PAPER - 1 2013 Main
1) ∫⁴₁ [x - 0.4] dx equals, (where [ . ] Denotes fractional part of x)
a) 1.3 b) 6.3 c) 1.5 d) 7.5
2) The value of ∫ 8ˣ/₂[3x] dx. (Where [.] denotes GIF) is equal to
a) [x]/ln2 b) [x]/ln4 c) 2[x]/ln2 d) [x]/ln8
3) If Sₖ ∫ sin x (i⁴x) where i= √-1 then ⁴ⁿ⁻¹ₖ₌₁∑ Sₖ (n∈N) is
a) -cos x+ c b) cosx + cc) 0 d) i⁴ⁿ sinx + c
4) let f(xy)= f(x) ∀ x > 0, y> 0 and f(1+x)= 1+ x {1+ g(x)} where lim ₓ→₀g(x)=0 then ∫ f(x)/f'(x) dx =
a) x²/2+ c b) x³/3 + c c) x + c d) x⁴/4 + c
5) ∫ (x + x²⁾³ + x¹⁾⁶)/{x(1+ x¹⁾³) dx equals
a) (3/2) x²⁾³ + tan⁻¹(x¹⁾⁶) + c
b) (2/3) x²⁾³ + 6 tan⁻¹(x¹⁾⁶) + c
c) (3/2) x²⁾³ - 6 tan⁻¹(x¹⁾⁶) + c
d) (3/2) x²⁾³ + 6tan⁻¹(x¹⁾⁶) + c
6) ∫ dx/{(1+ √x) √(x - x²)} equals
a) (1+√x)/(1- x)² + c
b) (1+√x)/(1+ x)² + c
c) (1- √x)/(1- x)² + c
d) 2(√x -1)/√(1- x) + c
7) ∫ dx/{x²²(x⁷-6)} = A{logu⁶+ 9u²- 2u³- 18u}+ c, then
a) A= 1/9072, u= {(x⁷-6)/x⁷}
b) A= 1/54432, u= {x⁷/(x⁷-6)}
c) A= 1/54432, u= {(x⁷-6)/x⁷}
d) A= 1/9072, u= {x⁷/(x⁷-6)}
8) If y(x - y)²= x, then ∫ dx/(x - 3y)=
a) (1/2) log|(x + y)²- 1|+ c
b) (1/2) log|(x - 2y)²- 1|+ c
c) (1/2) log|(x + 2y)²- 1|+ c
d) (1/2) log|(x - y)²- 1|+ c
9) ∫ eˣ(2- x²)dx/{(1- x)√(1- x²)}=
a) eˣ √{(1- x)/(1+ x)}+ c
b) eˣ √{(1 + x)/(1- x)}+ c
c) eˣ/(1- x) + c
d) eˣ /(1+ x)}+ c
10) ∫ (cos²x + sin2x)/(2cosx - sinx)² dx = cosx/(2cosx - sinx)+ Ax + B ln|2cosx - sinx|+ c then (A, B) =
a) (-1/5,-2/5) b) (1/5, 2/5) c) (2/5,1/5) d) (-1/√5,-2/√5)
11) The area bounded by y= x², y= {x +1}, 0 ≤ x < 1 and the y-axis is (where {.} denotes the GIF)
a) 1/3 sq.unit
b) 2/3 sq.unit
c) 1 sq.unit
d) 7/3 sq.unit
12) Let f(x)= x + sinx and g(x) be the inverse of g(x). Then area bounded by g(x) and the ordinates at x= 0, x = π is
a) π²/2 +2 b) π²+2 c) π²/2 - 2 d) π² - 2
13) The area enclosed between the curves y= logₑ(x + e), x= logₑ(1/y) and x-axis is
a) 2 b) 1 c) 1/e d) 4
14) If f(x) and ∅(x) are continuous functions on the interval [0,4] satisfying f(x)= f(4-x),
∅(x) + ∅(4- x) = 3 and ⁴₀∫ f(x) dx= 2 then ⁴₀∫ f∅(x) dx=
a) 0 b) 1 c) 2 d) 3
15) The value of ∫ [cosx] dx at (π,0), (where [,] is the greatest integer funy) is
a) π/2 b) 0 c) π d) -π/2
16) The value of ∫ (sinx + cosx). √{eˣ/sinx} dx is equal to
a) 2 √(ₑπ/2) b) √(ₑπ/2) c) 2 √(ₑπ/2) . cos1 d) (1/2) (ₑπ/4)
17) If the solution of differential equation dy/dx = (ax +3)/(2y + f) representa a circle of non zero radius then
a) a= 2, 9 + 4f² > 4c
b) a= - 2, 9 + 4f² < 4c
c) a= 2, 9 + 4f² < 4c
d) a= - 2, 9 + 4f² > 4c
18) solution of (x eʸ⁾ˣ - y sin(y/x))dx + x sin(y/x) dy=0
a) logx = c + (1/2) ₑ⁻ʸ⁾ˣ (sin(y/x)+ cos(y/x))
b) logx = c + (1/2) ₑʸ⁾ˣ (sin(y/x)- cos(y/x))
c) logx = c + (1/2) ₑʸ⁾ˣ (sin(y/x)+ cos(y/x))
d) logx = c + (1/2) ₑ⁻ʸ⁾ˣ (sin(y/x)- cos(y/x))
19) The the solution of differential equation
xdy + y dy + (xdy - ydx)/(x²+ y²)= 0 is
a) y= tan{(c - x - y)/2}
b) y= xtan{(c - x² - y²)/2}
c) y= xtan{(c + x² +y²)/2}
d) y= xtan{(c + x² - y²)/2}
20) The differential equation of all circles in a plane must be
a) y₃(1+ y²₁) - 3y₁y₂²= 0
b) of order 3 and degree 3
c) of order 3 and degree 2
d) y₃(1- y²₁) - 3y₁y₂²= 0
21) If the population of a country doubles in 50 years in how many years will it becomes the original. assume the rate of increase is proportional to the number of inhabitants
a) 75 b) 50log₂3 c) 50 logy₃2 d) 100
22) The equation of the curve not passing through the origin and having the portion of the tangent included between the co-ordinate axis is bisected at the point of contact is
a) a parabola
b) an ellipse or a straight line
c) a circle or an ellipse
d) a hyperbola or a straight line
23) if the differential equation corresponding to the family of curves, y= (A + Bx)e³ˣ is given by d¹y/dx⅖ = 4 dy/dx + by then (a - b) equals
