PU CET PG 2019 Question Paper MSc_HS__Mathematics_

MSc(HS)(Mathematics)
1. If x is real, then the largest interval in which the expression assumes values, is
A) [-1, 1] B) [ , 2] C) [ , 3] D) [ , 4]

2. The equation − − 4 = 0 has
A) No real root B) One real root
C) Two real roots D) Infinitely many roots

3. If = 2 + 2 + 2 , then the value of −6 + 6 , is
A) 1 B) 2 C) 3 D) 4

4. The product 2 . 4 . 8 . 16 … … .. is equal to
A) 1 B) 2 C) 3/2 D) 5/2

5. The solution of " = # cos with '(0) = 0 is
"#

A) ' = sin + 1 B) # = sin + 1 C) ' = + 2 D) ' = sin

6. Bisection method is
A) Always convergent B) Always divergent
C) Not always convergent D) None of these

1 ., is rational 3
7. Let , = -
0 ., is irrational
, then
A) f is continuous everywhere on real line
B) f is discontinuous everywhere on real line
C) f is continuous on (0, 1) otherwise discontinuous
D) None of the above

A) = 4 sin 5 cos 6 , ' = 4 sin 6 cos 6, 7 = 4 cos 5
8. Relation between Cartesian and polar spherical coordinates are

B) = 4 sin 5 sin 6 , ' = 4 cos θ sin 6, 7 = 4 cos 5
C) = 4 sin 5 cos 6 , ' = 4 sin 5 sin 6, 7 = 4 cos 6
D) = 4 sin 5 cos 6 , ' = 4 sin 5 sin 6, 7 = 4 cos 5

+ ' + 7 = tan 8 + ' = 7 tan 8
9. Equation of right circular cone in standard form is

+ ' = 7 tan 8 + ' − 7 = tan 8
A) B)
C) D)

10. The probability that the 13th day of a randomly chosen month is a second Saturday, is
A) 1/7 B) 1/12 C) 1/84 D) 19/84

11. The limit point of (9, :) 9, : ∈ ℝ
A) is a B) is b C) Both a and b D) Every element of [a,b]

12. lim>→@
(> !)B
>
A) E B) 1/e C) 1 D) C

13. If D = +' = H D, where K is
# FG FG
√ # F F#
, then
A) 2 B) ½ C) 3/2 D) 2/3

14. The curvature of a circle at any point on it is
A) Constant and equal to radius of the circle
B) Variable
C) Constant and equal to reciprocal of the radius of the circle
D) Proportional to the radius of the circle

15. Weight is a
A) Scalar quantity B) Vector quantity
C) Scalar and vector both D) Mass contained in the body

16. The vectors 8 = (9 , 9 ) and I = (: , : ) in J (K) are linearly dependent if
A) 9 : − 9 : = 0 B) 9 : − 9 : = 0
C) 9 9 − : : = 0 D) None of these

17. The vector space L0M is defined to have dimensions
A) Zero B) One C) ∞ D) Not defined

18. A mapping ,: ⟶ which is defined as ,( ) = sin ; ∈ is
A) One-one only B) Onto only
C) One-one onto D) Neither one-one nor onto

19. Let R = L Ι ∈ S, .T UDVW.XV Y, 3 9[\ ≤ 100M and
^ = L Ι ∈ S, .T UDVW.XV Y, 5 9[\ ≤ 100M What is the number of elements in
(R × ^) ∩ (^ × R)
A) 36 B) 25 C) 49 D) 16

(A) lim>→∞ (>!) = 27
( >)!
20. Consider the statements

(B) lim>→∞ (> =
>B C
)(> )…(> >)
Then
A) (A) is true (B) is false B) (A) is false (B) is true
C) (A) and (B) both are false D) (A) and (B) both are true

21. The value of ‘C’ of Lagrange’s mean value theorem, if ,( ) = ( − 1)( − 2);
9 = 0, : = is

c √( ) c √( )
c c
A) 1/4 B) 1/3 C) D)

22. At t=0, the function ,(W) =
d
e
has
A) A minimum B) A discontinuity
C) A point of inflection D) A maximum

' sin , ≠ 03
23. If ,( , ') = f
0, = 0
then
A) , (0,0) = 1 = ,# (0,0) B) , (0,0) = 0 = ,# (0,0)
C) , (0,0) and ,# (0,0) does not exist D) None of these

24. Given the function ,( , ') = −2 '+' + −' + i
, then
A) Has maximum value at origin
B) Has minimum value at origin
C) Has maximum value but no minimum value at origin
D) Has neither maximum nor minimum value at origin

