CUSAT CAT 2025 Question Paper Mathematics

MATHEMATICS - PG LEVEL
1. The function f ( z ) = z + 2iz is

(A) nowhere analytic
(B) analytic everywhere
(C) analytic except at the origin
(D) analytic only at origin

2. Let n denote the number of solutions of the equation z 2 + 2 z = 0, where z is a
∞ 1
complex number. Then, the value of ∑ k is equal to
k =0 n

(A) 1
3
(B)
2
4
(C)
3
(D) 2

100
3. If ( 3 + i) = 299 ( p + iq ) , then p and q are roots of the equation

(A) ( 3 − 1) x − 3 = 0
x2 −

(B) x + ( 3 − 1) x − 3 = 0
2

(C) x + ( 3 − 1) x + 3 = 0
2

(D) x − ( 3 − 1) x + 3 = 0
2

∞ ∞
4. If R is the radius of convergence of the series ∑ an z n , then the series ∑ nan z n−1 has
n =1 n=1
the radius of convergence

1
(A)
R
(B) R −1
(C) R
(D) 0
5. The area of the polygon, whose vertices are the non-real roots of the equation z = iz 2 , is

3 3
(A)
4
3 3
(B)
2
3
(C)
2
3
(D)
4

1
6. If z is a complex number such that z ≥ 2, then the minimum value of z +
2

5
(A) is equal to
2
(B) lies in the interval (1, 2)
5
(C) is strictly greater than
2
3
(D) is strictly greater than but less than 2
2

3
1 3 4
7. The continued product of all the values of  + i  is
2 2 


(A) 1
(B) 0
(C) 4π
π
(D)
6

8. The value of ( ̶ ii) is

(A) i
(B) 0
π
(C) e2
π
(D) ie 2
1
9. If z is a complex number such that z ≤ 1, then the minimum value of z + (3 + 4i ) is
2

3
(A)
2
5
(B)
2
1
(C)
2
7
(D)
2

ez
10. The value of the integral ∫ 2
dz is
1− z =1 z − 1

(A) 0
(B) π ie
(C) π e + π ie−1
(D) e + e −1

dz
11. The value of ∫ where C is the boundary of z − i = 1 is
C z2 + 6

(A) 2π i
(B) 4π i
(C) 0
(D) π i

12. Which of the following functionf(z)of the complex variable z is NOT analytic at all
the points of the complex plane?

(A) f (z) = z2
(B) f (z) = log z
(C) f (z) = ez
(D) f (z) = sin z

dz
13. The value of ∫ z + 3 is
z =2

(A) 0
π
(B) −
2
(C) 2π i
π
(D)
2

∞ n!
14. The radius of convergence R of the series ∑ n z n is equal to
n =1 n

(A) 1
(B) ∞
(C) e
(D) 2e

15. The value of ∫ z dz, where C is the left half of the unit circle z = 1 from z = −i to
C
z = i is

(A) 0
(B) i
(C) 2i
(D) 2πi

1 
16. The value of lim  + cot x  equals
x→0  x 

(A) ̶1
(B) 0
(C) 1
(D) ∞

17. Let P ( x) be the particular solution of the differential equation y "+ y = x sin x, then
π 
the value of P   is
2
π
(A)
2
π
(B)
3
(C) 2π
π
(D)
8

dy
18. Let y = y (t ) be a solution of the differential equation + py = re − qt , where
dt
p > 0, q > 0 and r > 0. Then, lim y (t )
t →∞

(A) is 0
(B) is 1
(C) is −1
(D) does not exist

19. The number of positive roots of the equation 10x3 – 7x – 4 = 0 is

(A) 1
(B) 2
(C) 0
(D) 3

20. A person goes to the office either by car, bike, bus or train, probability of which being
1 3 2 1
, , , respectively. Probabilities that he reaches office late, if he takes car, bike,
7, 7 7 7
2 1 4 1
bus or train are , , and respectively. Given that he reaches the office on time,
9 9 9 9
then the probability he travelled by car is
1
(A)
7
2
(B)
7
2
(C)
9
4
(D)
7

21. The area of the curve f ( x ) = 2 x , bounded by x = 1, x = 4 and the X-axis in first
quadrant is equal to the area of the rectangle bounded by x = 1, x = 4 and X-axis in first
quadrant. Then the height of the rectangle is

(A) 3 units
28
(B) units
3
28
(C) units
6
28
(D) units
9

θ 
22. If a and b are unit vectors and θ is the angle between them, then sin 2   is
2