a) 9 b) 12 c) 15 d) 18
24) The order of differential equation whose general solution is given by
y= (c₁ + c₂) sin(x + c₃) - c₄ ₑx+c₅ is
a) 5 b) 4 c) 2 d) 3
25) If ∆= a² -(b - c)², where '∆' is area of triangle ABC, then tanA=
a) 15/16 b) 8/15 c) 8/17 d) 1/4
26) In ∆ ABC, if 'r' is in-radius and 'R' is circum-radius, then
a) R≥ 2r b) R≤ 2r c) R> 2r d) R< 2r
27) AD is a median of the ∆ ABC. If AE and AF are medians of the triangle ABD and ADC respectively, and AD= m₁, AE= m₂, AF= m₃, BC= a then a²/8 is equals to
a) m₁²+ m₃²- 2m₁² b) m₁²+ m₂² - 2m₃²
c) m₂¹+ m₃¹ - m₁² d) m₁²- m₂² - 3₃²
28) If cosB cosC + sinB sinC sin²A = 1, then
a) angle A= 45°
b) angle A= 60°
c) angle A= angleB
d) angleA= 90°, angleB= angle C
29) in triangle ABC, a²+ b²+ c²- ac - √3 ab =0, then the triangle is necessarily
a) isosceles b) right angled c) obtuse angled d) equilateral
30) The sides of a triangle are in AP and its area is 3/5 th of an equilateral triangle of same perimeter then its sides are in the ratio.
a) 1:2:√7 b) 2:3:5 c) 1:6:7 d) 3:5:7
31) in a triangle ABC, if (a+ b + c)(b + c - a)= k be then
a) k< 0 b) k>6 c) 0<k <4 d) k> 4
32) In ∆ ABC if a= √3, b= 2, angle C=30° then r=
a) (√3-1)/2 b) (√3 +1)/2 c) (√3-1)/4 d) √2 +1
33) In a ∆ ABC, if c= (a - b) secθ, then tanθ is equal to
a) 2 √(ab)/(a - b) cos(C/2)
b) 2 √(ab)/(a - b) sin(C/2)
c) 2 √(ab)/(a - b) tan(C/2)
d) 2 √(ab)/(a + b) cot(C/2)
34) In a ∆ ABC, if cosA+ cosB + cosC = 7/4 then R/r=
a) 4/3 b) 3/4 c) 2/3 d) 3/2
35) If O is the circumcenter of the triangle ABC and R₁, R₂, R₃ and R are the radii of the circumcircles of the triangle OBA,OCA,OAB and ABC respectively, then a/R₁ + b/R₂ + c/R₃ is equal to
a) abc/R b) abc/R³ c) abc/R⁴ d) abc/R²
36) If p,q are the length ps of the internal bisectors of the angles A, B, C of a ∆ ABC respectively, then (1/p) cos(A/2) + (1/q) cos(B/2) + (1/r) cos(C/2)=
a) 1/a + 1/b + 1/c
b) 1/c + 1/a - 1/b
c) 1/a + 1/b - 1/c
d) 1/b + 1/c - 1/a
37) If in a ∆ abcr, angles A, B, C are in AP, then (a+ c)/√(a²- ac + c²)=
a) 2sin{(A - C)/2}
b) sin{(A - C)/2}
c) 2cos{(A - C)/2}
d) cos{(A - C)/2}
38) Three circles , whose radii are a,b,c touch each other externally and the tangents at their points of contact meet in a point. Then the distance of this point from either of points of contact is
a) √{abc/(a+ b+ c)}
b) √{abc/(a- b- c)}
c) 2 √{abc/(a+ b+ c)}
d) √{(a+ b+ c)/abc}
39) The equation of the line x+ y+ z -1=0, 4x+ y - 2z + 2 =0, written in symmetrical form is
a) (x+1)/1= (y-2)/-2=(z - 0)/1
b) x/1= y/-2=(z - 1)/1
c) (x-(1/2))/2 = (y+2)/-2=(z - 2)/2
d) (x-1)/2= (y+2)/-1=(z - 0)/2
40 A plane which passes through the point (3,2,0) and the line!(x-4)/1= (y-7)/5 =(z - 4)/4 is
a) x- y+ z -1=0
b) x+ y+ z -5=0
c) x- 2y- z -1=0
d) 2x- y+ z -5=0
41) Equation of the line of shortest distance between the lines , x/2 = y/-3=z/1 and (x -2)/3 = (y- 1)/-5 =(z +2)/2 is
a) 3(x -21) = 3y + 92) = 3z - 32
b) (x - 62/3)/1/3 = (y- 31)/1/3 =(z + 31/3)/1/3
c) (x - 21)/1/3 = (y- 92/3)/1/3 =(z + 32/3)/1/3
d) (x -2)/1/3 = (y +3)/1/3 =(z - 1)/1/3
42) If a line with direction ratios 2: 2: 1 intersects the line (x-7)/3 = (y-5)/2=(z - 3j)/1 and A and B then AB=
a) √2 b) 2 c) √3 d) 3
43) The length of the projection of the line segment joining P(-1,2,0) and (1,-1,2) on the plane 2x - y - 2z =4 is
a) 1 b) 17 c) 5 d) 4
44) The corner of a tetrahedron are A(3,4,2), B(1,2,1), C(4,1,3), D(-1,-1,3). The height of A above the base BCD is
a) 27/√237 b) 23/√237 c) 20/√237 d) 27/√247