−5 −8 0
25. If j = k 3 5 0 l, then j is
1 2 −1
A) Nilpotent B) Periodic C) Idempotent D) Involutory

1 2 −2
26. The square matrix A is defined as j = k 1 2 1 l, the diagonal matrix D of A is
−1 −1 0
1 0 0 0 0 0 1 0 0 3 0 0
A) k0 2 0l B) k0 −1 0l C) k0 −1 0l D) k0 1 0 l
0 0 3 0 0 2 0 0 3 0 0 −3

27. Dim V, where J = L9 , 9 , … 9 mm ∶ 9 + 9 = 0, 9 + 9 = 0 M is
A) 100 B) 102 C) 2 D) 98

0 '
28. If C is a non-singular matrix and o = p k0 0 7l p
0 0 0
, then

A) o = 1 B) o = 1 C) o = 0 D) o = 0

29. The dimension of the subspace of spanned by (−3,0,1), (1,2,1)9[\ (3,0, −1) is
A) 0 B) 1 C) 2 D) 3

4 + −3 − =0
30. The system of equations

2 + 3 + − 5 = 0 has
−2 −2 +3 =0
A) No solution

B) Only one solution (0,0,0,0)
C) Infinite no of solutions
D) Only two solutions (0,0,0,0) and (3/5,1,4/5,1)

31. Let A be [ × [ matrix which is both Hermitian and unitary. Then
A) j = q
B) A is real
C) Eigen values of A are 0,1,-1
D) Characteristics and minimal polynomials are the same

32. Which of the following is correct?
A) R is a vector space over N B) R is vector space over Z
C) R is vector space over Q D) R is vector space over C

A) No solution in c B) A unique solution in c
33. A homogeneous system of 5 linear equations in 6-variables admits

C) Infinite no of solutions in ℝc D) Finite, but more than two solutions in ℝc

34. Minimal polynomial U( ) of j>×> , each of whose element is 1, is
A) − 1 B) + C) +[ D) −[

2 3 4
35. Let k0 5 6l, then
0 0 7
A) j is diagonalizable but not j B) j is diagonalizable but not j
C) Both j and j are diagonalizable D) Neither j nor j is diagonalizable

36. Let r: ⟶ R be a linear transformation given by r( , )=( + , − , )
then Rank T is
A) 0 B) 1 C) 2 D) 3

1 9 t
37. Let s = k0 2 : l , 9, :, t ∈ , then M is diagonalizable, if and only if
0 0 1
A) 9 = :t B) : = 9t C) t = 9: D) 9 = : = t

38. The limu→m u is
u̅

A) 0 B) 1 C) ½ D) Does not exists

39. The real part of the complex number (1 + .)> is
A) 2 cos B) 2> cos C) 2 cos [x D) 2 > cos
B
>w >w B
>w

40. The value of m so that 2 − + U' may be harmonic is
A) 0 B) 1 C) 2 D) 3

41. Let G be a group of order 77. Then, the centre of G is isomorphic to

A) ℤ B) ℤz C) ℤzz D) ℤ

42. The number of zeros at the end of 100! is
A) 22 B) 24 C) 26 D) 20

43. The unit digit of 2 mm is
A) 2 B) 4 C) 6 D)8

44. The largest integer n such that 33! is divisible by 2> is
A) 29 B) 28 C) 33 D) 31

45. The number of generators in cyclic group of order 10 are
A) 1 B) 2 C) 3 D) 4

46. For the differential equation ' ′ − ' = 0, which of the following is not an integrating
factor
A) B) # C) # D) #

47. The solution of X + '{ = 7 is
A) ,( , ' ) = 0 B) ,( ', '7) = 0 C) ,( , ') = 0 D) , # , u = 0
#

48. The equation De = t D is classified as
A) Elliptic B) Hyperbolic C) Parabolic D) None of these

49. Simpson’s rule for integration gives exact result, when f(x) is a polynomial of degree
A) 1 B) 2 C) 3 D) All of these

50. The Newton-Raphson method converges fast, if , ′ (8) is (8 is the exact value of the root)
A) Small B) Large C) Zero D) None of these

51. If 3 is taken as an approximate root of the equation − 30 = 0, then a better
approximation by Newton-Raphson method is
A) 10/3 B) 28/9 C) 8/3 D) 26/9

