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).
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.
TRIGONOMETRY (MIXED)
R-1
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.
2) If cosθ - sinθ= √2 sinθ show sinθ + cosθ= √2 cosθ
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).
21) If tan(α +β) + tan(α -β)=4 and tan(α -β) tan(α +β)= 1 find α, β.
R-2
1) Show that sin(α +β)/ sin(α -β)= (tanα tanβ)/(1+ tanα tanβ).
2) Show that cosecx(x +y)= (cosecx cosecx y)/(cotx + cot y).
3) Show that tan40+ tan20=√3(tan45- cot 50 cot70).
4) If tanA + tanB= x and cotA + cotB= y show that cot(A+ B)= 1/x + 1/y.
5) Find the maximum and minimum values of cosθ + √3 sinθ .
6) If tanα = n/(n +1) and tanβ = 1/(2n +1) show that
a) (α +β)= π/4
b) (α +2β)= 1+ 1/n.
7) If tan β)= sin2α/(9+ cos2α) show that 5 tan(α -β)= 4 tanα.
8) If tan²β = tan(α +θ) tan(α - θ ) show that tan² θ = tan(α+ β) tan(α- β).
9) cos²(α- β) + cos²β - 2 cos(α- β) cosα cosβ = sin²α .
10) If cosα + cos(α+ β) + cos(α+ β+γ ) =0 and sinα + sin(α+ β) + sin(α+ β+γ ) =0 show that β = γ =2π/3.
11) If a sin( θ + α)= b sin (θ+ β) show that tanθ = (b sinβ - a cosα)/(a cosα - b cosβ).
12) Simplify: cot(β - γ) cot(γ-α)+ cot(γ -α ) cot(σ -β) + cot(α - β) cot(β- γ).
13) if cos(α -β)/( cos(α +β) + cos(γ+ δ)/cos(γ- δ)= 0 show that tanα tanβ tanγ tanδ= -1.
14) If 9x =π find cosx cos2x cos3x cos4x.
15) Show 4 cos θ cos(60- θ) cos(60+ θ)= cos3θ.
16) Show that sinπ/12 sin3π/12sin5π/12 sin9π/12sin11π/12 = 1/32.
17) Show tan5 tan 55tan65 tan 75= 1
18) 4 sin40 - tan 40= √3.
19) (sin9θ cosθ - cos5θ sin3θ)/(sin16θ cos6θ + cos12θ cos10θ) = tan6θ.
20) If cost = m cosx show that tan{(x - y)/2}= (m -1)/(m +1). cot{(x +y)/2}.
21) If sinθ = n sin(2α- θ) show tan(θ- α) = (n -1)/(n +1) Tan α.
R-3
I) If (1+ m) sin(θ + α)= (1- m) cos(θ - α) show that tan(π/4- θ)= m cot(π/4 - α).
2) If sin(3α+ θ)= 7 sin(α- θ) show tan θ =sinα(1+ sin²α)/{cosα(1+ cos²α).
3) If tan(α -β)= sin2β/{(2n +1) - cos2β)}, show that tanα / tanβ = 1+ 1/n.
4) If cosx + cost + cosx = 0 and sinx + siny + sinx = 0 show that cos{(x +y)/2}= ± 1/2.
5) If cosα + cosβ = -27/65, sinnα + sinβ = 21/65 and π< α -β<3π. Find the values of sin(α +β/2) and cos(α +β/2).
6) If (cosα )/a = (cos(α +β)/b = cos(α +2β)/c = cos(α +3β)/d, then show b(b + d)= c(c + a).
7) Prove: sec2α + tan2α= tan(π/4+ α).
8) Show that cosec50+ √3 sec50= 4.
9) Show: (3- 4 cos2α + cos4α)/(3+ 4 cos2α + cos4α)= tan⁴α.
10) Show that cos(π/7) cos(2π/7)cos(4π/7)= -1/8.
11) If 13 θ= π show that cosθ cos2θ cos3θ cos4θ cos5θ cos6θ = 1/2⁶.
12) If. Tan3A/tanA = k show that sin3A/sinA = 2k/(k -1) and the value of k does not lie between 1/3 and 3.
13) 16 cos⁵θ= cos5θ+ 5 cos3θ+ 10 cosθ.
14) 16 sin⁵θ= sin5θ - 5 sin3θ + 10 sinθ.
15) sin²x cos⁴x = (1/32)(2+ cos2x - 2 cos4x - cos6x).
16) If α and β are positive acute angles and cos2α = (n cos2β -1)/(n - cos2β). Show that √(n -1) tanα = √(n +1) tanβ. (n> 1).
17) If n tanα = (n +1) tanβ, show that tan(α -β)= sin2β/{(2n +1)- cos2β}
18) Show (2cosθ -1)(2 cos2θ -1)(2 cos 2²θ -1) ....(2 cosⁿ⁻¹θ -1)= (2 cos2ⁿθ +1)/(2 cosθ +1).
19) If cos2β = cos(α +γ) sec(α -γ), show that tanα, tanβ, tanγ are in GP.
20) If α and β are two different values of θ lying between 0 and 2π which satisfy the equation 6 cosθ + 8 sinθ = 9, find the value of sin(α + β).
21) Show that value of cot3x/cotx does not lie between 1/3 and 3.
R- 4
1) Show that 2 cosec4θ = sec2θ = (1- tanθ)/(1+ tanθ) cosec2θ.
2) Show that √{(1+ sinθ)/(1- sinθ) = tan(π/4+θ). (0< θ<π/2).
3) Show that cos315/2= (-1/2) √{2+√2}.
4) Show that (sin²24 - sin²6)(sin²42- sin²12)= 116.
5) Show that sin45/4= √{2 - √2+ √2}.
6) Simplify sin(144- x) - sin(144+ x)+ sin(72- x) - sin(72+ x).
7) Show sin(β -γ)+ sin(γ -α) + sin(α -β)+ 4 sin{(β- γ)/2} sin{(γ-α)/2} sin{(α-β)/2} =0.
8) If 270< A< 360 show that 2 sin(A/2)= √(1- sinA) - √(1+ sinA).
9) If α and β are two roots of the equation a cosθ + b sin θ= c show that tan{(α+ β)2}= b/a.
10) Show that sin5 - sin67 + sin 77 - sin139+ sin149= 0.
11) Evaluate cot(15/2)= tan(15/2) - tan(75/2)+ cot(75/2)
12) If cosθ = (cos u - e)/(1- e cosu) show tan(u/2) = √{(1- e)/(1+ e)} tan(θ/2).
13) If A+ B+ C=π. Show that
a) sin²A + sin²B - sin²C = 2 sinA sinB cosC.
b) cotB cotC + cotC cotA + cotA cotB = 1
c) (cotB + cotC)/(tanB+ tanC) + (cotC + cotB)/(tanC + tanA) + cotA+ cotB)/(tanA + tanB)= 1.
d) cosA/(sinB sinC + cosB/sinC sinA + cosC/sinA. SinB = 2
e) cos(A/2) + cos(B/2) + cos(C/2)= 4 cos{(π-A)/4} cos{(π-B)/4} cos{(π- C)/4}.
14) If α +β -γ)=π, show that sin²α + sin²β - sin²γ = 2 sinα sin β cosγ.
15) If A,B,C are the angle of ∆ then show cos²(A/2) - sin²(B/2) - sin²(C/2)= 2 sin(A/2) sin(B/2) sin(C/2).
16) If A+ B+ C= π/2 and cotA, cotB, cotC are in AP show that cotA cotC = 3.
R- 5
Solve:
1) cosmx + cosnx = sin mx + sin mx, m ≠ n.
2) sin(3θ - 30) = cos(2θ+ 10)
3) sin7θ + sin4θ + sinθ = 0, 0≤θ≤π/2.
4) cos3θ + cos2θ= sin(3θ/2) + sin(θ/2), 0≤θ< 2π.
5) sin5x - sin3x - sinx = 0, 0< x <360
6) cot(θ/2) + cosecx(θ/2)= cotθ.
7) 8 cosx cos2x cos4x = (sin6x)/sinx.
8) cotx - 2 sin2x= 1
9) 4 sin2θ cosθ = cpsecθ, 0≤θ≤π.
10) sinx + cosx = 1+ sinx cosx.
11) 3 sinx + 4 cosx = 5.
12) tan²θ + sec2θ= 1.
13) eᶜᵒˢˣ⁺ ˢᶦⁿˣ⁻¹ = 1
14) cosx - sinx = cosθ + sinθ
15) tanx + secx = 2 cosx.
16) tan3θ= tanθ tan(x - θ) tan(x + θ).
17) 2 tan2x + tan3x = tan5x.
18) tan(π/4+ θ) - tan(π/4 - θ)= 2 tanθ tan(π/4 - θ) tan(π/4+ θ)
19) cosx + cos y = 1 and cosx cost= 1/4.
20) If sec ax + sec bx = 0, show that the values of x form two AP.
21) If sinx + sin y =√3(cos y - cosx) show that sin3x + sin3y= 0.
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) Verify 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.
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.
COORDINATES AND LOCUS
1) The polar coordinates of the point whose Cartesian coordinates are P(2, -2), are
a) (2√2,π/4) b) (2√2,3π/4) c) (2√2, -3π/4) d) (2√2, -π/4)
2) Rhe polar or coordinates of the point whose Cartesian coordinates are (-√3,1), are
a) (2, 2π/3) b) (2, 5π/6) c) (2, -5π/6) d) (2, -2π/3)
3) The Cartesian coordinates of the point whose polar coordinates are (√3, -3π/4)
a) (√6/2, -√6/2) b) (- √6/2, √6/2) c) (-√6/2, -√6/2) d) none
4) The polar coordinates of the centroid of the Triangle formed by the points A(3,2), B( -6,-3) c) (0,-2) are
a) (√2,π/4) b) (-√2, -π/4) c) (√2, 3π/4) d) (√2, -3π/4)
5) The cartesian coordinates of A and the polar polar coordinates of B are respectively A(2,3) and B(2,60°). The coordinates of the point at which AB is divided internally in the ratio 2:1 are
a) (4/3, √3(2+√3)/3)
b) (5/3, √3(2+√3)/3)
c) (4/3, √3)
d) (5/3, √3)
6) If the coordinates of the centroid of the Triangle formed by the points A(2x, 3x), B(y, 2y), C(-1,-3) are (2,3), then the coordinates of A are
a) (6,6) b) (9,4) c) (4,9) d) none
7) The points (0,-2), (2,4), (-1,-5) form
a) an isosceles triangle b) a right angle triangle c) an equilateral triangle d) none
8) The point (1,2) divides the line segment AB joining the points A (3,2) and B(-2,2) internally in the ratio k: 1. Then k is equals to
a) 2/3 b) 3/2 c) 1/2 d) 2
9) The x-axis divides the line segment AB, where A≡ (2,-3) B≡(5,6), in the ratio
a) 2:3 b) 1:2 c) 2:1 d) 3:2
10) The coordinates of two points A and B are (3,-3) and (-5,7) respective. The line y= x divides AB in the ratio
a) 2:1 b) 1:2 c) 2:3 d) 3:2
11) A(1,6)B(3,-4) and C(x,y) are 3 collinear points such that AB= BC, then the value of x and y satisfy
a) x²+ y²= 220
b) 14y + 5x = 0
c) 13x + 5y+5 = 0 d) none
12) The extremities of the diagonal of a parallelogr are (3,-4) and (-6,5). If the third vertex is the point (-2,1), then the coordinates of the 4th vertex are
a) (-1,0) b) (1,0) c) (0,1) d) (0,-1)
13) The coordinates of the points A, B, C are (-1,-1), (5,7) and (1, -15). The length of the median through A is
a) √41 unit b) √29 unit c) 5 unit d) none
14) The coordinates of the orthocentre of the Triangle, formed by the straight lines given by (x -2)(y -2)(x + y -1)= 0, are
a) (2,-1) b) (- 1,2) c) ( 2,2 ) d) (3/2,3/2).
15) The coordinate of the four points A, B, C, D are (0,0),(0,10),(8,16), (8,6) respectively. If the points are joined in order, then which one is the most appropriate statement ?
a) ABCD forms a square
b) ABCD forms a rhombus
c) ABCD forms a paralellogram
d) none of the statement is true
16) The points (2a,0),(0, 2b) and (1,1) are collinear if
a) 1/a + 1/b = 2
b) 2/a + 2/b = 1
c) 1/a + 1/b = 1 d) none
17) If two vertices of an equilateral triangle have co-ordinates , then for the third vertex which one is most applicable ?
a) the coordinates are integral
b) the coordinates arw rational
c) the coordinates are irrational
d) at least one coordinates is irrational
18) If the coordinates of the midpoints of a triangle ABC are (0,0),(2,-1) and (-1,3), then the coordinations of the centroid of ∆ ABC are
a) (1/3,2/3) b) (-4,-3) c) (-3,-4) d) none
19) The coordinates of the three vertices of a triangle ABC are (-2,1), (-1,-3) and (3,-2); then the coordinates of its circumcentre are
a) (0,0) b) (0,-4) c) (0,-4/3) d) none
20) The equation of the three sides of a triangle are x -2y +4=0, 2x + y - 7 = 0, x + y +3=0. The coordinates of the orthocentre of the triangle are
a) (3/2, 17/12) c) (2,3) c) (3,2) d) none
21) The area of a triangle 5 square unit. Two of the vertices are (2,1) and (3,-2), the third vertex lies on the line y= x +3. The coordinates of the third vertex are
a) (7/2,13/2),(3/2,-3/2)
b) (7/2,13/2),(-3/2,3/2)
c) (-7/2,13/2),(-3/2,3/2) d) none
22) The coordinates of two vertices are (2,2) and (3,1), the third vertex lies on the line y+ 3x = 0. If the coordinates of the centroid of the triangle are (2,0), then the coordinates of the third vertex are
a) (1/3, 1) b) (-1,3) c) (1,-3) d) none
23) The co-ordinates of the the 3 vertices of a triangle are (2,7),(5,1), (x,3) and the area of the triangle 18 sq unit. The values of is/are
a) 10 b) 2,-10 c) 10,-2 d) none
24) The area of the coordinates whose vertices are (a,0), (-B,0), (0,a), (0,-B) (with a,b > 0) is
a) 0 b) (a+ b)²/2 sq unit c) (a²+ b²+ ab)/2 sq unit d) none
25) If the coordinates of the points A, B, C, D are (6,3), (-3,5),( 4,-2) and +x,3x) respectively, and area of ∆ DBC/area of ∆ ABC= 1/2, then the value of x is
a) 8/11 b) 11/8 c) 3/11 d) 11/3
26) The circumcentre of the the triangle formed by the points (-3,1),(1,3) and (3,0) lies on 2x + y= 0. The coordinates of the circumcentre are
a) (1/16,-1/8) b) (-1/8,1/4) c) (-1/16),1/8) d) (1/8,-1/4)
27) A rectangle has two opposite vertices at the points (1,2) and (5,5). If the other vertices lie on the line x = 3, then the coordinates of the other vertices are
a) (3,2),(3,6) b) (3,1), (3,6) c) (3,1),( 3,5) d) (3,-1) ,(3,-6)
28) The locus of the centroid of the triangle whose vertices are (a cosθ, b sinθ), (a sinθ, - b cosθ) and (1,2), where θ is a parameter, is
a) {(3x+1)/a}² + {(3y +2)/b}²= 2
b) (3x-1)² + (3y -2)²= a²- b²
c) (3x+1)² + (3y +2)²= a²- b²
d) {(3x-1)/a}² + {(3y -2)/b}²= 2
29) The locus of the centroid of a triangle whose vertices are (1,0), (a sec θ, a tan θ) and (b sec θ, - b tan θ), where θ is a parameter, is
a) (3x -1)²/(a+ b)² - 9y²/(a - b)²= 1
b) (3x -1)² - 9y²= (a + b)²
c) (3x +1)²/(a+ b)² - 9y²/(a - b)²= 1
d) (3x +1)² - 9y²= a ²- b²
30) If the equation of the locus of a point equidistant from the point (a₁, b₁) and (a₂, b₂) is (a₁ - a₂)x + (b₁ - b₂)y + c= 0, then the value of 'c' is
a) (1/2)(a₁²+ b₁²+ a₂²+ b₂²)
b) (1/2)(a₂²+ b₂² - a₁²- b₁²)
c) (1/2)(a₁² - a₂²+ b₁²- b₂²)
d) √(a₁²+ b₁²- a₁²- b₂²).
