1) Let the equation of an ellipse x²/144 + y²/25 = 1. then the radius of the circle with centre (0,√2) and passing through the foci of the ellipse is
A) 9 B) 7 C) 11. D) 5
2) If y= 4x+3 is parallel to a tangent to the parabola y²= 12x, then to its distance from the normal parallel to the given line is
A) 213/√17 B)219/√17. C) 211/√17 D) 210/√17
3) In a ∆ABC, tanA and tanB are roots of pq(x²+1)= r²x then ∆ABC is
A) a right angled triangle.
B) an acute angled triangle
C) an obtuse angled triangle
D) an equilateral triangle
4) Let the number of elements of the sets A and B be p and q respectively. Then the number of the relations from the set A to the Set B is..
A) 2ᵖ⁺ᑫ B) 2ᵖᑫ. C) p+q D) pq
5) The function f(x)={tan{π(x- π/2)}}/(2+[x]²), where [x] denotes the greatest integer ≤ x, is
A) continuous for all values of x.
B) discontinuous at x= π/2
C) not differentiable for some values of x
D) discontinues at x=2
6) Let z₁, z₂ be two fixed complex numbers in the Argand plane and |z - z₁| + |z - z₂|= 2|z₁ - z₂|. Then the locus of z will be
A) an ellipse
B) a straight line joining z₁ and z₂
C) a parabola
D) a bisector of the line segment joining z₁ and z₂
7) Let S= 2/1 ⁿC₀+ 2²/2 ⁿC₁ + 2³/3 ⁿC₂ +.....+ 2ⁿ⁺¹/(n+1) ⁿCₙ. Then S equals.
A) (2ⁿ⁺¹ -1)/(n+1)
B) (3ⁿ⁺¹-1)/(n+1).
C) (3ⁿ -1)/n D) (2ⁿ -1)/n
8) Out of 7 consonants and 4 vowels, the number of words(not necessarily meaningful) that can be made, each consisting of 3 consonants and 2 vowels, is
A) 24800 B) 25100
C) 25200. D) 25400
9) The remainder obtained when 1! + 2! + 3! + ....+ 11! is divided by 12 is...
A) 9. B) 8 C) 7 D) 6
10) Let S denotes the sum of the infinite series 1+ 8/2! + 21/3! + 40/4! + 65/5! + ...., Then
A) S < 8 B)S> 12 C) 8< S<12. D) S= 8
11) For every real number x, let f(x)= x/1! + 3x²/2! + 7x³/3! + 15x⁴/4! +.....
Then the equation f(x)= 0 has
A) no real solution
B) exactly one real solution.
C) exactly two real solutions
D) infinite number of real solutions.
12) the coefficient of x³ in the Infinite series expansion of 2/{(1-x)(2-x)} , for |x|< 1, is
A) -1/16 B)15/8. C) -1/8 D) 15/16
13) if a,b are the roots of the quadratic equation x²+px+ q= 0, then the value of a³+ b³ and a⁴+ a²b² + b⁴ are respectively.
A) 3pq - p³ and p⁴-3p²q+ 3q²
B) -(3q - p²) and (p²- q)(p²+ 3q)
C) pq - 4 p⁴- q⁴
D) 3pq - p³ and (p²- q)(p²- 3q).
14) A fair six faced die is rolled 12 times. The probability that each face turns up twice is equal to
A) 12!/(6!6!6¹²) B) 2¹²/(2⁶.6¹²)
C) 12!/(2⁶.6¹²). D) 12!/(6².6¹²)
15) Let f(x) be differentiable function in [2,7]. If f(2)=3 and f'(x) ≤5 for all x in (2,7), then the maximum possible value of f(x) at x= 7 is..
A) 7 B) 15 C) 28 D) 14
16) The value of tan π/5 +2tan 2π/5 + 4 cot 4π/5 is
A) cot π/5 B) cot 2π/5
C) cot 4π/5 D) cot 3π/5
17) Let R be the set of all real numbers and f: R --> R be given by f(x)= 3x² +1. Then the set f⁻¹([1,6]) is
A) {-√(5/3, 0, √(5/3)}
B) {-√(5/3, √(5/3)}
C) {-√(1/3, √(1/3)}
D) {√(5/3, -√(5/3)}
18) The area of the region bounded by the curves y= x² and x = y² is ..