45) a mirror and a source of light are situated at the origin O and at a point on OX respectively. a ray of light the source strikes the mirror at O and is reflected. if the D. R's of the normal to the plane mirror are (1,-1,1) then D. C's for the reflected ray are
a) 1/3,2/3,2/3
b) 1/3,-2/3,-2/3
c) -1/3,-2/3,-2/3
d) -1/3,-2/3,2/3
46) The plane lx + my =0 is rotated about its line of intersection with xOy plane through an angle θ. Then the equation of the plane is lx + my + nz=0 where n is
a) ±√(l²+ m²) cosθ
b) ±√(l²- m²) sinθ
c) ±√(l²+ m²) tanθ
d) ±√(l²- m²) secθ
47) If the position vectors of the vertices of an equilateral triangle are 72i + 56 √3j, xi + yj (y>0) and the third vertex is the original itself, then the values of x and y are respectively---
a) 32, 8√3
b) -32, 32√3
c) -48, 64√3
d) 48, 16√3
48) In a ∆ ABC , angle A= 30°, H is the orthocentre and D is the midpoint of BC, segment HD is produced to T, such that HD= DT. The ratio of AT/BC equals to
a) 1:2 b) 2:1 c) 3:2 d) 2:3
49) If 2a+ 3b + 5c=0 then the area of the triangle whose sides are represented by the vectors sides are represented by the vectors a, b, c is
a) 0 b) 3 c) 5 d) 8
50) If i, j, k are unit ortho-normal vectors and a is a vector, such that a x r = j, then a . r is equal to, for any vector r
a) 0 b) 1 c) -1 d) arbitrary scalar
51) If a, b are perpendicular vectors, then the projection of the vector { la/|a| + mb/|b| + n(axb)/|axb|} along the angle bisector of the vector a and b may be given as
a) (l²+ m²)/√(l²+ m²+ n²)
b) √(l²+ m²+ n²)
c) √(l²+ m²)/√(l²+ m²+ n²)
d) (l+ m)/√2
52) If a= i + j + k, a.b=1 and axb = j - k, then b=
a) I b) i - j + k c) 2j - k d) 2i
53) In a parallelogram ABCD , |AB|= a, |AD|= b and|AC|= c, the value of DB. AB is
a) (3a²+ b²- c²)/2
b) (a²+ 3b²- c²)/2
c) (a²- b²+ 3c²)/2
d) (a²+ 3b² + c²)/2
54) Let a= i + j and b= 2i - k the point intersection of lines rxa= bxa and rxb= axb is
a) - i + j + k b) 3i - j + k c) 3i + j - k d) i - j - k
55) If a,b and c are three non coplaner unit vectors each inclined with other at an angle of 30° then the volume of tetrahedron whose edges are a,b, c is (in cubic units)
a) √{3(√3-5)}/12
b) (3√3+5)/12
c) (5√2+3)/12
d) 3√3/8
56) Let a,b,c be three mutually perpendicular vectors with same magnitude. If x satisfies the relation ax {(x - b) x a} + b x {(x - c) x b}+ c x {(x - a) x c}=0 then x=
a) (1/3) (a+ b + c)
b) (1/2) (a+ b + c)
c) (1/2) (a+ b -2c)
d) (1/2) (2a+ b - c)
57) For any vector u and v
(1- u.v)²+ 1u+ v+ (ux v) l²=
a) (1- |u|²)(1- |v|²)
b) (1 + |u|²)(1+ |v|²)
c) (u+ v)²
d) |u + v|²
58) If a,b,c are unit vector equally inclined to each other angle θ(≠0) then the angle between a x b and the plane containing b and c is
a) sin⁻¹(tan(θ/2). |Cotθ|)
b) cos⁻¹(tan(θ/2). |Cotθ|)
c) cos⁻¹(cot(θ/2). |Tanθ|)
d) sin⁻¹(cot(θ/2). |tanθ|)
59) If a.a= b.b= c.c= 1; a.b= 1.2, b.c =1/√2; c.a=√3/2 then [a b c] is
a) (√3-1)/2√2
b) (√3+1)/2√2
c) √(√6+2))/2
d) √(√6-2))/2
60) let AD be the angular bisector of the angle A of ∆ abcr, then AD= θ AB + β AC, where
a) θ = |AB|/|AB+ AC|, β= |AC|/|AB+ AC|
b) θ = (|AB| + |AC|)/|AB|, β=(|AB + |AC|)/|AC|
c) θ = |AC|/|AB+ AC|, β= |AB|/|AB+ AC|
d) θ = |AB|/|AC|, β= |AC|/|AB|
61) OABC is a tetrahedron in which O is the origin and position vector of points A, B, C are i+ 2j + 3k; 2i + θ j + k and I + 3j + 2k respectively. A value of θ for which shortest distance between OA and BC is √(3/2) is
a) 1/2 b) 4/3 c) 3/2 d) 3
62) Let a= 3i + 2βj+ 5k, b= i+ 2j + 5k; c= i+ βj - 3k, d= i+ j + 4k. L₁ and K₂ be the line passing through respectively the points with position vectors a and c and parallel to respectively b and d. If they do not meet, then the can not teke the value
a) 3 b) 1 c) 2 d) 4