31) The locus of the point which divides the line segment joining the points (a,0) and (0,b) in the ratio 2:3 is
a) 2bx - 2ay =0
b) 3bx + 2ay =0
c) 2bx - 3ay =0
d) 2bx + ay =0
32) The locus of the point which divides the line segment joining the points (a cosθ, 0) and (0, b sinθ) in the ratio m: n is
a) x²/a²n² + y²/b²m² = 1/(m + n)²
b) x²/a²m² + y²/b²n² = 1/(m + n)²
c) x²/b²n² + y²/a²m² = 1/(m + n)² d) none
33) A( a cosα , a sinα) and B(b cosβ, b sin β) are 2 points, M(x,y) is another point such that M divides the line segment AB internally in the ratio a: b. Then {tan(α + β)/2} is equals to
a) x/y b) y/x c) (x + y)/(x - y) d) - x/y
34) The locus of the mid-point of the portion of the line x cosθ + y sinθ= p intercepted between the axis is
a) x²+ y²= 4/p² b) x²+ y²= p²/4 c) 1/x²+ 1/y²= p²/4 d) 1/x²+ 1/y²= 4/p²
35) A( a cosα , a sinα) and B(b cosβ, b sin β) are 2 points, M(x,y) is another point such that M divides the line segment AB externally in the ratio a: b. Then {tan(α + β)/2} is equals to
a) x/y b) y/x c) (x + y)/(x - y) d) - x/y
36) The coordinates of the points at a distance 2√2 unit from the point (2,3) in the direction making angle 45° with the positive direction of the x-axis, are
a) (0,1),(4,1) b) (0,5),(4,5) c) (4,5),( 0,1) d) (0,5),(4,1)
37) if the area of the Triangle formed by x cosθ + y sinθ = p with the coordinate axes is always k², then the locus of the midpoint of the segment of the line intercepted between the axes is
a) 2xy = ± k²/p²
b) xy = ± 2k²/p²
c) 2xy = ± k²
d) xy = ± 2k²
38) The parametric coordinates of a point P are {(t+1)/(t -1), 2t+3}, where t is a parameter, then the locus of P is
a) x(y -3)= y -1
b) x(y -5)= y -1
c) x(y -5)= y -3 d) none
39) If θ be a parameter, then the locus of the point P{2/(1+ sinθ), 3 cosθ) is
a) x²y²= 36(x -1) b) x²y²= 36(x +1) c) x²y²+ 36(x -1) = 0 d) x²y²= - 36(x +1)
40) If t be a parameter, then the locus of the point P(2t - 3/t, 2t + 3/t) is
a) x² - y²= 6 b) - x² + y²= 6 c) x² - y²= 24 d) -x² + y²= 24
41) Let P(1,22) and Q(3,4) be two points . The point R on the x-axis is such that PR+ RQ is minimum. The coordinates of R are
a) (3/5,0) b) (5/3, 0) c) (-3/5, 0) d) (-5/3, 0)
42) The Four Points (-a, -B), (0,0),(a, b) and (a², ab) form a
a) parallelogram b) square c) a quadrilateral of area (1/2) ab √(a²+ b²) square unit d) none
43) The base of a triangle lies along the line apx= a and is of length a. If the area of the triangle is a², then its vertex lies on the line
a) x= -2a b) x= 2a c) x= 3a d) x= -3a
44) The locus of the point (2t²+ t +1, t²- t +1) is
a) (x - 2y +1)²= 3(x + y -2)
b) (x - 2y +1)²= 3(x + y +2)
c) (x - 2y +1)²= 3(x - y +2)
d) (x - 2y +1)²= 3(x - y -2)
45) The polar equation of y= x tan with respect to the origin as pole and the+ve y-axis as the initial line is
a) θ=α b) θ= π/2 -α c) θ = -α d) none
46) The cartesian equation of r²= a² cos2θ is
a) (x²- y²)²= a²(x²+ y²)
b) (x²- y²)²/(x²+ y²)= a²
c) (x²- y²)²= a²(x²- y²)
d) (x²- y²)/(x²+ y²)= a²
47) The cartesian equation of θ=α is
a) y = x sinα b) y = x cosα c) y = x tanα d) x = y tanα
48) The Cartesian equation of r cos²(θ/2) =1 is
a) y²= 4(1- x)
b) y²= 4(x - 1)
c) x²= 4(1- y)
d) x²= 4(1+ y)
49) The cartesian equation of √r = √a cos(θ/2) is
a) (2x²+ 2y²+ ax)²= a²(x²+ y²)
b) (2x²+ 2y²- ax)²= a²(x²+ y²)
c) (2x²+ 2y²+ ax)²= a²(x²- y²)
d) (2x²+ 2y²- ax)²= a²(x²- y²)
50) The coordinates of the points A, B, C, P are (6,3),(-3,5),( 4,-2) and (x,y) respectively; then area of ∆ PBC/area of ∆ ABC is equals to
a) |(x + y+2)/7| b) |(x - y+2)/7| c) |(x + y-2)/7| d) none
51) The new coordinates of the point (4,3) when the coordinate axes are translated by shifting the origin to (-2,1) are
a) (6,2) b) (2,4) c) (6,4) d) (2,2)
52) Without changing the direction of the axes, the origin is transferred to the point (-4, -7). The coordinates of a point P in new system, are (5,-2). The coordinates of P in the origin of system, are
a) ( 9,5) b) (1,5) c) ( 9,-9) d) (1,-9)
53) Let P be the image of the point (2,-3) with respect to the x-axis. Then the coordinates of the point P in the new system of coordinates obtained by the translation of axes in which the origin is shifted to (-3,2) are
a) ( 1,1) b) (-1,5) c) (5,1) d) (-1,1)
54) The co-ordinate axes are translated by shifting the origin to the point (-3,4). In the new system of coordinate axes, the respective x and y-intercepts of a straight line l of which the equation in the original system is 2x + 3y= 5, are
a) -1/2, -1/3 b) 11/2,11/3 c) 23/2,23/3 d) none
55) By a translation of Axes if the origin be transferred to (α, β)so that the linear terms in the equation (x + y)(x - y -2)= 4 are eliminated, then the point (α, β) is
a) (1,-1) b) (-1,1) c) (- 1,-1) d) (1,1)
56) If the origin (0,0) is shifted to the point (2,3), by a translation of axes, the equation x²+ y²- 4x - 6y +9=0 changes to
a) x²+ y² + 4 =0 b) x²+ y²- 4=0 c) x²+ y²- 8x - 12y +48=0 d) none
COORDINATES AND STRAIGHT LINES
1) If two vertices of an equilateral triangle have integral co-ordinates then the third vertex will have
a) integral coordinates
b) co-ordinate which are rational
c) at least one co-ordinate irrational
d) coordinates which are irrational.
2) If the line sigment joining (2,3) and (-1,2) is divided internally in the ratio 3:4 by the line x+ 2y= k then k is
a) 41/7 b) 5/7 c) 36/7 d) 31/7
3) The polar coordinates of the vertices of a triangle are (0,0), (3,π/2 and (3,π/6). Then the triangle is
a) right angle b) isosceles c) equilateral d) none
4) The points (a, b+ c),(b + c+ a), (c, a+ b) are
a) vertices of an equilateral triangle
b) collinear c) concyclic d) none
5) The incentre of the triangle formed by the axes and the line x/a + y/b = 1 is
a) (a/2, b/2)
b) {ab/(a+ b +√(ab)),ab/(a+ b +√(ab)}
c) (a/3, b/3)
d) {ab/(a+ b +√(a²+ b²)),ab/(a+ b +√(a²+ b²)}
6) In the ∆ ABC, the coordinates of B are (0,0), AB=2, angle ABC=π/3 and the middle point of BC has the coordinates (2,0). The centroid of the triangle is
a) (1/2,√3/2) b) (5/3, 1/√3) c) ((4+√3)/3, 1/3) d) none
7) The coordinates of these consecutive vertices of a parallelogram (1,3),(- 1,2) and(2,5). The coordinates of the fourth vertex are
a) (6,4) b) (4,6) c) (-2,0) d) none
8) The area of the pentagon whose vertices are (4,1),(3,6),(- 5,1),(-3,3) and (-3,0) is
a) 36 unit² b) 60 unit² c) 120 unit² d) none
9) A point moves in the xy plane such that the sum of its distances from two mutually perpendicular lines is always equal to 3. The area enclosed by the locus of the point is
a) 18 unit² b) 9/2unit² c) 9unit² d) none
10) Let A(=1,2), B=(3,4) and let C=(x,y) be a point such that (x -1)(x -3)+ (y -2)(y -4)= 0. If area (∆ ABC)= 1 then maximum number of positions of C in the xy plane is
a) 2 b) 4 c) 8 d) none
11) The point (α,β),(γ,δ),(α,δ) and (γ,β) taken in order, where α, β, γ, δ are different real numbers are
a) collinear b) vertices of a square c) vertices of a rhombus d) concyclic
12) The diagonals of a parallelogram PQRS are along the lines x+ 3 y= 4 and 6x - 2y =7. Then PQRS must be
a) rectangle b) square c) cyclic quadrilateral d) rhombus
13) The coordinates of the 4 vertices of the quadrilateral are (-2,4),(- 1,2) and (2,4) taken in order. The equation of the line passing through the vertex (-1,2) and dividing the quadrilateral in two equal areas is
a) x+1=0 b)x+y-1=0 c) x - y+3=0 d) none
14) The equation of the straight line which passes through the point (-4,3) such that the portion of the line between the axes is divided internally by the point in the ratio 5:3 is
a) 9x - 20y+ 96=0 b) 9x - 20y -24=0 c) 20x +9y+ 53=0 d) none
15) The equation of the straight line which bisects the intercepts made by the axes on the line x + y =2 and 2x + 3y = 6 is
a) 2x= 3 b) y= 1 c) 2y=3 d) x=1
16) The equation of a straight line passing through the point (-2,3) and making intercepts of equal length on the axes is
a) 2x +y+ 1=0 b) x -y-5=0 c) x -y +5=0 d) none
17) The foot of the perpendicular to the line 3x + y = λ drawn from the origin is C. if the line cuts the x-axis the y-axis at A and B respectively then BC: CA is
a) 1:3 b) 3:1 c) 1:9 d) 9:1
18) The distance of the line 2x - 3y=4 from the point (1,1) in the direction of the line x + y= 1 is
a) √2 b) 5√2 c) 1/√2 d) none
19) The 4 sides of a quadrilaterals are given by the equation xy(x -2)(y -3)=0. The equation of the line parallel to x - 4y=0 that divides the quadrilaterals in two equal areas is
a) x -4y= -5 b) x -4y= 5 c) 4y = x +1 d) 4y+1= x
20) The coordinates of two consecutive vertices A and B of a regular hexagon ABCDEF are (1,0) and (2,0) respectively. The equation of the diagonal CE is
a) √3x + y= 4 b) x + √3 y= -4 c) x + √3 y= 4 d) none
21) ABC is an isosceles triangle in which A is (-1,0), angle A=2π/3, AB= AC and AB is along, the x-axis. If BC=4√3 then the equation of the line BC is
a) x + √3 y= 3 b) √3x + y= 3 c) x + y= √3 d) none
22) The graph of the function cos (x +2) - cos²(x +1) is a
a) straight line passing through the point (0, sin²1) with slope 2
b) straight line passing through the origin
c) parabola with vertex (1, - sin²1)
d) straight line passing through the point (π/2, - sin²1) and parallel to the x-axis .
23) if the points (-2,0),(-1,1/√3) and (cosθ , sinθ ) are collinear then the number of values of θ belongs to [0,2π] is
a) 0 b) 1 c) 2 d) infinite
24) The limiting pisition of the point of intersection of the line 3x + 4y= 1 and (1+c)x + 3c²y= 2 as c tends to 1 is
a) (-5,4) b) (5,- 4) d) (4,-5) d) none
25) The coordinates of the point on the x-axis which is equidistant from the points (-3,4) and (2,5) are
a) (20,0) b) (-23, 0) c) (4/5, 0) d) none
26) The distance between the lines 3x + 4y=9 and 6x + 8y +15= 0 is
a) 3/10 b) 33/10 c) 33/8 d) none
27) If a vertex of an equalateral triangle is the origin and the opposite to it has the equation x + y= 1 then the orthocentre of the triangle is
a) (1/3,1/3) b) (√2/3,√2/3) c) (2/3,2/3) d) none
28) The equation of the three sides of a triangle are x= 2y +1=0 and x+ 2y=4. The coordinates of the circumcentre of the triangle are
a) (4,0) b) (2,-1) c) ( 0,4) d) none
29) L is a variable line such that the algebraic sum of the distances of the points (1,1),(2,0) and (0,2) from the line is equals to zero. The line L will always pass through
a) (1,1) b) (2,1) c) (1,2) d) none
30) ABC is an equilateral triangle such that the vertices B and C lie on two parallel lines at a distance 6. If A lies between the parallel lines at a distance 4 from one of them then the length of the side of the equilateral triangle is
a) 8 b) √(88/3) c) 4√7/√3 d) none
31) If p and p' are the perpendiculars from the origin upon the lines x sec θ+ y cosecθ= a and x cosθ - y sinθ = a cos2θ respectively then
a) 4p²+ p'²= a² b) p²+ 4p'²= a² c) p²+ p'²= a² d) none
32) let the perpendiculars from any point on the line 2x + 11y= 5 upon the lines 24x + 7y= 20 and 4x -3y= 2 have the lengths p and p' respectively. then
a) 2p= p' b) p= p' c) p= 2p' d) none
33) If P(1+ t/√2 , 2+ t/√2) be any points on a line then the range of the values of t for which the point P lies between the parallel lines x+ 2y= 1 and 2x + 4y= 15 is
a) -4√2/5< t< 5√2/6
b) 0< t< 5√2/6
c) 4√2/5< t< 0 d) none
34) There are two parallel lines, one of which has the equation 3x + 4y=2. If the lines cut an intercept of length 5 on the line x+ y= 1 then the equation of the other line is
a) 3x+ 4y= (√6-2)/2
b) 3x+ 4y= (√6+2)/2
c) 3x+ 4y= 7 d) none
35) if the intercept made on the line y= mx by lines y=2 and y= 6 is less than 5 then the range of the value of m is
a) (-∞, -4/3) U (4/3, ∞) b) (-4/3,4/3) c) (-3/4,3/4) d) none
36) If a,b,c are any terms of an AP then the line ax + by + c=0
a) has a fixed direction
b) always passes through a fixed point
c) always cuts intercepts on the axes such that their sum is zero
d) forms a triangle with the axes whose area is constraint
37) If a,b,c are in GP then the line ax+ by + c= 0
a) has a fixed direction
b) always passes through a fixed point
c) forms a triangle with the axes whose area is constant
d) always cuts intercepts on the axes such that their sum is zero
38) The number of the real values of k for which the lines x - 2y +3=0, Kx + 3y +1=0 and 4x - ky +2=0 are concurrent is
a) 0 b) 1 c) 2 d) infinite
39) A family of lines is given by (1- 2k)x + (1- λ)y + λ= 0, λ being the parameter. The line belonging to this family at the maximum distance from the point (1,4) is
a) 4x - y +1=0 b) 33x +12y +7=0 c) 12x +33y -7=0 d) none
40) The members of the family of lines (x + μ)x + (2λ+ μ)y + λ+ 2μ , where λ≠ 0, μ≠ 0, pass through the point
a) (3,1) b) (-3,1) c) (1,1) d) none
41) The equation of of the sides AB, BC and CA of the ∆ ABC are y - x = 2, x+ 2y=1 and 3x + y+5=0 respectively . The equation of the altitude through B is
a) x -3y= -1 b) x -3y= -4 c) 3x - y= -2 d) none
42) The range of values of the ordinate of a point moving on the line x= 1, and always remaining in the interior of the triangle formed by the lines y= x, the x-axis and x + y= 4, is
a) (0,1) b) (0,1] c) [0,4) d) none
43) If the point (a,a) fails between the lines|x + y|= 2 then
a) |a|= 2 b) |a|= 1 c) |a|<1 d) |a|< 1/ 2
44) If A(sinα, 1/√2) and B(1/√2, cosα), -π≤α ≤π, are two points on the same side of the line x - y= 0 then α belongs to the interval
a) (-π/4,π/4)U (π/4,3π/4) b) (-π/4,π/4) c) (π/4,3π/4) d) none
45) The straight line L₁ ≡ 4x - 3y +2= 0, L₂≡ 3x + 4y - 4=0 and L₃≡ x - 7y +6=0
a) form a right angled triangle
b) form a right angled isosceles triangle
c) are concurrent d) none
46) The equation of the bisector of the acute angles between the lines 2x - y +4=0 and x - 2y -1=0 is
a) x +y +5=0 b) x - y +1=0 c) x - y -5=0 d) none
47) The equation of the bisectors of that angle between the lines x +y -3=0 and 2x -y -2=0 which contains the point (1,1) is
a) (√5-2√2)x +(√5+√2)y =3√5- 2√2
b) (√5 +2√2)x +(√5 -√2)y =3√5 +2√2
c) 3x=10 d) none
48) two lines 2x - 3y -1=0 and x +2y +3=0 divide the xy plane in four compartment which are named as shown in the figure.
