MH SET 2024: 7 April, Paper II, Mathematical Sciences
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Test Booklet Code & Serial No.
A
MATHEMATICAL SCIENCE
Signature and Name of Invigilator Seat No.
1. (Signature) ......................................... (In figures as in Admit Card)
(Name) ................................................ Seat No. ..............................................................
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(Name) ................................................ OMR Sheet No.
APR - 30224 (To be filled by the Candidate)
Time Allowed : 2 Hours] [Maximum Marks : 200
Number of Pages in this Booklet : 48 Number of Questions in this Booklet : 180
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APR - 30224/II—A 2
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APR - 30224/II—A
Mathematical Science
Paper II
Time Allowed : 120 Minutes] [Maximum Marks : 200
Note : This Paper contains One Hundred Eighty (180) multiple choice questions
in THREE (3) sections, each question carrying TWO (2) marks. Attempt
all questions either from Sections I & II only or from Sections I & III
only. The OMR sheets with questions attempted from both the Sections viz.
II & III, will not be assessed.
Number of questions, sectionwise :
Section I : Q. Nos. 1 to 20, Section II : Q. Nos. 21 to 100,
Section III : Q. Nos. 101 to 180.
SECTION I 3. Let K R n be with the property
that any real valued continuous
1. The set given by
function on K is bounded. Then K
{1 ( 1)n |n N} is :
is : (A) Bounded
(A) {1, –1} (B) Compact
(B) N (C) Derise
(C) {0, 1} (D) Connected
(D) {0, 2} 4. Suppose a bounded function
f : [0, 1] R is Riemann integrable.
2. The number of non-empty subsets
Then which of the following is not
of X = {1, 2, 3, 4, 5, 6, 7} containing true ?
only even integers is : (A) f is Lebesgue integrable
(A) 7 (B) Riemann integral of f is same
as Lebesgue integral of f
(B) 8
(C) f is not Lebesgue integrable
(C) 1 whenever f is discontinuous
(D) 3 (D) |f| is Lebesgue integrable
3 [P.T.O.
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APR - 30224/II—A
5. Let f : R R be a function. Then 8. Suppose that { an , n 1} is a sequence
which of the following is not true ?
of real numbers such that each term
(A) If f is continuous then so is |f| of the sequence is either 0, 1, 2, 3.
(B) If f is integrable then so is |f| Let S be the set of all such sequences.
(C) If f is differentiable then so Then, which of the following is
is |f| true ?
(D) If f is bounded then so is |f|
(A) S is finite
6. For n 1 , let An be the set of all
irrational numbers in the interval (B) S is countable infinite
[1 1 / n, 1 1 / n] . Then, which of (C) S is equivalent to the interval
the following is true ?
(0, 1)
(A) lim sup A n {1}
(D) S is equivalent to the set of all
(B) lim inf A n {1}
positive integers
(C) lim A n
9. Suppose that f 1 , f 2 and f 3 are
(D) lim A n {0}
functions defined by :
7. Suppose that f is a function defined
by : f1 ( x) x2 , 0 x
x if 0 x ½ f 2 ( x) cos( x), 0 x /2
f ( x)
x ½ if ½ x 1 sin( x) / x if x 0
f3 ( x )
1 if x 0
1 1 3
If M , , N ,1 and
4 2 4 Which of the above functions is /are
1 1 uniformly continuous on their
Q x| f ( x) , , then which
4 2 respective domains ?
of the following is true ? (A) None of the three functions
(A) Q = M
(B) Only f1 and f2
(B) Q N
(C) Only f1 and f3
(C) Q M N
(D) Q M N (D) Only f2 and f3
4
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APR - 30224/II—A
10. Consider the function f : R 2 R 12. Let A be a 4 × 3 matrix and B a
defined by : 3 × 4 matrix. Which of the following
is always true ?
x2 y 2 , if x 0, y 0
f ( x, y)
1, otherwise . (A) det (AB) = 0, det (BA) need not
be zero
Which of the following statements
(B) det (AB) and det (BA) need not
is/are true ?