63) Let P be the point of intersection of the three planes r. n₁= 0, r.n ₂ =1 and r.n₃=2 where n₁,n ₂ and n₃ are along 2i+ k, 5i -12j and 3i+ 4k respectively then the projection of OP on z axis (O being origin)
a) 3/2 b) 5/2 c) 7/2 d) 11/2
64) Let A be vector parallel to line up intersection of planes P₁ and P₂ through. P₂ is parallel to the vectors 2i + 3k and 4j - 3k and P₂ is parallel to j - k and 3i + 3j, then the angle between vector A and 2i + j - 2k is
a) π/2 b) π/4 c) π/6 d) π/3
65) Factors acting on a particle have magnitude of 5, 3, and 1 units and act in the direction of the vectors 6i+ 2j+ 3k, 3i- 2j + 6k and 2i - 3j - 6k respectively. They remain constant while the particle is displaced from the point A(2,-1,-3) to B( 5,-1,1). The work done is equal to
a) 31 units b) 33 units c) 34 units d) 44 units
66) The vector OA= -2i + j + 2k turned by a right angle about origin so that it passes through j+ k, then the vector in new position is
a) 2i - j + 2k b) (3/√5) (-2j + k) c) 3(2i + 2j +k) d) (3/√2) (i + k)
67) Let a= i + 2j + k , b= i - j + k; c= i + j - k. A vector coplanar to a and b has a projection along c of magnitude 1/√3, then the vector is
a) 4i - j + 4k b) 4i + j - 4k c) 2i + j -2k d) 2i - j -2k
68) Considered a tetrahedron with faces F₁, F₂, F₃, F₄. Let u₁, u₂ ,u₃, u₄ be the vector whose magnitudes are respectively equal to areas of F₁, F₂, F₃, F₄ and whose directions are perpendicular to these faces in outwards direction. Then |u₁ + u₂ + u₃ + u₄| equal
a) 1 b) 4 c) 0 d) 2
69) The shortest distance between the lines r= (3i -15j + 9k) + β(2i - 7j + 5k) and r= (-i +j + 9k) + K(2i +j -3k) is
a) √34 b) √3 c) 4√3 d) 48
70) Image of the point P with position vector 7i - j + 2k in the line whose vector equation is r= (9i +5j + 5k) + β(i +3j + 5k) has the position vector.
a) -9i +5j + 2k b) (9i + 5j - 2k) c) 9i -5j - 2k d) 9i +5j + 2k
71) Let points P and Q correspondent to the correspond complex numbers α and β respectively in the complex plane . If |α|= 4 and 4α²- 2αβ + β², then the area of the ∆ OPQ, O being the original equals.
a) 8√3 b) 4√3 c) 6√3 d) 12√3
72) If f"(x)> 0, ∀x ∈ R, f'(3)=0 and g(x)= f(tan²x - 2 tanx +4), 0< x <π/2, then g(x) is increasing in
a) (0,π/4) b) (π/6,π/3) c) (0,π/3) d) (π/4,π/2)
73) Let g(x)= 2gpf(x/2)=0+ f(2- x) and f"(x)< 0 ∀x ∈ (0,2). Then g(x) increase in
a) (1/2,2) b) (4/3,2) c) (9,2) d) (0,4/3)
74) The function f(x)= tan⁻¹ (sinx + cosx) is an increasing function in
a) (-π/2,π/4) b) (0,π/2) c) (00-π/2,π/2) d) (π/4,π/2)
75) The function f(x)= ln(π+x)/ln(e+ x) is
a) increasing (0, ∞)
b) decreasing
c) increasing in (0, π/e), decreasing in (π/e, ∞)
d) decreasing in (0, π/e), increasing in (π/e, ∞)
BOOSTER - WB JEE -1Directions: Each question has one correct option and carries 1 mark. For each wrong answer, 1/4 mark will be deducted.
1) Let a,b, c and d be any four real numbers. Then , aⁿ + bⁿ = cⁿ + dⁿ holds for any natural numbers n, if
a) a+ b= c + d
b) a- b= c - d
c) a+ b= c + d, a²+ b²= c²+ d²
d) a- b= c - d, a²- b²= c²- d²
2) If α and β are the roots of x²- px +1=0 and γ is a root of x²+ px + 1= 0, then (α+ γ) (β+ γ) is
a) 0 b) 1 c) -1 d) ρ
3) The number of irrational terms in the binomial expansion of (3¹⁾⁵ + 7¹⁾³)¹⁰⁰ is
a) 90 b) 88 c) 93 d) 95
4) The Quadratic expression
(2x +1)² - px + q ≠ 0 for any real x, if
a) p²- 16p - 8q< 0
b) p²- 8p + 16q< 0
c) p²- 8p - 16q< 0
d) p²- 16p + 8q< 0
5) In a certain town, 60% the families own a car, 30% own a house and 20% own both car and house. if a family is randomly chosen, then what is the probability that this family owns a car or a house but not both ?