Consider the location of the points (2,-1), (3,2) and (-1,-2). We get
a) (2,-1) belongs to IV
b) (3, 2) belongs III
c) (-1,-2) belongs to II d) none
49) If the lines y- x = 5, 3x+ 4y=1 and y= mx +3 are concurrent then the value of m is
a) 19/5 b) 1 c) 5/19 d) none
50) if the point (cos θ, sinθ) does not fall in that angles between the lines y=|x -1| in which the origin lies then θ belongs to
a) (π/2,3π/2) b) (-π/2,π/2) c) (0,π) d) none
51) The points (-1,1) and (1,-1) are symmetrical about the line
a) y+ x =0 b) y= x c) x+ y=1 d) none
52) The equation of the line segment AB is y= x. If A and B lie on the same side of the line mirror 2x - y= 1, the image of AB has the equation
a) x + y= 2 b) 8x + y= 9 c) 7x - y= 6 d) none
53) Let P= (1,1) and Q=(3,2). The point R on the x-axis such that PR+ RQ is the minimum is
a) (5/3, 0) b) (1/3,0) c) (3,0) d) none
54) If a ray travelling along the line x= 1 gets reflected from the line x + y= 1 then the equation of the line along which the reflected ray travels is
a) y= 0 b) x - y= 1 c) x=0 d) none
55) The point P(2,1) is shifted by 3√2 parallel to the line x + y= 1 in the direction of increasing ordinate , to reach Q. The image of Q by the line x + y= 1 is
a) (5,-2) b) (-1,-2) c) (5,4) d) (-1,4)
56) Let A=(1,0) and B (2,1). The line AB turns about A through an angle π/6 in the clock wise sense, and the new position of B is B'. B' has the coordinates
a) {(3+√3)/2, (√3-1)/2}
b) {(3-√3)/2, (√3+1)/2}
c) {(1-√3)/2, (1+ √3)/2} d) none
57) a line intercepts a, b on the co-ordinate axes. if the axes are rotated about the origin through an angle α then the line has intercepts p, q on the new position of the axes respectively. then
a) 1/p²+ 1/q²= 1/a²+ 1/b²
b) 1/p²- 1/q²= 1/a²- 1/b²
c) 1/p²+ 1/a²= 1/q²+ 1/b² d) none
58) Two points A and B move on the x-axis and the y-axis respectively such that the distance between the two points is always the same. The locus of the middle point of AB is
a) a straight line
b) a pair of straight line
c) a circle d) none
59) Three vertices of a quadrilateral in order are (6,1),(7,2) and (-1,0). If the area of the quadrilateral is 4 unit² then the locus of the fourth vertex has the equation
a) x - 7y= 1 b) x - 7y= -15 c) (x - 7y)²+ 14(x - 7y)- 15 = 0 d) none
60) A variable line through the point (a,b) cuts the axes of reference at A and B respectively. The line through A and B parallel to the y-axis and the x-axis respectively meet at P. Then the locus of P has the equation
a) x/a + y/b = 1 b) x/b + y/a = 1 c) a/x + b/y = 1 d) b/x + a/y = 1
Multiple choice
61) If the coordinates of the vertices of a triangle are rational numbers then which of the following points of the triangle will always have rational coordinates ?
a) centroid b) inCentre c) circumcenter d) orthocenter
62) Two consecutive vertices of a rectangles of area 10 unit² are (1,3) and (-2,-1). Other two vertices are
a) (-3/5,21/5),(-18/5,1/5)
b) (-3/5,21/5),(-2/5,-11/5)
c) (-2/5,-11/5),(13/5,9/5)
d) (13/5,9/5),(-18/5,1/5)
63) The ends of a diagonal of a square are (2,-3) and (-1,1). Another vertex of the square can be
a) (-3/2,-5/2) b) (5/2,1/2) c) (1/2,5/2) d) none
64) If each of the vertices of a triangle has integral coordinates then the triangle may be
a) right angled b) equilateral c) isosceles d) none
65) if (-1,2),(2,-1) and (3,1) are any three vertices of a parallelogram then the fourth vertex (a,b) will be such that
a) a=2, b=0 b) a=-2, b=0 c) a=-2, b=6 d) a=6, b=-2
66) If (α, β) be an end of a diagonal of a square and the other diagonals has the equation x- y=α then another vertex of the square can be
a) (α - β,α) b) (α,0) c) (0, α) d) (α + β, β)
67) A point on the line y= x whose perpendicular distance from the line x/4 + y/3=1 is 4 has the coordinates
a) (-8/7,-8/7) b) (32/7,32/7) c) (3/2,3/2) d) none
68) The parametric equation of a line is given by
x= -2 + r/√10 and y= 1+3r/√10. then, for the line
a) intercept on the x-axis = 7/3
b) intercept on the y axis = -7
c) slope of the line = tan⁻¹(1/3)
d) slope of the line= tan⁻¹3
69) One side of a square of length a is inclined to the x-axis at an angle α with one of the vertices of the square of the origin. The equation of a diagonal of the square is
a) y( cosα - sinα)= x(cosα + sinα)
b) y( cosα + sinα)= x(cosα - sinα)
c) y (cosα + sinα)- x(cosα - sinα)= a
d) y (cosα -0- sinα)+ x(cosα - sinα)= a
70) If the equation of the hypotenuse and a side of a right angle isosceles triangle be x+ my = 1 and x= k respectively then
a) m= 1 b) m= k c) m= -1 d) m + k= 0
71) The centroid and a vertex of an equilateral triangle are (1,1) and (1,2) respectively. Another vertex of the triangle can be
a) {(2-√3)/2, 1/2}
b) {(2 + 3√3)/2, 1/2}
c) {(2 + √3)/2, 1/2} d) none
72) if one vertex of an equilateral triangles of side 2 is the origin and another vertex lies on the line x= √3 y then the third vertex can be
a) (0,2) b) (√3,-1) c) (0,-2) d) (√3,1)
73) A line passing through the point (2,2) and the exes enclose an area λ. The intercept on the axes made by the line are given by the two roots of
a) x²-2|λ|x + |λ|= 0
b) x² + |λ|x + 2|λ|= 0
c) x²-|λ|x + 2|λ|= 0 d) none
74) A line passes through the origin and making an angle π/4 with the line y - 3x = 5 has the equation
a) x+ 2y=0 b) 2x= y c) x= 2y d) y+ 2x=0
75) The coordinate of a point on the line x+ y = 3 such that the point is at equal distance from the lines |x|=|y| are
a) (3,0) b) (0,3) c) (-3,0) d) (0,-3)
76) A line perpendicular to the line 3x - 2y= 5 cuts off an intercept 3 on the positive side of the x-axis bank. Then
a) the slope of the line is 2/3.
b) the intercept on the y-axis is 2
c) the area of the triangle formed by the line with the axis is 3 unit² d) none
77) One diagonal of a square is the portion of the line √3 x + y= 2√3 intercepted by the axes. Then the extremely of the other diagonal is
a) (1+√3,√3-1) b) (1+√3,√3+1) c) (1-√3,√3-1) d) (1-√3,√3+1)
78) if bx + cy = a, where a, b, c are of the same sign , be a such line that the area enclosed by the line and the axes of reference is 1/8 unit² then
a) b,a,c are in GP
b) b, 2a, c are in GP
c) b, a/2, c are in AP
d) b, -2a, c are in GP
79) The side of the triangle are x + y = 1, 7y= x and √3 y + x=0. then the following is an interior point of the triangle.
a) circumcenter b) centroid c) incentre d) orthocenter
80) If (x, y) be a variable points on the line line y= 2x lying between the lines 2(x +1)+ y= 0 and x + 3(y -1)=0 then
a) x∈ (-1/2,6/7) b) x∈ (-1/2,3/7) c) y∈ (-1,3/7) d) y∈ (-1,6/7)
81) if the equation of the three sides of a triangle are x+ y= 1, 3x + 5y=2 and x - y=0 then the orthocentre of the triangle lies on the line
a) 5x - 3y= 2
b) 3x - 5y= -1
c) 2x - 3y= -1
d) 5x - 3y= 1
82) A ray travelling along the line 3x - 4y= 5 after being reflected from a line l travels along the line 5x + 12y= 13. then the equation of the line l is
a) x +8y= 0 b) x = 8y c) 52x + 4y= 65 d) 32x - 4y= -65
83) A ray of light travelling along the line x +y= 1 is incident on the x-axis and after refraction it enters the other side of the x-axis by turning π/6 away from the x-axis. The equation of the line along which the refracted ray travels is
a) x +(2- √3)y= 1 b) (2- √3)x +y= 1 c) (2+√3)x = 2+√3 d) none
1c 2a 3c 4b 5d 6b 7b 8a 9a 10b 11b 12d 13c 14a 15b 16c 17d 18a 19a 20c 21a 22d 23b 24a 25d 26b 27a 28a 29a 30c 31a 32b 33a 34d 35a 36b 37c 38a 39c 40a 41b 42a 43c 44a 45c 46b 47a 48a 49c 50b 51b 52c 53a 54a 55d 56a 57a 58c 59c 60c 61acd 62 ac 63 ab 64 acd 65 bd 66 bd 67 ab 68 d 69 ac 70 ac 71 ac 72 ab 73c 74cd 75ab 76 bc 77 bc 78 bd 79 bc 80 bd 81 bd 82 bc 83 ac
1) Let X be the universal set for sets A and B. If n(A)= 200, n(B)= 300 and n(A∩B)= 100, then n(A' ∩ B') is equal to 300 provided n(X) is equal to
a) 600 b) 700 c) 800 d) 900
2) The sum of n terms of the infinite series 1.3²+ 2.5²+ 3.7²+...... ∞ is
a) (n/6)(n+1)(6n²+14n+7) b) (n/6)(2n+1)(3n+1) c) 4n³+ 4n²+ n d) none
3) If x²ᵏ occurs in the expansion of (x + 1/x²)ⁿ⁻³, then
a) n - 2k is a multiple of 2.
b) n - 2k is a multiple of 3.
c) n = 0 d) none
4) If y= 1- x + x²/2! - x³/3! + x⁴/4! - ........, then d²y/dx² is
a) x b) - x c) - y d) y
5) The value of ∫ secx dx/√{sin(2x+θ)+ sinθ} is
a) √{(tanx + tanθ) secθ}+ c
b) √{2(tanx + tanθ) secθ}+ c
c) √{2(sinx + tanθ) secθ}+ c d) none
6) The solution of the equation x² d²y/dx² = log x when x=1, y=0 and dy/dx = -1 is
a) y= (1/2)(logx)²+ logx
b) y= (1/2)(logx)²- logx
c) y= -(1/2)(logx)²+ logx
d) y= - (1/2)(logx)²- logx
7) If C²+ S²= 1 then (1+ C + iS)/(1+ C - iS) is equal to
a) C+ iS b) C- iS c) S+ iC d) S- iC
8) The number of real roots of (x + 1/x)³+ (x + 1/x)= 0 is
a) 4 b) 6 c) 2 d) 0
9) If Tₚ , Tq , Tᵣ are pᵗʰ , qᵗʰ and rᵗʰ terms of an AP, then
Tₚ Tq Tᵣ
p q r
1 1 1 is equal to
a) p+ q+ r b) 0 c) 1 d) -1
10) If A is a square Matrix of order n x n and k is a scalar then adj.(kA) is equal to
a) kⁿ⁻¹ adj. A b) kⁿ⁺¹ adj. A c) kⁿ adj. A d) k adj. A
11) 6 persons A, B, C, D, E and F are to be seated at a circular table. The number of ways this can be done if A must have either B or C on is right and B must have either C or D on his right is
a) 24 b) 12 c) 18 d) 36
12) For what values of α
lim ₓ→∞ √(2α²x²+ αx+7) - √(2α²x²+7) will be 1/2√2
a) any value of α b) α ≠ 0 c)α = 1 d) α = -1
13) A window is in the form of a rectangle with the semicircular bend on the top. If the perimeter of the window is 10m, the radius in metres of the semicircular bend that maximize the amount of light admitted is
a) 20/(4+π) b) 10/(4+π) c) (10 -π) d) none
14) There is line with positive slop λ through origin which cuts off a segment of length √10 between the parallel line 2x - y + 5 = 0 and 2 x - y + 10 = 0. Then λ should be
a) 1/2 b) 1/3 c) 1/5 d) none
15) Let C₁ and C₂ be the circles given by equations x²+ y²- 4x -5= 0 and x²+ y² + 8y +7 = 0. Then the circle having the common chord of C₁ and C₂ as its diameter has
a) centre at (- 1, - 1) and radius 2
b) centre at (1, - 2) and radius √5
c) centre at ( 1, - 2) and radius 2
16) The equation of common tangent to the parabola y²= 16x and the circle x²+ y²= 8 are
a) y= x+ 4; y= - x -4 b) y= 2x+ 4; 2y= - x +9 c) y= x+ 9; y= - x -4 d) none
17) An ellipse has OB as semi minor axis. F and F' its foci and the angle FBF' is a right angle. Then the eccentricity of the ellipse is
a) 1/√2 b) 1/2 c) 1/4 d) 1/√3
18) The angle between the lines 2 x= 3y= - z and 6x = - y = -4z is
a) 0° b) 30° c) 45° d) 90°
19) A vector parapendicular to the plane containing the vectors i - 2j - k and 3i - 2j - k is inclined to the vector i + j + k at an angle
a) tan⁻¹√14 b) sec⁻¹√14 c) tan⁻¹√15 d) none
20) Three integers are chosen at random from the first 20 integers. The probability that their product is even, is
a) 2/19 b) 3/29 c) 17/19 d) 4/29
21) If cotθ + tanθ = m and secθ - cosθ = n, then which of the following is correct?
a) m(mn²)¹⁾³ - n(nm²)¹⁾³ = 1
b) m(m²n)¹⁾³ - n(mn²)¹⁾³ = 1
c) n(mn²)¹⁾³ - m(nm²)¹⁾³ = 1
d) n(m²n)¹⁾³ - m(mn²)¹⁾³ = 1
22) If 3 sin⁻¹{2x/(1+ x²)} - 4 cos⁻¹{(1- x²)/(1+ x²)} + 2 tan⁻¹{2x/(1- x²)} =π/3, then value of x is
a) √3 b) 1/√3 c) 1 d) none
23) The A. M of ²ⁿ⁺¹C₀ , ²ⁿ⁺¹C₁ , ²ⁿ⁺¹C₂ , .... ²ⁿ⁺¹Cₙ is
a) 2ⁿ/n b) 2ⁿ/(n +1) c) 2²ⁿ/n d) 2²ⁿ/(n +1)
24) Let p and q be two statements, then (p∪q) ∪ - p is
a) tautology b) contradiction c) Both a and b d) none
25) Let f(x)= x - [x], for every real number of x, where [x] is the integral parts of x. Then ¹₋₁∫ f(x) dx is equal to
a) 1 b) 2 c) 0 d) -1/2
Directions (26-30): This section contains 5 questions numbered 26 to 30. Each question contains statement-1 (Assertion ) and statement-2 (Reason). Each question has four choices (a), (b), (c) and (d) out of which ONLY ONE is correct.
a) Statement-1 is True , Statement-2 is True , Statement-2 is a correct explanation for Statement -1.
b) statement-1 is true , statement-2 is true; statement-2 is not a correct explanation for Statement -1.
c) Statement -1 is True , Statement-2 is false .
d) statement-1 is false, statement -2 is true.
26) statement -1: 5/3 and 5/4 are the eccentricity of two conjugate hyperbolas.
Statement -2: If e and e₁ are the eccentricities of two conjugate hyperbolas, then ee₁ > 1.
27) Statement -1: The maximum area of triangle formed by the point (0,0), (a cosθ , b sinθ), (a cosθ , - b sinθ) is (1/2) |ab|.
Statement -2: Maximum value of sinθ is 2.
28) Statement -1: The Coefficient of xⁿ in the binomial expansion of (1- x)⁻² is (n +1).
Statement-2: The Coefficient of xʳ in (1- x)⁻ⁿ when n ∈N is ⁿ⁺ʳ⁻¹C ᵣ.
29) Statement -1: 20 persons are sitting in a row. Two of these persons are selected at random, The probability that two selected person are not together is 0.7.
Statement-2: if A is an event, then
P(not A)= 1- P(A).
30) Statement -1: A flagstaff of length 100m stands on tower of height h. if at a point on the ground the angle of elevation of the tower and top of the flagstaff be 30°, 45°, then h= 50(√3 +1)m.
Statement -2: A flagstaff of length 'd' stands on tower of height h. If at a point on the ground the angle of elevation of the tower and top of the flagstaff be α, β then h= d cotβ/(cot α - cotβ).
1) How many odd numbers of 6 digits can be formed with the digits 0, 1, 4, 5, 6, 7, none of the digit being repeated in any numbers ?
2) From 10 different 6 things are taken at a time so that a particular thing is always included. Find the number of such permutations .
3) How many different arrangements of the letters of the word FAILURE can be made so that the vowels are always together ?
4) In how many ways can the letters of the word STATION be arranged so that the vowels are always together ?
5) How many words can be formed by the letters of the word PEOPLE taken all together so that the two Ps are not together ?
6) In how many ways 6 books be arranged on a shelf of an almirah so that 2 prticular books will not be together?
7) Find the values of n and r when ⁿPᵣ = 336 and ⁿCᵣ = 56.
8) A box contains 10 electric lamps of which 4 are defective . Find the number samples of 6 lanpa taken at random from the box which will contain two detective lamps.
