawn, and parallel line segments joining points on the adjacent sides are drawn on both sides of the diagonal. The length of the diagonal is n √2 cm. If the distance between consecutive line segment be 1/√2 cm then the sum of the lengths of all possible line segments and the diagonal is
a) n(n +1)√2 cm b) n² cm c) n(n +2)cm d) n²√2 cm
109) ABCD is a square of length a, a ∈ N, a > 1. Let L₁, L₂, L₃, ...be points on BC such that BL₁ = L₁L₂ = L₂L₃ = ....= 1 and M₁, M₂, M₃, ....be points on CD such that CM₁ = M₁M₂ = M₂M₃ = ....= 1. Then ᵅ⁻¹ₙ₌₁∑ (AL²ₙ + LₙM²ₙ) is equal to
a) (1/2) a(a -1)² b) (1/2) a(a -1)(4a -1)
c) (1/2) a(a -1)(2a -1)(4a -1) d) none
110) The sum of infinite terms of a decreasing GP is equal to the greatest value of the function f(x)= x³+ 3x - 9 in the interval [-2,3] and the difference between the first two terms is f'(0). Then the common ratio of the GP is
a) -2/3 b) 4/3 c) 2/3 d) -4/3
111) The lengths of three unequal edges of a rectangular solid block are in GP. The volume of the block is 216 cm³ and the total surface area is 252 cm². The length of the longest edge is
a) 12 cm b) 6cm c) 18cm d) 3 cm
112) ABC is a right angled triangle in which angle B =90° and BC= a. If n points L₁, L₂, ....Lₙ in AB are such that AB is divided in n+1 equal parts and L₁M₁ , L₂M₂, ....LₙMₙ are line segments parallel to BC and M₁, M₂, ....Mₙ are on AC then the sum of the lengths of L₁M₁, L₂M₂, ....LₙMₙ is
a) a(n +1)/2 b) a(n -1)/2 c) an/2 d) impossible to find from the given data
113) If AM of the numbers 5¹⁺ˣ and 5¹⁻ˣ is 13 then the set of possible real values of x is
a) {5,1/5} b) {1,-1} c) {x : x²-1= 0, x ∈ R} d) none
114) If the AM of two positive numbers between three times their geometric mean then the ratio of the numbers is
a) 3± 2√2 b) √2±1 c) 17+ 12√2 d) (3- 2√2)⁻²
115) If a,b,c are in HP then 1/(b - a) + 1/(b - c) is equal to
a) 2/b b) 2/(a+ c) c) 1/a + 1/c d) none
116) Sᵣ denotes the sum of the first r terms of an AP. Then S₃ₙ : (A₂ₙ - Sₙ) is
a) n b) 3n c) 3 d) independent of n
117) If aˣ= vʸ = cᶻ and x, y,z are in GP then log꜀v is equal to
a) logᵥa b) logₐv c) z/y d) none
118) The value of ⁿᵣ₌₁∑ 1/[√(a+ rx) + √{a + (r -1)x}] is
a) n/{√a + √(a+ nx)}
b) {√(a+ nx) - √a}/x
c) n/{√(a+ nx) - a}/x d) none
119) Let ⁿₙ₌₁∑r⁴ = f(n). Then ⁿᵣ₌₁ ∑ (2r -1)⁴ is equal to
a) f(2n) - 16 f(n) for all n ∈N
b) f(n) - 16 f{(n-1)/2} when n is odd
c) f(n) - 16 f(n/1) when n is even d) none
120) If 1. ⁿP₁, ⁿP₂, ⁿO₃ are three consecutive terms of an AP then they are
a) in GP b) in HP c) equal d) none
121) In a GP the product of the first four terms is 4 and the second term is the reciprocal of the fourth term. The sum of the GP up to infinite terms is
a) 8 b) -8 c) 8/3 d) -8/3
122) If ⁿ ₖ₌₀ ∑(ᵏₘ₌₀ ∑ m¹)= an⁴ + bn³+ cn²+ dn + e then
a) a=1/12 b= 1/6 c) d= 1/6 d) e=0
123) If a,b,c, d are four positive numbers then
a) (a/b + b/c)(c/d + d/e)≥ 4 √(a/e)
b) (a/b + c/d)(b/c+ d/e)≥ 4 √(a/e)
c) a/b + b/c + c/d + d/e + e/a ≥ 5
d) b/a+ c/b + d/c + e/d + a/e ≥ 1/5
124) Let f(x)= +1- xⁿ⁺¹)/(1- x) and g(x)= 1- 2/x + 3/x² - ...... + (-1)ⁿ (n+1)/xⁿ. Then the constant term in f'(x) x g(x) is equal to
a) n(n²-1)/6, when n is even
b) n(n+1)/2, when n is odd
c) -n(n+1)/2, when n is even
d) -n(n -1)/2, when n is odd
125) Let aₙ = product of the first n natural numbers. Then for all n∈N
a) nⁿ ≥ aₙ b) {n+1)/2}ⁿ≥ n! c) nⁿ ≥ aₙ₊₁ d) none
126) Let the set A={2,4,6,8,....} And B={3,6,9,12,....} and n(A)= 200, n(B)= 250. Then
a) n(A∩B)= 67 b) n(A∪B)= 450 c) n(A∩B)= 66 d) n(A∪B)= 384
127) Let a, x, b be in AP; a, y, b be in GP and a, z, b in HP. If x= y +2 and a= 5z then
a) y²= xz b) x> y > z c) a= 9, b= 1 d) a= 1/4, b= 9/4
128) S₁, S₂, S₃, ......be squares such that for each n ≥ 1, the length of a side of Sₙ equals the length of a diagonal of Sₙ₊₁. If the length of a side of S₁ is 10cm the pn for which of the following values of n is the area of Sₙ less than 1 cm² ?
a) 7 b) 8 c) 9 d) 10
129) Three positive numbers form a GP. If the middle number is increased by 8, the three numbers form an AP. If the last number is also increased by 64 along with the previous increase in the middle number, the resulting numbers form a GP again. Then
a) common ratio= 3
b) first number= 4/9
c) common ratio= -5
d) first number= 4
130) If a, b, c are in GP and a, p, q are in AP such that 2a, b+ p, c+ q are in GP then the common difference of the AP is
a) √2 a b) (√+1)(a - b) c) √2(a+ b) d) (√2-1)(b - a)
131) If x, y, z are positive numbers in AP then
a) y²≥ xz b) y≥ 2√(xz)
c) (x + y)/(2y - x) + (y+ z)/(2y - z) has the minimum value 2
d) (x + y)/(2y - x) + (y+ z)/(2y - z)≥ 4
132) Between two unequal numbers, if a₁, a₂ are two AMa; g₁, g₂ are two GMs and h₁, h₂ are two HMs then g₁. g₂ is equal to
a) a₁h₁ b) a₁h₂ c) a₂h₂ d) a₂h₁
133) The number 1,4,16 can be three terms (not necessarily consecutive) of
a) no AP
b) only one GP
c) infinite number of APs
d) infinite number of GPs
DETERMINANTS
1) a+ x a x
If a- x a x= 0
a- x a -x
Then x is
a) 0 b) a c) 3 d) 2a
2) 0 p- q p- r
q - p 0 q - r
r -p r- q 0 is equal to
a) p+ q+ r b) 0 c) p - q - r d) -p + q + r
3) If a≠ b ≠ c such that
a²-1 b³-1 c³ -1
a b c = 0 then
a² b² c²
a) ab+ bc+ ca= 0 b) a+ b+ c=0 c) abc=1 d) a+ b+ c=1
4) 1+ x 1 1
1 1+ x 1 is equal to
1 1 1+x
a) x²(x+3) b) 3x³ c) 0 d) x³
5) 6i - 3i 1
If 4 3i -1 = x + iy then find x,upy
20 3 i
a) 3,1 b) 1,3 c) 0,3 d) 0,0
6) xp+ y x y
yp+ z y z= 0
0 xp+ y yp+z
foe all p ∈ E if
a) x,y,z are in AP
b) x,y,z are in GP
c) x,y,z are in HP
d) xy, yz, zx are in AP
7) a a+ d a+ d
∆= a² (a+d)² (a+2d)²=0. Then
2a+3d 2(a+d) 2a+d
a) d=0 b) a+ d=0 c) d=0 or a+ d= 0 d) none
8) The value of the determinant
bc ca ab
p q r
1 1 1
Where a,b,c are pth, qth and rth terms of a HP is
a) ap+ bq+ cr
b) (a+ b+ c)(p+ q+ r)
c) 0 d) none
9) The sum of two non integral roots of
x 2 5
3 x 3 =0
5 4 x is
a) 5 b) -5 c) -18 d) none
10) If x,y,z are integers in AP lying between 1 and 9 and x51, y41 and z31 are three digit numbers then the value of
5 4 3
x51 y41 z31
x y z is
a) x+ y+ z b) x - y + z c) 0 d) none
11) 1 1 1 1 bc a
If ∆₁=a b c & ∆₂= 1 ca b
a² b² c² 1 ab c then
a) ∆₁+ ∆₂=0 b) ∆₁+ 2∆₂=0 c) ∆₁= ∆₂ d) none
12) Two non-zero distinct numbers a,b are used as elements to make determinants of the third order. The number of determiners whose values is zero for all a,b is
a) 24 b) 32 c) a+ b d) none
13) The value of
a₁x+ b₁y a₂x+ b₂y a₃x + b₃y
b₁x+ a₁y b₂x+ a₂y b₃x+ a₃y
b₁x+ a₁ b₂x+ a₂ b₃x+ a₃
is equal to
a) x² + y² b) 0 c) a₁a₂a₃x²+ b₁b₂b₃y² d) none
14) x₁ y₁ 1 1 1 1
If x₂ y₂ 1 = b₁ b₂ b₃
x₃ y₃ 1 a₁ a₂ a₃
Then the two triangles whose vertices are (x₁, y₁), (x₂, y₂),(x₃, y₃ ) and (a₁,b₁),(a₂, b₂), (,a₃, ,b₃) are
a) congruent b) similar c) equal in area d) none
15) If α,β are nonreal numbers satisfying x³-1=0 then the value of
λ+1 α β
α λ+β 1
β 1 λ+ α
is equal to
a) 0 b) λ³ c) λ³+1 d) none
16) The value of
¹⁰C₄ ¹⁰C₅ ¹¹Cₘ
¹¹C₆ ¹¹C₇ ¹²Cₘ₊₂
¹²C₈ ¹²C₉ ¹³Cₘ₊₄
is equal to zero when m is
a) 6 b) 4 c) 5 d) none
17) If x> 0 and ≠ 1, y> 0 and ≠ 1, z> 0 and ≠ 1 and the value of
1 logₓy logₓk
logᵧx 1 logᵧk
logₖx logₖy 1
is
a) 0 b) 1 c) -1 d) none
18) The value of
1 1 1
(2ˣ+2⁻ˣ) (3ˣ+3⁻ˣ)² (5ˣ+ 5⁻ˣ)²
(2ˣ -2⁻ˣ)² (3ˣ - 3⁻ˣ)² (5ˣ - 5⁻ˣ)² is
a) 0 b) 30ˣ c) 30⁻ˣ d) none
19) The value of the determinant
⁵C₀ ⁵C₃ 14
⁵C₁ ⁵C₄ 1
⁵C₂ ⁵C₅ 1 is
a) 0 b) -(6!) c) 80 d) none
20) cosC tanA 0
sinB 0 -tanA
0 sinB cosC
has the value
a) 0 b) 1 c) sinA sinB sinC d) none
21) The value of
x x² - yz 1
y y² - zx 1
z z²- xy 1
a) 1 b) -1 c) 0 d) -xyz
22) If √-1= i and ω is a nonreal cube root of unity then the value of
1 ω² 1+ i+ ω²
-i -1 -1- i + ω
1- i ω²-1 -1
is equal to
a) 1 b) I c) ω d) 0
23) 1 x x+1
If f(x)= 2x x(x -1) x(x+1)
3x(x-1) x(x-1)(x-2) x(x²-1)
then f(100) is equal to
a) 0 b) 1 c) 100 d) -100
24) The value of
iᵐ iᵐ⁺¹ iᵐ⁺²
iᵐ⁺⁵ iᵐ⁺⁴ iᵐ⁺³
iᵐ⁺⁶ iᵐ⁺⁷ iᵐ⁺⁸ where i=√-1, is
a) 1 if m is a multiple of 4
b) 0 for all real m
c) - I if m is a multiple of 3 d) none
25) 7 x 2 x 2 7
If ∆₁= -5 x+1 3 , ∆₂= x+1 3 -5
4 x 7 x 7 4
Then ∆₁ - ∆₂ = 0 for
a) x=2 b) all real x c) x=0 d) none
26) 10 4 3 4 x+5 3
If ∆₁=17 7 4, ∆₂=7 x+12 4
4 -5 7 -5 x-1 7
such that ∆₁+ ∆₂= 0 then
a) x=5 b) x has no real value c) x=0 d) none
27) λ²+3λ λ-1 λ+3
Let λ+1 -2λ λ-4
λ-3 λ+4 3λ
= pλ⁴+ qλ³+ rλ²+ sλ + t be an identity in λ, where p,q,r,s,t are independent of λ. Then the value of t is
a) 4 b) 0 c) 1 d) none
28) 1+ x x x²
Let x 1+x x²
x² x 1+x
= ax⁵+ bx⁴ + cx³ + dx² + λx + μ be an identity in x, where a,b,c,d, λ, μ are independent of x. Then the value of λ is
a) 3 b) 2 c) 4 d) none
29) Using the factor theorem it is found that b+ c, c+ a and a+ b are three factors of the determinant
-2a a+ b a+ c
b+ a -2b b+ c
c+ a c+ b -2
The factor in the value of the determinant is
a) 4 b) 2 c) a+ b+ c d) none
30) If the determinant
Cos2x sin²x cos4x
Sin²x cos2x cos²x
Cos4x cos²x cos2x is independent in powes of sinx then the constant term in the expansion is
a) 1 b) 2 c) -1 d) none
31) 1 cosx 1- cosx
If ∆(x)= 1+ sinx cosx 1+sinx - cosx
Then ∫ ∆(x) dx at (π/2,0) is equal to
a) 1/4 b) 1/2 c) 0 d) -1/2
32) If I= √-1 and ⁴√1= α, β, γ, δ then
α β γ δ
β γ δ α
γ δ α β
δ α β γ
is equal to
a) I b) - I c) 1 d) 0
33) The roots of
x a b 1
λ x b 1 = 0
λ μ x 1
λ μ v 1
are independent of
a) λ, μ, v b) a,b c) λ, μ, v, a, b d) none
34) The value of
1 0 0 0 0
2 2 0 0 0
4 4 3 0 0
5 5 5 4 0
6 6 6 6 5 is
a) 6! b) 5! c) 1.2².3.4³.5⁴.6⁴ d) none
35) b²+ c² ab ac
If. bc c²+a² bc
ca cb a²+ b²
= square of a determinant ∆ of the third order then∆ is equal to
a) 0 c b b) a b c
c 0 a b c a
b a 0 c a b
c) 0 -c b
c 0 -a
-b -a 0 d) none
36) The system of equations ax+ 4y + z=0, bx+ 3y+ z=0, CX + 2y+ z=0 has non-trivial solutions if a, b,c c are in
a) AP b) GP c) HP d) none
37) If the equation a(y+ z)= x, b(z + x)= y and c(x + y)= z, where a≠ 1, b≠ -1, c≠ -1, admits of non-trivial solutions then
(1+ a)⁻¹ + (1+ b)⁻¹ + (1- c)⁻¹ is
a) 2 b) 1 c) 1/2 d) none
38) The system of equations 2x- y + z=0; x - 2y + z =0; λx - y + 2z =0 has infinite number of non-trivial solutions for
a) λ= 1 b) λ= 5 c) λ= -5 d) non real value of λ
39) The equation x+ y+ z = 6; x+ 2y + 3z = 10, x+ 2y + mz = n give infinite number of values of the triplet (x,y,z) if
a) m= 3, n ∈R b) m= 3, n ≠ 10 c) m= 3, n = 10 d) none
40) The system of equations 2x+ 3y= 8; 7x - 5y +3=0; 4x - 6y + λ =0 is solvable if λ is
a) 6 b) 8 c) -8 d) -6
41) If the system of equations ax+ by+ c=0; bx+ cy+ a=0; cx+ ay + b=0 has a solution then the system of equations
(b + c)x + (c + a)y + (a+ b)z =0
(c + a)x + (a + b)y + (b+ c)z =0
(a+ b)x + (b+ c)y + (c+ a)z =0 has
a) only one solution
b) no solution
c) infinite number of solutions d) none
42) Let {∆₁, ∆₂, ∆₃, .....∆ₖ} be the set of third order determinants that can be made with the distinct non-zero real numbers a₁, a₂, a₃, ....a₉. then
a) k=9! b) ᵏᵢ₌₀∑ ∆ᵢ = 0
c) atleast one ∆ᵢ= 0 d) none
43) x² (y+ z)² yz
y² (z +x)² zx
z² (x +y)² xy is divisible by
a) x² + y²+ z² b) x - y c) x - y - z d) x+ y+ z
44) The equation
1 x x²
x² 1 1 = 0 has
x x² 1
a) exactly two distinct roots
b) one pair of equal real roots
c) modulus of each root 1
d) three pairs of equal roots
45) n n+1 n+2
If f(n)= ⁿPₙ ⁿ⁺¹Pₙ₊₁ ⁿ⁺²Pₙ₊₂
ⁿCₙ ⁿ⁺¹Cₙ₊₁ ⁿ⁺²Cₙ₊₂
Where the symbols have their usual meanings. The f(n) is divisible by
a) n² + n+ 1 b) (n +1)! c) n! d) none
46) Let x≠ 1 and let a,b,c be non-zero real numbers. Then the determinants
a(1+ x) b c
a b(1+x) c
a b c(1+ x) is divisible by
a) abc b) (1+ x)² c) (1+ x)³ d) x(1+ x)²
47) The arbitrary constant on which the value of the determinant
1 α α²
cos(p - d)a cos pa cos(p - d)a
sin(p - d)a sin pa sin(p -d)a
does not depend is
a) α b) p c) d d) a
48) x+ a x+ b x+ a+ c
Let ∆(x)= x+b x+ c x- 1
x+ c x+ d x- b+d and
²₀∫ ∆(x) dx = -16, where a,b,c,d are in AP, then the common difference of the AP is
a) 1 b) 2 c) -2 d) none
49) If A+ B + C=π, eⁱᶿ = cosθ + i sinθ and
e²ⁱᴬ e⁻ⁱᶜ e⁻ⁱᴮ
z= e⁻ⁱᶜ e²ⁱᴮ e⁻ⁱᴬ
e⁻ⁱᴮ e⁻ⁱᴬ e⁻²ⁱᶜ then
a) Re(z)= 4 b) I'm(z)=0 c) Re(z)= -4 d) Im(z)= -1
50) a+ x a- x a- x
If a-x a+ x a- x =0
a- x a- x a+ x
Then x is
a) 0 b) a c) 3a d) 2a
51) A value of c for which the system of equations
x+ y=1;
(c + 2)x + (c +4)y = 6
(c+2)²x + (c +4)²y = 36 is solvable (consistent) is
a) 1 b) 2 c) 4 d) none
52) Eliminating a,b,c from x= a/(b - c), y= b/(c - a), z= c/(a - b) we get
a) 1 -x x b) 1 - x x
1 -y y= 0 1 1 -y =0
1 -z z 1 z 1
c) 1 -x x
y 1 -y= 0
-z z 1 d) none
53) The system of equations : 6x + 5y + λz= 0; 3x - y + 4z= 0; x + 2y - 3z= 0 has
a) only a trivial solution for λ ∈ R
b) exactly one non-trivial solution for some real λ
c) infinite number of non-trivial solutions for one value of λ
d) only one solution for λ ≠ -5
PAPER-1(T)
1) If m and n are positive integers such that (m+ n)/(m²+ mn+ n²)= 4/49 then (m+ n) must be
a) 10 b) 16 c) 18 d) 25
2) An equilateral triangle of side length 2 units is inscribed in a circle, The length of a chord of this circle which passes through the mid points of two sides of this triangle is
a) 2 b) √2 c) 5 d) √5
3) How many ways are there of walking up a flight of 10 stairs if you take either one or three stairs with each step.
a) 28 b) 32 c) 34 d) 40
4) A straight line joins two opposite vertices P and Q of a cube of side length 1m and M is any other vertex.