a) 0.5 b) 0.7 c) 0.1 d) 0.9
6) The letters of the word COCHIN are permuted and all the permutations are arranged in alphabetical order as an English dictionary. The number of words that appear before the word COCHIN, is
a) 360 b) 192 c) 96 d) 48
7) Let f: R---> R be a continuous function which satisfies f(x)= ˣ₀∫ f(t) dt. Then, the value of f(logₑ5) is
a) 0 b) 2 c) 5 d) 3
8) Let f: R---R> R be defined as f(x)= (x²- x +4)/(x²+ x +4). then, range of the function f(x) is
a) [3/,5/3] b) (3/5,5/3) c) (-∞, 3/5) U (5/3, ∞) d) [-5/3, -3/5]
9) The least value of 2x²+ y²+ 2xy + 2x - 3y + 8 for real numbers x and y, is
a) 2 b) 8 c) 3 d) -1/2
10) Let f: [-2,2] ---> R be a continuous function such that f(x) assumes only irrational values. If f(√2)= √2, then
a) f(0)= 0
b) f(√2 -1))= √2 -1
c) f(√2 -1))= √2 + 1
d) f(√2 -1))= √2
11) The minimum value of cosθ + sin θ + 2/sin2θ for θ∈ (0,π/2), is
a) 2+√2 b) 2 c) 1+ √2 d) 2√2
12) The value of lim ₓ→₂ ˣ₂∫ 3t²/(x -2) dt is
a) 10 b) 12 c) 8 d) 16
13) If cot(2x/3) + tan(x/3) = cosec(kx/3), then the value of k is
a) 1 b) 2 c) 3 d) -1
14) If θ ∈(π/2, 3π/2), then the value of √(4 cos⁴θ + sin²2θ+ 4 cot θ cos²(π/4 - θ/2) is
a) - 2 cotθ b) 2 cotθ c) 2 cosθ d) 2 sinθ
15) The number of real solutions of the equation (sinx - x)(cosx - x²)= 0 is
a) 1 b) 2 c) 3 d) 4
16) The value of {(1+ √3 i)/((1- √3 i)}⁶⁴ + {(1- √3 i)/((1+ √3 i)}⁶⁴ is
a) 0 b) - 1 c) 1 d) i
17) Find the maximum value of |z| when |z - 3/z|= 2, where z being a complex number.
a) 1 + √3 b) 3 c) 1 + √2 d) 1
18) Given that x is a real number satisfying (5x²- 26x +5)/(3x²- 10x +3) < 0, then
a) x < 1/5 b) 1/5< x <3 c) x> 5 d) 1/5< x <3 or 3< x < 5
19) The value of λ such that the system of equations
2x - y - 2z = 2; x - 2 y + z = -4; x + y + λz = 4, has no solution, is
a) 3 b) 1 c) 0 d) -3
20) 1 x x+1
If f(x)= 2x x(x -1) x(x +1
3x(x -1) x(x -1)(x -2) (x+1)(x-1)x
Then, f(100) is equal to
a) 0 b) 1 c) 100 d) 10
21) Let xₙ = (1- 1/3)²(1- 1/6)²(1- 1/10)².....(1- 1/{n(n+1)}/2, n ≥ 2. Then, the value of
lim ₓ→∞ xₙ is
a) 1/3 b) 1/9 c) 1/81 d) 0
22) The various of first 20 natural numbers is
a) 133/4 b) 279/12 c) 133/2 d) 399/4
23) A fair coin is tossed at a fixed number of times. If the probability of getting exactly 3 heads equals to the probability of getting exactly 5 heads, then the probability of getting exactly one head is
a) 1/64 b) 1/32 c) 1/16 d) 1/8
24) if the letters of the word PROBABILITY are written down at random in a row, then probability that two B's are together, is
a) 2/11 b) 10/11 c) 3/11 d) 6/11
25) Which of the following is not always true ?
a) |a + b|²= |a|²+ |b|², if a and b are perpendicular to each other.
b) |a + λb|≥ |a| for all λ ∈ R, if a and b are perpendicular to each other.
c) |a + b|²+ |a - b|² = 2(|a|² +|b|²)
d) |a + λb|≥ |a| for all λ ∈ R, if a is parallel to b
26) if the four points with position vectors - 2i + j + k, i + j + k, j - k and λj + k are coplanar, then λ is equal to
a) 1 b) 2 c) -1 d) 0
27) The least positive value of t, so that the lines x= t + α, y + 16= 0 and y= ax are concurrent, is
a) 2 b) 4 c) 16 d) 8
28) In ∆ ABC, if a² cos²A - b²- c²= 0, then
a) π/4< A< π/2
b) π/2 < A< π
c) A= π/2
d) A< π/4
29) { x ∈ R : |cosx|≥ sin x} ∩ [0, 3π/2] is equal to
a) [0,π/4] U [3π/4, 3π/2]
b) [0,π/4] U [π/2, 3π/2]
c) [0,π/4] U [5π/4, 3π/2]
d) [0,3π/2]
30) If sin⁻¹(x - x²/2 + x³/4 - x⁴/8 + ...)=π/8, where |x|< 2, then the value of x is
a) 2/3 b) 3/2 c) -2/3 d) -3/2
31) The area of the region bounded by the curve y= x³, its tangent at (1,1) and xaxr, is
a) 1/12 sq unit
b) 1/6 sq unit
c) 2/17 sq unit
d) 2/15 sq unit
32) if log₀·₂(x -1)> log₀·₀₄(x +5), then
a) -1< x < 4
b) 2 < x < 3
c) 1< x < 4
d) 1< x < 3
33) The number of real roots of equation logₑ x + ex = 0 is
a) 0 b) 1 c) 2 d) 3
34) The number of distinct real roots of determinant
Sinx cosx cosx
cosx sinx cosx = 0
cosx cosx sinx
in the interval -π/4≤ x ≤ π/4
a) 0 b) 2 c) 1 d) >2
35) Let x₁, x₂, .....x₁₅ be 15 distincts numbers chosen from 1,2,3,.....15. then, the value of (x₁ -1)(x₂- 1)(x₃ - 1).....(x₁₅ -1) is
a) always ≤ 0 b) 0 c) always even d) always odd
36) Let [x] denotes the greatest integer less than or equals to x. then, the value of α for which the function
Sin|-x²|/[-x²| , x ≠ 0
f(x)= α , x= 0
is continuous at x= 0 is
a) α= 0 b) α= sin(-1) c) α = sin(1) d) α = 1
37) For all real values a₀, a₁, a₂, a₃ satisfying
a₀ + a₁/2 + a₂/3 + a₃/4 = 0, the equation a₀ + a₁x + a₂x²+ a₃x³= 0 has a real root in the interval
a) [0,1] b) [-1,0] c) [1,2] d) [-2, -1]
38) Let f: R---R> R be defined as
f(x)= 0, x is irrar
sin|x|, x is rational
Then, which of the following is true ?