9) A committee of five is to be formed from 6 boys and 4 girls. How many different committees can be formed so that each committee contains at least two girls ?
10) An examinee has to answer 6 questions out of 12 questions . The questions are divided into two groups , each group containing 6 questions. The examinwe is not permitted to answer more than 4 questions from any group. In how many ways can he answer in all 6 questions ?
11) The Indian cricket eleven is to be selected out of fifteen players, five of them are bowlers. In how many ways the team can be selected so that team contains at least three bowlers?
12) In a plane 5 points out of 12 points are collinear, no three of the remaining points are callinear. Find the number of straight lines formed by joining these points.
13) Find the values of n and r, when ⁿPᵣ = ⁿPᵣ₊₁ and ⁿCᵣ = ⁿCᵣ₋₁.
14) which polygon has the same number of diagonals as sides ?
15) How many words can be made out of the letters of the word EQUATION taken all at a time such that each word will have one constant in the beginning and one at the end.
16) How many numbers of 7 digits can be formed with the digit 1,2, 3,4, 3, 2, 1 so that the odd digits will occupy odd places.
17) How many numbers greater than 1 lakh can be formed with the digits 0, 2, 5, 2, 4, 5 ?
18) In how many telephone numbers of 6 digits, two consecutive digits will be different?
19) m men and n women take seats in a row. If m > n and no two women sit together then show that they can take seats in m!(m+1)!/(m - n +1)! ways.
20) Two person A and B sit with other 10 persons in a row. In how many arrangements will there be three persons in between A and B ?
21) Find the rank of the word MOTHER when its letters are arrange as in a dictionary.
22) 18 guests have to be seated, half on each side of a long table. 4 particular guests desire to sit on one particular side and three others on other side. Determine the number of ways in which the sitting arrangementa can be made.
23) Show that the total number of permutations of n different things taking and more than r things at a time (repetition is allowed ) is n(nʳ -1)/(n -1).
24) If ⁿCᵣ₋₁/a = ⁿCᵣ/b = ⁿCᵣ₊₁/c, then show that,
n= (ab + 2ac + bc)/(b²- ac) and r= a(c + b)/(b²- ac)
25) If ⁿ⁺¹Cₘ₊₁ : ⁿ⁺¹Cₘ : ⁿ⁺¹Cₘ₋₁ = 5:5:3, find the values of m and n.
26) ⁴ⁿC₂ₙ/²ⁿCₙ = {1.3.5....(4n -1)}/{1.3.5....(2n -1)²}.
27) A committee of five persons is to be formed from 6 ladies and 4 gentlemen. How many committees containing at least two ladies can be formed where Mr A and Mrs B will
a) always remain
b) never remain.
28) Different arrangements are made by taking 3 vowels and 5 consonants out of 5 vowels and 10 consonants respectively so that the vowels always come together. Find the number of such arrangements.
29) In how many ways 4 or more men from 10 men can be selected ?
30) A box contains two white balls , 3 black balls and 4 red balls. In how many ways can three balls be drawn from the box if atleast one black ball is to be included in the draw ?
31) Find the total number of combinations taking at least one green ball and one blue ball from 5 different green balls, 4 different blue balls and 3 different red balls.
32) A student is allowed to select atmost n books from a collection of (2n +1) books. If the total number of selections of at least one book is 63, then find the value of n.
33) A student has to select even number of books from a collection of 2n books. if he can select books in 2047 different ways, find the value of n.
34) How many different selections can be done taking at least one letter from each of the words TABLES , CHAIR, BENCH
35) 2n out of 3n articles are alike and the rest are different. In how many ways 2n articles out of 3n articles can be selected.
36) A box contains 2 guavas , 3 orange ps, 4 apples of different shapes.
a) in how many ways one or more fruits can be selected ?
b) how many selections of fruits can be made taking at least one of each kind?
37) A question paper contains 10 questions . Four answers for each question are given of which one is correct. If one examinee gives answers for all of the 10 questions and select one answers for each question then in how many ways he can give correct answers for 5 questions?
38) If any 7 dates are named at random then in how many cases of them, there will be three Sundays .
39) There are two women participating in a chess competition. Every participant played 2 games with other participants. The number of games the men played between themselves exceed by 66 the number the meen played with the women . How many participated in the tournament and how many games were played?
40) 5 balls of different colours that we placed in three boxes of different sizes All the five balls can be placed in each box. In how many different ways the balls can be kept in the boxes so that no box will be empty ?
41) In how many ways 12 different fruits can be divided among three boys so that each one can get at least one fruit ?
42) There are four balls of four colours and 4 boxes of same colours with the balls. In how many different ways each box will contain one ball so that no ball will go to the box if its own colour.
43) a box contain 5 pair of shoes. In how many different ways can 4 shoes be selected so that there will be no complete pair of shoes?
44) A, B, C have 5,3 and 7 books of different types respectively. In how many different ways can they exchange the books so that number of books of everyone remains as before?
45) How many a) selections and b) arrangement can be made taking 4 letters from the word PROPORTION ?
46) How many numbers of 4 digit can be formed with the digits 1,1,2, 2, 3, 3, 4, 5 ?
47) Find the number of arrangements taking 5 letters at a time from the letters a,a,a,b,b,b,c,d.
48) How many different arrangements can be made by taking Four Pens from 3 same red colour pens, 2 same blue colours pens and 3 pens of other different colour ?
49) No 3 diagonals of a decagon are concurrent except at the vertices . Find the number of points of interaction of the diagonals.
50) AB and CD are two parallel straight lines. 20 points on AB and 20 points in CD are taken and the points on AB are connected with the points on CD . if no two straight lines are parallel and no three points are concurrent then how many Triangles can be formed so that one of the angular points of each triangle lies on AB and one on CD and another vertex does not lie on AB or CD?
51) How many Triangles can be obtained by joining the vertices of a polygon of 24 sides so that the sides of the triangle will not be the sides of the polygon ?
52) The length and breath of a parallelograms are cut by p lines being parallel to both length and breadth . Show that in all (1/4)(p+1)²(p+2)² parallelograms are formed.
53) In a place there are m parallel roads along north-south and n parallel roads along east -west. In how many different short routes can a man go from the junction of north- east to the junction of Southwest?
54) Find the number of different squares those can be formed on a chessboard.
55) There are n letters and n directed envelopes. In how many ways could all the letters be put into wrong envelopes ?
56) In how many ways can 3A's, 2B's and 1C's be arranged in one line so that 2A's never occur together?
57) The number of 5 digit telephone numbers , none of their digits being repeated is
a) 50 b) ⁵⁰P₅ c) 5¹⁰ d) 10⁵
58) The number of 10 digit numbers formed with the digit 1 and 2, is
a) ¹⁰C₁ + ⁹C₂ b) 2¹⁰ c) ¹⁰C₂ d) 10!
59) A, B, C, D and E have been asked to deliver a lecture in a meeting. In how many ways can their lectures be arranged so that C delivers lecture just before A?
60) How many different signals can be given by using any number of the flags from six flags of different colours ?
61) The number of ways in which five unlike rings a man can be wear on the four fingers of one hand
a) 120 b) 625 c) 1024 d) none
62) Show that the product of r successive natural number is divisible by r!.
63) If n> 7, prove that ⁿ⁻¹C₃ + ⁿ⁻¹C₄ > ⁿC₂.
64) The value of ⁴⁷C₄ + ⁵ᵢ₌₁∑⁵²⁻ⁱC₃ is
a) ⁴⁷C₄ b) ⁵²C₃ c) ⁵²C₄ d) none
65) The value of ¹⁵C₁+ ¹⁵C₃ + ¹⁵C₅+....+ ¹⁵C₁₅ =
a) 15!16! b) 15.2⁸ c) 2¹⁴ d) 2¹⁵
66) In a football championship, there were played 153 matches. Every two teams playex one match with each other. The number of teams, participating in the championship is
a) 17 b) 18 c) 9 d) none
67) Everybody in a room shakes handa with everybody else. The total number of hand shakes is 66. The total number of persons in the room is
a) 11 b) 12 c) 13 d) 14
68) The number of ways to form a team of 11 players out of 22 players where 2 particular players are included and 4 particular players are never included in the team is
a) ¹⁶C₁₁ b) ¹⁶C₅ c) ¹⁶C₉ d) ²⁰C₉
69) The total number of factors of 1998 (including 1 and 1998), is
a) 18 b) 16 c) 12 d) 10
70) In how many ways can 9 different things be divided into 3 groups of 2, 3 and 4 things respectively ?
71) In how many ways can 12 different things be divided equally into 4 groups ?
72) Of the four numbers 25, 150 , 170, 210 which one is the number of diagonals of a polygon of 20 sides ?
73) If a polygon has 54 diagonals, find the number of sides of the polygon.
74) In how many ways can the result (win or loss, or draw) of 3 successive football matches be decided?
75) There are 10 electric bulbs in a hall. Each of them can be lightened separately. The number of ways for lightning the ball is
a) 10² b) 1023 c) 2¹⁰ d) 10!
76) How many quadrilaterals can be formed with the seven sides of the length 1cm, 2 cms, 3 cms, 4 cms, 5 cms, 6 cms and 7 cms ?
77) How many different algebraic expression can be formed by combining the letters a, b, c, d, e, f with the signs '+' and '-', all the letters taken together?
78) Find the total number of ways in which six '+' and four '-' signs can be arranged in a line such that no two '-' signs occur together.
1) If ⁿCᵣ₋₁ = 56, ⁿCᵣ = 28 and ⁿCᵣ₊₁ = 8 then r is equal to
a) 8 b) 6 c) 5 d) none
2) The value of ⁴⁰C₃₁ + ¹⁰ⱼ₌₀ ∑ ⁴⁹⁺ʲC₁₀₊ⱼ is equal to
a) ⁵¹C₂₀ b) 2. ⁵⁰C₂₀ c) 2. ⁴⁵C₁₅ d) none
3) In a group of boys, the number of arrangements of 4 boys is 12 times the number of arrangements of 2 boys. The number of boys in the group is
a) 10 b) 8 c) 6 d) none
4) The value of ¹⁰∑ᵣ₌₁ r. ʳPᵣ is
a) ¹¹P₁₁ b) ¹¹P₁₁ -1 c) ¹¹P₁₁ +1 d) none
5) From a group of persons the number of ways of selection 5 persons is equals to that of 8 persons . The number of person in the group is
a) 13 b) 40 c) 18 d) 21
6) The number of distinct rational numbers x such that 0< x <1 and x= p/q, where p, q belongs to {1,2,3,4,5,6} is
a) 15 b) 13 c) 12 d) 11
7) The total number of 9 digit numbers of different digits is
a) 10. 9! b) 8.9! c) 9. 9! d) none
8) The number of 6 digit numbers that can be made with the digits 0, 1, 2, 3, 4 and 5 so that even digits occupy odd places is
a) 24 b) 36 c) 48 d) none
9) The number of ways in which 6 men can be arranged in a row so that 3 perpendicular men are consecutive, is
a) ⁴P₄ b) ⁴P₄ x ³P₃ c) ³P₃ x ³P₃ d) none
10) Seven different lecturers are to deliver lectures in 7 periods of a class on a particular day. A, B and C are three of the lecturers . The number of ways in which a routine for the day in the can be made such that A delivers his lectures before B, and B before C, is
a) 420 b) 120 c) 210 d) none
11) The total number of 5 digit numbers of different digits in which the digit in the middle is the largest is
a) ⁹ₙ₌₄∑ⁿP₄ b) 33(3!) c) 30. 3! d) none
12) A five digit number divisible by 3 to be formed using the digits 0, 1, 2, 3, 4 and 5 without repetition. The total number of ways in which this can be done is
a) 216 b) 600 c) 240 d) 3125
13) let A=x : x is a prime number and x< 30}. The number of different rational numbers whose numerator and denominator belongs to A is
a) 90 b) 180 c) 91 d) none
14) The total number of ways in which 6 '+' and 4 '-' signs can be arranged in a line such that no two '-' signs occur together is
a) 7!/3! b) 6! 7/3/ c) 35 d) none
15) The total number of words that can be made by writing the letters of the word PARAMETER so that no vowels is between two consonants is
a) 1440 b) 1800 c) 2160 e) none
16) The number of numbers are four different digits that can be formed from the digit of the number 1 2 3 5 6 such that the numbers are divisible by 4, is
a) 36 b) 48 c) 12 d) 24
17) let S be the set of all fractions from the set A to the set A. If n(A)= k then n(S) is
a) k! b) kᵏ c) 2ᵏ -1 d) 2ᵏ
18) Let A be the set of the four digit numbers a₁a₂a₃a₄ where a₁ >a₂>a₃>a₄ then n(A) is equal to
a) 126 b) 84 c) 210 d) none
19) The number of numbers divisible by 3 that can be formed by 4 different even digits is
a) 18 b) 36 c) 0 d) none
20) The number of 5 digit even numbers that can be made with the digit 0, 1 , 2 and 3
a) 384 b) 192 c) 768 d) none
21) The number of 4 digit number that can be made with the digit 1, 2, 3, 4 and 5 in which at least two digits are identical is
a) 4⁵ - 5! b) 505 c) 600 d) none
22) the number of words that can be made by writing down the letters of the word CALCULATE such that each word starts and ends with a constant is
a) 5.7!/2 b) 3.7!/2 c) 2.7! d) none
23) The number of numbers of 9 different digits such that all the digits in the first four places are less than the digits in the middle and all the digits in the last four places are greater than that in the middle is 2.4! b) (4!)² c) 8! d) none
24) In the decimal system of numeration the number of 6 digit numbers in which the digits in any place is greater than the digit to the left of it is
a) 210 b) 84 c) 126 d) none
25) the number of 5 digit numbers in which no 2 consecutive digits are identical is
a) 9² . 8³ b) 9x 8⁴ c) 9⁵ d) none
26) in the decimal system of numeration the number of 6 digit numbers in which the sum of the digits is divisible by 5 is
a) 180000 b) 540000 c) 5 x 10⁵ d) none
27) The sum of all the numbers of four different digits that can be made by the using 0, 1, 2,and 3: is
a) 26664 b) 39996 c) 38664 d) none
28) A teacher take three children from her class to the zoo at time as often as she can, but she does not take the same three childrens to the zoo more than once. she finds that she goes to the zoo 84 times more than a particular child goes to the zoo. The number of children in her class is
a) 12 b) 10 c) 60 d) none
29) ABCD is a convex quadrilateral 3, 4, 5 and 6 points are marked on the sides AB , BC, CD and DA respectively. The number of triangles with vertices on different sides is
a) 270 b) 220 c) 282 d) none
30) there are 10 points in a plane of which no 3 points are collinear and 4 points are concyclic. The number of different circles that can be drawn through at least 3 points of these points is
a) 116 b) 120 c) 117 d) none
31) in a polygon the number of diagonals is 54. The number of sides of the polygon is
a) 10 b) 12 c) 9 d) none
32) in a polygon no 3 diagonal concurrent. If the total number of points of intersection of the diagonals interior to the polygon be 70 then the number of diagonals of the polygon is
a) 20 b) 28 c) 8 d) none
33) n lines are drawn in a plane such that no two of them are parallel and no 3 of them are concurrent. The number of different points at which these lines will cuts is
a) ⁿ⁻¹∑ₖ₌₁ k b) n(n -1) c) n² d) none
34) The number of triangles that can be formed with 10 points as vertices, n of them being collinear , is 110. Then n is
a) 3, 4, 5, 6
35) There are three coplanar parallel lines. if any p points are taken on each of the lines, the maximum number of triangles with vertices at these points is
a) 3p²(p -1)+ 1 b) 3p²(p -1) c) p²(4p -3) d) none
36) Two teams are to play a series of 5 matches between them. A match ends in a way or loss or draw for a team. A number of peoples forecast the result of each match and no two people make the same forecast for the series of the matches. The smallest group of people in which one person forecast correctly for all the matches will contain n people, where n is
a) 81 b) 243 c) 486 d) none
37) A bag contain three black, 4 white and two red balls, all the balls bings different. The number of selections of atmost six balls containing balls of all the colours is