2
(A) a +b
4
2
(B) a −b
4
2
(C) a +b
2
2
(D) a −b
2

23. If the sum of the coefficients in the expansion of (a + b)n is 4096, then the greatest
coefficient in the expansion is

(A) 792
(B) 1594
(C) 924
(D) 1024

dy
24. The solution of the differential equation
dx
( )
= (1 + x ) 1 + y 2 is

(A) ( )
y = tan x 2 + x + c

(B) y = tan ( 2 x + x + c )
2

(C) y = tan ( x − x + c )
2

 x2 
(D) y = tan  + x + c
 2 
 

 n! 
The value of lim 
n→∞ (n + 1)! − n ! 
25. is
 

(A) ∞
(B) 1
1
(C)
2
(D) 0

2
26. (
If x ∈ [ −1,1] , then the value of 2sin −1 x + 2 cos −1 x ) is
(A) π2
π2
(B)
4
π2
(C)
2
(D) 4

27. The minimum value of the functionf (x, y) = p2x + q2y,wherexy = r2 is

(A) 2pqr
(B) 1
(C) p2q2r2
(D) p2q2r4

1
28. For all complex numbers z on the curve C1 : z = 4, let the locus of the point z + be
z
the curve C2 . Thenthe curves

(A) C1 and C2 intersect at 4 points
(B) C1 lies inside C2
(C) C1 and C2 intersect at 2 points
(D) C2 lies inside C1

29. Let A =  aij  , where aij ≠ 0 for all i, j and A2 = I . Let a be the sum of all
  2×2
diagonal elements ofA and b = A . Then 4a 2 + 7b 2 is equal to

(A) 4
(B) 3
(C) 7
(D) 14

30. If the centroid of tetrahedronOPQR, where P, Q, R are given by (a, 1, 2), (2, b, 3) and
(3, 1, c) respectively be(1, 2, −1), then the distance ofA(a, b, c) from origin is equal to

(A) 107
(B) 114
(C) 109
(D) 118

1 1
31. 3 9
The product (64)(64) (64) ......∞ is equal to

(A) 83
(B) 43
(C) 8
(D) 0

32. The number of solutions of the equation sin (ex) = 2x + 2–x is

(A) 2
(B) 0
(C) 1
(D) infinitely many

33. A Statistics book contains 200 pages. A page is selected at random. What is the
probability that the number on the page selected is a perfect square?

7
(A)
50
7
(B)
100
7
(C)
200
7
(D)
25

34. The solution of the system of equations
5x + y + z = 7,
x + 5y + z = 7,
x + y + 5z =7
is

(A) 1, 1, 1
(B) −1, −1, −1
(C) 1, −1, 1
(D) −1, 1, −1

35. When 5125 is divided by 13, the remainder obtained is

(A) 1
(B) 4
(C) 12
(D) 9

36. If the points with position vectors

α iˆ + 10 ˆj + 13kˆ,
6iˆ + 11 ˆj + 11kˆ,
9ˆ
i + β ˆj − 8kˆ
2

are collinear, then (19α − 6β )2 is equal to

(A) 49
(B) 36
(C) 25
(D) 16

37. If roots of the equation x2 + x + 1 = 0 are α and β, then the equation whose roots are α
7
and β 4 is

(A) x2−x− 1 = 0
(B) x2−x + 1 = 0
(C) x2 + x− 1 = 0
(D) x2 + x + 1 = 0

38. The curvature of the unit circle x2 + y2 = 9 is

(A) 3
3
(B)
2
(C) 0
3
(D)
4

39. ( )
If f ( x) = log e x + 1 + x 2 , then f −1 ( x) is equal to

e x − e− x
(A)
2
e x + e− x
(B)
2
e x − e− x
(C)
e x + e− x
e x + e− x
(D)
e x − e− x

10
 1 
40. If the term independent of x in the expansion of  ax 2 + 3  is 105, then a3 is
 2x 
equal to

(A) 4
(B) 8
(C) 6
(D) 9

2
41. The function f ( z ) = z + iz is differentiable at

(A) i
(B) −1
(C) −i
(D) 0

42. Let (X1, d1) and (X2, d2) be metric spaces and let f : X1 → X2 be a homeomorphism.
Consider the following statements:
Statement I: If X1 is compact, then X2 is compact.
Statement II: If X1 is complete, then X2 is complete.