a) f is discontinuous for all x
b) f is continuous for all x
c) f is discontinuous at x= kπ, where k is an integer
d) f is continuous at x= kπ, where k is an integer
39) A particle starts moving from rest from the fixed point in a fixed direction. The distance s from the fixed point at a time t is given by s= t²+ at - b + 17, where a and b are real numbers. If the particle comes to rest after 5s at a rof s = 25 units from the fixed point, then values of a and b are respectively
a) 10,-33 b) -10,-33 c) -8,-33 d) -10,33
40) lim ₙ→∞ {√1+ √2+....√(n -1)}/n√n is equal to
a) 1/2 b) 1/3 c) 2/3 d) 0
41) if lim ₓ→₀ {axeˣ - b log(1+ x)}/x²= 3, then the value of a and b are, respectively
a) 2,2 b) 1,2 c) 2,1 d) 2,0
42) if the vertex of the conic y²- 4y = 4x - 4a always lies between the straight lines x + y = 3 and 2x + 2y -1= 0, then
a) 2< a<4
b) 1/2< a<2
c) 0< a<2
d) -1/2< a<3/2
43) Number of intersecting points of the conics 4x²+ 9y²=1 and 4x²+ y²=4 is
a) 1 b) 2 c) 3 d) 0
44) The value of λ for which the straight line (x - λ)/3 = (y -1)/(2+ λ)= (z -43)/-1 may lie on the plane x - 2y = 0, is
a) 2 b) 0 c) -1/2 d) there is no such λ
45) Area of the region bounded by y= |x| and y = -|x|+ 2 is
a) 4 sq unit
b) 3 sq unit
c) 2 sq unit
d) 1 sq unit
46) Let d(n) denotes the number of divisors of n including 1 and itself. Then , d(225), d(1125) and d(640) are
a) in AP b) in HP c) in GP d) consecutive integers
47) The trigonometrical equation sin⁻¹ x = 2 sin⁻¹2a has a real solution, if
a) |a|> 1/√2 b) 1/2√2< |a| 1/√2 c) |a|> 1/2√2 d) |a|≤ 1/2√2
48) If (2+ i) and (√5- 2i) are the roots of the equation (x²+ ax + b)(x²+ cx + d) =0,where a, b, c and d are real constants , then product of all the roots of the equation is
a) 40 b) 9√5 c) 45 d) 35
49) If f: [0,π/2]---R is defined as
1 tanθ 1
f(θ)= - tanθ 1 tanθ
-1 -tanθ 1
Then the range of f is
a) (2, ∞) b) (-∞,-2) c) [2,∞) d) (-∞,2]
50) If A and B are two matrices such that AB= B and BA= A, then A²+ B² equals
a) 2AB b) 2 BA c) A+ B d) AB
51) If ω is an imaginary cube root of unity, then the value of the determinant
1+ ω ω² -ω
1+ ω² ω -ω² is
ω+ ω² ω -ω²
a) -2ω b) -3ω² c) -1 d) 0
52) Let [x] denotes the fractional part of a real numbers x. Then, in the value of
∫ f(x²) dx at (√3,0) is
a) 2√3 - √2 -1
b) 0 c) √2 - √3 + 1 d) √3 - √2 +1
Directions: Each question has one correct option and carries 2 marks. For each wrong answer, 1/2 mark will be deducted.
53) Let f: R--->R be differential at x=0. If f(0)=0 and f'(0)=2, then the value of
lim ₓ→₀ (1/x) [f(x)+f(2x)+ f(3x)+......+ f(2015x)] is
a) 2015 b) 0 c) 2015x2016 d) 2015 x 2014
54) If x and y are digits such that 17!= 3556xy428096000, then x+ y equals
a) 15 b) 6 c) 12 d) 13
55) A person goes to office by car, scooter, bus and train, probability of which are 1/7, 3/7, 2/7, 1/7, respectively. Probability that he reaches office late, if it takes car, scooter, bus or train is 2/9, 1/9, 4/9 and 19, respectively. Given that he reached office in time, the probability that he travelled by a car is
a) 1/7 b) 2/7 c) 3/7 d) 4/7
56) The value of ∫ (x -2)/{(x -2)²(x +3)⁷}¹⁾³ dx is
a) (3/20){(x -2)/(x+3)}⁴⁾³+ C
b) (3/20){(x -2)/(x+3)}³⁾⁴+ C
c) (5/12){(x -2)/(x+3)}⁴⁾³+ C
d) (3/20){(x -2)/(x+3)}⁵⁾³+ C
57) Let f: N---> R be such that f(1)=1 f(1) + 2f(2) + 3f(3) + ....+ nf(n) = n(n+1) f(n), for all n ∈N, n≥ 2, where N is the set of natural numbers and R is the set of real numbers. Then, the value of f(500) is
a) 1000 b) 500 c) 1/500 d) 1/1000
58) if 5 distinct balls are placed at random into 5 cells, then the probability that exactly one cell remains empty, is
a) 48/125 b) 12/125 c) 8/125 d) 1/125
59) A Survey of people in a given region showed that 20% were smokers. The probability of death due to the lung cancer, given that a person smoked, was 10 times the probability of death due to lungs cancer, given that a person did not smoke . If the probability of death due to lung cancer in the region is 0.006. What is the probability of death due to lung cancer given that a person is a smoker?