A) 1/3 B) 1/2 C) 1/4 D) 3
19) The point on the parabola y²= 64x which is nearest to the line 4x+3y+ 35= 0 has coordinates.
A) (9,-24) B) (1,81)
C) (4,-16) D) (-9,-24)
20) The equation of the common tangent with positive slope to the parabola y²= 8√(3x) and hyperbola 4x² - y²=4 is
A) y= √(6x) +√2 B) y= √(6x) - √2
C) y= √(3x) +√2 D) y= √(3x) - √2
21) Let p,q be real numbers. if a is the root of x²+ 3p²x+ 5q²= 0, b is a root of x²+ 9p²x+ 15q²= 0 and 0< a< b, then the equation x²+ 6p²x+ 10q²= 0 has a root c that always satisfies.
A) c= a/4 + b B) b< c
C) c= a/2 + b B) a< c< b
22) the value of the sum (ⁿC₁)² +(ⁿC₂)²+ .....+ (ⁿCₙ)² is
A) (²ⁿCₙ)² B) ²ⁿCₙ C) ²ⁿCₙ+1 D) ²ⁿCₙ- 1
23) Ram is visiting a friend. Ram knows that his friend has 2 children and 1 of them is a boy. Assuming that a child is equally likely to be a boy or girl, then the probability that the other child is a girl, is..
A) 1/2 B) 1/3 C) 2/3 D) 7/10
24) Let n≥ 2 be an integer,
Cos(2π/n) sin(2π/n) o
A= - sin(2π/n) Cos(2π/n) 0
0 0 1
and I is the identity matrix of order 3. then
A) Aⁿ= I Aⁿ⁻¹≠ I
B) Aᵐ ≠ I for any positive integer m
C) A is not invertible
D) Aᵐ = 0 for positive integer m.
25) let I denote the 3x3 identity matrix and P be a matrix obtained by rearranging the columns of I. Then
A) there are six distinct choices for P and det(P)= 1
B) there are six distinct choices for P and det(P)= ±1
C) there are more than one choices for P and some of them are not invertible.
D) there are more than one choices for P and P⁻¹= I in each choice.
26) The sum of the series ∞ₙ₌₁⇒∑ sin(n!π/720) is
A) sin(π/180) +sin(π/360)+ sin(π/540)
B) sin(π/6) +sin(π/30)+ sin(π/120)+ sin(π/360)
C) sin(π/6)+ sin(π/360) + sin(π/120) +sin(π/360)+ sin(π/720)
D) sin(π/180) +sin(π/360)
27) Let a,b be the root of x² - x -1= 0 and Sₙ = aⁿ + bⁿ, for all integers n≥ 1. then for every integer n≥ 2, A) Sₙ + Sₙ₋₁= S ₙ₊₁
B) Sₙ - S ₙ₋₁= S ₙ₊₁
C) Sₙ₋₁= S ₙ₊₁
D) Sₙ + Sₙ₋₁ = 2 Sₙ₊₁
28) In a ∆ABC, a, b, c are the sides of the triangle opposite to the angles A, B, C respectively. then the value of a³ sin(B-C) + b³sin(C-A) + c³ sin(A-B) is equal to...
A) 0 B) 1 C) 3 D) 2
29) In the Argand plane, the distinct roots of 1+z+z³+z⁴= 0 (z is a complex number) represent vertices of
A) a square
B) an equilateral triangle
C) a rhombus
D) a rectangle.
30) the number of digit in 20³⁰¹
(Given log 2= 0.3010) is
A) 602 B) 301 C) 392 D) 391
31) If √y= cos⁻¹x, then it satisfies the differential equation (1-x²) d²y/dx² - x dy/dx = c, where c is equals to
A) 0 B) 3. C) 1 D) 2
32) the integrating factor of the differential equation (1+x²) dy/dx + y = ₑ tan⁻¹x is..