a) 42 (4!) b) 2⁶ x 4! c) (2⁶-1)(4!) d) none
38) in a room there are 12 bulbs of the same wattage, each having a separate switch . The number of ways to light the room with different amounts of illumination is
a) 12²-1 b) 2¹² c) 2¹²-1 d) none
39) In an examination 9 papers of candidates has to pass in more papers than the number of papers in which he fails in order to be successful. The number of ways in which he can unsuccessful is
a) 255 b) 256 c) 193 d) 319
40) The number of 5 digit numbers that can be made using the digit 1 and 2 and in which at least one digit is different, is
a) 30 b) 31 c) 32 d) none
41) In a club election the number contestants is one more than the number of maximum candidates for which a voter can vote . if the total number of ways in which a voter can vote be 62 then the number of candidates is
a) 7 b) 5 c) 6 d) none
42) The total number of selection of atmost n things from (2n +1) different things is 63. Then the value of n is
a) 3 b) 2 c) 4 d) none
43) let 1≤ m< n ≤ p. The number of subsets of the set A={1,2,3,....p} having m,n as the least and the greatest elements respectively, is
a) 2ⁿ⁻ᵐ⁻¹ -1 b) 2ⁿ⁻ᵐ⁻¹ c) 2ⁿ⁻ᵐ
44) The number of ways in which n different prizes can be distributed among m(<n) persons if each is entitled to receive atmost n -1 prizes ,is
a) nᵐ b) mⁿ c) mn d) none
45) The number of possible outcomes in a throw of n ordinary dies in which at least one of the dice shows an odd number is
a) 6ⁿ-1 b) 3ⁿ -1 c) 6ⁿ - 3ⁿ d) none
46) The number of different 6 digit numbers that can be formed using the three digit 0, 1, 2 is
a) 3⁶ b) 2x 3⁵ c) 3⁵ d) none
47) The number of different matrices that can be formedd with elements 0, 1, 2norn3, each Matrix are 4 element, is
a) 3x 2⁴ b) 2x 4⁴ c) 3 x 4⁴ d) none
48) Let A be a set of n(≥ 3) distinct elements . The number of triplets (x,y,z) of the elements of A in which at least two co-ordinates are equal is
a) ⁿP₃ b) n³ - ⁿP₃ c) 3n² - 2n d) 3²(n -1)
49) The number of different pairs of words (_ _ _, _ _ _) that can be made with the letters of the word STATICS is
a) 828 b) 1260 c) 396 d) none
50) Total number of 6 digit numbers in which all the odd digits and only odd digit appear, is
a) 5(6!)/2 b) 6! c) 6!/2 d) none
51) The number of divisors form 4n +2(n≥ 0) of the integer 240 is
a) 4 b) 8 c) 10 d) 3
52) in the next World Cup of cricket there will be 12 teams , divided equally in two groups. Teams of each group will play a match against each other. From each group 3 top team will qualify for the next round. In this round each team will play against other once. Four top teams of this round will qualify for the semi final round, where each team will play against the others once. Two top teams of this round will go to the final round, where they will play the best of 3 matches. The minimum number of matches in the next world cup will be
a) 54 b) 53 c) 38 d) none
53) The number of different ways in which 8 persons can stand in a row so that between two particular persons A and B there are always two persons, is
a) 60(5!) b) 15(4!(5!) c) 4! x 5! d) none
54) four couples (husband and wife) decide to form a committee of four members . The number of different committees that can be formed in which no couple finds a place is
a) 10 b) 12 c) 14 d) 16
55) from four gentlemen and 6 ladies a community of 5 to be selected. The number of ways in which the committee can be formed so that is gentlemen are in majority is
a) 66 b) 156 c) 60 d) none
56) There are 20 questions in a question paper. If no two students solve the same combination questions but solve equal number of questions then then the maximum number of student who appeared who appeared in the examination is
a)²⁰C₉ b) ²⁰C₁₁ c)!²⁰C₁₀ d) none
57) 9 hundred distinct n-digit positive numbers are to formed using only the digits 2, 5 and 7. The smallest value of n for which this is possible is
a) 6 b) 7 c) 8 d) 9
58) The total number of integral solutions (x, y, z) such that XYZ= 24 is
a) 36 b) 90 c) 120 d) none
59) The number of ways in which the letters of the word ARTICLE can be rearranged so that the even places are always occupied by consonants is
a) 576 b) ⁴C₃. 4! c) 2(4!) d) none
60) a cabinet ministers consist of 11 ministers, one ministers being the chief minister. A meeting is to be held in a room having a round table and 11 chairs round it, one of them being meant for the chairman. The number of ways in which the ministers can take their chairs, the chief minister occupying the chairman's place is
a) 10!/2 b) 9! c) 10! d) none
61) The number of a ways in which a couple can sit around a table with 6 guests if the couple take consecutive seats is
a) 1440 b) 720 c) 5040 d) none
62) The number of ways in which 20 different pearls of two colours can be set alternatively on a necklace there, being 10 pearls of each colour, is
a) 9! x 10! b) 5(9!)² c) (9!)² d) none
63) If r> p > q, the number of different selections of p+ q things taking r at a time, where p things are identical and q things are identical is
a) p+ q+ r b) p+ q- r +1 c) - p- q+ r +1 d) none
64) There are four mangoes , three apples, two oranges and one each of 3 other varieties of fruits. The number of ways of selecting at least one fruit of each kind is
a) 10! b) 9! c) 4! d) none
65) The number of proper divisors of 2ᵖ. 6ᑫ. 15ʳ is
a) (p+ q+1)(q+r+1)(r+1)
b) (p+ q+1)(q+r+1)(r+1) -2
c) (p+ q)(q+r)r -2
d) none
66) The number of proper divisors of 1800 which are divisible by 10, is
a) 18 b) 34 c) 27 d) none
67) The number of odd proper devisors of 3ᵖ. 6ᑫ. 21ʳ is
a) (p+1)(m+1)(n+1)-2
b) (p+ m+ n+1)(n+1) -1
c) (p+ 1)(m+1)(n+ 1)
d) none
68) The number of even proper devisors of 1008 is
a) 23 b) 24 c) 22 d) none
69) in a test there were n questions. In the test 2ⁿ⁻ⁱ students of wrong answers to i questions where i= 1, 2, 3,.....n. if the total number of wrong answers given is 2047 then n is
a) 12 b) 11 c) 10 d) none
70) The number of ways to give 16 different things to 3 persons A, B and C so that B gets one more than A and C gets two more than B, is
a) 16!/(4!5!7!)
b) 4!5!7!
c) 16/(3! 5!8!) d) none
71) The number of ways to distribute 32 different things equally among 4 persons is
a) 32/(8!)³ b) 32!/( 8!)⁴ c) 32!/4 d) none
72) if 3n different things can be equally distributed among 3 persons in k ways then the number of ways to divide 3n things in 3 equal groups is
a) k x 3 b) k/3! c) (3!)ᵏ d) none
73) in a packet there are m different books, n different pens and p different pencils. The number of selections of at least one articles of each type from the packet is
a) 2ᵐ⁺ⁿ⁺ᵖ -1 b) (m+1)(n +1)(p+1) -1 c) 2ᵐ⁺ⁿ⁺ᵖ d) none
74) The number of 6 digits number that can be made with the digit 1, 2, 3 and 4 and having exactly two pairs of digit is
a) 480 b) 540 c) 1080 d) none
75) The number of words of 4 letters containing equal number of vowels and consonants , repeatation being allowed , is
a) 105² b) 210 x 243 c) 105 x 243 d) none
76) The number of ways in which 6 different balls can be put in two boxes of different sizes so that no box remains empty is
a) 62 b) 64 c)!36 d) none
77) A shopkeeper sells three varieties of perfumes and he has a large number of bottles of the same size of each variety in his stock. There are five places in a row in his showcase. The number of different ways of displaying the 3 varieties of perfumes in the showcase is
a) 6 b) 50 c) 150 d) none
78) The numbers of arrangement of the letters of the word BHARAT taking 3 at a time is
a) 72 b)!120 c) 14 d) none
79) The number of ways to fill each of the four cells of the table with a distinct natural numbers such that the sum of the number is 10 and the sums of the numbers placed diagnolly are equal is
a) 2!2! b) 4! c) 2. 4! d) none
80) in the figure, 4 digit numbers are to be formed by filling the places with digits . The number of different ways in which the places can be filled by digits so that the sum of the numbers forme is also a 4 digit number and in no place the addition will carrying is
a) 555⁴ b) 220 c) 45⁴ d) none
81) The number of positive integral solutions of x+ y+ z= n n belongs to N, n≥ 3, is
a) ⁿ⁻¹C₂ b) ⁿ⁻¹P₂ c) n(n -1) d) none
82) The number of non negative integral solution of a+ b+ c+ d= n, n belongs to N, is
a) ⁿ⁺³P₂ b) {(n+1)(n+2)(n+3)}/6 c) ⁿ⁻¹Cₙ₋₄ d) none
83) The number of points (x,y, z)) in space, whose each co-ordinate is a negative integer such that x+ y+ z+12=0, is
a) 385 bb) 55 c) 110 d) none
84) If a,b,c are 3 natural numbers in AP and a+ b+ c= 21, then the possible number of values of the order ped triplet (a, b, c) is
a) 15 b) 14 c) 13 d) none
85) if a,b,c,,d are odd natural numbers that a + b + C + d = 20 then the number of values of the ordered quadruplet (a,b,c,d) is
a) 165 b) 455 c) 310 d) none
86) if x, y, z are integers and x≥0, y≥1, z≥2, x+ y+ z= 15 then the number of values of the ordered triplet (x,y,z) is
a) 91 b)) 455 c) ¹⁷C₁₅ d) none
87) If a,b,c are positive integer such that a + b + c≤ 8 then the number of possible values of the ordered triplet (a,b,c) is
a) 84 b) 56 c)) 83 d) none
88) The number of different ways of the distributing 10 marks among 3 questions, each question carrying atleast one mark is
a) 72 b) 71 c) 36 d) none
89) The number of ways to give away 20 apples in 3 boys, each boy receiving at least 4 apples, is
a) ¹⁰C₈ b) 90 c) ²²C₂₀ d) none
90) The position vector of a point P is r= xi + yj+ zk, where x ,y, z belongs to N and a= i+ j+ k, if r. a = 10, the number of possible position of P is
a) 36 b) 72 c) 66 d) none
**
91) If P= n(n²-1)²(n²-2²)(n²-3²).....(n²- r²), n> r, n belongs to N, then P is divisible by
a) (2r+2)! b) (2r -1)! c) (2r+1)! d) none
92) ⁿ⁺⁵Oₙ₊₁ = 11(n -1)/2. ⁿ⁺³Pₙ then the value of n is
a) 7 b) 8 c) 6 d) 9
93) If ⁿC₄, ⁿC₅ and ⁿC₆ are in AP then n is a
a) 8 b) 9 c) 14 d) 7
94) The product of r consecutive integers is divisible by
a) r b) ʳ⁻¹ₖ₌₁∑ k c) r! d) none
95) There are 10 bags B₁, B₂, B₃, .....B₁₀, which contain 21, 22, ....30 different articles respectively. The total number of ways to bring out 10 articles from a bag is
a) ³¹C₂₀ - ²¹C₁₀ b) ³¹C₂₁ c) ³¹C₂₀ d) none
96) If the number of arrangements of n-1 things taken from n different things is k times the number of arrangements of n-1 things taken from n things in which two things are identical then the value of k is
a) 1/2 b) 2 c) 4 d) none
97) Kanchana has 10 friends among whom two are married to each other. she wishes to invite five of them for a party. if the married couple refuse to attend separately then the number of different ways in which she can invite 5 friends is
a) ⁸C₅ b) 2x ⁸C₃ c) ¹⁰C₅ - 2 x ⁵C₄ d) none
98) In a plane there are two families of lines y= x+ r, y= - z+ r, where r belongs {0, 1, 2, 3, 4}. The number of squares of the diagonals of the length 2 formed by the lines is
a) 9 b) 16 c) 25 d) none
99) there are n seats round a table numbered 1, 2, 3,....n. The number ways in which m(≤ n) persons can take seats is
a) ⁿPₘ b) ⁿCₘ x (m -1)! c) ⁿ⁻¹Pₘ₋₁ d) ⁿCₘ x m!
100) Let a= i+ j + k and let r= r be a variable vector such that r.j, r.j and r.k are positive integers. If r.a≤ 12 then the number of values of r is
a) ¹²C₉ -1 b) ¹²C₃ c) ¹²C₉ d) none
101) The total number of ways in which a beggar can be given at least one rupee from four 25 paise coins, three 50 paise coins and 2 one rupee coins, is
a) 54 b) 53 c) 51 d) none
102) for the equation x+ y+ z+ w= 19, the number of positive integral solutions is equals to
a) the number of ways in which 15 identical things can be distributed among 4 persons
b) the number of ways in which 19 identical things can be distributed among four persons
c) coefficient of x¹⁹ in (x⁰+ x¹+ x²+....+ x¹⁹)⁴
d) coefficient of x¹⁹ in (x + x²+ x³+....+ x¹⁹)⁴
1) If ⁿP₄ = 6 x ⁿ⁻¹P₄ then the value of n is
a) 22 b) 24 c) 48 d) 50
2) ⁿ⁺¹P₄ : ⁿ⁻¹P₃ = 72:5 Then the value of n is
a) 8 b) 9 c) 10 d) 11
3) If ¹⁰Pᵣ = 5040 then r is equal to
a) 2 b) 3 c) 4 d) 5
4) If ⁿPᵣ = ⁿPᵣ₊₁ and ⁿCᵣ = ⁿCᵣ₋₁ then the value of n and r are respectively equal to
a) 2,5 b) 3,2 c) 6,3 d) 7,4
5) If ¹²Pᵣ = ¹¹P₆ + 6, ¹¹P₅ then the value of r is
a) 3 b) 4 c) 5 d) 6
6) If (n +2)!= 2550. n! Then the value of n is
a) 47 b) 48 c) 49 d) 50
7) If ¹⁰Pᵣ = ⁹P₅ + 5. ⁹P₄ then the value of r is
a) 2 b) 3 c) 4 d) 5
8) If ⁴⁻ˣP₂ = 6 then the value of x is
a) 1 b) 2 c) 3 d) none
9) If ²ⁿ⁺¹Pₙ₋₁ : ²ⁿ⁻¹Pₙ = 3:5 Then the value of n is
a) 4 b) 3 c) 2 d) none
10) ⁿCᵣ₊₁ + ⁿCᵣ₋₁ + 2 ⁿCᵣ is equal to
a) ⁿ⁺²Cᵣ b) ⁿ⁺¹Cᵣ₊₁ c) ⁿ⁺¹Cᵣ d) ⁿ⁺¹Cᵣ₊₁
11) If ¹⁶Cᵣ = ¹⁶Cᵣ₊₂ then ʳC₄ is equal to
a) 21 b) 27 c) 35 d) 39
12) If 3 ˣ⁺¹C₂ + ²P₂ x= 4. ˣP₂, then the value of x is
a) 2 b) 3 c) 5 d) 7
13) ²ⁿC₃ : ⁿC₂ = 44:3 then n is equal to
a) 3 b) 4 c) 5 d) 6
14) If ⁿP₅ = 60. ⁿ⁻¹P₃ then n is equal to
a) 8 b) 10 c) 12 d) 14
15) The value of ²⁰C₅ + ⁵ⱼ₌₂∑²⁵⁻ʲC₄ is
a) 24504 b) 44502 c) 42504 d) 45042
16) If ⁿC₄ = 21. ⁿ⁾²C₃ then the value of n is
a) 6 b) 10 c) 12 d) 14
17) The value of ⁴⁰C₃₁ + ¹⁰ⱼ₌₀∑⁴⁰⁺ʲ₁₀₊ⱼ is equal to
a) ⁷²C₃₁ b) ²⁷C₈ c) ¹⁹C₁₁ d) ⁵¹C₂₀
18) If ²⁸C₂ᵣ : ²⁴C₂ᵣ₊₄ = 225:11 then value of r is
a) 7 b) 9 c) 14 d) none
19) If ⁵⁶Pᵣ₊₆ : ⁵⁴Pᵣ₊₃ = 30800:1, then the value of r is
a) 11 b) 21 c) 31 d) 41
20) If ⁿ⁻¹C₃ + ⁿ⁻¹C₄ > ⁿC₃, then
a) n<6 b) n>7 c) n < 5 d) n> 6
21) If ⁿ⁺²C₈ : ⁿ⁻²C₄ = 171:2 then the value of n is
a) 18 b) 19 c) 20 d) 21
22) The value of ⁴⁷C₄ + ⁵ᵣ₌₁∑⁵²⁻ʳC₃ is
a) 277025 b) 275027 c) 507227 d) 270725
23) ⁿPᵣ = 504 and ⁿCᵣ = 84 then the value of n is equal to
a) 3 b) 6 c) 9 d) 12
24) The number of different algebraic expressions that can be made by combining the letters p,q,r,s and t in this order with the signs '+' and '-' taking all the letters together is
a) 21 b) 23 c) 31 d) 32
25) The number of different factors of 2160 is
a) 29 b) 39 c) 49 d) none
26) 8 different chocolates can be distributed equally between two boys in
a) 70 ways b) 35 ways c) 38 ways d) 19 ways
27) From a group of person the number of ways of selecting 5 persons is equals to that of 8 persons. The number of person in the group is
a) 29 b) 25 c) 13 d) 11
28) a man has 5 oranges and 4 mangoes. How many different selections having at least one orange is possible?
a) 25 b) 30 c) 35 d) 40
29) A man has 6 friends . The number of ways in which he can invite one or more them to his house is
a) 6! b) 6! - 1 c) 2⁶! d) none
30) A 5 digited number is divisible by 3 and it is formed by using 0, 1, 2, 3, 4 and 5 without repetition. The total number of ways in which such a number can be formed is
a) 126 b) 216 c) 621 d) 261
31) all the letters of the string AEPRAB are arranged in all possible ways. The number of such arrangements in which two vowels are not adjacent to each other is
a) 220 b) 115 c) 72 d) 65
32) The number of ways in which the letters of the string ANRTIPF can be arranged so that the vowels may appear in the odd place is
a) 1230 b) 1350 c) 1440 d) 1570
33) if there are 10 persons in a gathering and if each of them shakes hand with everyone else , then the number of hand shakes that takes place in the gathering is
a) 20 b) 45 c) 2¹⁰ d) 10²
34) The number of parallelograms that can be formed from a set of 4 parallel lines intersecting another set of 3 parallel lines is
a) 21 b) 20 c) 18 d) 16
35) The number of students to be selected at a time from a group of 16 students , so that the number of selection is the greatest is
a) 16 b) 14 c) 8 d) none
36) The number of different arrangement with the letters of the word ALGEBRA so that 2A's are not together is
a) 1800 b) 2520 c) 720 d) none
37) The number of odd integer of 6 significant digits can be formed with the digit 0, 1, 4, 5, 6, 7 without repetition of the digit is
a) 96 b) 108 c) 266 d) 288
38) The number of words that can be made by writing down the letters of the word CALCULATE such that each word starts and ends with a constant is
a) 7! b) 7!/2 c) 5(7!)/2 d) 9(7!)/2
39) The number of triangles that can be formed with 10 points as vertices, K of them them being collinear, is
a) 3 b) 5 c) 7 d) none
40) The number of ways in which 5 '+' sign and 3 'X' sign can be arranged in a row is
a) 56 b) 65 c) 72 d) 81
41) The number of ways in a which 15 class XI students and 12 class XII student be arranged in a line so that no 2 class. XII students may occupy consecutive positions is
a) 12! x 16!/4! b) 15! x 13!/4! c) 16! x 13!/ 4! d) 15! x 16!/ 4!