a) 1/140 b) 1/70 c) 3/140 d) 1/10
60) In a ∆ ABC , if Angle C= 90°, r and R are the inradius and circumradius of the ∆ ABC respectively, then 2(r + R) is equal to
A) b+ c b) c+ a c) a+ b d) a+ b + c
61) Let α and β be two distinct roots of a cosθ + b sinθ = c, where a,b,c are three real constants and θ ∈ [9,2π]. Then α + β is also a root of the same equation, if
a) a+ b = c b) b+ c = a c) c+ a = b d) c= a
62) For a matrix
1 0 0
A= 2 1 0
3 2 1 if U₁, U₂ and U₃ are 3 x 1 column matrices satisfying
1 2 2
AU₁ = 0 & AU₂= 3 & AU₃=3
0 0 1 and U is 3 x 3 matrix whose column are U₁, U₂ and U₃. Then, sum of the elements of U⁻¹ is
a) 6 b) 0 c) 1 d) 2/3
Directions: Each question has one or more correct option/s, choosing which will fetch maximum 2 marks on pro rata basis, however, choice of any wrong option/s will fetch zero mark for the question.
63) Let f be any continuously differentiable function on [a, b] and twice differentiable on (a,b) such that f(a)= f'(a)= 0 and f(b)= 0. Then,
a) f"(a)= 0
b) f'(x)= 0 for some x ∈ (a,b)
c) f"(x)=0≠ 0 for some x ∈ (a,b)
d) f'''(x)= 0 for some x ∈ (a,b)
64) A relation ρ on the set of real number R is defined as {xρy : xy> 0}. Then, which of the following is/are true ?
a) ρ is reflexive and Symmetric
b) ρ is Symmetric but not Reflexive
c) ρ is symetric and transitive
d) ρ is equivalence relation.
65) if cos x and sin x are solution of the differential equation
a₀ d²y/dx² + a₁ dy/dx + a₂y = 0.
Where a₀, a₁ and a₂ are real constants, then which of the following is/are always true ?
a) A cosx + B sinx is a solution , where A and B are real constants
b) A cos(x +π/4) is a solution , where A is a real constant.
c) A cosx sinx is a solution , where A is a real constant.
d) A cos(x + π/4) + B sin(x -π/4) is a solution , where A and B are real constants.
66) Which of the following statements is/are correct for 0 <θ < π/2 ?
a) √cos θ≤ cos(θ/2)
b) (cosθ)³⁾⁴≥ cos(3θ/4)
c) cos(5θ/6)≥ (cosθ)⁵⁾⁶
d) cos(7θ/8) ≤ (cosθ)⁷⁾⁸
67) Let 16x²- 3y²- 32x - 12y= 44 represents a hyperabola. Then
a) length of the transverse axis is 2√3
b) length of each latus rectum is 32/√3
c) eccentricity is √(19/3)
d) equation of a directrix is x= √19/3
68) For the function f(x)= [1/[x]], where [x] denotes the greatest integer less than or equal to x, which of the following statements are true ?
a) the domain is (-∞,∞)
b) the range is (0) U (-1) U {1}
c) the domain is (-∞,0) U [1, ∞)
d) the range is {0} U {1}
69) Which of the following is/ are always false?
a) a quadratic equation with rational co-efficients has zero or two irrational roots.
b) a quadratic equation with real co-efficients has zero or two non real roots.
c) a quadratic equation with irrational co-efficients has zero or two irrational roots.
d) A quadratic equation with integer co-efficients has zero or two irrational roots.
70) If the straight line (a -1)x - by + 4=0 is normal to the hyperbola xy= 1, then which of the following does not hold ?
a) a >1, b > 0
b) a >1, b < 0
c) a <1, b < 0
d) a <1, b > 0
71) Suppose a machine produces metal parts that contains some defective parts that contain some defective parts with probability 0.05. How many parts should be produced in order that the probability of at least one part being defective is 12 or more ? (Given that log95= 1.977 and log2= 0.3)
a) 11 b) 12 c) 15 d) 14
72) Let f: R---R> R be such that f(2x -1)= f(x) for all x ∈ R. If f is continuous at x = 1 and f(1)= 1, then
a) f(2)=1
b) f(2)=2
c) f is continuous only at x= 1
d) f is continuous at all points.
1) function
2) If y= (1+x)(1+x²)(1+x⁴), then dy/dx at x= 1 is ...
A) 20 B) 28 C) 1 D) 0
3) If y= (tan⁻¹x)², then (x²+1)² d²y/dx² + 2x(x²+1) dy/dx is..
A) 4. B) 0 C) 2 D) 1
4) function+ mean theorem
5) which of the following is not a correct statement ?
A) mathematics is interesting
B) √3 is a prime
C) √2 is irrational
D) the sun is star.
6) If the function f(x) satisfies lim ₓ→₁ {f(x) -2}/(x² -1)=π, then lim ₓ→₁ f(x)=
A) 1 B) 2 C) 0 D) 3
7) The tangent to the curve y= x³+1 at (1,2) makes an angle ¢ with y-axis, then the value of tan ¢ is..