A) tan⁻¹x B) 1+x² C)ₑ tan⁻¹x D) log(1+x²)
33) The solution of the equation log₁₀₁ log₇{√(x+7)+ √(x)}= 0 is
A) 3 B) 7 C) 9 D) 49
34) If m,n are the roots of ax²+bx+ c= 0(a≠0) and m+ h, n+ h are the roots of px²+ qx +r= 0 (p≠0) then the ratio of the squares of their discriminants is
A) a²: p² B) a:p² C) a²: p D) a: 2p
35) Let f(x)= 2x²+ 5x+1. If we write f(x) as f(x)=a(x+1)(x-2)+ b(x-2)(x-1)+ c(x-1)(x+1) for real numbers a,b,c, then
A) there are infinite number of choices for a,b,c
B) only one choice for a but infinite number of choices for b and c
C) exactly one choice for each of a,b,c
D) more than one but finite number of choices for a,b,c.
36) Let f(x)=x+ 1/2. then the number of real values of x for which the three unequal terms f(x), f(2x), f(4x) are in H. P. is
A) 1. B) 0 C) 3 D) 2
37) The function f(x)= x²+ bx + c, where b and c real constants, describes
A) one to one mapping
B) onto mapping
C) not one to one but onto mapping
D) neither one to one nor onto mapping
38) suppose that the equation f(x)= x²+ bx + c= 0 has two distinct real roots m, n. the angle between the tangent to the curve y= f(x) at the point ((m+n)/2, f(m+n)/2) and the positive direction of the x-axis is..
A) 0° B) 30° C) 60° D) 90°
39) The solution of the differential equation y dy/dx= x[y²/x² + ¢ (y²/x²)/¢'(y²/x²)] is (where c is a constant)
A) ¢ (y²/x²)= cx
B) x¢ (y²/x²)= x
C) ¢ (y²/x²)= cx²
D) x²¢ (y²/x²)= c
40) Let f(x) be a differentiable function and f'(4) = 5. then
lim ₓ→₂ {f(4) - f(x²)}/(x-2) equals
A) 0. B) 5 C) 20. D) -20
41) The value of lim ₓ→ₐ ∫ Cos(t²)/x sinx dt at (x²,0) is .
A) 1. B) -1 C) 2 D) log 2
42) The range of a function y= 3 sin{√(π²/16 - x²)} is
A) (0,√3/2), B) (0,1) C) 0,3/√2) D) (0,∞)
43) There is a group of 265 persons who like either singing or dancing or painting. In this group 200 like singing, 110 like dancing and 55 like painting. If 60 persons like both singing and dancing, 30 like both singing and painting and 10 like all three activities, then the number of persons who like only dancing and painting is
A) 10 B) 20 C) 30 D) 40
44) The curve y= (cosx +y)¹⁾² satisfy the differential equation
A) (2y-1)d²y/dx² +2(dy/dx)²+ cosx= 0
B) d²y/dx² -2y(dy/dx)²+ cosx= 0
C)(2y-1)d²y/dx² -2(dy/dx)²+ cosx= 0
D) (2y-1)d²y/dx² - (dy/dx)²+ cosx= 0
45) suppose that z₁, z₂, z₃ are three vertices of an equilateral triangle in the Argand plane. Let a= 1/2 (√3 +i) and b be a non-zero complex number, The points az₁ + b, az₂+ b, az₃ + b will be..
A) the vertices of an equilateral triangle.
B) the vertices of an isosceles triangle
C) collinear
D) the vertices of a scalene triangle.
46) if lim ₓ→₀ (2a sinx - sin2x)/tan³x exist and is equals to , then the value of a is...
A) 2 B) 1 C) 0 D) -1
47) If f(x)= 2x²+1, x ≤1
4x³ -1, x > 1 then ²₀∫f(x) dx is
A) 47/3 B) 50/3 C) 1/3 D) 47/2
48) The value of |z|² + |z -3|² + |z-i|² is minimum when z equals.
A) 2 - 2i/3. B) 45+3i
C) 1+ i/3 D) 1 - i/3
49) The number of solution/s of the equation √(x+1) - √(x-1)= √(4x-1) is are..
A) 2. B) 0 C) 3. D) 1
50) the value of λ for which the curve (7x+5)²+ (7y+3)²= λ²(4x+3y-24)² represent a parabola is..
A) ± 6/5 B) ±7/5 C) ±1/5 D)±2/5
51) If sin⁻¹(x/13)+ cosec⁻¹13/12 = π/2, then the value of x is
A) 5 B) 4 C) 12 D) 11
52) The straight lines x+y= 0, 5x +y= 4 and x+5y= 4 form
A) an isosceles triangle
B) an equilateral triangle
C) a scalene triangle
D) a right angled triangle
53) if ²₀ ∫ ₑx⁴ (x - k)dx= 0, then k lies in the interval.