Then which of the following is true?
(A) Only Statement I
(B) Only Statement II
(C) Neither Statement I nor Statement II
(D) Both Statement I and Statement II

43. Let G be a cyclic group such that G has an element of infinite order. Then the number
of elements of finite order in G is

(A) 0
(B) 1
(C) 2
(D) infinite
5cos n 2
44. lim =
n→∞ n5

(A) 5
(B) 0
(C) ∞
(D) 1

45. Which of the following principal ideals is a maximal ideal of ℤ ?

(A) 9
(B) 8
(C) 15
(D) 11
3z − 4
46. Consider the bilinear transformation f ( z ) = . Then f is
z −1

(A) parabolic

(B) hyperbolic
(C) elliptic
(D) not parabolic

47. The sum of residues of f(z) = tan z at its poles inside | z | = 2 is

(A) 0
(B) −2
(C) 2
(D) −1

48. ConsiderX = [0, 1] with the discrete metric. Then which of the following is not true?

(A) X is disconnected
(B) X is complete
(C) X is compact
(D) X is closed

49. If f ( z ) = x3 − 3xy 2 + 2 y + iv( x, y ) is analytic, then

(A) v ( x, y ) = 3 x 2 y − y 3 + 2 x

(B) v ( x, y ) = 3 x 2 y − y 3 + 2 y

(C) v ( x, y ) = y 3 − 3 x 2 y + 2 y

(D) v( x, y ) = x3 − 3xy 2 + 2 x

50. A particular solution ofy′′′ + y′′−y + 1 = −e−x is a constant multiple of

(A) xe− x
(B) −x2e x
(C) −xe x
(D) x2e x

 1 1 1 
51. Let K = 1, , ,..., ,... ⊂ ℝ . If I and L denote the set of interior points and the
 2 3 n 
limit points of K respectively, then

(A) I ∩ L = {0}

(B) I=L
(C) I ∩L =φ
(D) I ∪ L = {0}

52. The bilinear transformation which maps the points 0, 1, ∞ in the z-plane onto the
points−i, ∞, 1 in w-plane is

z −1
(A)
z +i
z +i
(B)
z −1
z −i
(C)
z +1
z +1
(D)
z −i

53. { }
Let S = ( a, b) ∈ ℝ 2 | ab ∈ ℚ . Then S is

(A) countable
(B) bounded
(C) finite
(D) uncountable

54. Let f : ℝ → ℝ be defined as f ( x) = 2025 x 2 . Then the function is

(A) continuous and bounded
(B) uniformly continuous
(C) unbounded but not uniformly continuous
(D) not differentiable at zero

55. The differential equation whose linearly independent solutions are
sin 2 x, cos 2 x and e2 x is

(A) ( D3 + D 2 − 4 D ) y = 0
(B) ( D − 2 D + 4 D + 8 ) y = 0
3 2

(C) ( D3 + 2 D 2 − 4 D + 8 ) y = 0
(D) ( D − 2 D + 4 D − 8 ) y = 0
3 2

cos z
56. Let f ( z ) = . Then ∫ f ( z )dz =
(
2 z z 2 − 27 ) z =2

(A) 0
1
(B) −
27
πi
(C) −
27
2π i
(D) −
27

2025
57. The function f : ℝ → ℝ defined by f ( x) = x 2 + 1 ( ) is

(A) onto but not one-one
(B) one-one but not onto
(C) both one-one and onto
(D) neither one-one nor onto

58. The sequence 5, 5 + 5 , 5 + 5 + 5 ,... converges to

1 + 33
(A)
2
1 − 21
(B)
2
1 + 21
(C)
2
(D) 5

59. If G is a simple graph of order n which is alson −2 regular, then

(A) G always connected
(B) n is even

(C) n is odd
(D) G does not exist

1 2
60. Let A =   be the matrix over ℤ 7 . Then the inverse of A is
0 1

 1 1
(A)  
 0 1
1 5 
(B)  
 0 −1 
 1 −5 
(C)  
0 1 
 1 5
(D)  
 0 1

61. If G is a cyclic group of order 2025, then which of the following is not an order of a
subgroup of G?

(A) 81
(B) 25
(C) 35
(D) 45

62. ( )
A solution of 1 + x 2 y '+ 2 xy − 4 x 2 = 0 satisfying the initial condition y (0) = 0 is

4 x3
(A)
(
3 1 + x2 )
2 x2
(B)
(
8 1 + x2 )
(C) (
4 x3 1 + x 2 )
(D) 12 x + 4 x3

63. Which of the following statement(s)is/are NOT true?

Statement I: Binomial Distribution is a continuous distribution.
Statement II: In a Poisson distribution, the mean and standard deviation are equal

(A) Only Statement I
(B) Only Statement II
(C) Neither Statement I nor Statement II
(D) Both Statement I and Statement II

64. Which of the following is the real part of an analytic function?

(A) u(x, y) = x2 – y2 – 2xy – 2x + 3
(B) u(x, y) = x3 + y3
(C) u(x, y) = x4 – y4 – xy
(D) u(x, y) = x2+y2+ 2xy

65. Which of the following statements about sequence ( an ) are true?

Statement I: If an2 → a 2 , then an → a .