42) The number of strings of 3 letters that can be formed with the letter chosen from CALCUTTA is
a) 48 b) 62 c) 96 d) 102
43) The number of permutation of the letters of the MADHUBANI where the arrangementa do not begin with M but end with I is
a) 16740 b) 17460 c) 14670 d) none
44) The number of ways in which a committee of 5 persons may be formed out of 6 men and 4 women under the condition that least one woman has to be selected necessarily is
a) 252 b) 246 c) 242 d) none
45) Given that balls of the same colour are identical, the number of ways in which 18 white balls of 19 red balls may be arranged in a row so that no two white balls may come together is
a) 180 b) 190 c) 200 d) 210
46) In an examination there are 3 multiple choice questions and each question has at 4 choices. The number of ways in which one can fail to get all answers correct is
a) 12 b) 21 c) 36 d) 63
47) The number of diagonals that can be drawn by joining the vertices of an octagon is
a) 28 b) 20 c) 18 d) 16
48) out of 6 given point 3 are collinear. The number of different straight lines that can be drawn by joining any two points from those 6 given point is
a) 12 b) 10 c) 9 d) none
49) The total number of selections of at least one red ball from 4 red balls and 3 blue balls, if the balls of the same colour are different is
a) 95 b) 105 c) 120 d) 125
50) In an examination of 9 papers a candidate has to pass in more papers than the number of papers in which he fails in order to be successful. The number of ways in which he can unsuccessful is
a) 265 b) 255 c) 256 d) 625
51) The number of integers greater than 50000 that can be formed by the using the digits 3, 5, 6, 6, 7 is
a) 54 b) 40 c) 32 d) none
52) The number of arrangement that can be formed from the letters of the word VIOLENT, so that the vowels may occupy only odd positions is
a) 576 b) 574 c) 572 d) none
53) in a group of 15 boy there are 7 boys-scouts . The number of ways in which 12 boys can be selected from the group so as to include at least 6 boys-scouts is
a) 125 b) 127 c) 252 d) 255
54) 15 distinct objects may be divided into three groups 4, 5 and 6 objects in
a) 230230 ways b) 320320 ways c) 360360 ways d) 630630 ways
55) The number of different ways in which 1440 can be expressed as the product of two factors is
a) 18 b) 16 c) 14 d) none
56) The number of different rectangles ( regarding every square as a rectangle as well) that are there on a chess board is
a) 1280 b) 1284 c) 1296 d) 1300
57) The number of arrangements which can be made out of the letters of the word ALGEBRA without changing the relative position of the vowels and consonant is
a) 54 b) 64 c) 70 d) 72
58) The number of factors of 420 is
a) 22 b) 23 c) 24 d) none
59) A boat has a crew of 10 man of which three can row only on one side and 2 only on the other. The number of ways the crew can be arranged in the boat is
a) 142000 b) 144000 c) 124000 d) none
60) There are 10 points in a plane of which no 3 points are collinear and 4 points are concyclic. The number of different circles that can be drawn through at least 3 points of these point is
a) 117 b) 120 c) 122 d) 124
61) The number of 6 digited integers that can be made using the digits 3 and 4 and in which at least 2 digits are different is
a) 60 b) 61 c) 62 d) none
62) The sum of the digits in unit place of all the four digited numbers formed with the help of 2,3,4,5, taken all at time is
a) 54 b) 108 c) 84 d) none
63) The number of different ways in which 15 distinct objects may be divided into three groups of 5 objects each is
a) 216216 b) 126126 c) 216612 d) 126612
64) the number of different arrangements that can be made out of the latest of the word ALLAHABAD , such that the vowels may occupy the even positions only is
a) 70 b) 50 c) 60 d) 120
65) The number of ways in which 4 letters can be posted in 3 post boxes is
a) 256 b) 81 c) 12 d) none
66) At an election, a voter may vote for any number of candidates, not greater than the number to be elected p. There are 10 candidates and 4 are to be elected. if a water votes for at least one candidate , then the number of ways in which he can vote is
a) 385 b) 1110 c) 5040 d) 6210
67) How many ways are there to arrange the letters in the word GARDEN with the vowels in alphabetic order?
a) 360 b) 240 c) 120 d) 480
68) The numbers of different solutions (x,y,z) of the equation x+ y + z= 10, where each of x, y, z is a +ve integer, is
a) 36 b) ¹⁰C₃ - ¹⁰C₂ c) 10³ -10 d) none
69) If ²ⁿC₁ + ²ⁿC₂ +....+ ²ⁿCₙ₋₁ + (1/2) ²ⁿCₙ = 127, then n is equal to
a) 4 b) 5 c) 3 d) none
1b 2a 3c 4b 5d 6c 7d 8a 9a 10d 11c 12b 13d 14b 15c 16b 17d 18a 19d 20b 21b 22d 23b 24d 25b 26a 27c 28a 29d 30b 31c 32c 33b 34c 35c 36a 37d 38c 39b 40a 41d 42c 43d 44b 45b 46d 47b 48d 49c 50c 51d 52a 53c 54d 55a 56c 57d 58b 59b 60a 61c 62d 63b 64c 65b 66a 67a 68b 69a 70a
1) The sum of all the numbers of four different digits that can be made by the using the digits 0, 1, 2 and 3 is
a) 26664 b) 39996 c) 38664 d) none
2) The sum of all four digit numbers that can be formed by using the digit 2, 4, 6, 8( when repetition of digits is not allowed) is
a) 133320 b) 533280 c) 53328 d) none
3) The number of ordered pairs of integers (x,y) satisfing the equation x²+ 6x + y²= 4 is
a) 2 b) 8 c) 6 d) none
4) Among 10 persons , A, B, C are to speak at a function. The number of ways in which it can be done if A wants to speak before B and B wants to speak before C is
a) 10!/24 b) 9!/6 c) 10!/6 d) none
5) In how many ways can a team of 11 players be formed out of 25 players, if six out of them are always to be included and 5 always to be excluded
a) 2020 b) 2002 c) 2008 d) 8002
6) The number of ways in which the letters of the word PERSON can be placed in the squares of the given figure
so that no rows remains empty is
a) 24. 6! b) 26. 6! c) 26 . 7! d) none
7) The number of words of 4 letters that can be formed from the letters of the word EXAMINATION is
a) 1464 b) 2454 c) 1678 d) none
8) The number of even divisors of the number N= 12600= ³3²5²7 is
a) 72 b) 54 cc) 18 d) none
9) In an election, the number of candidates is one greater than the persons to be elected. If a voter can vote in 254 ways , the number of candidates is
a) 7 b) 10 c) 8 d) 6
10) A person predicts the outcome of 20 cricket matches of his home team. Each match can result in a either win, loss or tie for the home team . Total number of ways in which he can make the predictions so that exactly 10 predictions are correct is equals to
a) ²⁰C₁₀ x2¹⁰ b) ²⁰C₁₀x 3²⁰ c) ²⁰C₁₀x 3¹⁰ d) ²⁰C₁₀x 2²⁰
11) There are 10 points in a plane of which no 3 points are collinear and 4 points are concyclic. The number of different circles that can be drawn through at least 3 points of these points
a) 116 b) 120 c) 117 d) none
12) There are three coplaner planet parallel lines. if any p points are taken on each of the lines, the maximum number of triangles with vertices on these points is
a) 3p²(p -1) +1
b) 3p²(p -1)
c) p²(4p -3) d) none
13) The maximum number of points of intersection of the five lines and four circles is
a) 60 b) 72 c) 62 d) none
14) The number of integral solutions of x + y + z= 0 with x≥ 5, y≥-5, z≥ -5 is
a) 134 b) 136 c) 138 d) 140
15) The number of ways in which 12 books can be put in three shelves with four on each shelf is
a) 12!/(4!)³ b) 12!/(3!)(4!)³ c) 12!/(3!)³(4!) d) none
16) Number of ways in which 25 identical things be distributed among 5 persons if each gapets odd numbers of things is
a) ²⁵C₄ b) ¹²C₅ c) ¹⁴C₁₀ d) ¹²C₃ e) none
17) The total number of divisor of 480, that are of the form 4n +2, n≥ 0, is equals to
a) 2 b) 3 c) 4
18) The number of 3 digit numbers of the form xyz such that x< y and z≤ y is
a) 276 b) 285 c) 240 d) 244
19) A man has three friends. The number of ways he can invite one friend everyday for dinner on 6 successive nights so that no friends is invited more than three times is
a) 640 b) 320 c) 420 d) 510
20) A bag contains four one rupee coins, two twenty five paise coins and five ten paise coins. In how many ways can an amount, not less than Rs 1 be taken out from the bag ? (consider coins of the same denomination to be identical)
a) 71 bb) 72 c) 73 d) 80
1) There are 10 points in a plane of which no three points are collinear but 4 points are concyclic. The number of different circles that can be drawn through atleast 3 points of these points are
a) 116 b) 120 c) 117 d) none
2) In an examination of 9 papers a candidate has to pass in more papers than the number of papers in which he fails in order to be successful. The number of ways in which he can be unsuccessful is
a) 255 b) 256 c) 193 d) 319
3) Let 1≤ m < n ≤ p. The number of subsets of the set A={1,2,3,....p) having m,n as the least and the greatest elements respectively, is
a) 2ⁿ⁻ᵐ⁻¹ -1 b) 2ⁿ⁻ᵐ⁻¹ c) 2ⁿ⁻ᵐ d) none
4) The number of even proper divisors of 1008 is
a) 121110 d) none
5) If a,b,C be three natural numbers in AP and a+ b+ c = 21, then the possible number of values of a,b,c is
a) 15 b) 14 c) 13 d) 16
6) The number of selections of four letters from the letters of the word ASSASSINATION is
a) 72 b) 71 c) 66 d) 52
7) The number of times the digits 5 will written while listing the integers from 1 to 1000 is
a) 271 b) 275 c) 285 d) 300
8)
1) In how many ways can clean & clouded (overcast) days occur in a week assuming that an entire day is either clean or clouded.
2) Four visitors A, B C , D arrive at a town which has 5 hotels. In how many ways can they disperse themselves among 5 hotels, if 4 hotels are used to accommodate them.
3) If the letters of the word VARUN are written in all possible ways and then are arranged as in a dictionary, then the rank of the word VARUN is
a) 98 b) 99 c) 100 d) 101
4) How many natural numbers are their 1 to 1000 which have none of their digits repeated.
5) 3 different railway passes are allotted to 5 students. The number of ways this can be done is
a) 60 b) 20 c) 15 d) 10
6) There are 6 roads between A& B and 4 roads between B and C.
a) In how many ways can one drive from A to C by way of B?
b) In how many ways can one drive from A to C and back to A, passing through B on both trips?
c) In how many ways can one drive the circular trip described in (b) without using the same road more than once.
7)a) How many car number plates can be made if each plate contains 2 different letters of English alphabet, followed by 3 different digits.
b) Solve the problem, if the first digit can not be 0.
8) a) Find the number of four letters word that can be formed from the letters of the word HISTORY. (each letter to be used atmost once)
b) How many of them contains only consonants?
c) How many of them begin with a vowel?
d) How many contains the letters Y ?
e) How many begin with T and end in a vowel?
f) How many begin with T and also contain S ?
g) How many contain both vowels?
9) If repetition are not allowed
a) How many 3 digit numbers can be formed from the six digits 2,3,5,6,7, 9
b) How many of these are less than 400?
c) How many are even?
d) How many are odd ?
e) How many are multiples of 5?
10) How many two digit numbers are there in which the tens digit and the unit digit are different are odd ?
11) Every telephone number consists of 7 digits. How many telephone numbers are there which do not include any other digits but 2,3,5, 7
12)a) In how many ways can four passengers be accommodated in three railway carriages, each carriage can accommodate any number of passengers.
b) In how many ways four persons can be accommodated in different chairs if each person can occupy only one chair.
13) How many odd numbers of five distinct digits can be formed with the digits 0,1,2,3,4?
14) Number of natural numbers between 100 and 100 such that atleast one of their digits is 7, is
a) 225 b) 243 c) 252 d) none
15) How many four digit numbers are there which are divisible by 2?
16) The 120 permutation of MAHES are arranged in dictionary order, as if each were an ordinary five letters word. The last letter of the 86th word in the list is
a) A b) H c) S d) E
17) Find the number of 7 lettered palindromes which can be formed using the letters from the English alphabets.
a)
18) Number of ways in which 7 different colours in a rainbow can be arranged if green is always in the middle.
19) Number of 4 digit numbers of the form N= abcd which satisfy three conditions:
i) 4000≤ N<6000 b) N is multiple of 5 iii) 3≤ b < c ≤ 6 is equal to
a) 12 b) 18 c) 24 d) 48
20) Find the number of ways in which the letters of the word MIRACLE can be arranged if vowels always occupy the odd places.
21) The number of 10 digit numbers such that the product of any two consecutive digits in the number is a prime, is
a) 1024 b) 2048 c) 512 d) 64
22) A letter lock consist of three rings each marked with 10 different letters. Find the number of ways in which it is possible to make an unsuccessful attempts to open the lock.
23) How many 10 digits numbers can be made with odd digits so that no two consecutive digits are same.
24) How many natural numbers are there with the property that they can be expressed as the sum of the cubes of two natural numbers in two different ways.
TEST PAPER ON TRIGONOMETRY
1) If sinα, cosα,tanα are in GP, show that cot⁶α- cot²α = 1.
2) The value of sin3x/sinx - cos3x/cosx is
a) -2 b) 2 c) -3 d) 3
3) If α + β = tan⁻¹m, α - β = tan⁻¹n, express tan2α in terms of m,n.
4) The value of (6 cos40° - 8 cos³40° is
a) 1 b) -1 c) -√3 d) √3
5) Evaluate: tan((1/2)tan⁻¹3 + (1/2) tan⁻¹ (1/3)).
6) If sinθ + cosθ = √2 sinθ, the value of tan2θ is
a) -1 b) 0 c) 1 d) √2
7) The least value of 3 sin²θ+ 4 cos²θ is
a) 3 b) 4 c) 5 d) none
8) Two equal arcs of two circles subtend angles of 60° and 75° at the centre. Find the ratio of the radii of the two circles.
9) If 6α = 11π, then the value of 2 cosα + 3 tanα is
a) 1 b) 0 c) √3 d) 2√3
10) If cos(α+ β)=0, then sin(α +2β) equals
a) sinα b) cosα c) sinβ d) cosβ
11) If (1- tanα)(1- tanβ)= 2, find the value of α + β.
12) If sinα = -3/5 (π<α <3/2), then cos(α/2) is
a) -1/√10 b) 1/√10 c) 3/√10 d) none
13) Evaluate: (2 cos40° - cos20°)/sin20°.
14) Show that, tan(π/8)= √2 -1.
15) Solve: tan⁻¹x + tan⁻¹2x + tan⁻¹3x=π.
16) In a ∆ ABC, if A+ C= 2B, evaluate (cosA - cosC)/(sinC - sinA).
17) Find the value of (tan75° + cot75°).
18) In a ∆ ABC, if r₁, r, r₂ are in HP, show that a,b,c are in AP.
19) If 3 cosα + 4 sinα= 5, the value of tanα is
a) 3/4 b) 3/5 c) 4/5 d) none
20) Find the value of the expression
cosx + cos(x +72°)+ cos(x + 144°)+ cos(x - 72°)+ cos(x - 144°).
21) Find the maximum and minimum values of sin²θ + cos⁴θ.