What is the distance, in metres, from M to the closest point on the line PQ ?
a) √6 b) 3 c) √6/3 d) √3
5) In a soccer tournament eight teams play each other once , with two points awarded for a win, one point for a draw and zero for a loss. How many points must a team score to ensure that it is in a top four (i.e., has more points than atleast four other teams)?
a) 10 b) 11 c) 12 d) 13
Com- paper
1) The value of sec(- 1680°) sin330° is
a) -1 b) 0 c) 1 d) none
2) If cosx + sinx = √2(0< x <π/2), then the value of cos3x is
a) 1/2 b) -1/2 c) 1/√2 d) -1/√2
3) sec²θ = 4xy/(x + y)² is true if and only if
a) x+ y ≠ 0 b) x=y, x≠ 0 c) x= y d) x≠ 0 y= 0
4) If sinx + sin²x = 1, the value of cos¹²x + 3 cos¹⁰x + 3 cos⁸x + cos⁶x +1 is
a) -1 b) 0 c) 1 d) 2
5) If tan 1°= t, the value of cos2° + t sin2° is
a) t b) 1 c) 1/2 d) none
6) If α + β=π/2 and β + γ = α, then tanα =
a) 2(tanβ + tanγ)
b) tanβ + tanγ
c) tanβ + 2 tanγ
d) 2 tanβ + tanγ
7) If sinB= (1/5) sin(2A + B), the pn tan(A+ B)/tanA is equal to
a) 5/3 b) 2/3 c) 3/2 d) 3/5
8) If sinx + cos y= (1/3) and cosx + sin y = 1/2, then tan{(x - y)/2} is equal to
a) -5 b) -1/5 c) 1/5 d) 5
9) The value of 1/cos20° - √3/sin20° is
a) 1 b) 4 c) -1 d) -4
10) The value of cos(π/8) cos(3π/8)cos(5π/8)cos(7π/8) is
a) 1 b) 1/8 c) -1/8 d) none
11) (√2 - sinx - cosx)/(sinx - cosx) equals to
a) tan(x/2 - π/8)
b) tan(x/2 + π/8)
c) tan(x/2 - π/4)
d) tan(x/2 + π/4)
12) If cosα = a/(b + c), cosβ = b/(c + a), cosγ = c/(a+ b), then tan²(α/2) + tan²(β/2) + tan²(γ/2) is equal to
a) 1 b) a+ b+ c c) cos(α+β+γ) d) 2
13) If tan x + sec x = 2 cosx, wjerr0≤ x ≤ 2π, the number of solution of the equation is
a) 0 b) 1 c) 2 d) 3
14) The general value of θ satisfying equation tan²θ+ sec2θ = 1 is
a) nπ b) nπ + π/3 c) nπ - π/3 d) none
15) The equation sin⁶x + cos⁶x = a is solvable if
a) 1/2≤ a ≤1 b) 1/4 ≤ a ≤1 c) -1≤ a ≤1 d) 0 ≤ a ≤1/2
16) If log₀.₅sinx = 1- log₀.₅cosx, then the number of values of x in -2π≤ x ≤ 2π is
a) 4 b) 2 c) 3 d) 1
17) If tan(cos⁻¹x)= sin(cot⁻¹(1/2), then x equals
a) 1/√5 b) 2/√5 c) 3/√5 d) √5/3
18) The value of cot{cosec⁻¹+5/3) + tan⁻¹(2/3)} is
a) 5/17 b) 6/17 c) 3/17 d) 4/17
19) The trigonometric equation sin⁻¹x = 2 sin⁻¹a has a solution for
a) |a|≥ 1/√2 b) 1/2≤|a|≤ 1/√2 c) ∀ a∈ R d) |a|≤ 1/√2
20) If cos⁻¹x - cos⁻¹(y/2)= α, then 4x²- 4xy cosα+ y² equals
a) 2 sin2α b) 4 c) 4 sin²α d)- 4 sin²α
21) In a triangle ABC , (a+ b + c)(b + c - a)=λbc, if
a) λ< 0 b) 0<λ<4 c) 0≤λ<4 d) 0 <λ≤4
22) In a ∆ ABC, (cosA + cosC)/(a+ c) + (cosB)/b is equal to
a) 1/a b) 1/b c) 1/c d) 1/(a+ b+ c)
23) Two sides of a triangle are given by the roots of the equation x²- 2√3+2=0. If the angle between the side is π/3, then the third side is
a) 6 b) 3 c) 4 d) none
24) The perpendicular AD to the base of a triangle divides it into segments such that BD, CD, AD are in the ratio 2: 3: 6. The angle A of the triangle is
a) 90 b) 60 c( 45 d) 30
25) If θ and φ are acute angles satisfying sinθ = 1/2, cos φ= 1/3, then
a) π/3< θ + φ ≥0≤π/2
b) π/3p2 < θ + φ <2π/3
c) 2π/3< θ + φ < 5π/6
d) 5π/6 < θ + φ < π
26) If u= (1+ cosθ)(1+ cos2θ) - sinθ sin2θ and v= sinθ(1+ cosθ)(1+ cos2θ)+ (1+ cosθ) sin2θ, then u²+ v²= k(1+ cosθ)(1+ cos2θ). The value of k is
a) 1 b) 2 c) 3 d) 4
27) If the equation sin(θ+ x)= (1/2) sin2x is satisfied by two different values α and β of θ[0≤α, β ≤2π], then (sinα + sinβ)/ (cosα + cosβ)=
a) cosx b) sinx c) cotx d) tanx
MATRIX
1) If A= 1 2 3 & B= 2
4
1 then AB is
a) 2 8 3 b) 2 c) 13 d) none
8
3
2) If A= cosx sinx
- sinx cosx then AᵩAᵦ equals
a) Aᵩᵦ b) Aᵩ₊ᵦ c) Aᵩ₋ᵦ d) none
3) If C= i 0 & B= i 2
3 -i 3 4+ i
With the relation C+ A = B - A, then A equals to
a) 0 1 B) 0 -1 c) 1 0 d) none
0 2+ i 3 i 0 2- i
4) If A= ab b²
- a² -ab then A² equals
a) 1 b) O c) -1 d) none
5) If A= 1 2. 1
0 1 -1
3 -1 1 then A³- 3A²- A + 9I equals
a) I b) O c) A d) A²
6) If A= 1 2 & B= -2 5 & C= 1
3 2 2 find A(BC) is
a) [23] b) [22] c) [122] d) none
7) If A= 5 -1 3 & B= 0 2 3
0 1 2 1 -1 4 then (AB')' is
a) -7 8 b) 7. 8 c) 7 8 d) none
0 7 18 7 7 0
8) If A= 1 0 2
5 1 x
1 1 1 is a singular matrix, then x is
a) 5 b) 11 c) 3 d) 9
9) If A is a square matrix such that A²+ I= O, then A is
a) 1 0 b) 1 2 c) -1 0 d) i 0
0 1 -1 1 0 -i 0 i
10) If A= 3 -4
1 -1 then Aⁿ (n ∈N) equals
a) n+2 5-n b) 3n -4n c) 3ⁿ (-4)ⁿ
n - n n -n 1 (-1)ⁿ d) none
11) If A= 3 -3 4
2 -3 4
0 -1 1 then inverse of A is
a) A b) A³ c) A² d) A⁴
12) If A= 1 2 3
1 2 3
-1 -2 -3 then A is a milopotent of index
a) 5 b) 4 c) 3 d) 2
13) If A= 3 2
0 1 then A⁻³ is
a) 1/27 -26/27 b) -1/27 26/27
0 1. 0 -1
c) 1/27 -26/27 b) -1/27 -26/27
0 1. 0 -1
14) If A= 2 3- i -i
3+ i π 7+ i
i 7- i e
a) Hermitian b) Skew Hermitian c) symmetric d) none
15) Which of the following matrices is not invertible
a) 1 1 b) 3 3 c) -1 -1 d) 2 -2
0 1 5 5 -1 2 1 1
16) If A= i 0
0 I then A⁴ⁿ (n ∈ N) equals
a) 0 i b) i 0 c) 1 0 d) 0 0
i 0 0 i 0 1 0 0
17) If A = B + C such that B is a symmetric matrix and C is a skew symmetric matrix, then B is given by:
A+ A' b) A - A' c) (1/2) (A+ A') d) (1/2) (A - A')
18) If A(Adj. A)8I, for a 3 x 3 matrix A, then det. A is
a) 1 b) 2 c) 4 d) 8
19) If A and P are 3x3 matrices with integral entries such that P'AP= A, then det P is
a) -1 b) 1 c) ±1 d) ±1, provided A is non singular
20) If each element of a 3x3 matrix A is multiplied by 3, then the determinant of the newly formed matrix is
a) 3 det. A b) 9 det. A c) (det. A)³ d) 27 det. A
21) If A and B are two square matrices of order 3, then
a) (AB)' = A'B'
b) AB= O => A= O or B= O
c) AB= O => |A|= 0 and|B|= 0
22) If A and B are two matrices of order 3, such that |A|= 1, |B|= 2, then the determinant of the matrix 2AB is
a) 4 b) 8 c) 16 d) 32
23) Let A be a real matrix such that A⁶⁷ = A⁻¹, then
a) |A|= ±1 b) |A|=1 c) A= I, I being unit matrix
d) A is diagonal matrix.
24) If A and B are two matrices such that A+ B = λ I, where I is the identity matrix, then
a) A= μI for some μ
b) B= μ I for some μ
c) A= μ I for some μ, B= μ' I forgot some μ'
d) A= μ I, if B= μ' I, for some μ'.
25) Let A and B be 3x3 matrices such that A' = - A. B'= B, then matrix λAB + 3BA is a skew symmetric matrix for:
a) λ= 3 b) λ=- 3 c) λ= 3 or λ= - 3 d) λ= 3 and λ= - 3
26) If A is a 3x3 non-singular matrix, then det.[adj. A] is equal to
a) (det A)² b) (det A)³ c) det A d) (det A)⁻¹
27) If A and B are matrices such that A+ B and BA are both defined, then
a) A and B can be any matrices
b) A, B are square matrices not necessarily of same order
c) A, B are square matrices of same order
d) number of columns of A= number of rows of B
28) If α, β and γ are roots of the equation x³+ px + q=0 then the value of
α β γ
Det β γ α
γ α β is
a) p b) Q c) p²- 2q d) 0
29) If A= 1 2 -1
-1 1 2
2 -1 1
Then det (adj (adj. A)) is
a) 14¹ b) 14² c) 14³ d) 14⁴
30) If α β
γ -α is to be square root of the two rowed unit matrix, then α, β and γ should satisfy the relation:
a) 1+ α²+ β γ = 0
b) 1- α²- β γ = 0
c) 1- α²+ β γ = 0
d) 1+ α²- β γ = 0
31) If A= 1 - tanx & B= 1 tanx
tanx 1 -tanx 1
And C= a - b
b a
with the relation AB⁻¹ = C then
a) a= 1, b= 1
b) a= cos2x , b= sin2x
c) a= sin2x , b= cos2x d) none
32) If A and B are square matrices of order 3, such that |A|= -1, |B|= 3, then the determinants of 3AB equal
a) -9 b) -27 c) -81 d) 81
33) If A is a square matrix of order 3, then det(λA), λ being a scalar is
a) λ det. A b) λ² det. A c) 0 d) λ³ det. A
34) If A is a square matrix such that det. A = 2, then det(A'), where A' is the transpose of A, is equal to
a) 0 b) -2 c) 1/2 d) 2
35) If A= a b
c d such that ad - bc ≠ 0, then A⁻¹ is
a) d/(ad - bc) b/(ad - bc)
-c/(ad - bc) a/(ad - bc)
b) d - b
-c a
c) d/(ad - bc) -b/(ad - bc)
-c/(ad - bc) a/(ad - bc)
d) none
36) If Iₙ is the identity matrix of order n, then(Iₙ)⁻¹:
a) does not exist b) = Iₙ c) = O d) = nIₙ
37) If A is a square matrix such that A² = I, then A⁻¹ is
a) 2A b) O c) A d) A+ I
38) The value of a in order that
2 3 5
1 a 2
0 1 -1 is singular
a) -5/3 b) 5/3 c) 2 d) none
39) If A= 3 4 & B= -2 -2
2 3 0 -1 then (A+ B)⁻¹
a) = A⁻¹+ B⁻¹ b) does not exist
c) is a skew symmetric d) none
40) If A, B are square matrices of order 3, then:
a) Adj.(AB)= (adj. A)(adj. B)
b) (A+ B)⁻¹= A⁻¹+ B⁻¹
c) AB = O => |A|= 0 or |B|= 0
d) AB = O => |A|= 0 and |B|= 0
41) If A= 2 3
1 -2 and A⁻¹= αA, then the value of α is
a) 7 b) -7 c) 1/7 d) -1/7
42) The transformation due to the reflection of (x,y) through the origin is described by the matrix:
a) 0 0 b) -1 0 c) 0 -1 d) 1 0
0 1 0. -1 -1 0 0 1
43) If A= 0 c -b & B= a² ab ac
-c 0 a ab b² bc
b -a 0 ac bc a² then AB is
a) A b) B c) I d) O
44) If A= a 0. 0
0 a 0
0 0 aⁿ
Then Aⁿ is
a) aⁿ 0 0 b) aⁿ 0 0
0 a 0 0 aⁿ 0
0 0 a 0 0 a
c) na 0 0 d) aⁿ 0 0
0 na 0 0 aⁿ 0
0 0. na 0 0 aⁿ
45) The value of x, so that the matrix
x+ a b c
a x+ b c
a b x+ c has rank 3 is
a) x≠ 0 b) x= a+ b + c c) x≠ 0, x≠ -(a+ b+ c) d) x= 0 and x= a+ b+ c
46) If A= 1 2 & B= 1 0
3 -5 0 2
and X be a matrix such that A= BX, then X is
a) 2 4 b) 1 2 c) -1 2 d) none
3 -5 3/2 -5/2 3/2 5/2
47) Inverse of 1 2 3
2 3 4
3 4 6 is
a) 2 0 -1 b) 1 2 3
0 -3 2 2 3 4
-1 2 -1 3 4 6
c) -2 0 1
0 3 -2
1 -2 1 d) none
48) if A and B are any 2x2 matrices , then det.(A+ B)=
a) det. A+ det. B
b) det. A= 0 or set. B= 0
c) det. A= 0 and det. B= 0 d) none
49) cosα - sinα 0
let F(α)= sinα cosα 0
0 0 1
G(β)= cosβ 0 sinβ
0 1 0
- sinβ 0 cosβ
Then [F( α) G(β) ⁻¹ equals
a) F( α) - G(β)
b) - F( α) - G(β)
c) [F( α)]⁻¹ [G(β)] ⁻¹
d) [G( α)]⁻¹ [F(β)] ⁻¹
50) let A be an invertible Matrix. Which of the following is not true ?