Statement II: If an is bounded, then it has a convergent subsequence.

Statement III: If lim
( a1 + a2 + ...an ) = l , then lim = l
n→∞ n n→∞

(A) Both Statement I and Statement II
(B) Only Statement II
(C) Both Statement II and Statement III
(D) Statement I, Statement II and Statement III
66. If G is a connected planar graph with 20 vertices and 25 edges, then the number of
bounded faces in any embedding of G on the plane is

(A) 7
(B) 6
(C) 5
(D) 3

67. The equation of plane through the intersection of 2x – y + 2z = 3 and x + 2y + z = 2
and passing through (1, 2, 1) is

(A) 7x + 5y + 4z = 8
(B) 13x −6y + 13z = 22
(C) 8 x −6 y + 5 z = − 2
(D) 9x −2y + 9z = 14

68. Let A be a n × n complex matrix. Then which of the following statement(s) is/are
true?

Statement I: If λ is an eigen value of A, then zero is an eigen value of (A −λI)n.

Statement II: If all the eigen values of A are zero, then A is a zero matrix.
Statement III: If A – 7I = I , then 6 is an eigen value of A

(A) Only Statement I
(B) Both Statement II and Statement III
(C) Both Statement I and Statement III
(D) Only Statement II

69. The number of permutations σ in S7satisfying σ(2) = 7 and σ(4) = 5is

(A) 24
(B) 120
(C) 110
(D) 720

∞ 2n + 1
70. ∑ n2 (n + 1)2 is convergent to
n=1

(A) 0
2
(B)
3
(C) 1
(D) 4

71. Which of the following is NOT a metric on ℝ ?

(A) d1 ( x, y ) = x − y

x− y
(B) d 2 ( x, y ) =
1+ x − y

(C) d 3 ( x, y ) = x 3 − y 3

(D) d 4 ( x, y ) = min {− x, y}

72. Let R be a ring with characteristics n wheren ≥ 2. If M is the ring of all 2 × 2 matrices
over R, then the characteristic of M is

(A) n
(B) 2
(C) 2n
(D) 4n

73. Which of the following differential equations is exact?

(A) (y3 – 3xy) dx + (x2 – xy) dy = 0
(B) (x7y2 + 3y) dx + (3x8y – x) dy = 0
(C) (x2− 2y2) dx + (y4 – 4xy) dy = 0
(D) 2xydx + (3x2 – y2) dy = 0

1 0 2 3 2
 
2 2 −1 3 1
74. The rank of the matrix  is
 1 5 −2 4 3
 4 −1 4 
7 3 


(A) 4
(B) 3
(C) 2
(D) 5

0, n is prime
75. Let f : ℤ → ℤ be defined as f (n) =  . If D denote the set of points at
 n, otherwise
which f is discontinuous, then D is

(A) ℤ
(B) φ
(C) set of all primes
(D) 2 ℤ

76. If G is a connected graph with 10 vertices and 27 edges, then

(A) the maximum degree of G is 2
(B) G is non-planar
(C) G is 6-connected
(D) G is a tree

77. Let P = {− p : p is prime} ⊂ ℤ. For any x ∈ P , the number of divisors of xin ℤ is

(A) one
(B) two
(C) three
(D) four

78. Let F = x3y2z be a scalar valued function. Then the directional derivative of Fat(1, 2,
3) is maximum along the direction

(A) 4 i + 3 j + 2k
(B) 18i + 9 j + 3k
(C) 9i + 3 j + k
(D) 18i + 6 j + 2k

 a 2
79. If A =   is a matrix with eigenvalues 6 , then a and b are respectively
1 b 

(A) 2 and −1
(B) 2 and −2
(C) 2 and 1
(D) −2 and 1

∞ 3n
80. The infinite series ∑ n+3
n=1 2

(A) diverges to ∞
1
(B) converges to
4
1
(C) converges to
32
1
(D) converges to
5