TEST PAPER ON ALGEBRA
1) Find the smallest positive integer n, for which {(1- i).(1+ i)}ⁿ is purely real.
2) If 2¹⁰ is approximate 10², then log₂10 is approximately.
a) 0.3 b) 4/3 c) 3 d) 1/3
3) If ˣ√a = ʸ√b= ᶻ√c and a,b,c be in GP , show that x,y,z are in AP.
4) if the roots of ax²+ bx + c=0 are both positive, then
a) a>0, b>0, c<0
b) a>0, b<0, c>0
c) a<0, b>0, c<0
d) a<0, b<0c,>0
5) Resolve into factors : p²- pq + q².
6) If 2x = a+ 1/a and 2y= b + 1/b (a>1, b>1), then find the value of xy - √{(x²-1)(y²-1)}.
7) The sum of the n terms of a GP is (1/3)(2²ⁿ⁺¹ -2); find its common ratio.
8) Prove that 1/4< log₁₀ 2 < 1/3.
9) If logₐv = p and logₐᵥv= q, then p and q are connected by
a) p+ q - pq b) p+ q = p/q c) p- q = p/q d) p - q = pq
10) Of the numbers 2, x, y, 12 (x>0, y> 0), if first three are in GP and last 3 in AP, find x and y.
11) If S be the sum and R the sum of the reciprocal of n terms of a GP, whose first term is 4 and last term 36. Find the value of S/R.
12) For what value of m, the expression x²+ 4x + log₂m will be perfect square ?
13) Find the amplitude of (1+ sinα + i cosα).
14) Express (1+ ix)(1+ iy)(1- ix)(1- iy) as sum of two squares.
15) If z= x + iy be a complex number, than |z²| equals
a) z² b) (conjugate of z)² c) z x conjugate of z d) none
16) simplify: 1/(1+ logₐvc) + 1/(1+ logᵥca) + 1/(1+ log꜀av).
17) For what value of k, does the equation 8x²+ 2x + k= 0 have its one root square of the other ?
18) In an examination a candidate has to pass in each case of the four subjects. The number of ways he can fail in the examination is
a) 1 b) 15 c) 16 d) 24
19) Find the coefficient of x⁴ in (1+ 2x + 3x²+ 4x³+.....∞)².
20) Which term in (x + 1/x)⁹ is independent of x?
a) 2nd b) 3rd c) 4th d) 5th
21) calculate √2 correct to 3 places of dec by binomial expansion.
22) prove that the sum of the coefficient of xʳ and xʳ⁻¹ in the expansion of (1+ x)ⁿ is equal to the coefficient of xʳ in (1+ x)ⁿ⁺¹.
23) If α, β are the roots of equation ax²+ bx + c=0, evaluate
1/(aα + b) + 1/(aβ + b).
24) For what value of p, the expression x²+ 4x + 4 and x²+ px + 6 for the common factor ?
25) For what value of m, the roots of x² -(m -1)x + m + 1/4= 0 are real and equal.
26) For what values of k, the roots of Kx(1- x)= 1 will not be real ?
27) In a GP sum of n terms is 255, the last term is 128 and the common ratio is 2. Find n.
28) Find the square root of 9i.
29) The sum of n terms of a series is (n/3)(4n²-1). The 5th term of the series is
a) 81 b) 84 c) 121 d) 165
30) At an election, a voter may vote for any number of the candidates not greater than the number of to be chosen . There are 6 candidates and four of them are to be chosen. Find the number of ways in which a voter may vote.
31) Find the greatest value of |z +1| if |z +4|≤ 3. (z = x + iy).
32) Find the middle terms in the expansion of (1+ x)⁹?
33) prove that, log(2+ 2+2²+2³+.....2⁹) = 10 log2.
34) Show that e⁻¹ = 2/3!+ 4/5! + 6/7! +.....∞.
35) The middle term of the progression 3,7,11,.....147 is
a) 71 b) 73 c) 75 d) 77
36) If α, β are the roots of x²- 2x +2= 0, the least integer n (>0) for which αⁿ/βⁿ = 1, is
a) 2 b) 3 c) 4 d) none
37) The number of terms in expansion (x + y+ z)¹² is
a) 13 b) 78 c) 91 d) none
38) A polygon has 44 diagonals; the number of its side is
a) 8 b) 7 c) 11 d) none
PERMUTATION & COMBINATION
1) If 5 x ⁿP₄ = 6 x ⁿ⁻¹P₄ then the value of n is
a) 22 b) 24 c) 48 d) 50
2) ⁿ⁺¹P₄ : ⁿ⁻¹P₃ = 72: 5 then the value of n is
a) 8 b) 9 c) 10 d) 11
3) If ¹⁰Pᵣ = 5040 then r is equal to
a) 2 b) 3 c) 4 d) 5
4) If ⁿPᵣ = ⁿPᵣ₊₁ and ⁿCᵣ = ⁿCᵣ₋₁ then the values of n and r are respectively equal to
a) 2,5 b) 3,2 c) 6,3 d) 7,4
5) If ¹²Pᵣ = ¹¹P₆ + 6. ¹¹O₅ then the value of r is
a) 3 b) 4 c) 5 d) 6
6) If (n +2)!= 2550. n! then the value of n is
a) 47 b) 48 c) 49 d) 50
7) If ¹⁰Pᵣ = ⁹P₅ + 5. ⁹P₄ then the value of r is
a) 2 b) 3 c) 4 d) 5
8) If ⁴⁻ˣP₂ = 6 then the value of x is
a) 1 b) 2 c) 3 d) none
9) If ²ⁿ⁺¹Pₙ₋₁ : ²ⁿ⁻¹Pₙ = 3:5 then the value of n is
a) 4 b) 3 c) 2 d) none
10) ⁿCᵣ₊₁ + ⁿCᵣ₋₁ + 2 ⁿCᵣ is equal to
a) ⁿ⁺²Cᵣ b) ⁿ⁺²Cᵣ₊₁ c) ⁿ⁺¹Cᵣ d) ⁿ⁺¹Cᵣ₊₁
11) If ¹⁶Cᵣ = ¹⁶Cᵣ₊₂ then ʳC₄ is equal to
a) 21 b) 27 c) 35 d) 39
12) If 3 ˣ⁺¹C₂ + ²P₂. x = 4 ˣP₂, then the value of x is
a) 2 b) 3 c) 5 d) 7
13) If ²ⁿC₃ : ⁿC₂ = 44: 3 then n is equal to
a) 3 b) 4 c) 5 d) 6
14) If ⁿP₅ = 60. ⁿ⁻¹P₃ then n is equal to
a) 8 b) 10 c) 12 d) 14
15) The value of ²⁰C₅ + ⁵ⱼ₌₂∑²⁵⁻ʲC₄ is
a) 24504 b) 44502 c) 42504 d) 45042
16) If ⁿC₄ = 21. ⁿ⁾²C₃ then the value of n is
a) 6 b) 10 c) 12 d) 14
17) The value of ⁴⁰C₃₁ + ¹⁰ⱼ₌₀∑⁴⁰⁺ʲC₁₀₊ⱼ is equal to
a) ⁷²C₃₁ b) ²⁷C₈ c) ¹⁹C₁₁ d) ⁵¹C₂₀
18) If ²⁸C₂ᵣ : ²⁴C₂ᵣ₋₄ = 225 : 11 then value of r is
a) 7 b) 9 c) 14 d) none
19) If ⁵⁶Pᵣ₊₆ : ⁵⁴Pᵣ₊₃ = 30800: 1, then the value of r is
a) 11 b) 21 c) 31 d) 41
20) If ⁿ⁻¹C₃ + ⁿ⁻¹C₄ > ⁿC₃, then
a) n< 6 b) n > 7 c) n <5 d) n > 6
21) If ⁿ⁺²C₈ : ⁿ⁻²C₄ = 171:2 then value of n is
a) 18 b) 19 c) 20 d) 21
22) The value of ⁴⁷C₄ + ⁵ᵣ₌₁∑⁵²⁻ʳC₃ is
a) 277025 b) 275027 c) 507227 d) 270725
23) If ⁿPᵣ = 504 and ⁿCᵣ = 84, then the value of n is equal to
a) 3 b) 6 c) 9 d) 12
24) The number of different algebraic expressions that can be made by combining the letters p,q,r,s and t in this order with the signs '+' and '-' taking all the letters together is
a) 21 b) 23 c) 31 d) 32
25) The number of different factors of 2160 is
a) 29 b) 39 c) 49 d) none
26) 8 different chocolates can be distributed equally between two boys in
a) 70 ways b) 35 ways c) 38 ways d) 19 ways
27) From a group of persons the number of ways of selecting 5 persons is equal to that 8 persons . The number of persons in the group is
a) 29 b) 25 c) 13 d) 11
28) A man has 5 oranges and 4 mangoes. How many different selections having at least one orange is possible ?
a) 25 b) 30 c) 35 d) 40
29) a man has 6 friends. The number of ways in which he can invite one or more than one of them to his house is
a) 6! b) 6!-1 c) 2⁶! d) none
30) A 5 digited number is divisible by 3 and it is formed by 0, 1, 2, 3, 4 and 5 without repetition. The total number of ways in which such a number can be formed is
a) 126 b) 216 c) 621 d) 261
31) All the letters of the string AEPRAB are arranged in all possible ways. The number of such arrangements in which two vowels are not adjacent to each other is
a) 220 b) 115 c) 72 d) 65
32) The number of ways in which the letters of the string ANRTIPF can be arranged so that the vowels may appear in the odd places is
a) 1230 b) 1350 c) 1440 d) 1570
33) if there are 10 persons in a gathering and if each of them shakes hand with everyone else, then the number of hand shakes that takes place in the gathering is
a) 20 b) 45 c) 2¹⁰ d) 10²
34) The number of parallelograms that can be formed from a set of 4 parallel lines intersecting another set of 3 parallel lines is
a) 21 b) 20 c) 18 d) 16
35) The number of students to be selected at a time from a group of 16 students, so that the number of selections is the greatest is
a) 16 b) 14 c) 8 d) none
36) The number of different arrangements with the letters of the word ALGEBRA so that the two As are not together is
a) 1800 b) 2520 c) 720 d) none
37) The number of odd integer of 6 significant digits that can be formed with the digits 0, 1, 4, 5, 6, 7 without repetition of the digits is
a) 96 b) 108 c) 266 d) 288
38) The number of words that can be made by writing down the letters of world CALCUTTA such that each word starts and ends with a constant is
a) 7! b) 7!/2 c) 5.7!/2 d) 9.7!/2
39) The number of triangles that can be formed with 10 points as vertices, k of them being collinear, is 110. Then value of k is
a) 3 b) 5 c) 7 d) none of
40) The number of ways in which 5 '+' sign and 3 'X' sign can be arranged in a row is
a) 56 b) 65 c) 72 d) 81
41) The number of ways in a which 15 class XI students and 12 class XII students be arranged in a line so that no two class XII students may occupy connective positions is
a) 12! x 16!/4! b) 15! x 13!/4! c) 16! x 13!/4! d) 15! x 16!/4!
42) The number of strings of 3 letters that can be formed with the letters chosen from CALCUTTA is
a) 48 b) 62 c) 96 d) 102
43) The number of permutations of the letters of the word MADHUBANI where the arrangementa do not begin with M but end with I is
a) 16740 b) 17460 c) 14670 d) none
44) The number of ways in which committee of 5 perss may be formed out of 6 men and four women under the condition that at least one woman has to be selected necessarily is
a) 252 b) 246 c) 242 d) none
45) Given that balls of the same colour identical, the number of a ways in which 18 white balls and 19 red balls may be arranged in a row so that no two white balls may come together is
a) 180 b) 190 c) 200 d) 210
46) in an examination there are 3 multiple choice questions and each question has four choices. The number of ways in which one can fail to get all answers correct is
a) 12 b) 21 c) 36 d) 63
47) The number of diagonals that can be drawn by joining the vertices of an octagon is
a) 28 b) 20 c) 18 d) 16
48) Out of six given points 3 are collinear . The number of different straight lines that can be drawn by joining any two points from those 6 given points is
a) 12 b) 10 c) 9 d) none
49) The total number of selections of at least one red ball from 4 red balls and 3 blue balls. If the balls of the same colour are different, is
a) 95 b) 105 c) 120 d) 125
50) In an examination of 9 papers a candidate has to pass in more papers than the number of paper in which he fails in order to be successful. The number of ways in which he can be unsuccessful is
a) 265 b) 255 c) 256 d) 625
51) The number of integers greater than 50000 that can be formed by using the digits 3, 5, 6, 6, 7 is
a) 54 b) 46 c) 32 d) none
52) The number of arrangements that can be formed from the letters of the word VIOLENT, so that the voeel may occupy only odd positions is
a) 576 b) 574 c) 572 d) none
53) in a group of 15 boys there are 7 boys scouts. The number of a ways in a which 12 boys can be selected from the groups so as to include at least 6 boys-scouts is
a) 125 b) 127 c) 252 d) 255
54) 15 distinct objects may be divided into 3 groups of 4,5 and 6 objects in
a) 230230 b) 320320 c) 360360 d) 630630
55) The number of different ways in which 1440 can be expressed as the product of two factors is
a) 18 b)) 16 c)) 14 d) none
56) The number of different rectangles (regarding every square as a rectangle as well) that are there on a chess board is
a) 1280 b) 1284 c) 1296 d) 1300
57) The number of arrangements which can be made out of the letters of the word ALGEBRA without changing the relative positions of the vowels and consonants is
a) 54 b) 64 c) 70 d) 72
58) The number of factors of 420 is
a) 22 b) 23 c) 24 d) none
59) a boat as a crew of 10 men of which 3 can row only one one side and 2 only on other. The number of ways the crew can be arranged in the boat is
a) 142000 b) 144000 c) 124000 d) none
60) There are 10 points in a plane of which no 3 points are collinear and 4 points are concyclic. The number of different circles that can be drawn through atleast three points of these points is
a) 117 b) 120 c) 122 d) 124
61) The number of 6 digited integers that can be made using that digit 3 and 4 and in which at least two digits are different, is
a) 60 b) 61 c) 62 d) none
62) The sum of the digit in unit place of all the 4 digited numbers formed with the help of 2,3,4,5, taken all at a time, is
a) 54 b) 108 c) 84 d) none
63) The number of different ways in which 15 distinct objects may be divided into 3 groups of 5 objects each, is
a) 216216 b) 126126 c) 216612 d) 126612
64) The number of different arrangements that can be made out of the letters of the word ALLAHABAD, such that the vowels may occupy the even positions only is
a) 70 b) 50 c) 60 d) 120
65) The number of ways in which 4 letters can be posted in 3 post boxes is
a) 256 b) 81 c) 12 d)) none
66) At an election a voter may vote for any number of candidates, not greater than the number to be selected. There are 10 candidates and 4 are to be elected. if a voter votes for at least one candidate, then the number of a ways in which he can vote is
a) 385 b) 1110 c) 5040 d) 6210
67) How many ways are there to arrange the letters in the word GARDEN with the vowels in alphabetical order ?