a) A⁻¹ = |A|⁻¹
b) (A²)⁻¹ = (A⁻¹)²
c) (A')⁻¹ = (A⁻¹)' d) none
51) If A= a 0 0
0 a 0
0 0 a
then the value of |A| |adj. A| is
a) a³ b) a⁶ c) a⁹ d) a²⁷
52) The multiplicative inverse of
A= cosx - sinx
sinx cosx
a) - cosx sinx
- sinx - cosx
b) cosx sinx
- sinx cosx
c) -cosx -sinx
sinx -cosx
d) cosx sinx
sinx - cosx
53) If A is a square Matrix such that |A|= 2, then for any +ve integer n, |Aⁿ| is equal to
a) 0 b) 2n c) 2ⁿ d) n²
54) If A= 1 - tanx & B= 1 tanx
tanx 1 -tanx 1
C= a - b
b a with the the relation AB⁻¹= C then
a) a= cos2x, b= sin2x
b) a= sin2x, b= cos2x
c) a= cos2x, b= 1 d) none
55) If A is a non zero column matrix of order mx1 and B is a non zero matrix of order 1xn, then rank of AB is equal to
a) 1 b) 2 c) m d) n
56) If A, B are two square Matrix such that AB= A and BA= B, then
a) only A is idempotent
b) A, B are idempotent
c) only B is idempotent d) none
57) The sum of the idempotent matrices A and B is idempotent if:
a) AB = BA= O
b)AB = BA= I
c) AB = BA= B
d) AB = BA= A
58) If A= 1 1
1 1 and n ∈N, then Aⁿ equals
a) nA b) 2ⁿA c) 2ⁿ⁻¹A d) none
59) If A= 2 1 & B= -3 2 & C= 1 0
3 2 5 -3 0 1 with relation AXB= C then find matrix X
a) 0 1 b) 1 0 c) 1 1 d) 1 1
1 1 1 1 0 1 1 0
60) If A= cosx - sinx
sinx cosx then
a) A is a symmetric matrix
b) A is a skew-symmetric matrix
c) A is an orthogonal matrix d) none
61) If A= a² ab ac & B= 0 c - b
ab b² bc -c 0 a
ac bc c² b -a 0 then the product AB is
a) I b) O c) A d) B
62) If A is an orthogonal matrix, then inverse of A is
a) AA' b) A' c) A d) none
63) If A is an invertible matrix and B is an orthogonal matrix of the same order as that of , than C= A⁻¹ BA is
a) symmetric matric
b) skew- symmetric matrix
c) orthogonal matrix d) none
64) Let E(x)= cos²x cosx sinx
cosx sinx sin²z
If α, β differ by an odd multiple of π/2, then E(α) E( β) is
a) a unit Matrix
b) a null matrix
c) an orthogonal Matrix
d) a diagonal matrix
65) Let A and B be skew-symmetric metrices of order n. Then
a) AB is a symmetric matrix
b) AB is a skew-symmetric matrix
c) AB is a symmetric metrix if A and B commute d) none
66) The inverse of a symmetric matrix is
a) a symmetric matrix
b) a skew-symmetric matrix
c) a diagonal Matrix d) none
67) the inverse of a skew&symmetric matrix is
a) a symmetric matrix
b) a skew-symmetric matrix
c) a diagonal Matrix d) none
68) The inverse of a skew-symmetric matrix of odd order
a) is a symmetric matrix
b) is a skew-symmetric matrix
c) is a diagonal matrix
d) does not exist
69) If A is an orthagonal Matrix, then |A| is
a) 0 b) -1 c) 1 d) ±1
70) if w is a complex cube root of unity, than the matrix
A= 1 w² w
w² w 1
w 1 w² is a
a) symmetric matric Qp
b) skew- symmetric matric4
c) singular Matrix
d) non singular Matrix
71) If A and B are two matrices such that AB= B and BA= A, then A²+ B⅖ equals
a) 2AB b) 2BA c) AB d) A+ B
72) If A and B are symmetric matrix, then ABA is
a) symmetric
b) skewo
c) diagonal
d) triangular
73) If A, B are symmetric matrices of same order, then AB - BA is a
a) null Matrix
b) unit matrix
c) symmetric matrix
d) skew-symmetric matrix
74) If A is a non singular square matrix, then |adj. A| is
a) |A| b) |A|ⁿ⁻² c) |A|ⁿ⁻¹ d) |A|ⁿ
75) If A= cosx sinx
- sinx cosx, then
lim ₙ→∞ (1/n) Aⁿ is
a) an identity Matrix
b) a null Matrix
c) 0 1 d) none
-1 0
76) For a square Matrix A and a non singular matrix B of the same order, the value of det(B⁻¹AB) is
a) |A| b) |A⁻¹| c) |B| d) |B⁻¹|
77) If α β
γ - α is a square root of I, then α, β and γ satisfy :
a) 1+ α²+ βγ= 0
b) 1- α²+ βγ= 0
c) 1- α²- βγ= 0
d) 1+ α²- βγ= 0
78) For a 3x3 Matrix A, if |A|= 4, then |adj. A| equals
a) 4 b) -4 c) 16 d) 64
79) rank of null matrix is
a) I b) 0 x) does not exist d) none
80) For a matrix of rank r,
a) rank (A')< r
b) rank (A')= r
c) rank (A')> r d) none
81) A=0 0 -1
0 -1 0
-1 0 0
The only correct statement about Matrix is A is
a) A is zero Matrix
b) A= (&1)= I, where I is a unit Matrix
c) A⁻¹ does not exist
d) A²= I
82) A= 1 -1 1 & B= 4 2 2
2 1 -3 -5 0 a
1 1 1 1 -2 3
If B is the inverse of A, than a is
a) -2 b) -1 c) 2 d) 4
ⁿ ⁿ λ μ ⁻¹ α β γ
SAP-1
1) If A= 2 -1
-1 2 and I is the unit matrix of order 2, then A² is equal to
a) 4A - 3I b) 3A - 4I c) A - I d) A + I
2) The multiplicative inverse of
2 1
7 4 is
a) 4 -1 b) 4 -1 c) 4 -7 d) -4 -1
-7 -2 -7 2 7 2 7 -2
3) For a real number α, let A(α) denote the matrix
cosα sinα
-sinα cosα
Then for real numbers α₁ and α₂, the value of A(α₁) A(α₂) is
a) A(α₁α₂) b) A(α₁ + α₂) c) A(α₁ - α₂) d) A(α₂ -α₁)
4) If the system of equations x+ 2y + 3z= 1, 2x + ky + 5z= 1, 3x+ 4y + 7z= 1 has no solutions, then
a) k= -1 b) k= 1 c) k= 3 d) k= 2.
5) Assuming that the sums and products given below are defined, which of the following is not true for matrices?
a) AB= AC does not imply B= C
b) A+ B= B+ A
c) (AB)'= B'A'
d) AB= O implies A= O or B= O.
6) If A=1. 0 2 & Adj A= 5 a -2
-1 1 -2 1 1 0
0 2 1 -2 -2 b
Then the values of a and b are
a) a= -4, b= 1 b) a= -4, b= -1 c) a= 4, b= 1 d) a= 4, b= -1
7) If A= -1 0
0 2 then the value of A³- A² is
a) I b) A c) 2A d) 2I.
8) If 1, w, w² are the cube roots of unity then the value of m for which the matrix
1. w m
w m 1
m 1 w is singular, is
a) 1 b) -1 c) w d) w².
9) If A= -x - y
z t, then transpose of adj A
a.) t z b) t y c) t -z d) none
-y -x -z -x y -x.
10) If A square metrix of order 3x3 and λ is a scalar, then adj(λA) is equal to
a) λ adj A B) λ² adj A c) λ⅗ adj A d) λ³ adj A.
11) The inverse of 5 -2
3 1
a) -2/13 5/13 b) 1 2 c) 1/11 2/11 d) 1 3
1/13 3/13 -3 5 -3/11 5/11 -2 5
12) If A=3 5 & B= 1 17
2 0. 0 -10
a) 80 b) 100 c) -110 d) 92.
Sap-2
1) If A is a singular matrix of order n then A(adj A) is equal to
a) a null matrix
b) a row matrix
c) a column matrix d) none.
2) If the determinant of the matrix
a₁ b₁. c₁
a₂ b₂ c₂
a₃ b₃ c₃ is denoted by D, then the determinant of the matrix
a₁+ 3b₁- 4c₁ b₁ 4c₁
a₂+ 3b₂- 4c₂ b₂ 4c₂
a₃+ 3b₃- 4c₃ b₃ 4c₃ will be
a) D b) 2D c) 3D d) 4D.
3) If A and B are two square matrices and if A⁻¹ and B⁻¹ exist, then (AB)⁻¹ is
a) A⁻¹B⁻¹ b) AB⁻¹ c) A⁻¹ B C) B⁻¹ A⁻¹
4) If A= 3 -5
-4 2 then the value of A²- 5A is
a) I b) 14I c) O d) none.
5) If A= 5 6 -3
-4 3 2
-4 -7 3 then the cofactors of the elements of second row are
a) 3,3,11 b) 3, - 3,11 c) -39 ,3, -11 d) 39 , -3, 11.
6) If A= a b & A²= α β
β α
a) α = 2ab, β= a²+ b²
b) α = a²+ b², β= ab
c) α = a²+ b², β= 2ab
d) α = a²+ b², β= a²- b².
7) If A= 1 2 & B= 1 2
2 3 2 1
3 4
a) both AB and BA exit
b) neither AB nor BA exist
c) AB exists but BA does not exist
d) AB does not exist but BA exit.
8) If A= 2. -1 & B= 1 0
0 1 -1 -1 then (A+ B)² is not equal to
a) A²+ AB + BA + B²
b) A²+ AB + BA + B²I
c) A²I+ AB + BA + B²
d) A²+ 2AB + B².
9) If A be an n x n matrix and k any scalar, then det kA is equal to
a) k det A B) nᵏ det A c) kⁿ detA d) kn det A.
10) The value of determinant
b²c² bc b+ c
c²a² ca c+ a
a²b² ab a+ b is
a) abc (a²+ b²+ c²) b) 0 c) abc (bc+ ca+ ab) d) (a+ b+c) (a²+ b²+ c²)(ab+ bc+ ca).
11) If x - 3 i 1
y 1 i = 6+ 11 i
0 2 i -i
Then the values of x and y are
a) -3,4 b) 3, 4 c) 3, -4 d) -3, 0.
12) If A= 1 2
3 -5, then inverse of A is
a)-5 -2 b)-5/11 -2/11 c)5/11 2/11 d) 5 2
-3 1. -3/11 1/11 3/11 -1/11 3 -1
CONICS
1. Conic sections:
A conic section, or conic is the locus of a point which moves in a plane so that its distance from a fixed point is in a constant ratio to its perpendicular distance from a fixed straight line.
a) The fixed point is called the focus.
b) The fixed straight line is called the directrix.
c) The constant ratio is called the eccentricity denoted by e.
d) The line passing through the focus & perpendicular to the directrix is called the axis.
e) A point of intersection of a conic with its axis is called vertex.
2. General equation of a conic: Focal Directrix Property:
The general equation of a conic with focus (p,q) & directrix lx + my + n=0 is
(l²+ m²)[(x - p)²+ (y - q)²]= e²(lx + my + n)² ≡ ax¹+ 2hxy+ by²+ 2gx + 2fy + c=0
3. Distinguish between the Conic:
The nature of the conic section depends upon the position of the focus S w.r.t. the directrix and also upon the value of the eccentricity e. Two different cases arise.
Case(i): when the focus lie on the directrix:
In this case D≡ abc+ 2fgh - af²- bg² - ch²=0 and the general equation of a conic represent a pair of straight lines and if:
e> 1 the line will be real and distinct intersecting at S.
e= 1 The lines will be coincident.
e < 1 The lines will be imaginary.
Case(ii): When the focus does not lie on the directrix:
e=1; D≠ 0 and h²= ab a parabola
0< e < 1; D≠ 0 and h²< ab an ellipse
D≠ 0; e> 1 and h²> ab. a hyperbola
e> 1; D≠ 0 and h²> ab; a+ b = 0 . A rectangular hyperbola
4. PARABOLA
A parabola is the locus of a point which moves in a plane, such that its distance from a fixed point (focus) is equal to its perpendicular distance from a fixed straight line (directrix).
Standard equation of a parabola is y²= 4ax. For this parabola:
i) Vertex is (0,0)
ii) Focus is (a,0)
iii) Axis is y=0
iv) Directrix is x+ a=0
a) Focal distance:
The distance of a point on the parabola from the focus is called the focal distance of the point.
b) Focal chord:
A chord of the parabola, which passes through the focus is called a focal chord.
c) Double ordinate:
A chord of the parabola perpendicular to the axis of the symmetry is called a double ordinate.
d) Latus rectum:
A double ordinate passing through the focus of a focal chord perpendicular to the axis of parabola is called the latus rectum. For y²= 4ax.
• Length of the latus rectum= 4a.
• Length of the semi latus rectum= 2a.
• Ends of the latus rectum= L(a, 2a) & L'(a, -2a)
Notes that:
i) Perpendicular distance from focus on directrix= half the latus rectum.
ii) Vertex is middle point of the focus and the point of intersection of directrix and axis.
iii) Two parabola are said to be equal if they have the same latus rectum.
5. Parametric Representation:
The simplest and the best form of representing the coordinates of a point on the parabola is (at², 2at). The equation x= at² & y= 2at together represents the parabola y²= 4ax, t being the parameter.
6. Type of PARABOLA
Four standards forms of the parabola are y²= 4ax; y²=- 4ax; x²= 4ay; x²= -4ay
When parabola y²= 4ax
Vertex: (0,0)
Focus: (a,0)
Axis: y= 0
Directrix: x= - a
Length of latus rectum: 4a
Ends of Latus rectum: +a, ±2a)
Parametric equation: (at²,2at)
Focal length: x + a
When parabola y²= - 4ax
Vertex: (0,0)
Focus: (- a,0)
Axis: y= 0
Directrix: x= a
Length of latus rectum: 4a
Ends of Latus rectum: (-a, ±2a)
Parametric equation: (-at²,2at)
Focal length: x - a
When parabola x²= 4ay
Vertex: (0,0)
Focus: (0,a)
Axis: x = 0
Directrix: y= - a
Length of latus rectum: 4a
Ends of Latus rectum: (±2a, a)
Parametric equation: (2at², at²)
Focal length: y + a
When parabola x²= - 4ay
Vertex: (0,0)
Focus: (0, - a)
Axis: x = 0
Directrix: y= a
Length of latus rectum: 4a
Ends of Latus rectum: (±2a, - a)
Parametric equation: (2at², - at²)
Focal length: y - a
When parabola (y - k)²= 4a(x - h)
Vertex: (h+ a, k)
Focus: (0,a)
Axis: y = k
Directrix: x+ a - h= 0
Length of latus rectum: 4a
Ends of Latus rectum: (h+ a, k± 2a)
Parametric equation: (h+ at², k+ 2at)
Focal length: x - h + a
When parabola (x - o)²= 4b(y - q)
Vertex: (p,q)
Focus: (p, b+ q)
Axis: x = p
Directrix: y+ b - q = 0
Length of latus rectum: 4apb
Ends of Latus rectum: (p± 2a, q+ a)
Parametric equation: (p+ 2at, q+ at²)
Focal length: y - q + b
7. Position of a point relative to a parabola:
The point (x₁, y₁) lies outside, on or inside the parabola y²= 4ax according as the expression y₁²- 4ax₁ is positive, zero or negative.
8. Chord joining two points:
The equation of a chord of the parabola y²= 4ax joining its two points P(t₁) and Q(t₂) is
y(t₁ + t₂) = 2x + 2at₁t₂
Note:
i) If PQ is focal chord then t₁t₂= -1.
ii) Extremities of focal chord can be taken as (at², 2at) and (a/t², -2a/t).
9. LINE AND A PARABOLA :
a) The line y= mx + c meets the parabola y²= 4ax in two points real, coincident or imaginary according as > = < cm => condition of tangency is, c= a/m.
NOTE: Line y= mx + c will be tangent to parabrx²= 4ay if c= - am².
b) Length of the chord intercepted by the parabola y²= 4ax on the line y= mx + c is: (4/m²) √=a(1+ m¹)(a - mc)}.
NOTE
Length of the focal chord making an angle α with x-axis is 4a coesec²α.
10. LENGTH OF SUBTANGENT & SUBNORMAL
PT and PG are the tangent and normal respectively at the qP to the parabola y½= 4ax. Then
TN= length of subtangent= twice the abscissa of the point P (Subtangent is always bisected by the vertex)
NG= length of subnormal which is constant for all points on the parabola and equal to its semilatus rectum (2a).
11. TANGENT TO THE PARABOLA y²= 4ax:
a) Point form:
Equation of tangent to the given parabola at its point (x₁ y₁) is
yy₁ = 2a(x + x₁)
b) Slope form:
Equation of tangent to the given parabola whose slope is 'm', is
y= mx + a/m, (m≠ 0)
Point of contact us (a/m², 2a/m)
c) Parametric form:
Equation of tangent to the given parabola at its point P(t), is ty= x + at²
NOTE:
Point of intersection of the tangents at the point t₁ & t₂ is [at₁t₂, a(t₁ + t₂)]
12. NORMAL TO THE PARABOLA y²= 4ax:
a) Point form:
Equation of normal to the given parabola at its point (x₁, y₁) is
y- y₁ = - y₁(x - x₁)/2a.
b) Slope form:
Equation of normal to the given parabola whose slope is 'm', is
y= mx - 2am - am³
foot of the normal is (am², - 2am)
c) Parametric form:
Equation of normal to the given parabola at its point P(t), is
y+ tx = 2at + at²
NOTE
i) Point of intersection of normals at t₁ & t₂ is [a(t₁²+ t₂²+ t₁t₂+2), - at₁t₂ (t₁ + t₂))
ii) If the normal to the parabola y²= 4ax at the point t₁, meets the parabola again at the point t₂ then t₂ = - (t₁ + 2/t₁).
iii) If normals to the parabola y²= 4ax at the point t₁ & t₂ intersect again on the parabola at the point 't₃' then t₁ t₂ = 2; t₃ = -(r₁ + t₂) and the line joining t₁ and t₂ passes through a fixed point (-2a, 0).
i.e., am³+ m(2a - h)+ k =0
This gives m₁ + m₂+ m₃= 0; m₁ m₂+ m₂m₃+ m₃m₁= (2a - h)/a; m₁m₂m₃= - k/a
Where m₁ , m₂, m₃ are the slopes of the three concurrent normas:
• Algebraic sum of slopes of the three concurrent normals is zero.
• Algebraic sum of ordinates of the three co-normal points on the parabola is zero.
• Centroid of the ∆ formed by three consecutive normal points lies on the axis of parabola (x-axis).
13. AN IMPORTANT CONCEPT:
If a family of straight lines can be represented by an equation λ²P+ λQ+ R= 0 where λ is a parameter and P,Q,R are linear functions of x and y then the family of lines will be tangent to the curve Q²= 4PR.
14. PAIR OF TANGENTS:
The equation of the pair of tangents which can be drawn from any point P(x₁ , x₂) outside the parabola to the parabola y²= 4ax is given by: SS₁ = T² where
S= y²- 4ax; S₁= y₁² - 4ax₁; T ≡ yy₁ - 2a(x + x₁).
15. DIRECTOR CIRCLE:
Locus of the point of intersection of the perpendicular tangents to the parabola y²= 4ax is called the director circle. It's equation is x+ a = 0 which is parabola's own directrix.
16. CHORD OF CONTACT:
Equation of the chord of contact of tangents drawn from a point P(x₁, y₁) is yy₁ = 2a(x + x₁)
NOTE
The area of the triangle formed by the tangents from the point (x₁, y₁) and the chord of contact us (y₁²- 4ax₁)³⁾²/2a i.e., (S₁)³⁾²/2a, also note that the chord of contact exists only if the point P is not inside.
17. CHORD WITH A GIVEN MIDDLE POINT
Equation of the chord of the parabola y²= 4ax whose middle point is x₁, y₁ is
y - y₁= 2a(x - x₁)/y₁.
This reduced to T= S₁ where T≡ 2a(x + x₁) and S₁ = y₁²- 4ax₁.