81. The number of generators of the cyclic group ℤ 25 is

(A) 20
(B) 24
(C) 5

(D) 6

82. The Laplace transform of e4t is

1
(A)
s+2
1
(B)
s−2
1
(C)
s+4
1
(D)
s−4

83. Let f : ℝ \ {0} → ℝ be a continuous function such that f '( x ) = 0 for all x ∈ ℝ \ {0}.
Then the range of f has

(A) uncountable number of points
(B) countably infinite number of points
(C) at most two points
(D) at most one point

1
ez
84. Let f ( z ) = . Then f has
( )
9 + z 2 ( z − i)

(A) an essential singularity at zero
(B) removable singularity at i
(C) pole of order two at 3
(D) simple pole at 3i

85. Let Vbe an n dimensional vector space over Fand T : V →V be a linear
transformation. Then which of the following condition implies T is one-to-one?

(A) ker(T ) = φ
(B) T is onto
(C) nullity of T is one
(D) nullity of T is n

86. If (3,3,3,3,3) is the degree sequence of a graph G, then

(A) G is connected
(B) G is regular

(C) G is Eulerian
(D) no such Gexists

87. Which of the following equivalents are true for a graph G?

Statement I: G is complete if and only if diam (G ) = 1
Statement II: G is 2-colorable if and only if G is bipartite
Statement III: G is connected if and only if G is disconnected

(A) Only Statement II
(B) Both Statement I and Statement II
(C) Both Statement II and Statement III
(D) Both Statement I and Statement III
dy 2 y
88. The ordinary differential equation = with the initial condition y (0) = 0 , has
dx x

(A) infinitely many solution
(B) no solution
(C) more than one but only finitely many solution
(D) unique solution

1 2 3 4 5 6 7 13
89. Let σ =   be a permutation on S7 . Then σ is
3 1 4 7 2 5 6

1 2 3 4 5 6 7
(A)  
3 5 4 2 7 1 6
1 2 3 4 5 6 7
(B)  
 2 5 1 3 6 7 4
1 2 3 4 5 6 7
(C)  
 2 1 5 3 6 7 4
1 2 3 4 5 6 7
(D)  
7 1 4 3 2 6 5

90. { }
Let S = ( x, y ) ∈ ℝ 2 : xy > 0 . Then S is

(A) connected but not compact
(B) not connected but compact
(C) neither connected nor compact
(D) both connected and compact

91. Which of the following is NOT a vector space?

(A) ℝ × ℝ over ℚ
(B) ℂ over ℚ
(C) ℚ[i ] over ℚ
(D) ℚ over ℝ

 a 1 0  
  
92. Let W =  0 b 1  : a, b, c ∈ ℝ  . Then choose the appropriate option.
 0 0 c  
  

(A) There exists a matrix A ∈W whose rank is 2
(B) Wis a subspace of M ( ℝ ) with dimension 4
3
(C) Wis a subspace of M ( ℝ ) with dimension 3
3
(D) W is a subspace

93. Let V = ℝ 3 be a real inner product space with the usual inner product. A basis for the
subspace U ⊥ , where U = (1, −3, 4) is

(A) {(3, 1, 0), (12, 4, 0)}
(B) {(3, −1, 0), (12, −4, 0)}
(C) {(−3, 1, 0), (4, 0, 1)}
(D) {(3, 1, 0), (−4, 0, 1)}

94. Which of the following permutation is an even permutation?

 1 2 3 4 5
(A)  
 2 3 5 4 1
1 2 3 4 5
(B)  
 3 2 1 4 5
 1 2 3 4 5
(C)  
 2 3 4 5 1
1 2 3 4 5 
(D)  
1 3 4 5 1 

95. If A = yzi + zyj + xyk , then the line integral ∫ A . dr from (0, 0, 0) to (2, 4, 8), along
C
2 3
the curve C; x = t , y = t , z = t is

(A) 0
(B) 64
(C) 32
(D) −1

96. If S = {a + ib : a, b ∈ ℤ, b is even} , then S is

(A) a subring and ideal of ℤ[i ]
(B) a subring but not an ideal of ℤ[i ]
(C) an ideal of ℤ[i ] but not a subring of ℤ[i ]
(D) neither a subring nor an ideal of ℤ[i ]