a) 360 b) 240 c) 12⁰ d) 480
68) The number of ways of distributing 8 identical balls in distinct boxes so that none of the boxes is empty is
a) 38 b) 21 c) 5 d) ⁸C₃
69) The number of different solutions (x,y,z) of the equation x+ y+ z = 10, where each of x, y, z is a+ve integer, is
a) 36 b) ¹⁰C₃ - ¹⁰C₂ c) 10³ d) none
70) If ²ⁿC₁ + ²ⁿC₂ + ....+ ²ⁿCₙ₋₁ + (1/2) ²ⁿCₙ = 127, then n is equal to
a) 4 b) 5 c) 3 d) none
1) α is the real cube root of 2 and β, γ are its imaginary cube roots,
then (xβ + yγ + zα)/(xγ + yα + zβ) is equal to
a) ³√2 b) ³√2 w² c) ω² d) none
2) The modulus amplitude form of the complex number -1 - I is
a) -√2(cos(π/4) + i sin(π/4))
b) √2(cos(3π/4) + i sin(3π/4))
c) √2(cos(-3π/4) + i sin(-3π/4)) d) none
3) The amplitude of √12 + 6{(1- i)/(1+ i)} is
a) π/3 b) 2π/3 c) -π/3 d) -2π/3
4) If x√2 = 1+ √-1, then the value of x⁶+ x⁴+ x² +1 is
a) 0 b) 4 c) -4 d) none
5) The value of √i + √-i is/are
a) √2 b) ±√2 c) ±√2 i d) ±√2, ±√2 i
6) If y= √(z² + 6x +8), then (1- iy)¹⁾² is equal to
a) ±(1/√2){i √(x +4) - √(x +2)}
b) ±(i/√2){√(x +4) - √(x +2)}
c) ±(1/√2){√(x +4) - i √(x +2)} d) none
7) The square roots of a² + 1/a² - 4i(a - 1/a) - 6 are
a) ±(a - i/a -2)
b) ±(a - 1/a -2i)
c) ±(a - 1/a +2i) d) none
8) If ³√(x + iy)= a + ib, where a,b, x, y are real, then x/a + y/b is equal to
a) 4(b² - a²) b) 4(a² + b²) c) 4(a² - b²) d) none
9) If n is a +ve integer, not a multiple of 3, the {(1+ √-3)/2}ⁿ + {(-1- √-3)/2}ⁿ is equal to
a) -1 b) 2 c) 0 d) none
10) If x + iy moves on the line 3x + 4y +5=0, then the least value of |x + it| is
a) 0 unit b) 1/5 unit c) 1 unit d) none
11) If z₁ = √3 i and z₂= -1+ √3 i, then amp(z₁z₂) is equal to
a) 7π/6 b) -5π/6 c) 5π/6 d) none
12) If {(2- i)x + i}/(1+ i) + {(1+ 2i)y + i}/(1- i) = -1/2 + 5i/2, where x and y are real, then x - y is equal to
a) 1 b) -1 c) 6 d) 8
13) If {(1+ i)x - 2i}/(3+ i) + {(2- 3i)y + i}/(3- i)= i, where x and y are real, then 4x + 9y is equal to
a) 10 b) -10 c) 3 d) -3
14) The modulus of {(1- i)/(3+ i)} + 4i/5 is
a) √5 unit b) √11/5 unit c) √5/5 unit d) none
15) The least positive integer n such that {2i/(1+ i)}ⁿ is a positive integer, is
a) 2 b) 4 c) 8 d) 16
16) If (√3 + i)¹⁰⁰ = 2¹⁰¹ (a + ib), then a is equal to
a) 4 b) -4 c) 1/4 d) -1/4
17) If (a + 1)²/(2a - i) = p + iq, then p² + q² is equal to
a) (a² +1)²/(4a² -1) b) (a²+1)²/(2a² -1) c) (a² +1)²/(4a² +1) d) none
18) The complex numbers z= x + iy which satisfy the equation|(z - 5i)/(z + 5i)|= 1, lie on
a) x = -5 b) y= 6 c) the x-axis d) the y-axis
19) For any complex number z= x + it, if the imaginary part of (2z +1)/(iz + 1) is -2, then the locus of z is
a) a straight line
b) a circle
c) an ellipse d) none
20) If the complex number z= x + iy satisfies the condition |(z - k)/(z + ki)|= 1, where k is any real number, then the locus of z is
a) a straight line b) a circle c) an ellipse d) none
21) The complex number z= x + iy satisfying the condition amp{(z - i)/(z + i)}=π/4 lies on
a) a straight line b) a circle c) an ellipse d) none
22) The complex number z= x + iy satisfying the condition amp{(z -1)/(z +1)}=π/6 lies on
a) a straight line b) a circle c) an ellipse d) none
23) If the arguments of (z - a)(conjugate of z - b) and {(√3+ i)(1+ √3 i)}/(1+ i) are equal where z is a complex number and a, b are real numbers, then the locus of z is
a) a straight line b) a circle c) a parabola d) none
24) The locus of the complex number z= x + it satisfying the condition 'real part of 1/z = 1/4' is
a) a straight line at a distance of 4 unit from the imaginary axis
b) a circle with radius 1 unit
c) a circle with radius 2 unit
d) a straight line not passing through the origin.
25) The equation conjugate of b .z + b. conjugate of z= c, where b is a non-zero complex constant and C is a real, represents
a) a circle b) a straight line c) a parabola d) an ellipse
26) If z = x + it and if z. conjugate of z - (2+ 3i)z - (2- 3i). conjugate of z+ 9 = 0, then the locus of z in the complex plane is
a) a straight line b) an ellipse c) a circle d) none
27) If z= x + iy and w= (1- iz)/(z - i), then |w|= 1 implies that, in the complex plane
a) z lies on the imaginary axis
b) z lies on the real axis
c) z lies on the unit circle with centre at origin
d) none of these happens.
28) If x+ iy = 1/(2+ cosθ + i sinθ), where x, y, θ are real, then as θ varies, in the complex plane the point z= x + iy moves on
a) a straight line b) an ellipse c) a parabola d) a circle
29) Let i² = -1; then
a) i and - i each has exactly one square root.
b) i has two square root but - i does not have any.
c) neither i and - i each has any square root.
d) i and - i each has exactly two square root.
30) The expression {(1+ i)ⁿ/(1- i)ⁿ⁻² is equal to
a) iⁿ⁺¹ b) - iⁿ⁺¹ c) 1 d) -2iⁿ⁺¹
31) The argument of the complex number z= (1+ i √3)²/4i(1- i √3) is
a) π/6 b) π/4 c) π/2 d) none
32) Let z and w be two non-zero complex numbers such that |z|= |w| and amp z + amp w =π, then z is equal to
a) w b) - w c) conjugate of w d) - conjugate of w
33) If z and w are two complex numbers satisfying the equation|(z + w)/(z - w)|= 1, Then z/w is a number which is
a) positive real
b) negative real
c) zero d) none
34) If z₁ and z₂ are two complex numbers such that
|z₁ + z₂|= |z₁|+ |z₂|, then
a) arg z₁ = arg z₂
b) arg z₁ + z₂ = 0
c) arg (z₁z₂)= 0 d) none
35) If z₁ and z₂ are two complex numbers such that |z₁|²+ |z₂|²= |z₁ + z₂|², then
a) Re(z₁/z₂)= 0
b) Im(z₁/z₂)= 0
c) Re z₁z₂ = 0
d) Um(z₁z₂)= 0
36) If z₁ and z₂ are two complex numbers, then|z₁ + √(z₁² - z₂²)| + |z₁ - √(z₁² - z₂²)| is equal to
a) |z₁| b) |z₂| c) |z₁ + z₂| d) |z₁ + z₂|+ |z₁ - z₂|.
37) If a² + b² = 1, then (1+ a+ ib)/(1+ a - ib) is equal to
a) a+ ib b) a- ib c) b+ ia d) b- ia
38) If z is an imaginary number and z/(1+ z) is purely imaginary, then z
a) can be neither real or not purely imaginary
b) is real
c) is purely imaginary
d) satisfies none of these properties.
39) In a GP the first term and the common ratio are both (1.2) (√3+ i), then the absolute value of the nth term of the progression is
a) 2ⁿ b) 4ⁿ c) 1 d) none
40) The complex number z satisfying |z - 1|= |z -3|= |z - i| is
a) 2+ i b) 3/2 + i/2 c) 2+ 2i d) none
41) The value of the expression
1(2- ω)(2- ω²)+ 2(3- ω)(3- w²)+ ....+ (n -1)(n - ω)(n - ω²), where ω is an imaginary cube root of unity, is
a) {n(n +1)/2}²
b) n²(n+1)².4
c) n²(n+1)²/4 + n d) none
42) The number of solutions of the equation z² = conjugate of z, where z is a complex number, is
a) 2 b) 3 c) 4 d) none
43) The solution of the equation|z| - z = 1+ 2i is
a) 2 - 3i/2 b) 3/2 - 2i c) 3/2+ 2i d) -2+ 3i/2
44) The number of solutions of the equation z² + |z|²= 0, where z is a complex number, is
a) 1 b) 2 c) 3 d) none
45) If x= cosα + i sin α, y = cosβ + i sinβ, z= cosγ + i sinγ and x+ y+ z= 0, then 1/x + 1/y + 1/z is equal to
a) xyz b) 1 c) Re x + Re y + Re z d) 0
46) If x + 1/x = 2 cosθ then for any integer n, xⁿ - 1/xⁿ is equal to
a) 2 cosnθ b) 2i 2i sin nθ c) - 2i sin nθ d) none
47) For all complex numbers z₁, z₂ satisfying |z₁|= 12 and |z₂ - 3 - 4i|= 5, the minimum value of |z₁ - z₂| is
a) 0 b) 2 c) 7 d) 17
48) Given that the equation z² + (p + iq)z + r + i s= 0 where p,q,r and s are non-zero real numbers, has a real root, then
a) pqr= r² + p²s
b) pqs= s² + q²r
c) prs = q²+ r²p
d) qrs= p² + s²q
49) The equation z² + conjugate of z ² - 2|z|²+ z + conjugate of z represents
a) a straight line
b) a circle
c) an ellipse
d) a parabola
50) The complex numbers sinx + i cos2x and cosx - i sin2x are conjugate to each other for
a) x= 0 b) x= nπ c) x= (2n+1)π/2 d) no value of x.
51) The greatest value of the modulii of the complex numbers z satisfying the equation |z - 4/z|= 2 is
a) √5 -1 b) √5 c) √5+ 1 d) none
52) For any complex number z, the minimum value of |z| + |z -1| is
a) 0 b) 1/2 c) 1 d) 3/2
53) Let z₁ and z₂ be two complex numbers such that z₁/z₂ + z₂/z₁ = 1, then
a) z₁, z₂ and the origin are collinear
b) z₁, z₂ and the origin form a right angled triangle
c) z₁, z₂ and the origin form an equilateral triangle d) none
54) The maximum values of |z| when z satisfies the condition |z - 2/z|= 2, is
a) √3- 1 b) √3 c) √3 + 1 d) √2+ √3
55) If α is a root of the equation x²+ x +1= 0, then α³ᵐ + α³ⁿ⁺¹ + α³ᵖ⁺² (where m,n,p are three integers) is equal to
a) -3 b) 3 c) 1 d) 0
56) The area of the triangle in the complex plane formed by the points z, iz and z + iz is
a) |z|² b) (1/4 |z|² c) (1/2) |z|² d) z²/2
57) z₁, z₂, z₃, z₄ are the four complex numbers represented by the vertices of a quadrilateral taken in order such that z₁ - z₄ = z₂ - z₃ and amp (z₄ - z₁)/(z₂ + z₁) =π/2. Then the quadrilateral is
a) a square b) a rectangle c) a rhombus d) none
58) The complex numbers z₁, z₂, z₃ are respectively the vertices A,B, C of a parallelogram ABCD, then the fourth vertex D is
a) (1/2)(z₁ + z₂)
b) (1/2) (z₁ + z₂ + z₃)
c) (1/3)(z₁+ z₂ + z₃)
d) z₁ + z₃ - z₂.
59) Suppose z₁, z₂, z₃ are the vertices of an equilateral triangle circumscribing the circle |z|= 2. If z₁= 1+ √3 i and z₁z₂, z₃ are are in the anticlockwise sense, then z₂ is
a) 1- √3 i b) 2 c) 1- √3 I d) -2
60) In the complex plane the points -2+ i, 1+ 2i, 4+ 5i and 1+ 4i form
a) a square b) a rectangle c) a parallelogram d) none
61) The statement 'a + ib> c + id where a,b,c,d are real numbers and i²= -1' is
a) correct when a> c and b> d
b) correct when a> c and b= d
c) correct when a> c and b= d=0 d) never true
62) The statement 'mi > ni where m,n are real numbers and i² = -1' is
a) correct when m> n and m,n are positive rational numbers
b) correct when m> n and m,n are positive prime integers
c) correct when m> n =π
d) meaningless for all real numbers m,n.
63) The statement 'i² = -1' is equivalent to
a) i= √-1, taking positive square root
b) i= - √-1, taking positive square root
c) √-1= ± i
d) all the statement in above three options.
64) If arg z< 0, then arg(-z) - argz is equal to
a) π b) -π c) -π/2 d) π/2
65) If z² + z +1= 0 where z is complex number, then the value of (z + 1/z)²+ (z² + 1/z²)²+ (z³+ 1/z³)²+....+ (z⁶ + 1/z⁶)² is equal to
a) 6 b) 12 c) 18 d) 54
66) The value of ¹⁰ₖ₌₁∑ (sin(2kπ/11) + i cos(2kπ/11) is equal to
a) -1 b) - I c) I d) 1.
67) The value of ⁶ₖ₌₁∑(sin(2πk/7 - I cos(2πk/7)) is equal to
a) -1 b) - I c) I d) 1
68) Let n(∈ N) be a multiple of 5 and x= cos(2π/5)+ i sin(2π/5) then
1+ xⁿ+ x²ⁿ + x³ⁿ + x⁴ⁿ is equal to
a) 0 b) 5 c) -5 d) 5 i
69) Let α = cos(4π/3) + i sin(4π/3) then the value of {(1+ α)/2}³ⁿ is equal to
a) (-1)ⁿ b) (-i)ⁿ c) (-1)ⁿ/2³ⁿ d) 1/2³ⁿ.
70) Let zₙ = cos(2nπ/7) + i sin(2nπ/7), n = 0,1,2,....6 then z₁z₂z₃....z₆ is equal to
a) 0 b) 1 c) -1 d) - i
71) Let zₙ = cos(π/2ⁿ)+ i sin(π/2ⁿ) then z₁z₂z₃....to ∞ is equal to
a) 0 b) -1 c) 1 d) i
72) The product of values of (1+ i √3)³⁾⁴ is equal to
a) 80 -8i -8 d) 8
73) Let α, β are the roots of the equation x²- 2x cosθ +1= 0, then the equation whose roots are αⁿ and βⁿ, is
a) x² + 2x cos nθ + 1
b) x² - 2x cos nθ + 1
c) x² + 2x cos nθ - 1
d) x² - 2x cos nθ - 1
74) If |z|= 1 and w= (z -1)/(z +1), z≠ -1 then Re w is equal
a) 0 b) z/|z +1|² c) |z/(z +1)| z/|z +1|² d) √2/|z +1|²
75) If one root of ax²+ bx - 2c = 0 (a,b,c are real) is imaginary and 8a + 2b > c, then
a) a> 0 and c< 0
b) a> 0 and c> 0
c) a< 0, c > 0
d) a< 0 and c< 0
76) If x²+1= √3 x, then (x³ - 1/x³)ⁿ⁺¹ for any n ∈ N is equal to
a) ω b)0 c) ±1 d) ± i
77) Let z be a complex number and z= (1- t²)+ i √(1+ t²), where t is a real parameter, then the locus of z in the complex plane is
a) a straight line b) a parabola c) a hyperbola d) an ellipse
78) Let z be a complex number and z= 1- t + i √(t²+ t +2), where t is a real parameter then the locus of z is
a) a straight line b) a parabola c) a hyperbola d) an ellipse
79) If the three complex numbers z₁, z₂, z₃ are in AP then z₁, z₂, z₃ lie on
a) a straight line b) a circle c) a parabola d) an ellipse
80) Let z₁ and z₂ are two complex numbers such that |z₁|= |z₂|= 1, then |(z₁ - z₂)/(1- z₁. Conjugate of z ₂)| is equal to
a) 2 b) 1/2 c) 1 d) none
81) The modulus of the complex number {(2+ i√5)/(2- i √5)}¹⁰ + {(2- i√5)/(2+ i √5)}¹⁰ is equal to
a) 2 sin(10 cos⁻¹(2/3))
b) 2 sin(20 cos⁻¹(2/3))
c) 2 cos(20 cos⁻¹(2/3))
d) 2 cos(10 cos⁻¹(2/3))
82) Let C be the set of all complex numbers and A, B be two subsets of C x C defined by A={(z,w): |z|= |w| and z, w ∈ C}
B={(z,w): z²= w², z, w ∈ C} then
a) A= B b) A⊆B c) B⊆A d) none
83) If |z²- 1|= |z|²+ 1, then z lies on
a) the real axis b) the imaginary axis c) a circle d) an ellipse
84) If x+ iy= |6i -3i 1
4 3i -1
20 3 i | then
a) x=3, y= 1 b) x=1, y= 3 c) x=0, y= 3 d) x=0, y= 0
85) If z and w are two complex numbers such that |zw|= 1 and arg z - arg w=π/2, then conjugate of z. w is equal to
a) -1 b) i c) 1 d) - i
86) If z is a complex number such that iz³+ z²- z + i =0, then |z|=
a) 2 b) 1 c) √2 d) none
87) (√3/2 + i/2)¹⁷⁷ is equal to
a) i b) - I c) -1 d) √3/2 - i/2.
88) Let z and w be two complex numbers such that conjugate of z+ i. conjugate of w = 0 and arg zw =π, then arg z is equal to
a) π/4 b) π/2 c) 3π/4 d) 5π/4
89) If ω is an imaginary cube root of 1 then determinant
1 1+ ω² ω²
1- i -1 ω² -1
i -1+ ω -1 is equal to
a) 0 b) 1 c) I d) ω
90) The sum ¹⁹ₖ₌₁∑ (sin(kπ/5)+ i cos(kπ/5)) is
a) purely real and positive
b) purely imaginary
c) purely real and negative d) none
91) If w is a given complex number outside the circle with centre at origin and radius |a - 1| (a is real), then the points z, satisfying z. Conjugate of z - 2 conjugate w. z - w. Conjugate of w - 2w. Conjugate of w + 5(a -1)⅖= 0, lie on
a) a circle b) a parabola c) a straight line d) none
92) If Z and W represent diagonally opposite vertices of a square, then the other two vertices are given by the complex numbers
a) Z + iW and Z - iW
b) (1/2) (Z + W)± (i/2)(Z - W)
c) (1/2) (Z - W)± (i/2)(Z - W)
d) (1/2) (Z - W)± (i/2)(Z - W)
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