18 DIAMETER:
The locus of the middle point of a system of parallel chords of a parabola is called a DIAMETER. Equation to the diameter of a parabola is y= 2a/m, where m= slope of parallel chords.
19. IMPORTANT HIGHLIGHTS:
a) If the tangent and normal at any point P of the parabola intersect the axis at T and G then ST= SG= SP where S is the locus. In other words the tangent and the normal at a point P on the parabola are the bisectors of the angle between the focal radius SP and the perpendicular from P on the directrix. From this we conclude that all rays emanating from S will become parallel to the axis of the parabola after reflection.
b) The portion of a tangent to a parabola cut off between the directrix and the curve subtends a right angle at the focus.
c) The tangents at the extremities of a focal chord intersect at right angles on the directrix, and a circle on any focal chord as diameter touches the directrix. Also a circle on any focal radii of a point P(at², 2at) as diameter touches the tangents at the vertex and intercepts a chord of length a√(1+ t²) on a normal at the point P.
d) Any tangent to a parabola and the perpendicular on it from the focus meet on the tangent at the vertex.
e) Semi latus rectum of the parabola y²= 4ax, is the harmonic mean between segments of any focal chord of the parabola is; 2a = 2bc/(b + c) i.e., 1/b + 1/c = 1/a.
f) If the tangent at P and Q meet in T, then:
i) TP and TQ subtend equal angles at the focus S.
ii) ST²= SP. SQ
iii) The triangles SPT and STQ are similar.
g) Tangents and Normals at the Extremities of the latus rectum of a parabola y²= 4ax constitue a square, their points of intersection being (-a,0) and (3a,0).
NOTE
i) The two tangents at the Extremities of focal chord meet on the foot of the directrix.
ii) Figure LNL'G is a square of side 2√2 a.
h) The circle circumscribing the triangle formed by any three tangents to a parabola passes through the focus.
Exercise -1
1) Find the locus of a point which moves such that its distance from the point (0,1) is twice its distance from the line 3x + 4y +1=0.
2) What conic does the equation 25(x²+ y²- 2x +1)= (4x -3y +1)² represent ?
Exercise -2
1) What conic does 13x²+ 37y²+ 2x +14y - 18xy -2=0 represent ?
2) What conic is represented by the equation √(ax)+ √(by)= 1?.
3) if the equation x² - y²- 2x +2y + K=0 represent a degenerate conic then find the value of K.
4) If the equation x²+ y²- 2x -2y + c =0 represents an empty set then find the value of c.
5) If the equation of conic 2x²+ 3y² -3x +5y + xy +K =0 represent a single point, then find the value of K.
6) For what value of K the equation of conic 4x -6y +2xy + K =0 represents two intersecting straight lines ? If K= 17 then , this equation represent ?
EXERCISE -3
1) Find the centre of the conic 14x²- 4xy + 11y²- 44x - 58y + 71=0.
Exercise - 4
Exercise - 5
1) Find the equation of the parabola whose focus is at (-1,-2) and directrix is the straight line x - 2y +3=0.
2) Find the equation of the parable whose focus is (4,-3) and vertex is (4,-1).
3) The focal distance of a point on a parabola y²= 8x is 8. find it.
4) QQ' is a double ordinate of a parabola y²= 4ax. Find the locus of the point of trisection .
5) Prove that area of the triangle inscribed in the parabola y²= 4ax is (1/8a)(y₁ - y₂)(y₂ - y₃)(y₃ - y₁), where y₁, y₂, y₃ are the ordinates of the vertices.
6) Find the length of the side of an equilateral triangle inscribed in the parabola y²= 4ax, so that one angular point is at the vertex.
7) Prove that the equation of the parabola whose focus is (0, 0) and tangent at the vertex is x - y+1=0 is x²+ y² -4x + 4y +2xy - 4=0.
8) Find the equation of the parabola whose latus rectum is 4 units , axis is the line 3x + 4y -4=0 and the tangent at the vertex is the line 4x - 3y +7=0.
9) Find the vertex, focus, latuce rectum , axis and the directrix of the parabola x²+ 8x +12y +4 =0.
10) Prove that the equation y²+ 2ay +2by + c=0 represents a parabola whose axis is parallel to the axis of x . Find its vertex.
11) The x and y coordinates of any point P are expressed as x= (V cosk) t, y= (V sink) t - (1/2) gt², where t is a parameter and V, k, g are constants. Show that the locus of the point P(x,y) is a parabola. Find the coordinates of the vertex of the parabola.
12) Find the equation of the parabola with its vertex at (3,2) and its focus at (5,2).
13) Find equation of the parabola with the latus rectum joining the point (3,6) and (3,-2).
14) Find the equation of the parabola whose axis parallel to the y-axis and which passes through the points (0,4),(1,9) and (4,5) and determine its latus rectum.
OBJECTIVE - 1
1) The vertex of the parabola y²+ 6x - 2y +13=0 is:
a) (-2,1) b) (2,-1) c) (1,1) d) (1,-1)
2) if the parabola y²= 4ax passes through (3,2) then the length of latuce rectum is:
a) 1/3 b) 2/3 c) 1 d) 4/3
3) The value of p such that vertex of y= x²+ 2px + 13 is 4 units above the x-axis is:
a) ±2 b) 4 c) ±3 d) 5
4) The length of the latus rectum of the parabola whose focus is (3,3) and directrix is 3x - 4y -2=0, is:
a) 1 b) 2 c) 4 d) 8
5) If the vertex and focus of a parabola are (3,3) and (-3,3) respectively, then its equation is:
a) x²- 6x + 24y -63=0
b) x²- 6x + 24y + 81=0
c) y²- 6y + 24x + 63=0
d) y²- 6y - 24x + 81=0
6) If the vertex of the parabola y= x²- 8x + c lies on x-axis, then the value of c is:
a) 4 b) -4 c) 16 d) -16
7) The parabola having its focus at (3,2) and directrix along the y-axis has its vertex at:
a) (3/2,1) b) (3/2,2) c) (3/2,1/2) d) (3/2,-1/2)
8) The directrix of the parabola x²- 4x - 8y +12 =0 is :
a) y=0 b) x=1 c) y=-1 d) x = -1
9) The equation of the latus rectum of the parabola x²+ 4x + 2y =0 is:
a) 3y -2=0 b) 3y + 2=0 c) 2y -3=0 d) 2y + 3 =0
10) trhe focus of the parabola x²- 8x + 2y +7 =0 is:
a) (0,,-1/2) b) (4,4) c) (4,9/2) d) (-4,-9/2)
11) The equation of the parabola with the focus (3,0) and directrix x +3=0 is:
a) y²= 2x b) y²= 3x c) y²= 6x d) y²= 12x
12) Equation of the parable whose axis is parallel to y-axis and which passes through the points (1,0), (0,0) and (-2,4) is:
a) 2x²+ 2x = 3y
b) 2x²- 2x = 3y
c) 2x²+ 2x = y
d) 2x²- 2x = y
Subjective - 1
1) Find the equation of the parabola whose focus is (3,5) and directors is to the line 3x - 4y +1=0.
2) Find the equation of the parabola is Focus is at (-6,-6) and vertex is at (-2,2).
3) Find the vertex, focus, axis , directrix and latus rectum of the parabola 4x²+ 12x - 20y + 67 = 0.
4) Find the name of the conic represented by √(x/a) + √(y/b)= 1.
5) Determine the name of the curve described parametrically by the equation x= t²+ t+1, y= t²- t +1.
6) Prove that equation of the parabola whose vertex and focus are on the x-axis at a distance a and A' from the origin respectively is y²= 4(A' - a)(x - a).
7) Find the equation is parabola whose axis is parallel to x-axis and which passes through the point (0,4),(1,9) and (-2,6). Also , find the letus rectum.
8) The equation ax²+ 4xy + y²+ ax + 3y +2 = 0 represents a parabola then find the value of a.
EXERCISE - 2
1) Show that the point (2,3) lies outside the parabola y²= 3x.
2) Find the position of the point (-2,2) with the respect to the parabola y²- 4y + 9x +13=0.
3) if the point (at², 2at) be the externalty of a focal chord of parabola y² 4ax then show that the length of the focal chord is a (t + 1/t)².
4) Prove that the semilatus rectum of the parabola y²= 4ax is the harmonic mean between the segments of any focal chord of the parabola.
5) Show that the focal chord of parabola y²= 4ax makes an angle α with the x-axis is of length 4a cosec²α.
6) Prove that the length of a focal chord of a parabola varies inversely as the square of its distance from the vertex.
7) Show that the straight line lx + my + n=0 touches the parabola y²= 4ax if ln = am².
8) Show that the line x cosα + y sinα = p touches the parabola y²= 4ax if p cosα + a sin²α =0 and than the point of contact is (a tan²α, -2a tanα).
9) Show that the line x/l + y/m = 1 touches the parabola y²= 4a(x + b) if m²(l + b) + al²= 0.
10) Find the equations of the straight lines touching both x²+ y²= 2a² and y²= 8ax.
11) Find the equation of the common tangents to the parabola y²= 4ax and x²= 4by.
12) The tangents to the parabola y²= 4ax make angle θ₁ and θ₂ with x-axis. Find the locus of their point of intersection if cotθ₁ + cot θ₂ = c.
13) Show that the locus of the points of intersection of the mutually perpendicular tangents to a parabola is the directrix of the parabola.
14) The tangents to the parabola y²= 4ax at P (at²₁, 2at₁) and Q(at²₂, 2at₂) intersect at R. Prove that the area of the triangle PQR is (1/2) a²|(t₁ - t₂)|³.
15) Show that normal to the parabola y²= 8x at the point (2, 4) meets it again at (18, 12). Find also the length of the normal chord.
16) Prove that the chord y- x √2 + 4a √2= 0 is a normal chord of the parabola y²= 4ax. Also , find the point on the parabola when the given chord is normal to the parabola.
17) If the normal to the parabola y²= 4ax, makes an angle with θ with the axis show that it will cut the curve again at an angle tan⁻¹((1/2) tanθ).
18) Prove that the normal chord to a parabola y²= 4ax at the point whose ordinate is equal to abscissa subtends a right angle at the focus.
19) Show that the locus of points such that two of the three normals drawn from them to the parabola y²= 4ax coincide is 27ay²= 4(x - 2a)³.
20) Find the locus of the point through which pass three normals to the parabola y² = 4ax such that two of them make angles θ₁ and θ₂ respectively with the axis such that tanθ₁ tan θ₂= 2.
21) If the three normal from a point to the parabola y²= 4ax cut the axis in points whose distance from the vertex are in AP, show that the point lies on the curve 27ay²= 2(x - 2a)³.
22) The normal at P, Q, R on the parabola y²= 4ax meet in the point on the line y= k. Prove that the sides of the triangle PQR touch the parabola x²- 2ky = 0.
23) Find the point on the axis of the parabola 3y²+ 4y - 6x +8=0 from when three distinct normals can be drawn.
24) A circle cuts the parabola y²= 4ax at right angles and passes through the focus, show that its centre lies on the curve y²(a + 2x)= a(a+ 3x)².
OBJECTIVE - 2
1) If2x + y + λ = 0 is a normal to the parabola y²= -8x then value of λ is:
a) -24 b) -16 c) - 8 d) 24
2) The slope of a chord of the parabola y²= 4ax which is normal at one end and which subtends a right angle at the origin is
a) 1/√2 b) √2 c) -1/√2 d) -√2
3) The common tangent to the parabola y²= 4ax and x²= 4ay is
a) x+ y + a=0
b) x+ y - a=0
c) x- y + a=0
d) x - y - a=0
4) The circle x²+ y²+ 4λx = 0 which λ ∈ R touches the parabola y²= 8x. The value of λ is given by:
a) λ∈ (0, ∞)
b) λ∈ (-∞, 0)
c) λ∈ (1, ∞)
d) λ∈ (-∞, 1)
5) If the normal at two points P and Q of a parabola y²= 4ax intersect at a third point R on the curve, then the product of ordinates of P and Q is
a) 4a² b) 2a² c) -4a² d) 8a²
6) The normals at three points P, Q, R of the parabola y²= 4ax meet in (h,k). The centroid of triangle PQR lies on
a) x= 0 b) y=0 c) x = - a d) y= a
7) The set of points on the axis of the parabola y²- 4x - 2y +5=0 from which all the three normals to the parabola are real is
a) (λ,0); λ > 1
b) (λ,1); λ > 3
c) (λ,2); λ > 6
d) (λ,3); λ > 8
SUBJECTIVE - 2
1) Show that any three tangents to a parabola whose slope are in harmonic progression enclose a triangle of constant area
2) A chord of parabola y²= 4ax subtends a right angle at the vertex. Find the locus of the point of intersection of tangents at its extremities.
3) Find the equation of the normal to the parabola y²= 4x which is:
a) parallel to the line y= 2x -5
b) Perpendicular to the line 2x + 6y +5=0.
4) The ordinates of a point P and Q on the parabola y²= 12yx are in the ratio 1:2. Find the locus of the point of intersection of the normal to the parabola at P and Q.
5) The normals at P, Q, R on the parabola y²= 4ax meet in a point on the line y= c. Show that the sides of the triangle PQR touch the parabola x²= 2cy.
6) The normals are drawn from (2λ,0) to the parabola y²= 4x. Show that λ must be greater than 1. One normal is always the x-axis. Find λ for which the other two normals are perpendicular to each other.
MARKS- 30 Time: 45 minutes 1x24= 24
1) If the determinant
P= 1 α 3
1 3 3
2 4 4 is adjoint of a 3x3 Matrix A and |A|= 4 then α is equals to
a) 11 b) 5 c) 0 d) 4
2) The term endependent of x in expansion of
{(x +1)/(x²⁾³ - x¹⁾³+1) - (x -1)/(x - x¹⁾²)}¹⁰ is
a) 120 b) 210 c) 310 d) 4
3) If the determinants
x 1 1 & B= x 1
A=1 x 1 1 x then dA/dx=
a) 3B+1 b) 3B c) -3B d) 1- 3B
4) The value of C₀ + 2C₁ + 3C₂ +....+ (n+1) Cₙ = 576, then n is
a) 7 b) 5 c) 6 d) 9
5) The remainder when, (10¹⁰+ 1)(10¹⁰+ 2) is divided by 6 is
a) 2 b) 4 c) 0 d) 6
6) If (1+ x + x²)ⁿ = 1+ a₁x + a₂x²+ ....+ a₂ₙx²ⁿ, then 2a₁ - 3a₂+ ... -(2n +1)a₂ₙ =
a) n b) -nl c) n +1 d) -nl -1
7) The value of x satisfying the equation of determinant
Cos2x sin2x sin2x
Sin2x cos2x sin2x= 0
Sin2x sin2x cos2x
And x ∈[0,π/4] is
a) π/4 b) π/2 c) π/16 d) π/8
8) If t₅, t₁₀, t₂₅ are 5ᵗʰ, 10ᵗʰ, and 25ᵗʰ terms of an AP respectively, then the value of determinant
t₅ t₁₀ t₂₅
5 10 25
1 1 1 is equal to
a) -40 b) 1 c) -1 d) 0
9) Five dice are tossed. What is the probability that five numbers shown will be different ?
a) 5/24 b) 5/18 c) 5/27 d) 5/81
10) if the events A and B are independent and if P(A')= 2/3, P(B)= 2/7 then P(A∩B) is equals to
a) 4/21 b) 3/21 c) 5/21 d) 1/21
11) Let P= [aᵢⱼ] be a 3 x 3 matrix and let Q= [bᵢⱼ], where bᵢⱼ = 2 ᶦ⁺ʲ aᵢⱼ for 1 ≤ i, j ≤ 3. If the determinant of P is 2, then the determinant of the matrix Q is
a) 2¹⁰ b) 2¹¹ c) 2¹² d) 2¹³
12) x(xⁿ⁻¹ - nαⁿ⁻¹) + αⁿ (n -1) is divisible by (x - α)² for
a) n> 1 b) n > 2 n ∈ N d) none
13) The sum of the series 1+ 3²/2! + 3⁴/4! + 3⁶/6!+....to ∞ is
a) e⁻³ b) e³ c) (1/2)(e³ - e⁻³) d) (1/2) (e³ + e⁻³)
14) Value of the series x/1.2 + x²/2.3 + x³/3.4 +.... is
a) 1- {(1- x)/x} log(1- x)
b) 1- {(1- x)/x} log(1+ x)
c) 1 + {(1- x)/x} log(1- x) d) none
15) Let the coefficient of powers of x in the 2ⁿᵈ, 3ʳᵈ and 4ᵗʰ terms in the expansion of (1+ x)ⁿ, where n is a positive integer, be in Arithmetic progression. The sum of the co-efficients of odd powers of x in the expansion is
a) 32 b) 64 c) 128 d) 256
16) The sum of the infinite series
1+ 1/3 + 1.3/3.6 + 1.3.5/3.6.9 + 1.3.5.7/3.6.9.12+ .....is equal to
a) √2 b) √3 c) √(3/2) d) √(1/√3))
17) The number of real values of K for which the system of equations
x+ 3y + 5z = Kx
5x+ y + 3z = Ky
3x+ 5y + z = Kz
has infinity number of solution is
a) 1 b) 2 c) 4 d) 6
18) Let Sₖ be the sum of an infinite GP series whose first term is K and common ratio is K/(K +1) (K> 0). Then the value of ∞ₖ₌₁∑ (-1)ᴷ/Sₖ is equal to
a) log4 b) log2 -1 c) 1- log2 d) 1- log4
19) Let A and B be two events with P(A')= 0.3, P(B)= 0.4 and P(A ∩B) = 0.5 then P(B/A U B') is equal to
a) 1/4 b) 1/3 c) 1/2 d) 2/3
20) Three numbers are choosen at random without replacement from {1, 2, 3, .....,8}. The probability that their minimum is 3, given that their maximum is 6, is
a) 1/4 b) 2/5 c) 3/8 d) 1/5
21) If C₀, C₁, C₂, C₃,.... are binomial coefficients in the expansion of (1+ x)ⁿ. Then C₀/3 - C₁/4 + C₂/5 - C₃/6+.... is equal to
a) 1/(n +1) - 2/(n+2) + 1/(n +3)
b) 1/(n +1) + 2/(n+2) - 1/(n +3)
c) 1/(n +1) - 1/(n+2) + 1/(n +3)
d) 2/(n +1) - 1/(n+2) + 2/(n +3)
22) If the matrix
A= a x
y a and xy=1. Then det(AA') is equal to
a) a²-1 b) (a²+1)² c) 1- a² d) (a²-1)²
23) Let A and B any two events. Which one of the following statements is always true ?
a) P(A'/B) = P(A/B)
b) P(A'/B) = P(B'/A)
c) P(A'/B) = 1- P(A/B)
d) P(A'/B) = 1- P(A/B')
24) The inverse of a symmetric matrix is
a) skew symmetric
b) symmetric
c) diagonal matrix d) none
Equation of Straight line
Sap-1
1) The number of points on x-axis which are at a distance c(c< 3) from the point (2,3) is
a) 2 b) 1 c) infinite d) no point
2) The distance between the points P(a cosα, a sinα) and Q(a cosβ, a sinβ) is
a) 4a sin{(α-β)/2} b) 2a sin{(α + β)/2} c) 2a sin{(α-β)/2} d) 2a cos{(α-β)/2}.