97. Let T : ℝ 3 → ℝ 2 be such that T (1, 0, 3) = (1,1) and T ( −2, 0, −6) = (2,1). Then

(A) T is a one-to-one linear transformation
(B) T is an onto linear transformation
(C) T is not a linear transformation
(D) T is linear but neither one-to-one nor onto
1
98. Let f : (ℤ, + ) → (ℚ*,.) be a group homomorphism such that f (3) = . Then the
3
value of f ( −9) is

(A) 1
(B) 27
(C) −1
1
(D)
27

99. If the probability that a batsman will hit a six is 60% and if 10 balls are bowled to
him, then the mean and variance are respectively

(A) 4, 2.6
(B) 0.6, 0.24
(C) 6, 2.4
(D) 0.4, 0.16

cos z
100. ∫ ( z − π )3 =
z =1

(A) 0
(B) 2πi
(C) πi
(D) 4πi

101. Let V be an n dimensional vector space over F and S ⊂ V . If S denotes the span of
S, then which of the following is true?

(A) Dimension of S > n
(B) If dimension of S < n, then S is linearly dependent
(C) If S is a subspace of V, then dim S ≥ S
(D) If dimension of S = n, then S is linearly independent

102. The number of ways in which 7 distinct objects can be placed in a circle is

(A) 720
(B) 360
(C) 380
(D) 840

103. Let V be a vector space of all n × n complex matrices over ℂ . Then which of the
followings are NOT a subspace of V?

(I) Set of all orthogonal matrices
(II) Set of all symmetric matrices
(III) Set of all diagonal matrices
(IV) Set of all hermition matrices

(A) Only I
(B) I, III and IV
(C) Both I and IV
(D) Both I and II

1 1 1
104. If a, b, c are sides of a triangle, then , , are also sides of a triangle is
a+b b+c c+a
(A) always true
(B) sometimes true
(C) true after some conditions
(D) never true

105. The product of three positive reals is 1 and their sum is greater than the sum of their
reciprocals. Then exactly one of them is greater than
(A) 0
(B) 1
(C) −1
(D) −2

106. If the fourth roots of unity are z1, z2 , z3 and z4 , then z12 + z22 + z32 + z42 is equal to

(A) 1
(B) −1
(C) 4
(D) 0

2 2
107. Let z − z1 + z − z2 = k be a circle. If z1 = 2 + 3i and z2 = 4 + 3i are the
extremities of a diameter, then k =

1
(A)
4
(B) 4
(C) 2
3
(D)
4
108. All the roots of the equationa1z3 + a2z2 + a3z + a4 = 3where |ai|≤ 1, i = 1, 2, 3, 4 lie
outside a circle with centre at origin. Then the radius of the circle is

(A) 1
1
(B)
3
2
(C)
3
4
(D)
3

109. The region of the argand diagram defined by |z – 1| + |z + 1| ≤4 is

(A) boundary and interior of a circle
(B) boundary and interior of an ellipse
(C) interior of an ellipse
(D) interior of a circle

110. The equation zz + az + az + b = 0, b ∈ R represents a circle, if

(A) |a|=b
(B) | a | ≠b
(C) | a |2>b
(D) | a |2<b

111. Let n be an integer which leaves remainder one when divided by 3. Then
n n
(1 + i 3 ) + (1 − i 3 ) equals
(A) −(−2)n

(B) 2n+1
(C) −2n+1
(D) −2n

112. The equations x3 + 5x2 + px + q = 0 andx3 + 7x2 + px + r = 0 have two roots in
common. If their third roots arek1 andk2, then (k1,k2) is

(A) (5, 7)
(B) ( −5, −7)
(C) ( −5, 7)
(D) (5, −7)

113. The difference between two roots of the equationx3− 13x2 + 15x + 189 = 0 is 2. Then
the roots of the equation are

(A) −3, 5, 7
(B) −3, −7, −9
(C) 3, −5, 7
(D) −3, 7, 9

114. 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

115. The number of triangles whose vertices are at the vertices of an octagon but none of
whose sides happen to come from the octagon, is

(A) 16
(B) 28
(C) 56
(D) 70

116. Let A be the set of 4-digit number a1, a2, a3, a4whereai∈ {1, 2, …, 9}, 1 ≤i≤ 4.
Thenn(A) is equal to

(A) 32
(B) 84
(C) 126
(D) 210

2 2
 1 1   1 1 
117. The value of 1 + + + ...  − 1 + + + ...  is
 2! 4!   3! 5! 