A) -1/3 B) 3. C) -3 D) 1/3
8) If the function of f(x) defined by f(x)= x¹⁰⁰/100 + x⁹⁹/99 + ... + x²/2 + x+1, then f'(0)=
A) 100 f'(0) B) 100 C)1 D) -1
9) integration
10) integration
11) integration
12) integration
13) relation
14) relation
15) function
16) function
17) The domain of the function f(x)=√cosx is ..
A) [3π/3,2π] B) [0,π/2]
C) [-π/2,π/2] D) [0,π/2]U [3π/3,2π]
18) In a class of 60 students, 25 students play cricket and 20 students play tennis and 10 students play both the games, then the number of students who play neither is..
A) 45 B) 0 C) 25 D) 35
19) given 0≤ x ≤ 1/2 then the value of tan[sin⁻¹{x/√2 +√(1-x²)/√2} - sin⁻¹x] is..
A) 1 B) √3. C) -1 D) 1/√3
20) The value of sin(2 sin⁻¹0.8) is
A) 0.48 B) sin 1.2° C) sin 1.6° D) 0.96
21) If A is 3x4 matrix and B is a matrix such that A'B and BA' are both defined then B is of the type
A) 4x4 B) 3x4 C) 4x3 D) 3x3
22) The symmetric part of the matrix A= 1 2 4
6 8 2
2 -2 7
A) 0 -2. -1 B) 1 4 3
-2 0 -2 2 8 0
-1 -2 0 3 0 7
C) 0 -2 1. D) 1 4 3
2. 0 2 4 8 0
-1 2 0 3 0 7
23) If A is a matrix of order 3, such that A (adj A)= 10 I, then |adj A| =
A) 1. B) 10 C) 100 D) 10I
24) consider the following statements:
I) if any two rows or columns of a determinant are identical, then the value of the determinant is zero.
ii) if the corresponding rows and columns of a determinants are interchanged, then the value of the determinant does not change
iii) if any two rows(or columns) of a determinant are interchanged, then the value of determinant changes in sign. which of these are correct
A) i and ii B) i and ii
C) i,ii and iii D) ii and iii
25) The inverse of the matrix
A= 2 0 0
0 3 0
0 0 4 is
A) 1/12 0 0
0 1/8 0
0 0 1/6
B) 2 0 0
0 3 0
0 0 4
C) 1/24 0 0
0 1/24 0
0 0 1/24
D) 1/2 0 0
0 1/3 0
0 0 1/4
26) If a, b and c are in AP, then the value of
x+2 x+3 x+a
x+4 x+5 x+b
x+6 x+7 x+c
A)0 B) x-(a+b+c)
C) (a+b+c) D) 9x²+a+b+c
27) The local minimum value of the function f' given by f(x)= 3+|x| x belongs to R is
A) -1 B) 3 C) 1 D) 0
28) increasing and decreasing
29) rate measure
30) definite integration (area)
31) definite integration (area)
32) differential equations
33) differential equations
34) 3D
35) 3D
36) 3D
37) 3D
38) 3D
39) probability
40) probability
41) probability
42) probability
43) vector
44) vector
45) vector
46) vector
47) If sinA= sin B, then
A) (A+B)/2 is any multiple of π/2 and (A-B)/2 is any odd multiple of π
B) (A+B)/2 is any odd multiple of π/2 and (A-B)/2 is any multiple of π
C) (A+B)/2 is any multiple of π/2 and (A-B)/2 is any even multiple of π
D) (A+B)/2 is any even multiple of π/2 and (A-B)/2 is any odd multiple of π
48) if tanx =3/4, π< x<3π/2, then the value of cos(x/2) is .
A) -1/√10 B)3/√10 C) 1/√10 D)-3/√10
49) in a triangle ABC, a[b cosC - c cos B]=
A) 0 B) a² C) b²- c² D) b²
50) If a and b are two different complex numbers with |b|=1, then |(b-a)/(1-ab)|
A) 1/2 B) 0 C) -1 D) 1 E) none
51) The set A={x:|2x +3|<7} is equal to the set
A) D= {x: 0<x +5<7}
B) B= {x: -3<x <7}
C) E= {x: -7<x <7}
D) C= {x: -13<2x <4}
52) How many 5 digit telephone numbers can be constructed using the digits 0 to 9, If each numbers starts with 67 and no digit appears more than once.
A) 335 B) 336 C) 338 D) 337
53) If 21st and 22nd terms in the expansion of (1+x)⁴⁴ are equal, then x is equal to
A) 8/7 B) 21/22 C) 7/8 D) 23/24
54) consider an infinite geometric series with first term 'a' and common ratio 'r'. If the sum is 4 and the second term is 3/4, then
A) 2,3/8 B) 4/7,3/7 C) 3/2,1/2 D) 3, 1/4
55) A straight line passes through the point (5,0) and (0,3). The length of the perpendicular from the point (4,4) on the line is
A) 15/√34 B)√17/2 C) 17/2 D) √(17/2)
56) equation of the circle (a-,-b ) and radius √(a²-b²) is
A) x²+y²+2ax+2by+2b²=0
B) x²+y²-2ax-2by-2b²=0
C) x²+y²-2ax-2by+2b²=0
D) x²+y²-2ax+2by+2b²=0
57) The area of the triangle formed by the lines joining the vertex of the parabola x²= 12y to the end of latus rectum is
A) 20 sq.units B) 18 sq.units
C) 17 sq.units D) 19 sq.units
58) If the coefficient of variation and standard deviation are 60 and 21 respectively, the arithmetic mean of distribution is
A) 60 B) 30. C) 35 D) 21
59)
60) 3 sin πx/5x, x≠ 0
If f(x)= 2K ,. x= 0
is continuous at x= 0, then the value of K is..
A) π/10 B) 3π/10 C) 3π/2 D) 3π/5