3) Determine the ratio in which y - x + 2 divides the line joining (3,-1) and (8,9).
4) If (1,4) is the centroid of a triangle and its two vertices are (4,-3) and (-9,7) then third vertices is
a) (7,8) b) (8,8) c) (8,7) d) (6,8).
5) The vertices of a triangle are A(0.-6), B(-6,0) and C(1,1), respectively, then coordinates of the excentre opposite to vertex A is.
a) (-3/2,-3/2) b) (-4,3/2) c) (-3/2,3/2) d) (-4,6).
6) If the vertices of a triangle are (1,2),(4,-6) and (3,5) then the area is
a) 25/2 b) 12 c) 5 d) 25.
7) The point A divides the join of the points (-5.1) and (3,5) in the ratio k: 1 and coordinates of points B and C are (1,5) and (7,-2) respectively. If the area of ∆ ABC be 2 units, then k equals to
a) (7,9) b) (6,7) c) 7,31/9 d) 9,31/9.
8) Show that the coordinates of the vertices of an equilateral triangle can not be rational.
9) The ends of the rod of length l moves on two mutually perpendicular lines, find the locus of the point on the rod which divides it in the ratio m₁: m₂
a) m₁²x²+ m₂²y²= l²/(m₁ + m₂)²
b) (m₂x)²+ (m₁y)²= {(m₁m₂l)/(m₁ + m₂)}²
c) (m₁x)²+ (m₂y)²= {(m₁m₂l)/(m₁ + m₂)}²
d) none.
10) If A(a,0) and B(-a,0) are two fixed points of ∆ ABC. If its vertex C moves in such way that cotA + cotB= λ, where λ is a constant, then the locus of the point C is
a) yλ = 2a b) y= λa c) ya = 2λ d) none
11) The equation of the lines which passes through the point (3,4) and the sum of its intercept on the axes is 14 is
a) 4x - 3y= 24, x - y= 7
b) 4x + 3y= 24, x + y= 7
c) 4x + 3y=- 24, x + y=- 7
d) 4x - 3y= -24, x - y=- 7.
12) Two points A and B move on the positive direction of x-axis and y-axis respectively, such that OA+ OB= K. Show that the locus of the foot of the perpendicular from the origin O on the line AB is (x + y)(x²+ y²)= Kxy.
13) Find the equation of the straight line on which the perpendicular from origin makes an angle 30° with x-axis and which forms a triangle of area (50/√3) square. units with the coordinates axes.
14) Equation of a line which passes through point A(2,3) and makes an angle of 45° with x-axis. If this line meet the line x+ y+1=0 at point P then distance AP is
a) 2√3 b) 3√2 c) 5√2 d) 2√5.
15) A variable line is drawn through O, to cut two fixed straight lines L₁ and L₂ in A₁ and A₂ respectively. A point A is taken on the variable line such that (m+ n)/OA = m/OA₁ + n/OA₂.
Show that the locus of A is a straight line passing through the point of intersection of L₁ and L₂ where O is being the origin.
16) A straight line through P(-2,-3) cuts the pair of straight line x²+ 3y²+4xy - 8x - 6y - 9= 0 in Q and R. Find the equation of the line if PQ. PE = 20.
17) If the line y - √3 x +3=0 cuts the parabola y²= x + 2 at A and B, then find the value of PA. PB (where P=(√3,0).
18) If x + 4y -5=0 and 4x + ky +7=0 are two perpendicular lines then k is
a) 3 b) 4 c) -1 d) -4.
19) A line L passes through the points (1,1) and (2,0) and another line M which is perpendicular to L passes through the point (1/2,0). The area of the triangle formed by these lines with y-axis is
a) 25/8 b) 25/16 c) 25/4 d) 25/32.
20) If the straight line 3x + 4y+ 5 - k(x + y +3)= 0 is parallel to y-axis, then the value of k is
a) 1 b) 2 c) 3 d) 4
21) If the algebraic sum of perpendiculars from n given points on a variable straight line is zero then show that the variable straight line passes through a fixed point.
22) Show that no line can be drawn through the point (4,-5) so that its distance from (-2,3) will be equal to 12.
23) Three lines x+ 2y+3=0, x + y= 7, 2x - y= 4 form 3 sides of two squares. Find the equation of remaining sides of these squares.
24) Find the equation to the sides of an isosceles right angled triangle, the equation of whose hypotenuse is 3x + 4y= 4 and the opposite vertex is the point (2,2).
25) Let P(sinθ, cosθ)(0≤θ≤2π) be a point and let OAB be a triangle with vertices (0,0), ((√3/2),0) and (0,√(3/2)). Find θ if P lies inside the ∆ OAB.
26) Through what angles should the axes be rotated so that the equation 9x² - 2√3xy = 10 may be changed to, 3x² + 5y²= 5?
27) For the straight lines 4x + 3y= 6, 5x +12y +9= 0, find the equation of the
a) bisector of the obtuse angle between them.
b) bisector of the acute angle between them.
c) bisector of the angle which contains origin.
28) Show that each member of the family of straight lines
(3sinθ + 4 cosθ)x + (2 sinθ - 7 cosθ)y + (sinθ + 2 cosθ)= 0 (θ is a parameter) passes through a fixed point.
29) λx²- 10xy + 12y²+ 5x - 16y -3=0 represents a pair of straight lines, then λ is equal to
a) 4 b) 3 c) 2 d) 1.
30) Show that the two straight lines x²(tan²θ+ cos²θ) - 2xy tanθ + y² sin²θ = 0 represented by the equation are such that the difference of their slopes is 2.
31) If pair of straight lines x¹- 2pxy - y²= 0 and, x² - 2qxy - y²= 0 be such that each pair bisects the angle between the other pair, show that pq= -1.
32) The chord √6y = √8 px + √2 of the curve py²+ 1= 4x subtends a right angle at origin then find the value of p.
EXERCISE - A
1) Find the cartesian co-ordinates of the point whose polar coordinates are
a) (5, π- tan⁻¹(4/3)).
b) 5√2, π/4).
2) Find the polar coordinates of the points whose cartesian co-ordinates are
a)!(-2,-2).
b) (-3,4).
3) Transform the equation r²= a² cos2θ into cartesian form.
4) transform the equation x²+ y²= ax into polar form.
OBJECTIVE - A
1) The polar coordinates of the point whose cartesian coordinates are (- 1, - 1) is :
a) (√2,π/4) b) (√2, 3π/4) c) (√2, - π/4) d) (√2, -3π/4)
2) The cartesian co-ordinates of the point whose polar coordinates are (13, π - tan⁻¹(5/12)) is:
a) (12,5) b) (-12,5) c) (- 12,-5) d) (12, - 5)
3) The transformation equation of r² cos²θ = a² cos2θ to cartesian form is (x²+ y²)x²= a²λ, then the value of λ is:
a) y²- x² b) x²- y² c) xy d) x²y²
4) The coordinate of P' in the figure is:
a) (3, π/3) b) (3, -π/3) c) (-3, -π/3) d) (-3, π/3)
5) The cartesian coordinates of the point Q in the figure is:
a) (√3,1) b) (-√3,1) c) (-√3,-1) d) (√3, -1)
SUBJECTIVE - A
1) A point lies on x-axis at a distance 5 units from y-axis. What are the coordinates ?
2) A point lies on y-axis at a distance of 4 units from x-axis. What are its coordinates ?
3) A point line on negative direction of x-axis at a distance 6 units from y-axis. What are its coordinates ?
4) Transform the equation y = x tanβ to polar form .
5) Transformation the equation r= 2β cosθ to cartesian form.
EXERCISE - B(1)
1) Prove that the distance of the point (a cosβ, a sinβ) from the origin is independent of β.
2) Find the distance between the points (a cosβ, a sinβ) and (a cosγ, a sinγ) where a >0.
3) If the point (x, y) be equidistant from the points (6,-1) and (2, 3), prove that x - y = 3.
4) Using distance formula, show that the points (1,5), (2,4) and (3,3) are collinear .
5) An equilateral triangle has one vertex at the point (0,0) and another at (3, √3). Find the coordinanates of the third vertex.
6) Show that the four points (1,-2),(3,6),(5, 10) and (3,2) are the vertex of a parallelogram.
7) Let the opposite angular points of a square be (3,4) and (1, -1). Find the coordinates of the remaining angular points.
8) Find the circumcenter of the triangle whose vertices are (-2,3),(-1,0) and (7,-6). Also find the radius of the circumcircle.
9) if the line segment joining the point A(a, b) and B(c, d) subtends an angle θ at the origin O, prove that
Cosθ = (ac + bd)/√{(a²+ b²)(c²+ d²)}
EXERCISE - B(2)
1) Show that the triangle, the coordinates of whose vertices are given by integers, can never be an equilateral triangle.
2) In any triangle ABC, show that
AB²+ AC²= 2(AD²+ BD²) Where D is the middle point of BC.
3) Let ABCD be a rectangle and P be any point in its plane.
Show that PA²+ PC²= PB²+ PD².
Distance between two points in polar coordinates
EXERCISE - B(3)
1) Prove that the points (0,0), (3, π/2) and (3, π/6) are the vertices of an equilateral triangle.
OBJECTIVE - B
1) If the distance between the point (a, 2) and (3,4) be 8, then a =
a) 2+ 3√15 b) 2- 3√15 c) 2± 3√15 d) 3 ± 2√15
2) The three points (-2,2), (8,-2) and (-4,-3) are the vertices of:
a) an isosceles triangle
b) an equilateral triangle
c) a right angle triangle d) none
3) The distance between the point (3, π/4) and (7,5π/4) is:
a) 8 b) 10 c) 12 d) 14
4) Let A(6, -1), B(1,3) and C(x,8) be three points such that AB= BC then the value of x are:
a) 3, 5 b) -3, 5 c) 3, -5 d) -3, -5
5) The points (a+1,1),(2a+1,3) and (2a+2, 2a) are collinear, if:
a) a= -1,2 b) a= 1/2 ,2 c) a= 2,1 d) a= -1/2,2
6) If A= (3,4) and B is a variable point in the line |x|= 6. If AB ≤ 4 then the number of positions of B with integral coordinanates is
a) 5 b) 6 c) 10 d) 12
7) The number of points on x-axis which are at a distance c units (c < 3) from (2,3) is :
a) 1 b) 2 c) 0 d) 3
8) The points on the axis of y which its equidistant from (-1, 2) and (3,4) is:
a) (0,3) b) (0,4) c) (0,5) d) (0,-6)
SUBJECTIVE -B
1) Find the distance between the points (at₁², 2at₁) and (at₂², 2at₂), where t₁, and t₂ are the roots of the equation x²- 2√3 x + 2 =0 and a> 0.
2) If P(at², 2at), Q(a/t², 2a/t) and S(a,0) be any the three points , show that 1/SP + 1/SQ is independent of t.
3) Prove that the points (3,4), (8,-6) and (13,9) are the vertices of a right angled triangle.
4) show that the points (0,-1),(6,7), (-2,3) and (8,3) are the vertices of a rectangle.
5) Find the circumcentre and circumradius of the triangle whose vertices are (-2,3),(2,-1) and (4,0).
6) The vertices of a triangle are A(1,1), B(4,5) and C(6,13). Find cos A.
7) Two opposite vertices of a square are (2,6) and (0,-2). Find the coordinanates of the other vertices.
8) If the point (x,y) is equidistant from the points (a+ b), b- a) and (a- b, a+ b), prove that bx = ay.
9) if a and b are real numbers between 0 and 1 such that the points (a,1),(1, b) and (0, 0) form an equilateral triangle, find a and b.
10) An equilateral triangle has one vertex at (3,4) and another at (-2,3). Find the coordinanates of the third vertex.
11) If P be any point in the plain of square ABCD, prove that PA²+ PC²= PB²+ PD².
SECTION FORMULA
EXERCISE - C
1) Find the coordinates of the point which divides the line segment joining the points (5, -2) and (9,6) in the ratio 3:1.
2) Find the length of median through A of a triangle whose vertices are A(-1,3), B(1,-1) and C(5,1)
3) Determine the ratio in which y - x + 2= 0 divides the line joining (3, -1) and (8, 9).
4) The coordinanates of three consecutive vertices of a parallelogram are (1,3), (-1, 2) and (2,5). Then find the coordinates of the fourth vertex.
5) In water ratio does x-axis divide the line segment joining (2,-3) and (5,6)?
6) The mid point of the sides of a triangle are (1,2),(0 ,1) and (2,-1). Find the coordinanates of the vertices of a triangle with the help of two unknowns.
7) Prove that in a right angled triangle the mid point of the hypotenuse is equidistant from its vertices.
8) Show that the line joining the midpoints of any sides of a triangle is half the third side.
EXERCISE - D
1) Find the coordinates of a point which divides the externally the line joining (1,-3) and (-3,9) in the ratio 1 :3.
2) The line segment joining A(6,3) to B(- 1,-4) is doubled in the length by having its length added to each end. Find the coordinates of the new events.
3) Using section formula show that the points (1, -1), (2,1) at(4, 5) are collinear .
4) Find the ratio in which the point (2, y) divide the line segment joining (4,3) and (6, 3) and henr find the value of y.
EXERCISE - E
1) Find the harmonic conjugate of the point R(5,1) with respect to the points P(2,10) and Q(6, -2).
EXERCISE - F
1) Two vertices of a triangle are (-1,4) and (5,2). If its centroid is (0,-3), find the third vertex.
2) The vertices of a triangle are (1, 2), ( h, - 3) and (-4, k).
Find the value of √{(h +k)²+ (h + 3k)²}.
If the centroid of the triangle be at the point (5,-1).
3) If D(-2,3), E(4,-3) and F(4,5) are the midpoint of the sides BC, CA and AB of triangle ABC, then find √{|AG|²+ |BG|² - |CG|²} where G is the centroid of ∆ ABC .
4) If G be the centroid of the triangle ∆ABC and O be any other point in the plane of the triangle ABC, then show that OA²+ OB²+ OC²= GA²+ GB²+ GC²+ 3GO².
5) If G be the centroid of ∆ ABC, show that AB²+ BC²+ CA²= 3(GA²+ GB²+ GC²).
6) The vertices of a triangle (1, a), (2,b) and (c², -3)
a) Show that its centroid can not lie on the y-axis.
b) Find the condition that the centroid may lie on the x-axis.
EXERCISE - G
1) Find the co-ordinate the incentre of the triangle is vertices are (4,-2),(-2,4) and (5,5).
2) If (3/2, 0),(3/2, 6) and (-1,6) are mid points of the sides of a triangle, then find
a) Centroid of the triangle.
b) Incentre of the triangle.
EXERCISE - H
1) If a vertex of a triangle be (1,1) and the middle points of two sides through it be (-2, 3) and (5,2) then find the centroid and the incentre of the triangle. {(5√2- 5√17+ 9√13)/(5√2+ √17+ √13), (5√2+ 5√17+ 3√13)/(5√2+ √17+ √13)}
2) If G be the centroid and I be the incentre of the triangle with vertices A(-36, 7), B(20,7) and C(0,-8) and GI= (25√205)/3λ, then find the value of λ. 1/25
3) In a triangle ABC with vertices A(1,2), B(2,3) and C(3,1) and angle A= cos⁻¹(4/5), angle B= angle C = cos⁻¹(1/√10), then find the circumcentre of the triangle ABC .
4) if the co-ordinate of the midpoint of the sides of a triangle are (1,1), (2,-3) and (3,4) then find the excentre opposite to the vertex A.
5) If a triangle has its orthocentre at (1,1) and circumcentre at (3/2,3/4) then find the centroid and nine point centre. (4/3,5/6), (5/4,7/8)
6) The vertices of a triangle are A(a, a tan α), B(b, b tanβ) and C(c, c tanγ). If the circumcentre of ∆ ABC coincides with the origin and H(bar x, bar y) is the orthocentre, then show that
bar y/bar x = (sinα + sinβ + sinγ)/(cosα + cosβ + cosγ).
EXERCISE - I
1) The co-ordinates of ABC are (6,3),(-3,5) and (4,-2) respectively and P is any point (x,y). Show that the ratio of the areas of the triangle PBC and ABC is |(x+ y - 2)|/7.
2) Find the area of the Pentagon whose vertices are A(1,1), B(7, 21), C( 7, -3), D(12,2) and E(0,-3).
3) Show that the point (a,0), (0,b) and (1,1) are collinear , if 1/a + 1/b =1
4) Prove that the co-ordinates of the vertices of an equilateral triangle can not all be rational.
5) The value coordinanates of two points A and B are (3,4) and (5,-2) respectively. Find the Co-ordinates of any point P if PA= PB and area of ∆ APB is 10.
6) Find the area of the triangle formed by the straight line 7x - 2y + 10 = 0, 7x + 2y -10= 0 and 9x + y +2=0 (without solving the vertices of the triangle).
7) If ∆₁ is the area of the triangle with vertices (0,0), (a tanα, b cotα), (asinα, b cosα); ∆₂ is the area of the triangle with vertices (a,b), (a sec²α, b cosec²α), (a+ a sin²α, b + b cos²α) and ∆₃ is the area of the triangle with vertices (0,0), (a tanα, - b cotα), (a sinα, b cosα). Show that there is no value of α for which ∆₁ , ∆₂ and ∆₃ are in GP.