(A) 2
(B) −2
(C) 1
(D) −1

118. Let R be a relation over the set of real numbers defined bymRnifmn≥ 0. Then R is

(A) symmetric and transitive only
(B) reflexive and symmetric only
(C) a partial order relation
(D) an equivalence relation

119. Consider the relation R over the set of integers defined by (x, y) ∈R if and only if |x −
y| ≤ 1. Then R is

(A) reflexive and transitive
(B) reflexive and symmetric
(C) symmetric and transitive
(D) an equivalence relation

1 7
120. Let A and B be two independent events such that P ( A) = , P ( A ∪ B ) = . Then
5 10
P ( B ) is equal to
3
(A)
8
2
(B)
7
7
(C)
9
5
(D)
7

121. It has been found that if A and B play a game 12 times. A wins 6 times, B wins 4 times
and they draw twice. A and B takes part in a series of 3 games. The probability that
they will win alternatively, is

1
(A)
72
5
(B)
72
19
(C)
27
5
(D)
36

122. All the spades are taken out from a pack of cards. From these cards, cards are drawn
one by one without replacement till the ace of spades comes. The probability that the
ace comes in the 4th draw is

3
(A)
13
12
(B)
13
4
(C)
13
1
(D)
13

123. 6 ordinary unbiased dice are rolled. The probability that at least half of them will
show at least 3, is

24
(A) 41× 6
3
24
(B)
36
24
(C) 20 × 6
3
24
(D) 21× 6
3

1
124. If X follows a binomial distribution with parameters n = 8 and p = , then
2
P ( X − 4 ≤ 2 ) is equal to

117
(A)
128
118
(B)
128
119
(C)
128
120
(D)
128

 A 1 B 1
125. For two events A and B, it is given that P ( A) = P   = and P   = . Then
B 4  A 2

(A) A and B are mutually exclusive events
(B) A and B are dependent events

 A 3
(C) P  =
B 4

 A 1
(D) P  =
B 4

126. A box contains 24 identical balls of which 12 are white and remaining black. The
balls are drawn at random from the box one at a time with replacement. The
probability that a white ball is drawn for the 4th time on the 7th draw, is

5
(A)
64
27
(B)
32
5
(C)
32
1
(D)
2

127. If 3sinθ + 5cos θ = 5, then the value of 5sin θ−3cos θis equal to

(A) 5
(B) 3
(C) 4
(D) 2

128. Iflog|sinx||cosx| + log|cosx| |sin x| = 2, then |tan x| is equal to

(A) ∞
1
(B)
2
(C) 1
(D) 2

129. The value ofcos 12° + cos 84°+ cos 156° + cos 132°is

1
(A)
2
(B) 1
1
(C) −
2
1
(D)
8

130. If sinx + sin2x = 1, then cos6 x + cos12 x+ 3 cos10 x + 3 cos8 x is equal to

(A) 0
(B) ∞
(C) cos3 x sin3 x
(D) 1

131. If a and b are positive numbers such that a > b, then the minimum value of
 π
a sec θ − b tan θ , where  0 < θ <  is
 2

1
(A)
a 2 − b2
1
(B)
a2 + b2
1
(C)
a
(D) a2 − b2

132. The minimum value of 27cos 2x × 81sin 2x is

(A) 1
1
(B)
9
1
(C)
81
1
(D)
243

f '(1) f "(1) f '''(1) (−1)n f ( n) (1)
133. If f ( x) = x n , then f (1) − + − + ... + is
1! 2! 3! n!

(A) 2n
(B) 0
(C) 2n−1
(D) 1

134. Let f ( x ) and g ( x ) be two differentiable functions on [0, 2] such that
3 3
f "( x ) − g "( x ) = 0, f '(1) = 2 g '(1) = 4, f (2) = 3 g (2) = 9. Then f   − g   =
2 2

(A) 0
(B) 2
(C) 10
(D) 5

135. Let f ( x ) be a continuous double differentiable function such that g ( x) = f '( x) and
2 2
  x    x 
f "( x ) = − f ( x ). If f ( x ) =  f    +  g    and f (5) = 5, then f (10) is
  2    2 

(A) 0
(B) 5
(C) 10
(D) 25

136. The vectors (x, y, 0), (1, 0, z), (1, 1, 0) are linearly independent in R3 if

(A) x ≠ y and z ≠ 0
(B) x = y and z ≠ 0
(C) x=y=z
(D) x = y and z = 0

137. Let X = (xij) be a matrix if order s × tandxij=1 for all i, j. Then rank (A) is

(A) s
(B) s−t
(C) 1
(D) 0

138. The number of subspaces of R3 over R is

(A) 3
(B) 6
(C) 8
(D) 9

139. Let f be a monotone function on the open interval (a, b). Then fis
(A) continuous except possibly at a countable number of points in (a, b)
(B) continuous at all rational points in (a, b)
(C) continuous at all irrational points in (a, b)
(D) continuous at all points in (a, b)