A) (0,2) B) (-1,0) C) (2,3) D) (-2,-1)
54) If the coefficient of x⁸ in (ax² + 1/bx)¹³ is equals to the coefficient of 1/x⁸ in (ax- 1/bx²)¹³, then a and b will satisfy the relation.
A) ab+1= 0 B) ab= 0 C) a= 1-b D) a+b= -1
55) the function f(x)= a sin |x|+ bₑ|x| is differentiable at x= 0 when
A) 3a+b= 0 B) 3a-b= 0
C) a+b= 0 D) a- b= 0
56) If a ,b ,c are positive numbers in a GP, then the roots of the quadratic equation (loga)x² - (log b)x + (log c)= 0 are
A) -1 and (log c)/(log a)
B) 1 and (log c)/(log a)
C) 1 and (log c)
D) -1 and (log a)
57) Let R be the set of all real numbers and f: [-1, 1] --> R be defined by
f(x) = x sin(1/x), x ≠ 0
0, x= 0 then
A) f satisfy the conditions of Rolle's theorem on [-1,1]
B) f satisfy the conditions of lagrange mean value theorem on [-1,1]
C) f satisfy the conditions of Rolle's theorem [0, 1]
D) f satisfies the conditions of Lagrange mean value theorem on [0, 1]
58) let z₁ be a fixed point on the circle of radius 1 centred at origin in the Argand plane and z₁≠±1. considered an equilateral triangle inscribed in a circle with z₁, z₂, z₃ as the vertices taken in the counterclockwise direction. Then z₁z₂z₃ is equals to
A) z₁² B) z₁³ C) z₁⁴ D) z₁
59) Suppose that f(x) is a differentiable function such that f'(x) is continuous, f'(0)=1 and f"(0) does not exist. Let h(x)= xf'(x). Then
A) g'(0) does not exist
B) g'(0)= 0
C) g'(0)= 1
D) g'(0)= 2
60) Let [x] denote the greatest integer less than or equal to x for any real number is equals x. Then
lim ₓ→∞ [n √2]/n ie equal to
A) 0 B) 2 C) √2. D) 1
CATEGORY II
Q.61 to Q.75 carry two mark each, for which only one option is correct, any wrong answer will lead to deduction of 2/3 Marks.
61) We define a binary relation ¢ on the set of all 3x3 real matrices as A ¢ B if and only if there exist invertible matrices P and Q such that B= PAQ⁻¹x. The binary relation ¢ is..
A)neither reflexive nor symmetric
B)reflexive and symmetric but not transitive
C) symmetric and transitive but not reflexive
D) an equivalence relation
62) The minimum value of 2ˢᶦⁿ ˣ+ 2ᶜᵒˢ ˣ is..
A) ₂2 - 1/√2 B) ₂ 2+ 1/√2 C) ₂√2 D) 2
63) for any real numbers a and b, we define aRb if and only if sec²a - tan²b= 1. The relation R is
A) reflexive but not transitive
B) symmetric but not reflexive
C) both reflexive and symmetric but not transitive.
D) an equivalence relation.
64) A relation starting from a point A and moving with a positive constant acceleration along a straight line reaches another point B in time T. suppose that initial velocity of the particle is u> 0 and P is the midpoint of the line AB. if the velocity of the particle at point P is v₁ and if the velocity at time T/2 is v₂, then
A) v₁= v₂ B) v₁> v₂ C) v₁< v₂ D) v₁= 1/2 v₂
65) Let tₙ denote the nth term of the Infinite series 1/1!+10/2!+21/3!+34/4!+49/5!+... Then lim ₓ→∞ tₙ is..
A) e B) 0 C) e² D) 1
66) let a, b denote the cube roots of unity other than 1 and a≠ b, Let s= ³⁰²ₙ₌₀ ∑ (-1)ⁿ (a/b)ⁿ. Then the value of s is
A) either -2ω or -2ω²
B) either -2ω or 2ω²
C) either 2ω or -2ω²
D) either 2ω or 2ω²
67) The equation of hyperbola whose coordinates of the foci are (±8,0) and the length of the latus rectum is 24 unit, is
A) 3x²-y²= 48 B) 4x²-y²= 48
C) x²- 3y²= 48 D) x²- 4y²= 48
68) Applying Lagrange's mean value theorem for a suitable function f(x) in [0,h], we have f(h) = f(0) + hf'(¢h), 0< ¢< 1. then for f(x) = cosx, the value of lim h→0⁺ is..