MISCELLANEOUS EXERCISE- A
OBJECTIVE - A
1) The coordinanates of the middle points of the sides of the triangle are (4,2),(3,3) and (2,2), then the coordinates the centroids are
2) The incentre of the triangle whose vertices are (-36,7),(20,7) and (0,-8) is:
a) (0,-1) b) (-1,0) c) (1,1) d) (1/2,1)
3) If the orthocentre and centroid of a triangle are (-3,5) and (3,3) then its circumcentre is:
a) (6,2) b) (3,-1) c) (-3,5) d) (-3,1)
4) An equilateral triangle has each side to a. If the co-ordinates of its vertices are (x₁, y₁), (x₂,y₂), and (x₃, y₃) then the square of the determinants
x₁ y₁ 1
x₂ y₂ 1
x₃ y₃ 1 equals to
a) 3a⁴ b) 3a⁴/2 c) 3a⁴/4 d) 3a⁴
5) The vertices of a triangle are A(0,0), B(0,2) and C(2,0). The distance between circumcentre and orthocentre is:
a) √2 b) 1/√2 c) 2 d) 1/2
6) A(a,b), B(x₁, y₁), and C(x₂,y₂) are the vertices of the triangle. If a, x₁, x₂ are in GP with common ratio r , and b, y₁, y₂ are in GP with common ratio s, then area of ∆ ABC is:
a) ab(r-1)(s-1)(s- r)
b) (ab/2) (r+1)(s+1)(s- r)
c) (ab/2) (r-1)(s-1)(s- r)
d) ab(r+1)(s+1)(r - s)
7) The point (x+1, 2),(1, x+2), (1/(x+1), 2/(x+1)) are collinear then x is equals to:
a) -4 b) -8 c) 4 d) 8
8) The vertices of a triangle are (6,0),(0,6) and (6,6), then distance between its circumcentre and centroid, is:
a) 2√2 b) 2 c) √2 d) 1
9) The nine point centre of the triangle with vertices (1,√3),(0,0) and (2,0) is :
a) (1,√3/2) b) (2/3, 1/√3) c) (2/3, √3/2) d) (1,1/√3)
19) The vertices of a triangle are (0,0),(1,0) and (0,1). Then ex-centre opposite to (0,0) is:
a) (1- 1/√2, 1+ 1/√2)
b) (1+ 1/√2, 1+ 1/√2)
c) (1- 1/√2, 1+ 1/√2)
d) (1- 1/√2, 1- 1/√2)
MISCELLANEOUS SUBJECTIVE - A
1) α, β, γ are the real roots of the equation x³- 3px²+ 3qx -1= 0 then find the centroid of the triangle whose vertices are (α, 1/α), (β,1/β) and (γ,1/γ).
2) If centroid of a triangle be (1,4) and the Co-ordinates of its any two vertices are (4,-8) and (-9,7), find the area of the triangle.
3) Find the centroid and incentre of the triangle whose vertices are (1,2),(2,3) and (3,4).
4) Show that the area of the triangle with vertices (λ, λ-2),(λ+3, λ) and (λ+2, λ+2) is independent of λ.
5) Prove that the points (a, b+ c), (b, c + a) and (c, a+ b) are collinear.
6) Show that the points (a, b) and (a - c, b - d) are collinear, if ad = bc.
7) If the points (x₁,y₁),(x₂,y₂) and (x₃,y₃) are collinear, show that
∑{(y₁ - y₂)/(x₁x₂)}= 0, i.e., (y₁ - y₂)/x₁x₂ + (y₂ - y₃)/x₂x₃ + (y₃ - y₁)/x₃x₁ = 0.
8) The coordinanates of points A, B, C and D are (-3,5),(4,-2),(x, 3x) and (6,3) respectively and ∆ ABC/∆ BCD = 2/3, find x.
9) Find the area of the hexagon whose vertices taken in order are (5,0),(4,2),(1,3),(-2,2), (-3,-1) and (0,-4).
MISCELLANEOUS - B
1) Find the polar coordinates of the point whose cartesian coordinates are (-√3,1). (2,5π/6)
2) Find the cartesian coordinates of the point whose polar coordinates are (√2, 5π/4). (-1,-1
3) Express r= 2a cosθ in cartesian form. x²+ y²= 2ar.
4) Express x²- y²= 2ax in polar coordinates form.
5) Express the following relations in polar coordinates:
a) y= x tanα (α is a constant). sin( θ - α)= 0
b) y²= 4x +3. r² sin²θ = 4r cos θ +3
c) x²+ y²= a². r= a
d) xy= 9. r² sin2θ = 18
e) 4x²+ 3y²= 12. 1/r²= cos²(θ/3) + sin²(θ/4)
f) 2x - 3y = 8. 2 cosθ - 3 sinθ = 8/r
g) x= 3. Cosθ = 3/r
h) x²+ y²= 2ax. r= 2a cosθ
i) x²+ y²- 2x + 4y =0. r - 2 cosθ + 4 sinθ = 0
6) Express the following relations in cartesian co-ordinates:
a) r= 2a cosθ. x²+ y²- 2ax
b) r² - 6r cosθ +5= 0. x²+ y²- 6x +5=0
c) r= 5. x²+ y²= 25
d) r²= 4/(cos²θ - sin²θ). x²- y²= 4
e) l/r = 1+ e cosθ; (l, e are constant). x²+ y²= (ex - l)²
f) r= a sinθ . x²+ y²= ay
7) Find the lengths of the sides of the triangle whose vertices are the points (-2,3), (-2,-1), (4,-1). 4, 6, 2√13, right angle at B
8) Show that the following points lie on a straight line (-3,-2),(5,2),(9,4).
9) Prove that the distance between the point (at², 2at) and (a/t², -2a/t) is a(t + 1/t)².
10) prove that the distance between the point (a cosθ , a sinθ ) and (acosβ, a sinβ) is 2a sin{(θ-β)/2}. (If θ> β).
11) Find the distance between two points P₁ and P₂ whose polar coordinates are respectively:
a) (2, 30°) and (4 0,120°). 2√5
b) (a,π/2) and (3a, π/6). a√7
12) Prove that the points (0,0),(3,π/2) and (3,π/6) form an equilateral triangle.
13) Show that the quadrilateral with the vertices P₁: (-3/2,4), P₂: (-7/2,3), P₃: (1,0), P₄: (3,1) is a parallelogram.
14) The segment joining P₁: (1,3) and P₂: (5,-2) is trisected . Find the point of trisection P near to P₁.
15) The segment from P₁:(5,-4) to P₂: (7,-9) is extended beyond P₂ so that its length is doubled. Find the coordinates of the terminal point P. (9,-14)
16) A circle with centre at A(-4,1) has one end of a diameter at B(2,6). Determine the coordinanates C(x,y) of the other and. (-10,-4)
17) If the point (9,2) divides the segment of the line from A(6,8) to B(a,b) in the ratio 3:7, find the co-ordinate of B. (16,-12)
18) Determine the coordinates of the vertices of a triangle if the middle points of its sides are (3,2),(-1,-2) and (5,-4). (-3,4),(9,0),(1,-8)
19) Show that analytically that the lines joining the middle points of the adjacent sides of the quadrilateral A(-3,2), B(5,4), C(7,-6), D(-5,-4) form a second quadrilateral whose perimeter is equal to the sum of the diagonals of the first.
20) Find the area of a triangle with vertices are (3,1),(2k, 3k), (k, 2k) and show that the distinct points are collinear when k= -2.
21) Find the area of the triangle whose vertices are (a, b+ c),(b, c+ a), and (c, a+ b). 0
22) Show that the line joining the middle points of the sides of the triangle (2,-3),(4,2),(-5,-2), divide the triangle and four triangles whose areas are equal .
23) Show that area of the triangle with the vertices (t, t-2),(t+3, t),(t+2, t+2) is independent of t.
24) A, B, C, D are the points (3,1),(2,4),(2,2), (3- 2t, t²+ 5) and O is the origin 0,0); if area of ∆OAB = ∆OCD, (both in magnitude and in sign), find the possible values of t. 1 or -3
25) The vertices of a quadrilateral in order, are (-2,3),(-3,-2), (2,-1),(x,y); if its area is 14. Show that x+ y = 2.
26) Show that the point (-1,-2) is the centre of a circle passing through the points (11,3),(-1,-15),(-13,-7) and (4, -14(.
27) A, B are points (-8,0),(-2,0): a point P(x,y) in such that|bar PA|= 2 |bar PB|. Show that x²+ y²= 16.
28) A line is of length 10 units . Its one end is at the point (2,-3); if the abscissa of the other end be 10. Prove its ordinate is either 3 or -9.
29) if the coordinates of the three vertices triangle are (-2,5),(- 4,-3) and (6,-2), find the coordinates of the centroid of the triangle. (0,0)
30) The co-ordinates of the vertices of a triangle are (4,-3),(-5,2) and (x,y). If the centre of gravity of the triangle is at the origin, find x, y. 1,1
31) The points (1,2,),(2,4),(t,6) are collinear, find t. 3
32) If the three points (a,0),(0,b) ay(2,2) are collinear, prove that 1/a + 1/b = 1/2.
33) What are the coordinates of P if O be the origin and Q(-2,-4) is the point on OP such that OQ= (1/3) OP ? -6,12)
34) The coordinates of the vertices of a triangle are (0,0),(5,3) and (3,5) respectively, find the circumcentre and circumradius of the triangle. (17/8,17/8), 17√2/8
35) Triangle OAB has vertices (0,0),(b cosθ, - b sinθ) and (sinβ, cosβ). Show that area of the triangle OAB is maximum when θ = β and find the maximum area. b/2
36) The centre of a circle is at (5,3) and its radius is 5. Find the length of the chord which is bisected at (3,2). 4√5
37) A circle with centre (0,-13) has the axis of x as one of its tangents. Does the circle pass through (11,-6) ? Through (-5,-1)? No, yes
38) If the three points (a,b),(a+ k cosθ, b + k sinθ) and (a+ k cosβ, b + k sinβ) are the vertices of an equilateral triangle, then which of the following results is true and why ?
a) |α -β|=π/4
b) |α - β|= π/2
c) |α- β|=π/6
d) |α - β|=π/3.
MISCELLANEOUS - C
1) Show. that the centroid of the triangle ABC were A(-2,5), B(-4,-3), C(6,-2) is the origin.
2) Find the area of the triangle ABC with vertices A(x₁, y₁), B(x₂,y₂) and C(x₃, y₃). If ABCD be collinear , is it, in general , true that the three points have formed a triangle of zero area ?
3) Find the conditions that the three points A(x₁, y₁), B(x₂, y₂) and C(x₃, y₃) be collinear.
4) show that area of the triangle of the vertices (-5,-2),(2,2) and (3,4) will be 5 square units.
5) Show that (1,5),(3,14) and (-1,-4) are collinear.
6) Show that the points (2,0),(0,2),(√3+1, √3+1) are the vertices of an equilateral triangle.
7) Prove that the triangle whose vertices at the points (1,8),(3,2),(9,4) is an isosceles right angled triangle.
8) Show that the points (-2,-1), (1,0),(4,3) and (1,2) are the vertices of a parallelogram. Is the Parallelogram is rectangle ?
9) Show that the point (-7,1),(5,-4),(10,8) and (-2,13) the vertices of a square and find its area.
10) Prove that the points (2,-2),(8,4),(5,7) and (-1,1) are the angular points of rectangle.
11) Prove that the points (2,5),(5,9), (9,12) and (6,8) when joined in order form a rambus.
12) Prove that the point (-1/14,39/14) is the circumcentre of a triangle whose vertices are (1,4),(2,3) and (-2,2). What is the length of the circum-radius ?
12) A circle with centre (3,2) passes through (13,- 10). Does this circle pass through (-11, 9)?
13) A circle with centre (0,13) has the axis of x as one of its tangents.. Does this circle pass through (11,- 6)? Through (-5,1)?
14) The centre of a circle is at (5,3) and its radius is 5. Find the length of the chord that is bisected at (3,2). 4√5
15) Find the radius of a circle with centre at (1,1), if a chord of length 10 units is bisected at (2,0). 3√3
16) A(1,2), B(3,4). A point P divides AB in the ratio 3:1 internally show that P=(13/7,20/7).
17) A line segment directed from (-3,2) to (1,-4) is trebled. Find the coordinates of the terminal point. (9,-16)
18) The segment joining P(1,3) and Q(5,-2) is trisected. Find the point of trisection B nearer to Q. (7/3,4/3)
19) Find the area of the triangle with vertices A(3,1), B( 2K, 3K) C(K, 2K) and show that the three distinct points A, B, C are collinear when k=-2.
20) Show that the area of the triangle formed by the points (-3,4),(6,2) and (4,-3) is 24.5 square units.
21) The points (2,3/2),(-3,-7/2),(t, 9/2)are collinear ; find t. 5
22) Find the area of the triangle with vertices (2,-1),(a+1, a-3), (a+2,a) and show that they are collinear if a = 1/2
23) If A(1,5) and B(-4,7), find the point P which divides AB in the ratio 2:3 internally . (-1,29/5)
24) What are the coordinates of P if O be the origin and Q(-2,4) is the point on OP such that OQ= (1/3) OP ? (-6,12)
25) A(1,2), B(3,4) are two fixed points. P divides AB internally in the ratio 1:m. Show that P= {(3+m)/(1+m), (4+2m)/(1+m)}
26) a point divides internally the line segment joining (8,9) and (-7,4) in the ratio 2:3. Show that the coordinates of the points are (2,7).
27) Find the coordinates of the centroid of the triangle whose vertices are (0,0),(6,4),(4,6). (10/3,10/3)
28) Show that the centroid of the triangle with vertices at (4,-1),( 0,3) and (-4,-2) is the origin of the coordinates.
29) The coordinates of the vertices of a triangle are (4,-3),(- 5,2) and (x,y). if the centroid of the triangle is at the origin, find x and y. 1, 1
30) Find the area of the triangle having vertices at (1,4), (-1,2) and (-4,-1) interpret the results.
31) Show that the area the triangle having vertices at (a, 1/a),(b, 1/2), (c, 1/c) is {(b-c)(c- a)(a- b)}/2abc.
32) The point (1,2),( 2,4),(t,6) are collinear, find t. 3
33) if the points (a,0),(0,b) and 2,2) are collinear, prove that 1/a + 1/b = 1/2
34) If (a,b), (A', b'), (a- A', b - b') be collinear, prove that ab'= a'b.
35) Find the condition that the points (a,b),(b,a) and (a², - b²) are in straight line. a³+ b³- a²b - ab²- a²+ b²= 0
36) Show that the point (-1,-2) is the centre of a circle passing through the points (11,3),(- 1,15),(-13,-7) and (4,-14).
37) A, B are points (-8,0),(-2,0); a point P(x,y) is such that |PA|= 2|PB|. Show that x²+ y²= 16
38) a line is of length 10 units. Its one end is at (2,-3); if the abscissa of the other end be 10, prove that its ordinate is either 3 or -9.
39) The coordinates of the vertices of a triangle (0,0)(5,3) and (3,5) respectively, find the circumcentre and the circumradius of the triangle triangle. 17/8,17/8,17√2/8
40) Triangle OAB has vertices (0,0),(b cosx , - b sinx) and (siny, cosy). Show the area of the triangle OAB is maximum when x= y ay find the maximum area. b/2
41) if the three points (a,b),(a+ k cosx, b+ k sinx) and (a+ k cosy, b + k siny) are the vertices of an equilateral triangle, then which of the following results is true
a) |x - y|= π/4
b) |x - y|= π/2
c) |x - y|= π/6
d) |x - y|= π/3
42) Find the coordinates of the points which divide, internal and externally, the line joining the point (a+ b, a- b) to the point (a- b, a+ b) in the ratio a: b (0< b < a). {(a²+ b²)/(a+ b), (a²+2ab- b²)/(a+b)}; {(a²- 2ab- b²)/(a- b), (a²+ b²)/(a-b)}
53) If (3,-1),(-4,-3) and (1,5) are the three vertices of parallelogram and the 4th vertex lies in the first quadrant, find the coordinates of the fourth vertex. (8,7)
54) a point with abscissa 6 lies on the lines joining the two points (2,5),(8,2). Find the ordinatr. 3
55) if O be the origin and if the coordinates of anr two points X and Y be respectively (a,b) and (c,d) show that OX . OY cos XPY= ac+ bs.
56) Find the area of the triangles the coordinators of whose vertices are:
a) (am₁², 2am₁),(am₂², 2am₂) and (am₃², 2am₃). a²(m₁- m₂)(m₂- m₃)(m₃- m₁)
b) (a cosx₁, b sinx₁),(acosx₂, b sinx₂) at(a sinx₃, b sinx₃). 2ab sin(x₂- x₃)/2 . sin(x₃ - x₁)/2 . sin(x₁ - x₂)/2
57) three vertices of a parallelogram are (2,1); (4,-5),( 4,2). Find the area of the parallelogram and also fourth vertex. 14; (6,-4)
58) A, B, C are points (x,y),(- 3,2),(-4,-4). If the area of ∆ ABC is 35/2, show that 6x - y - 15=0.
59) The lines joining the midpoints of opposite sides of a quadrilateral bisect each other.
60) The diagonals of a parallelogram bisect each other.
61) The midpoint of the hypotenuse of a right angled triangle is equidistant from the three vertices.
62) if two medians of a triangle are equal show that the triangle is isosceles .
63) An isosceles triangle has two equal medians.
64) The sum of the squares of the distances of any point in the plane of a given rectangle to two opposite vertices equals the sum of the squares of the distances from it to the two other vertices.
65) Verify In any triangle ABC, AB²+ AC²= 2(AD²+ DC²)= 2(AD²+ BD²) where D is midpoint of BC.
66) If G be the centroid of a triangle ABC and P any other point on the same plane of the triangle, then 3(GA²+ GB²+ GC²)= (BC²+ CA²+ AB² and PA²+ PB²+ PC²= GA²+ GB²+ GC²+ 3GP²
67) The lines joining the middle points of opposite sides of a quadrilateral and the line joining the middle points of its diagonals meet in a point and bisect one other .
68) The line joining the midpoints of the two sides of a triangle of a triangle is half the third side.
69) If P be the point which divides the line segment XY internally in the ratio m: n and Q be a point not lying on the line joining X and Y, then
Area of ∆ QPX/area of ∆QPY = m/n
70) A:(3,0), B:(0,6) and C:(6,9) from a triangle ABC. A line cuts AB and AC at D and E respectively, so that D divides AB in the ratio 1:2 and E divides AC also in the same ratio. Prove that
The numerical measure of ∆ ABC/the numerical measure of ∆ ADE = 9/1.
71) Show that in two ways that three points (1,5),( 3,14) and (-1,-4) lie on a straight line.
72) A(6,3), B(-3,5), C(4,-2) and P(a,b) are four points on a plane. Prove that the numerical measure of the triangle PBC and ABC and the ratio (a+ b -2): 7.
73) The line segment joining A(b cosx, b sinx) and B (a cosy, a siny) is produced along AB to same point C(x,y) so that AC: CB = b : - a(a,b >0); prove that
y+ x cot{(x+ y)/2}= 0
LOCUS AND ITS EQUATION
Locus: The locus of a moving point is the path traced out by that point under one or more given conditions .