140. Let and be two subgroups of group . Then,
(A) ( ∩ ) = ∩ , ∈
(B) ( ∩ ) = ∩ , ∈
(C) ( ∩ ) ≠ ∩ , ∈

(D) ( ∩ ) = ∩ , ∈

141. If A = [[−5 −8 0][3 5 0][1 2 −1]] then A is

(A) idempotent
(B) nilpotent
(C) involutory
(D) periodic

142. What is the equation of the surface of the cone with vertex at the origin and whose
axis is along the line x = y = z?

(A) x2 + y2 = z2
(B) x2 + y2−z2 = 0
(C) x2 + y2 + z2 = 0
(D) x2 + y2−z2 = 1

143. Which of the following is an assumption of Jacobi’s methods?

(A) The coefficient matrix has zeroes on its main diagonal
(B) The coefficient matrix has no zeroes on its main diagonal
(C) The rate of convergence is quite slow compared with other methods
(D) Iteration involved in Jacobi’s method converges

144. A force = (10 + 8 − 5 ) acts on a point p(2, 5, 6). What will be the moment of the
force about the point Q(3, 1, 4)?

(A) –36i + 15j – 48k
(B) 18i + 19j – 37k
(C) 32i – 31j – 4k
(D) Zero

145. What is the smallest positive integer in the set {30 + 170 + 1500 | , , ∈ ℤ}?

(A) 12
(B) 14
(C) 10
(D) 8

146. Find the ratio in which the xy plane divides the line joining the points A(7, 4, −2) and
B(8, −5, 3).

(A) 2 : 5
(B) 2 : 3

(C) 5 : 3
(D) 3 : 2

147. Which of the following statement is false?

(A) The eigenvectors of a diagonal matrix are orthogonal
(B) The determinant of a matrix is equal to the product of its eigenvalues
(C) The eigenvectors of a symmetric matrix are orthogonal
(D) The eigenvectors of a diagonal matrix form a basis for the vector space

2 3
148. If the curve + = 7 and = , cut orthogonally at (1, 1), then the value of a is

(A) 1
(B) 0
(C) 6
(D) −6

149. The number of divisors of 2100 is

(A) 35
(B) 36
(C) 26
(D) 53

150. Let f be continuous for ≥ 0, ′( ) exists for > 0, f(0) = 0 and ′ is monotonically
f ( x)
increasing. Let g be defined as g( ) = ( > 0). Then g is
x

(A) constant
(B) monotonically decreasing
(C) monotonically increasing
(D) periodic

ANSWER KEY
Subject Name: MATHEMATICS PG

SI No. Key SI No. Key SI No. Key SI No. Key SI No. Key
1 A 31 A 61 C 91 D 121 D
2 C 32 B 62 A 92 A 122 D
3 A 33 B 63 D 93 D 123 A
4 C 34 A 64 A 94 C 124 C
5 A 35 C 65 B 95 B 125 C
6 B 36 B 66 B 96 B 126 C
7 A 37 D 67 D 97 C 127 B
8 C 38 A 68 A 98 B 128 C
9 A 39 A 69 B 99 C 129 C
10 B 40 B 70 C 100 A 130 D
11 C 41 C 71 D 101 D 131 D
12 B 42 A 72 A 102 A 132 D
13 A 43 B 73 C 103 C 133 B
14 C 44 B 74 B 104 A 134 D
15 C 45 D 75 B 105 B 135 B
16 B 46 A 76 B 106 D 136 A
17 D 47 B 77 D 107 B 137 C
18 A 48 C 78 C 108 C 138 C
19 A 49 A 79 B 109 C 139 A
20 A 50 A 80 A 110 C 140 B
21 D 51 C 81 A 111 A 141 C
22 D 52 B 82 D 112 B 142 B
23 C 53 D 83 C 113 D 143 B
24 D 54 C 84 D 114 B 144 A
25 D 55 D 85 B 115 A 145 C
26 A 56 C 86 D 116 C 146 B
27 A 57 D 87 B 117 C 147 A
28 A 58 C 88 A 118 D 148 C
29 C 59 B 89 B 119 B 149 B
30 D 60 D 90 C 120 A 150 C