52. If h is the interval of differencing, then (∆ − ∇) equals to
A) 2ℎ B) 2ℎ C) 2ℎ D) 4ℎ

53. The first term of the series whose second and subsequent terms are 8, 3, 0, −1, 0 is
A) 5 B) 10 C) 15 D) 20

54. In Simpson’s one-third rule the curve ' = ,( ) is assumed to be a
A) Ellipse B) Hyperbola C) Circle D) Parabola

55. The solution of the equation 6( ) = + •m (€ − )6(€)\€ is
A) sin B) cos C) tan D) None of the these

56. A perfectly flexible rope of uniform density per unit length is suspended with its end
points fixed. It assumes the shape of
A) Cycloid B) Straight line C) Parabola D) Catenary

57. In case of a rigid body, having S particles, the number of degrees of freedom is
A) ∞ B) 3 C) 3N D) N

58. Let • be a random variable whose density function , forms an isosceles triangle above
the unit interval q = [0,1] and 0 elsewhere. Then the formula for pdf is
, ., 0 ≤ ≤ 2 , ., 0 ≤ ≤
A) ,( ) = „− + 1, ., ≤ ≤ 1 3 B) ,( ) = „−2 + 2, ., ≤ ≤ 13
0, VT … 4 0, VT … 4
3 , ., 0 ≤ ≤
C) ,( ) = „−3 + 3, ., ≤ 3
≤1 D) None of these
0, VT …ℎ 4

59. If • is normal with mean 2 and standard deviation 3, then the distribution of † = • − 1
is
A) N(0,9/4) B) N(1,9/4) C) N(0,3/4) D) N(1,3/4)

60. If • and † be two independent random variables, each representing the number of
failures preceding the first success in a sequence of Bernoulli trials with X as probability
of success in a single trial and { as probality of failure, then X(• = †) is
‡ ˆ ‡ ˆ
A) ˆ B) ‡ C) ‡ D) ˆ

61. Let • , • , … •> be n independent and identically distributed random variables each with
‹‹‹>‹ be the sample mean, then variance of •
mean ‰ and variance Š , and let • ‹‹‹>‹ is
B) Š C) [Š
Œ
A) 0 D) >
62. The value of mode for the frequency curve ' = sin , 0 ≤ ≤ x .T
A) x
w w w
B) C) D)

3
63. For a given probability distribution ,( ) = • , = 0, 1, 2 ,3 for a random variable •,

A) • (1 + e ) C) (1 + e)
the moment generating function of this is
B) e D) e

64. A bag X contains 2 white and 3 black balls and another bag Y contains 4 white and 2
black balls. One bag is selected at random and a ball is drawn from it. Then, the
z •
probability for the ball chosen be white is to
A) i B) i C) i D) i

65. Consider the group ^Ž of all the permutations on a set with 9 elements. What is the largest
order of a permutation in ^Ž ?
A) 20 B) 21 C) 14 D) 30

66. Suppose V is a real vector space of dimension 3. Then the number of pairs of linearly

A) ∞ B) 1 D) 3
independent vectors in V is
C) 2

67. Let X be a connected subset of real numbers. If every element of X is irrational, then the

B) Countably infinite C) 2 D) 1
cardinality of X is
A) Infinite

40 −29 −11
68. Suppose the matrix j = k−18 30 −12l has a certain complex number • ≠ 0 as an
26 24 −50

A) • + 20 B) • − 20 C) 20 − • D) −20 − •
eigen value. Which of the following numbers must also be an eigen value of A?

69. Let A be a 3 × 3 matrix with real entries such that \ W(j) = 6 and trace of A is 0. If
det(j + q) = 0, where I denotes 3 × 3 identity matrix, then eigen values of A are
A) -1,2,3 B) −1, −2,3 C) −1,2, −3 D) −1, −2, −3

70. Let 9> = T.[ > . For the sequence 9 , 9 , … the supremum is
w

A) 0 and it is attained B) 0 and it is not attained
C) 1 and it is attained D) 1 and it is not attained

71. The power series ∑∞“m 3 > (7 − 1) > converges, if
A) |7| ≤ √3 B) |7| < √3 C) |7 − 1| < √3 D) |7 − 1| ≤ √3

72. The distance between the origin and the point nearest to it on the surface 7 2 = 1 + ' is
C) √3
√
A) 1 B) D) 2

, [ ] ≠ 03
sin[ ]
73. If ,( ) = – [ ] where [ ] denotes the greatest integer less than or equal to
0, [ ] = 0
x, then lim →0 ,( ) is
A) 1 B) 0 C) −1 D) None of these

74. If f( ) = ( 2 − 1)— 2 − 3 + 2— + cos(| |), then set of point of non-differentiability is
A) {0,1,2} B) L1,2M C) L0,2M D) {2}

75. The function y= |log | is

A) Discontinuous at x=1 B) Differentiable at x=1
C) Not differentiable at x=1 D) None of these

x-x-x