Equation of a Locus
A relation f(x,y)= 0 between x and y which is satisfied by each point on the locus and such that each point satisfying the equation is on the locus is called the equation of the Locus.
EXERCISE - A
1) Find the locus of a point which moves such that its distance from the point (0, 0) is twice its distance from the y-axis .
2) Find the locus of the moving point P such that 2PA = 3PB, where A (0, 0) and B is (4, -3).
3) A point moves so that the sum of the squares of its distance from two fixed points A(a,0) and B (-a,0) is constant and equal to 2c², find the locus of the point a point.
4) A point moves such that the sum of it distance from two fixed points (ae,0) and (-ae,0) is always 2a. Prove that the equation of the locus is x²/a² + y²/b²= 1 where b²= a²(1- e²)
5) Find the equation of the locus of a point which moves so that the difference of its distances from the points (3,0) and (-3,0) is 4 units.
6) The ends of the hypotenuse of a right angled triangle are (6,0) and (0,6). Find the locus of the third vertex.
7) Find the equation of the locus of a point which moves so that the sum of their distances from (3,0) and (-3, 0) is less than 9.
8) Find the locus of a point whose coordinate are given by x= t + t², y= 2t +1, where t is variable .
9) A stick of length L rests against the floor and a wall of a room. If the stick begins to slide on the floor, find the locus of its middle point.
10) Find the locus of the point of intersection of the lines x cos β + y sin β = a and x sin β- y cos β= b where β is variable.
11) A variable lines cuts x-axis at A, y-axis at B where OA= a, OB= b (O as origin) such that a²+ b²=1.
Find the locus of
a) centroid of ∆ OAB.
b) circumcenter of ∆ OAB.
12) Two points P and Q are given, R is a variable point on one side of the line PQ such that angle RPQ - Angle RQP is a positive constant 2β . Find the locus of the point R.
CHANGES OF AXES OR THE TRANSFORMATIONS OF AXES
EXERCISE - B
1) Find the equation of the curve 2x²+ y²- 3x + 5y - 8= 0, when the origin is transferred to the point (-1,2) without changing the direction of axas.
2) The equation of a curve referred to the new axes, axes retaining their direction and origin is (4,5) is x²+ y² = 36. Find the equation referred to the original axes .
3) Shift the origin to a suitable point so that the equation y² + 4y + 8x - 2 = 0 will not contain term in y and the constant.
4) At what point the origin be shifted , if the coordinates of a point (-1,8) become (-7,3) ?
EXERCISE - C
1) If the axes are turned through 45°, find the transformed form of the equation 3x²+ 3y²+ 2xy = 2.
2) Prove that if the axes be turned through π/4 the equation x²- y²= a² is transformed to the form xy = λ. Find the value of λ.
3) Through what angle should the axes be rotated so that equation 9x² - 2√3 xy + 7y²= 10 may be changed to 3x²+ 5y²= 5 ?
4) If (x, y) and (X, Y) be the coordinates of the same points referred to two sets of rectangular axes with the same origin and if ux + by, when u and v are independent of x and y become VX + UY, show that u²+ v²= U²+ V²
DOUBLE TRANSFORMATION (ORIGIN SHIFTED AND AXES ROTATED)
EXERCISE - D
1) What does the equation 2x²+ 4xy - 5y² + 20x - 22y - 14 = 0
becomes when referred to rectangular axes through the point (-2,-3), the new axes being inclined at an angle of 45° with the old ?
2) Find λ if (λ, λ +1) is an interior point of ∆ ABC where A≅ (0,3); B(-2,0) and C≅ (6,1).
OBJECTIVE - 1
1) The equation of the locus of the points equidistant from (-1,-1) and (4,2) is:
a) 3x - 5y -7=0 b) 5x +3y -9=0 c) 4x +3y +2=0 d) x - 3y +5=0
2) The equation of the locus of a point which moves so that its distance from the point (ak, 0) is k times its distance from the point (a/k, 0), (k≠ 1) is :
a) x² - y² =a² b) 2x² - y² = 2a² c) xy = a² d) x² + y² =a²
3) if the coordinates of a variable point P be (t + 1/t, t - 1/t) where t is the variable quantity, then the locus of P is :
a) xy= 8 b) 2x² - y² =8 c) x² - y² =4 d) 2x² + 3y² =5
4) If the coordinates of a variable point P be (cos θ+ sinθ, sin θ - cos θ), where θ is the parameter, then the locus of P is:
a) x² - y² =4 b) x² + y² =2 c) xy= 3 d) x² + 2y² =3
5) If a point moves such that twice its distance from the axis of x exceeds its distance from the axis of y by 2, then its locus is:
a) x - 2y= 2 b) x + 2y= 2 c) 2y - x = 2 d) 2y - 3x = 5
6) The equation 4xy - 3x²= a² become when the axes are turned through an angle tan⁻¹2 is:
a) x² + 4y² =a² b) x² -4y² =a² c) 4x² +y² =a² d) 4x² - y² =a²
7) Transform the equation x² -3xy + 11x - 12y +36 =0 to parallel axes through the point (-4,1) becomes ax² + bxy +1=0 then b²- a =
a) 1/4 b) 1/16 c) 1/64 d) 1/256
SUBJECTIVE - 1
1) find the equation of the locus of all points equidistant from the point (2,4) and the y- axis.
2) Find the equation of the locus of the point twice as far from (-a,0) as from (a,0).
3) OA and OB are two perpendicular straight lines. A straight line AB is drawn in such a manner that OA+ OB = 8. Find the locus of the mid point of AB.
4) The ends of a length l move on two mutually perpendicular lines. Find the locus of the point on the rod which divides it in the ratio 1:2.
5) The coordinanates of three O, A, B are (0,0),(0,4) and (6,0) respectively. A point P moves so that area of ∆ POA is always twice the area of ∆POB. Find the equation to both parts of the locus of P.
6) What does the equation (a- b)(x²+ y²)- 2abx =0 become, if the origin be moved to the point (ab/(a- b), 0)?
7) The equation x²+ 2xy +4=0 is transformed to the parallel axes through the point (6, λ). For what value of λ its new form passes through the new origin ?
8) Show that if the axes be turned through (15/2)°; the equation √3 x²+ (√3 -1)xy - y²= 0 become free of xy in its new form.
9) Find the angle through which the axes may be turned so that the equation Ax + By + C =0 may reduce to the form x= constant, and determine the value of this constant.
10) Transform 12x²+ 7xy - 12y²- 17x - 31y - 7 =0 to rectangular axes through the point (1,-1) inclined at an angle tan⁻¹(4/3) to the original axes.
MISCELLANEOUS - 2
1) A point moves in a plane such that its distance from (2,3) exceeds its distance from y axis by 2. Find the equation of the locus . y²-6y- 8x +9=0.
2) Find the locus of the point equidistant from x and y axis. y= ± x
3) A point moves in the xy-plane in such a way that its distance from the point (4,0) is always equal to its distance from the y-axis. Find the equation to the locus of the moving point. y² - 8x +16 =0.
4) A point moves in such a manners that the sum of the squares of the distance from it to the points (a,0) and (-a,0), is 2b². Find the locus of the point. x²+y²= b²- a²
5) Find the locus of the point which moves such that it forms a triangle of area 12 square units with (3,2) and (5,6). 2x - y =16 and 2x - y+8=0
6) Find the locus of a point whose distance from (2,4) and (1,-2) are in the ratio 2:3. 5x²+ 5y²-88y- 46x +205=0.
7) P is a variable point (t+3, 2t-1), when t may have any value. Find the locus to the locus of P. 2x - y=7
8) a point P(x, y) moves according to the law. Find the equation to the locus of following:
a) equidistant from (-4,-4) and (2,8). x+ 2y=3
b) Distance of P from (4,0) is two thirds of the distance from (9,0). x²+ y²=36
c) P is always at a distance 2 units from (-3,0). x²+ y²+ 6x +5=0.
d) distance of P from (0,5) is two -distance of its distance from the x-axis. 9x²+ 5y²- 90y +225=0
e) Distance of P from (3,2) is twice its distance from the y-axis. 3x²- y²+4y+ 6x 139=0
f) distance of P from (2,0) is equal to distance from y-axis . y²= 4(x -1)
g) distance of P from (2,3) exceeds its distance from y-axis by 2. y²-6y- 8x +9=0.
9) The sum of the squares of the distance of P from (0,a) and (0,-a) is 6a². x²+y² =2a²
10) The sum of the distances of P from (4,0) and (-4,0) is 10. 9x²+25y²=225
11) The difference of the distances of P from (2,0) and (-2,0) is 2. 3x²-y² -3=0
12) A(a,0) and B(-a,0) are two fixed points, obtain the equation to the locus of a moving point P where
a) PA²- PB²= 2k² (a constant quantity). 2ax + k²= 0
b) PA²= n. PB (n is a constant). (n²-1)(x²+ y²+ a²)+ 2ax(n²+1)=0
c) PB²+ PC²= 2PA², C being the point (e,0). (6a- 2c)x = a²- c².
13) Find the locus of the point equidistant from x and y axis. y= ± x
14) a point moves in the xy plane in such a way that its distance from (4,0) is always to its distance from the y-axis. Find the equation of the locus of the moving point. y²- 8x +16=0
15) A point moves in such a manner that the sum of the squares of the distances from it to the point (a,0) and (-a,0) is 2b². Find the locus of the point. x²+y²=b²- a²
16) it twice the abscissa of a point moving in the xy-plane always exceeds 3 times its ordinate by 1, show that the locus of the point is a straight line 2x= 3y +1.
17) A fixed point is at a perpendicular distance k from a fixed straight line and a point moves so that its distance from the fixed point is always equal to its distance from the fixed line. Prove that the equation to its locus is x²+ 2ky = k². The axes are to be chosen suitably .
18) Two points O(0,0) and A(3,4) are given. Find the equation the locus of B, if the area of ∆ OAB, the vertices being taken in this order, is 7 square units. 4x - 3y +14=0
19) Find the locus of a point which forms a triangle of area 21 square units with (2,-7) and (-4,3).
20) P is a variable point (t+3, 2t -1), where t may have any value. Find the equation to the locus of P. 2x - y -7=0
21) P is a variable point (at², 2at), where t may have any value, show that equation to the locus of P is y²= 4ax.
22) P is a variable point (a cosθ, b sinθ), where θ any assume any value, show that the equation to the locus of P is x²/a²+ y²/b²= 1.
23) A and B are two points having coordinanates (-5,3) and (2,4) respectively. Find the locus of a point P such that PA: PB = 3:2.
24) Find the locus of a point whose distance from (3,4) and (1,-2) are in ratio 2:3. 5x²+ 5y²- 76x - 48y +46=0
REMOVAL OF THE TERM xy FROM f(x,y)= ax²+ 2hxy + by² WITHOUT CHANGING THE ORIGIN
EXERCISE - E
1) Given the equation 4x²+ 2√3 xy + 2y²= 1, through what angle should the axes be rotated so that the term in xy be wanting from the transformed equation.
1) The four points A(α,0), B(β,0), C(γ,0) and D(δ,0) are such that α , β are the roots of the equation ax²+ 2hx + b=0 and γ, δ are those of the equation a'x²+ 2h'x + b'= 0. Show that the sum of the ratios in which C and D divide AB is zero, if ab' + a'b = 2hh'.
2) The circumcenter of a triangle with vertices (a, a tanα), B(b,b tanβ) and C(c, c tanγ) lies at the origin, where 0< α, β, γ < π/2 and α + β + γ =π. Show that its orthocentre lies on the line
4 cos(α/2) cos(β/2) cos(γ/2) x - 4 sin(α/2) sin(β/2) sin(γ/2) y = y
3) if m₁ and m₂ are the roots of the equation
x²+ (√3 +2)x + (√3-1)=0
Show that the area of the triangle formed by the lines y= m₁x, y= m₂x and y= c is c²(√33+√11)/4.
4) if x coordinanates of two points B and C are the roots of equation x²+ 4x +3= 0 and their y-cordinates are the roots of equation x²- x - 6 =0. If x co-ordinate of B is less than x coordinanates of C and y coordinanate of B is greater than the y coordinanate of C and coordinanates of a third point A be (3,-5), find the length of the bisector of the interior angle at A.
5) A line L intersects three sides BC, CA and AB of a triangle in P,Q,R respectively, show that
BP/PC . CQ/QA . AR/RB = -1.
6) The distance between two parallel lines is unity. A point P lies between the lines at a distance a from one of them. Find the length of a side of an equilateral triangle PQR, vertex Q of which lies on one of the parallel lines and vertex R lies on the other line.
7) In a ∆ ABC, A= (α , β), B(1,2), C=(2,3) and point A lies on the line y= 2x +3 where α , β ∈ I, if the area of ∆ ABC be such that [∆] = 2, where [ . ] denote the greatest integer function, find all possible coordinanates of A.
8) Let S be the square of unit area. Consider any quadrilateral which has one vertex on each side of S. if a, b, c and d denote the lengths of the sides of the quadrilateral, prove that : 2≤ a²+ b²+ c²+ d² ≤ 4.
9) If the points, {a³/(a-1), (a²-3)/(a-1)}, {b³/(b -1), (b²-3)/(b-1)} and {c³/(c -1), (c²-3)/(c -1)} are collinear for three distinct values a, b, c, and a≠ 1, b≠ 1 and c≠ 1, then show that
abc - (bc + ca+ ab) +3(a+ b + c)=0.
10) If A₁, A₂, A₃,.......Aₙ are n points in a plane whose coordinanates are (x₁, y₁), (x₂, y₂), (x₃,y₃), .......(xₙ, yₙ) respectively. A₁A₂ is bisected in the point G₁ ; G₁A₃ is divided at G₂ in the ratio 1:2; G₂A₄ is divided at G₃ in the ratio 1:3; G₃A₅ at G₄ in the ratio 1:4 and so on until all the points are exhausted. Show that the coordinanates of the final point so obtained are
(x₁+ x₂ +.....xₙ)/n and (y₁+ y₂+.....yₙ)/n
11) If by change of axes without change of origin, the expression ax²+ 2hxy + by² becomes a₁x₁²+ 2hx₁y₁ + b₁y₁² prove that
a) a+ b = a₁ + b₁.
b) ab - h²= a₁b₁ - h₁²
c) (a - b)²+ 4h² = (a₁ - b₁)² + 4h₁².
SLOPE OF A LINE
EXERCISE - 1
1) Verify that the points A(-1,3) B(0,5) And C(3,1) are the vertices of a right angled triangle.
2) A moving point A(x,y) remains always equidistant from A(-1,0) and B(0,-2). Express this fact by an algebraic relation in x and y. 2x - 4y -3=0
3) Verify by using the concept of slope, that the three points (5,7), (-3,1) and (-7,-2) are collinear .
4) A line L₁passes through the two points (3,4) and (2,1). If there is another line L₂ such that angle from L₁ to L₂ is 45°, show that the slope of L₂ is -2.
5) Find the interior angles of the triangle with vertices are (1,1), (5,2) and (4,3). 5/14, -5
6) Find the interior angles of a triangle with vertices (-3,-2), (2,5) and (4,2).
7) Find the slopes of the line joining the points:
a) (2,-5) and (5,4). 3
b) (-2,-7) and (-3,-1). -6
c) (2/3,5/2) at(-1/2, -1/3). 17/7
d) (-5/4, 4/3) and (3/4, 3/5). 11/30
8) Show that the three points in each of the following
a) (1,4), (3,-2) and (-3, 16)
b) (0 ,-2), (2,4) and (-1,-5)
are collinear
9) Find the angle between the straight lines y= mx + c and y₁= m₁x + c and hence deduce the condition of perpendicularity and parallelism of the lines.
10) Verufy the following statements :
a) The points (-1,1/2),(0,-5/2 and (5,5/2) are the vertices of a right triangle .
b) The four points (-4,0),(6,4),(5,0) and (0,-2) are the vertices of a trapezium.
11) The points (-3,3), (-1,-1) and (3,-3) are the vertices of an isosceles triangle.
12) The points (-5,-1), (0,-7),(18,9),(13, 15) are the vertices of a parallelogram.
13) The circle having the points (7,2) and (-3,2) as ends of a diameter also passes through (-1,6).
14) The perpendicular bisector of the line joining (-3,1) and (13, 3) passes through (7, -14).
15) The points (-4,0), (6,3) and (36,12) are collo.
16) Show that the quadrilateral with vertices (10,10),(-14, -2), (-10,- 10) and (4, -24) can be divided into two right triangles.
Prove that its area is 400 square units.
17) The point (x,y) is equidistant from (5,-2) and (-3,4 0). 4x - 3y -1=0
18) The point (x, y) lies on the circumference of a circle with the segment directed from (-3,6) to (2,5) as diameter. x²+ y²+ x - 11y +24 =0
19) A line L₁ passes through (-5,-3) and (2,6). Another line L₂ passes through (6,4) and (8,2). If θ be the angle from L₁ to L₂ , prove that tanθ = 8.
20) L₁ passes through (1,9) and (2,6); L₂ passes through (3,3) and (-1,5); to prove θ = π/4.
21) Find the interior angles of a triangle with vertices
a) (-10,-8),(11,6) and (37,23)
b) (7,12, (2,5) and (-5,-5). -1/105
22) A, B, C, D are the points (-2,0),(4,3),(1,-1) and (5,1); prove that AB|| CD.
23) A point P(x, y) lies on a line which passes through the point (2,3) and which is perpendicular to the line joining the points (-1,2) and (-5,4); show that 2x - y -1= 0.
THE STRAIGHT LINE:
ITS EQUATION IN DIFFERENT FORMS
EXERCISE - A
1) Find the equation of the line line passing through the point (2,-3) and perpendicular to the line joining the point (4,1) and (2,4). 3y - 2x +13=0
2) Find the equation of a straight line which passes through the (4,-8) and is inclined at an angle 135° to the positive x-axis. x+ y+ 4=0
3) Write down the equation of the straight lines:
a) inclined at an angle 60° to the positive x-axis and passing through (5,-2). y- √3 x + 2 + 5√3= 0
b) included at an angle whose trigonometric tangent is 3/4, and passing through (2,6). 4y - 3x = 18
c) of slope -4/5 and passing through the point (-2,4). 5y+ 4x -12=0
d) of slope m and passing through (0, a√(1+ m²)). y= mx + a √(1+ m²).