A) 1. B) 0. C) 1/2. D) 1/3
69) let Xₙ = {z= x+iy : |z²| ≤ 1/n} for all integers n≥ 1. Then ∞ₙ₌₀∩ Xₙ is...
A) a singleton set
B) not 1 finite set
C) an empty set
D) a finite set with more than one elements
70) suppose M= π/2₀ ∫ cosx/(x+2) dx, N= π/4₀∫ (sinx cosx)/(x+1)² dx. then the value of (M- N) equal
A) 3/(π+2). B) 3/(π -4)
C) 4/(π-2). D) 3/(π+4)
71) cos 2π/7 + cos 4π/7 + cos 7π/7
A) is equal to zero
B) lies between 0 and 3
C) is a negative number
D) lies between 3 and 6.
72) A student answers a multiple choice questions with 5 alternatives of which exactly one is correct. The probability that he knows the correct answer is p, 0< p < 1. if he does not know the correct answer, he randomly ticks one answer. Given that he has answered the question correctly, the probability that he did not take the answer randomly, is
A) 3p/(4p+3). B) 5p/(3p+2)
C) 5p/(4p+). D) 4p/(3p+1)
73) A poker hand consists of five cards are drawn at random from a well shuffled pack of 52 cards. then the probability that the poker hand consists of a pair and a triple of equal face values (for example, 2 sevens and 3 kings or 2 aces and 3 queens, etc) is..
A) 6/4165. B) 23/4165
C) 1797/4165 D) 1/4165
74) Let f(x) = max{x + |x|, x- [x], where [x] denotes the greatest integer ≤ x. Then the value of ³₋₃∫ f(x) dx is...
A) 0 B) 51/2. C) 21/2 D) 1
75) The solution of the differential equation dy/dx + y/(x logx) = 1/x under the condition y= 1 when x= e is
A) 2y= logx + 1/logx
B) y= logx + 2/logx
C) y logx= logx + 1
D) y= logx + e
CATEGORY III
Q.76 to Q.80 carry two marks each, for which one or more than one options will lead to maximum mark of two on pro rata basis. there will be no negative marking for these questions. however, any marking of wrong option will lead to award of zero mark against the respective question -- irrespective of the number of correct options marked.
76) Let f(x)= ˣ₀ ∫ |1- t| dt, x>1
x -1/2, x ≤ 1
A) f(x) is continuous at x=1
B) f(x) is not continuous at x=1
C) f(x) is differentiable at x= 1
D) f(x) is not differentiable at x=1
77) The angle of intersection between the curves y= [|sinx|+ |cosx|] and x²+y²= 10, where [x] denotes the greatest integer ≤ x, is
A) tan⁻¹(3). B) tan⁻¹(-3)
C) tan⁻¹(√3) D) tan⁻¹(1/√3)
78) If u(x) and v(x) are two independent solutions of the differential equation d²y/dx² + b dy/dx + cy = 0, then additional solution/s of the given differential equation is(are)
A) y= 5 u(x)+ 8 v(x)
B) y= c₁{u(x)- v(x)}+ c₂v(x), c₁ and c₂ are arbitrary constants.
C) y= c₁u(x) v(x)}+ c₂ u(x)/v(x), c₁ and c₂ are arbitrary constants.
D) y= u(x) v(x)
79) for the two events A and B, let P(A)= 0.7 and P(B)= 0.6. The necessarily false statement(s) is/ are
A) P(A ∩B)= 0.35
B) P(A ∩B)= 0.45
C) P(A ∩B)= 0.65
D) P(A ∩B)= 0.28
80) If the circle x²+ y² + 2gx + 2fy + c= 0 cuts the three circles x²+ y² -5 = 0 , x²+ y² - 8x -6y + 10= 0 and x²+ y² -4x + 2y -2= 0 at the extremities of the diameters, then
A) c= -5 B) fg= 147/25
C) g+ 2f= c+2 D) 4f= 3g.
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