4) Find the equation of the line through (3,-6) perpendicular to the line joining (4,1) and (2,5). x - 2y = 15
5) Write the equation of a line through (3,-4) parallel to the line through (0,-5) and (4,-3). 2y - x +10= 10
6) Find the equation of the following lines:
a) passing through (0,0) and making an angle of 60° with the positive direction of axis of x. y= √3 x
b) Passing through (-4,0) and making an angle of 45° with the positive direction of axis of y. y - x - 4=0.
EXERCISE - B
1) Find the equation of a line passing through two points (3,4) and (8,-15). 5y+ 19x =77
2) What does the equation y= mx + c represent in each of the following case ?
a) when m= 0 and c is an arbitrary constant.
b) when c(≠0) is a fixed constant but m is an arbitrary constant.
c) when c=0 and m is an arbitrary constant
d) when m is a fixed constant (≠0) and c is also a fixed constant.
e) when both m and c are arbitrary constants.
3) Write down the equations of the lines joining the following points:
a) (0,0) and (5,-3). 3x - 5y= 0
b) (5,-3) and (5,2). x = 0
c) (-5,2) and (3,2). y= 0
d) (-4,1) and (3,-5). 6x +7y+ 17= 0
e) (a/m², 2a/m) and (a/4m², a/m). 3y = 4mx + 2a/m
f) (a,b) and (b, a). x + y= a+ b
g) (ct, c/t) and (2ct, c/2t). x + 2t²y= 3at
4) Find the equation of the sides of the triangle (produced (, formed with vertices (-5,6),(-1,-4) and (3,2). Derive the equations of the three medians. 5x + 2y +13= 0, 2y -3x +5= 0, x + 2y -7= 0, -x + 6y -9= 0, 7x + 6y -1 = 0
5) Show that the equation
x y 1
x₁ y₁ 1 = 0
x₂ y₂ 1
can be expressed as an equation of first degree in x and y.
EXERCISE - C
1) Find the equation to the line passing through (3,-4) and cutting of intercepts, equal but opposite signs , from the two axes. x - y =7
2) Find the equation to the line passing through (-5,4) and is such that the portion of it between two axes is divided by the point in the ratio 1:2. 5y - 2x = 30
3) a straight line passes through the point (4,3) and makes on the axes intercepts which are equal in magnitude and also in sign. Find the equation of the line and also the Intercepts on the axes. x + y=7
EXERCISE - D
1) Construct each of the following lines where
a) p= 5, α= 30°.
b) p= 4, α= 240°.
c) p= 5, α= 314°.
d) p= 6, α= 120°.
Write down the equation of each of these lines in the normal form.
EXERCISE - E
1. Reduce each of the equations to the normal form find and find a and p:
a) √3x + y=9. 30°, 9/2
b) x+ y+ 8= 0. 225°, 4√2
c) 4y -7=0. 90°, 7/4
d) x +5=0. 180°, 5
2) Determine the intercepts the following lines on each of the co-ordinate axes, wherever they exist and draw the lines:
a) 2x + 3y -12=0.
b) x - y + 1=0
c) 5x + 7y +13=0
d) 2x - 3y =0
e) 2x + 3=0
f) x= 0.
3) The diagonals of a square lie along the coordinate axes, and each has length 2 units. Find the equations of four sides (produced). x + y = ±1 and x - y= ±
4) find the equation to the diagonals of the rectangle the equation whose sides are x= a, x= A' , y= b, y= b'. y(a'- a)- x(b' - b)= a'b - ab'; y(a'- a)+ x(b' - b)= a'b - ab';
5) find the equation to the straight line which go through the origin and trisect the portion of the 3x+ y=12 which is intercepted by the axes. y= 6x, 2y= 3x,
6) find the equation to the line which bisects the distance between the points (a,b) and (A', b') and bisects the distance between the points (-a, b) and (A', -b'). 2ay - 2b'x = ab - a'b
EXERCISE - F
1) In the triangle with vertices (0,6),(-2,-2),(4,2), find
a) the equations of the medians and their point of intersection. y+ 6x -6=0, 2y -3x -2=0, y=2. (2/3,2)
b) the equations of their altitude and their point of intersection. 2y+3x -12=0, 4y+ x -12=0,
c) the equation of the perpendicular bisectors of the side. 2y+ 3x -3=0
Verify that the 3 points of intersection so found lie on a straight line.
2) In the triangle with vertices (2,0),(3, 2),(4,-3) find
a) the equations of the medians and their point of intersection
b) the equations of their altitude and their point of intersection
c) the equation of the perpendicular bisectors of the sides and their point of intersection .
Verify that the 3 points so found lie on straight line.
EXERCISE - G
1) Find the equation of a straight line, which passes through the point (5,-6) and is
a) parallel to the line 8x + 7y +5= 0. 8x + 7y +2= 0
b) perpendicular to the line 8x + 7y +5= 0. 7x -8y -83 = 0
2) Find the angle between the lines
a) x - y√3= 5 and √3x + y= 7. 90°
b) x - 4y -3= 0 and 6x -y = 11. tan⁻¹(23/10)
c) y= 3x +7 and 3y - x = 8. tan⁻¹(4/3)
d) y= (2- √3)x + 5 and y= (2+ √3)x -7. 60°
3) Find the trigonometrical tangent of the angle between the lines whose intercepts on the axes are respectively a, - b and b, -a. tan⁻¹{(a²- b²)/2ab}
4) Prove that 4 points (2,1), (0,2),(2,3) and (4,0) form a parallelograms and that the angle between its diagonals is. tan⁻¹(2).
5) Find the equation to the straight line:
a) passing through (4,-5) and parallel to the line passing through 3x + 4y +5=0. 3x + 4y +8=0
b) passing through (4,-5) and perpendicular to the line 3x + 4y +5=0. 4x - 3y = 31
c) passing through (2,-3) and perpendicular the line joining the points (5,7) and (-6,3). 4y + 11x = 10
6) Find the equation to the straight line drawn at right angles to the straight line x/a - y/b =1 (a, b > 0) through the point where it meet the axis of x. ax + by = a²
7) Find the equation to the straight line which bisects , and is perpendicular to, the straight line joining the point (a,b) and (c,d). 2x(a- c)+ 2y(b - d)= a²- c²+ b²- d².
8) Find the equation of a line passing through (x₁, y₁) and Perpendicular to the lines respectively
a) yy₁ = 2a(x + x₁). 2ay + xy₁ = 2ay+ x₁y₁
b) xx₁ + yy₁ = a². x₁y - xy₁= 0
c) xx₁/a² + yy₁/b² = 1. a²xy₁ - b²x₁y = (a²- b²)x₁y₁
d) x₁y + xy₁ = a². xx₁ - yy₁ = x₁²- y₁²
9) Prove that the equation of a line which passes through (a cos³ θ, a sin³ θ) and is perpendicular to the line x secθ + y cosecθ = a is x cosθ - y sinθ = a cos2θ.
10) Find the equation of the line perpendicular to 2x - y -4=0 and cutting from the first quadrant a triangle whose area is 16 square units. x + 2y -8=0
11) A line is parallel to the line 3x + 2y -6=0, and forms a triangle in the first quadrant with the lines.
x - 2y=0, and 2x -y= 0, whose area is 21 square units. Find the equation of the line. 3x + 2y -28= 0
EXERCISE - H
1) Find the equations to the straight lines which pass through (2,3) and make acute angle 45° with the line 3x - y +5=0. 2y - x -4=0, y+ 2x -7=0
2) Prove that the equation to the straight lines passes through (3,-2) and inclined at 60° to the line √3x + y =1 are y+2= 0 and y - √3 x + 2 + 3√3=0.
3) Prove that the equations to the lines which pass through a given point (x', y') and makes acute angle α with the given line y= mx + c, are
y- y'= (m± tanα)(x - X')/(1± tanα).
4) Find angle measured from the straight line 3x= 4y + 7 to the line 5y = 12x +6 and also the equations to the straight lines which pass through the point (4,5) and make equal angles with the two given lines. tan⁻¹(33/56); 9x - 7y = 1, 7x + 9y = 73.
EXERCISE - I
1) Prove that the origin and (2,3) are on the opposite sides of the line 7x - 24y +8=0.
2) Show that the points P:(3,2), Q(3,-1) lie within adjacent angles formed by the lines x - 2y +2= 0 and x + y +1=0.
3) Show that the four points (0,0),(-1,1),(-7,4) and (9,6) are in the four different compartments made by the two straight lines 2x - 3y +1=0, and 3x - 5y +2=0.
4) Show that the points (3,2) and (7,3) lie on the opposite sides of the line: 2x - 5y +3=0.
5) Show that the point (-2,6) lies on the negative side of the line 3x + 2y -7= 0.
6) Show that the origin is within the triangle the equations of whose sides are x - 7y +25= 0, 5x + 3y +11= 0 and 3x - 2y -1= 0.
Perpendicular distance of a point from a line
EXERCISE - J
1) Find the distance of the given point from the given line:
a) (5,2) ; 3x - 4y +6=0. 13/5
b) (6,-1) ; 3x - y +1=0. 2√10
c) (3,4) ; 2x +5 =0. 11/2
d) (-2,-5) ; y =0. 5
e) origin; 3x +2y -6=0. 6√13/13
2) Find the lengths of the altitude of the triangle with vertices in the points (2,0),(3,5),(-1,2). 17√26/26,17/5,5√13/13
EXERCISE - K
1) Find the equation to the bisectors of the angles between the straight line 3x -4y -2=0 and 5x +12y +6=0. 16x +2y+1=0; 8y- x +4=0
2) Find the centre of the circle inscribed in the triangle whose angular points A, B, C are rspectively (1,2,(25,8),(9,21). (23/2,11)
3) Find the equation to the bisectors of the angles between the lines
a) 13x -9y =10 and 11x+ 2y =6. 2x -6y+5=0;9x +3y=20.
b) 2x +y =10 and x+ 3y =2. 21x +7y+3=0; x -3y=7.
c) y =3x and x+ y =1. (3±√5)x -(1±√5)y= ±√5
d) 3x -4y =-7 and 12x-5y =8. 99x -77y+51=0; 21x +27y=131.
e) y- b = 2m(x - a)/(1- m²) and y- b= 2m'(x -a)/(1- m'²).
4) Find the equations to the bisectors of the internal angles of the triangle the equations of the sides are 3x +5y =15 , x+ y =4 and 2x +y+5=6.
5) If the equation of sides 7x -y +11=0; x+ y =15; 7x +17y+ 65=0. (5,1)
PARAMETRIC EQUATION OF A STRAIGHT LINE
EXERCISE - L
1) A line of slope 3/4 passing through a point P(-2,-5). Find the coordinates of any point Q(x,y) on the line, given PQ= 10 units. (6,1) Or (-10,-11)
2) Find the direction in which a straight line must be drawn through the point (1,2) so that its point of intersection with the line x+ y =4 may be at a distance √6/3 from this point. 15° or 75°
3) A is the point (-5,-3) and B lies on the line x - 3y -1=0. AB is inclined to the x-axis at an acute angle whose tangent is 5/12. Find the length of AB. 13 units
4) A line of love 5/12 passes through A(3,-7/2), Find the coordinates of a point B on the line where AB = 13/2. (9,-1) Or (-3,-6)
5) Find the points on the following lines at which the x and the y co-ordinates are equal, and hence write each equation in the form (x - x₁)/cosθ = (y - y₁)/sinθ = r
a) 4x - 3y -1=0. (x-1)/3/5 - (y-1)/4/5= r
b) 5x +12y +17=0. (x+1)/12/13 = (y+1)/-5/13= r
c) x - √3y =0. x/√3/2 = y/1/2= r
EXERCISE - M
1) Find the equation to the straight line passing through the point of intersection of the lines x + 2y + 3 = 0 and 3x + 4y + 7=0 and perpendicular to the straight line x - y + 8=0. x+ y+2=0.
2) Show that the lines x - 4y+2=0, 4x- y+3 =0, x+ 2y=0 meets at a point.
MISCELLANEOUS - A
1) Find the equation of a straight line passing through (1, 2) and inclined an angle 135° with the positive direction of the x-axis. x+ y -3=0.
2) Find the equation of the straight line passing through the points of intersection of the curves (y+1)²+ 4x =0 and x²- 4(y+1)= 0. x+ y+1=0.
3) Find the perpendicular distance from the origin of the straight line 3x+ 4y= 5√2; also find the angle that this perpendicular makes with the positive direction of the x-axis . tan⁻¹(4/3)
4) The equation of a straight line is 5x+ 2y =0; this equation can be expressed in the intercept from --- discuss the validity or otherwise of the statement.
5) Obtain the equation to the lines which pass through the origin and trisect the segment of the line 3x+ y -12=0 intercepted between the axes. y= 6x
6) Find the area of the triangle formed by the straight line 2x+ 3y -12=0 with the x-axis of coordinates. 12
7) if a + b + c = 0, show that the three lines ax+ by + c =0, bx + c y + a=0, and cx + ay + b= 0 are concurrent.
8) Find the equation of the straight line which passes through the point of intersection of the line 2x+ 3y -5=0 and 3x+ 5y -7=0 and makes equal positive intercs upon the coordinate axes. x+ y=3
9) Find the distance between the parallel lines y= mx + c₁ and y= mx + c₂.
10) Find the distance measured along the line 4x -3y+2=0 from the point (1,2) to the line x- 2y-2=0. 5
11) Find the locus of the middle point of the portion of the line-segment made by the straight line x cosα + y sinα =4 and the axes of coordinates. 4(x²+ y²)= x²y²
12) Find the distance of the point (3,5) from the line 2x+ 3y-14=0 measured parallel to the line x -2y-1=0. √5
13) A variable line drawn through the point of intersection of the lines x/a + y/b =1 and x/a + y/a = 1 meets the coordinate axes at P and Q. Show the locus of the midpoint of PQ is the curve 2xy(a+ b)= ab(x + y).
14) Show that the diagonals of the parallelogram formed by the lines √3x+ y=0, √3x+ y -1=0, √3y+ x=0, √3 y+ x=1 are at right angles.
MISCELLANEOUS - B
1) What does the equation y= mx + c represent in each of the following cases:
a) When m= 0 and c is an arbitrary constant;
b) When c≠ 0 is a fixed constant but m is an arbitrary constant.
c) When c= 0 and m is an arbitrary constant.
d) When m≠ 0 is a fixed constant and c is also a fixed constant .
e) When both m and c are arbitrary constants.
2) Find the angle which the straight line perpendicular to the line √3x + y=1 makes with the positive direction of the x-ai.
3) A straight line makes Intercepts h and k upon the co-ordinate axes; find its equation. What is the length of the perpendicular from the origin upon the line ?
4) Is it possible to express the equation to a line parallel to a coordinate axis in terms of intercept form x/a + y/b =1?
5) State with reasons whether the equation of
a) straight lines through the origin.
b) straight line parallel to the x-axis can be expressed in the form of x/a + y/b= 1
6) The equation of a straight line is 5x+ 2y=0, this equation can be expressed in the intercept from -- discuss the validity or otherwise of the statement .
7) Prove that a linear equation of the form ax+ by+ c =0 always represents a straight line on a plane.
8) What does ax+ by+c=0 represent, when
a) a≠0, b≠ 0, c= 0
b) a= 0, b≠ 0, c≠ ó
c) a= 0, b≠ 0, c= ó
d) a ≠0, b= 0, c≠ ó
e) a≠ 0, b= 0, c= ó
f) a≠ 0, b≠ 0, c≠ ó
9) Find the co-ordinates of the point of intersection of the lines ax+ by+ c =0 and a'x+ b'y+ c' =0.
10) Represent any line passing through the point of intersection 2 lines ax+ by+ c =0 and a'x+ b'y+ c' =0 in the form of an equation.
11) The perpendicular distance of a straight line from the origin is P and the perpendicular make an angle α with the positive direction of the x-axis. Find the equation of the straight line.
12) Find the perpendicular distance of a point from a line.
13) Reducing the straight line 3x+ 4y+ 15 =0 to its normal form, find the perpendicular distance of the line from the origin.
14) Find the angle between the straight line y= m₁x + c₁ and y= mx₂ + c₂ and hence deduce the condition of perpendicularity and parallelism of the lines. Do the condition change when lines become y= m₁ and y= m₂x ? find also the conditions when lines are ax+ by+ c =0 and a'x+ b'y+ c' =0.
15) Find the equation of the bisectors of the angles between two lines ax+ by+ c =0 and a'x+ b'y+ c' =0.
16) Find the condition that the three lines a₁x + b₁y + c₁=0, a₂x + b₂y + c₂ =0 and a₃x + b₃y + c₃ =0be concurrent.
1) Two equal arcs of two circles subtends angle of 60° and 75° at the centre. Find the ratio of the radii of the two circles.
3) If 7 cosθ + 5sinθ= 5, find 5cosθ - 7sinθ.
4) If secθ+ tanθ= x, show that sinθ = (x²-1)/(x²+1).
5) If sinθ + cosecθ= 2 show that sinⁿθ + cosecⁿθ= .
6) If tan⁴θ+ tan²θ= 1, show that cos⁴θ+ cos²θ= 1.
7) If If cis⁴θ+ cos²θ= 1, show that tan⁴θ+ tan²θ= 1.
8) If sinα, cosα, tanα are in GP show that cot⁶α- cot²α= 1.
9) If (secx -1)(secy -1) secx -1)= (secx +1)(secy +1)(secx+1) show the value of each side ± tanx tan y tan z.
10) If tanθ+ sinθ= m and tanθ - sinθ= n show that m²- n²= 4√(mn).
11) If x=cosecα - sinα and y= secα - cosα, show that x²y²(x²+ y²+3)= 1.
12) If cosecα + cosec β + cosecγ = 0, then show (sinα sinβ sinγ)²= sin²α +sin² β + sin²γ.
13) Find the least value of 9 tan²θ + 4 co5²θ.
14) If θ lies in the 2nd quadrant and tanθ = - 5/2 find the value of 2 cosθ/(1- sinθ).
15) If sinθ = 8/15 and sinθ is negative find {sin(θ)+ cos(-θ)}{sec(-θ)+ tan(-θ)}.
16) If α= π/19, show that (sin23 - sin3α)/(sin16α + sin4α)= +1.
17) Evaluate {cot570°+ sin(-330°)/tan(-210°) + cosecx(-750°).
18) If n be any integer; show sin{nπ+(-1)ⁿ π/4}= 1/√2.
19) If α and β are positive acute angls and cos α = 1/√10 and sinβ= 1/√2 find α - β.
20) If x,y are positive acute angles and cosecx =√5, secy = √10/3 find cosecx(x -y).