be zero
S1 : f is discontinuous at (0, 0)
(C) det (AB) = det (BA) = 0
S2 : lim lim f ( x, y)
y 0 x 0
lim lim
x 0 y 0
(D) det (AB) = 0 only if either
f ( x, y) AB = 0 or BA = 0
S 3 : lim
x 0
f ( x, y) is continuous at 13. The system of linear equations is
y = 0 consistent :
(A) Only S1 and S2
x1
2 0 6 b1
(B) Only S1 x2
2 2 4 b2 .
x3
(C) Only S3
(D) Only S2 and S3 (A) Only if b1 = 4 and b2 = –12 and
has a unique solution
11. The dimension of the vector space
over R of all 5 × 5 upper triangular (B) For all values of b1 and b2 and
matrices with complex entries is : has a unique solution
(A) 30 (C) Only if b1 = 4 and b2 = –12 and
(B) 20 has infinitely many solutions
(C) 25 (D) For all values of b1 and b2 and
(D) 15 has infinitely many solutions
5 [P.T.O.
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APR - 30224/II—A
14. Let A be a 6 × 7 matrix of rank 6 1 2 0
and B be a 7 × 5 matrix of rank 4, 17. Let A . Then, singular
0 1 2
then the rank of AB is :
values of A :
(A) 5
(A) are 7 and 3
(B) 6
(C) 4 (B) are 3 , 2 and 1
(D) Cannot be determined exactly, (C) do not exist
but less than or equal to 4
(D) 7 and 3
15. Let T be a linear transformation
from R3 to R given by 18. Let A and B be similar matrices.
T[x, y, z]T = 2x – 5y + 9z, Consider the following statements :
where [x, y, z]T donote the transpose (i) tr A tr B
of the vector [x, y, z]. Then, the
(ii) det A = det B
dimension of the null space of T
is : (iii) Characteristic polynomial of A
(A) 0 is equal to the characteristic
(B) 1 polynomial of B
(C) 2 (iv) Minimal polynomial of A is
(D) 3 equal to the minimal polynomial
16. Let P : R 3 R 3 represent the
of B
projection of R3 onto xy-plane along
z-axis. Then : (A) All the statements are true
(A) P2 = I
(B) None of the statement is true
(B) P3 = P
(C) Only (i), (ii), (iii) are true
(C) P is invertible
(D) P is not diagonalizable (D) Only (ii), (iii), (iv) are true
6
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19. Let A be an n × n nilpotent non-zero 22. Which of the following is false ?
real matrix. Then : (A) log z ln| z| i arg z, z 0
and arg z
(A) In + A is invertible (B) cos2 z 1 sin 2 z for all z C
(B) In – A is invertible (C) cosh 2 z sinh 2 z 1 for all
z C
(C) A is invertible
(D) log ( z1 z2 ) log z1 log z2 for all
(D) A is diagonalizable z1 , z2 C \{0}
23. Suppose z10 1 and z 1 . Which of
20. Let A be a 4 × 4 real orthogonal
the following statements is false ?
matrix with determinant –1. Then
(A) 1 z z2 ...... z9 0
the determinant of I4 – A is : (B) 1 z2 z4 ...... z18 0
(A) 0 (C) 1 . z . z2 . z3 . ..... . z9 1
(D) 1 . z2 . z4 . z6 . ..... . z18 1
(B) 1
24. Consider the following two
(C) 2 statements :
(D) 5 (i) Every harmonic function u on
C has a harmonic conjugate.
SECTION II
(ii) If u and v are harmonic
1 conjugates of each other on a
21. The function f ( z) has primitive
z
disk D, then u and v are
in the domain :
constant on D.
(A) { z C |1 | z| 2} Then :
(B) { z (A) Only (i) is true
C || z| 2}
(B) Only (ii) is true
(C) { z C |1 | z|}
(C) Both (i) and (ii) are true
(D) { z x iy C| x 2} (D) Both (i) and (ii) are false
7 [P.T.O.
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25. Which of the following Möbius 27. Which of the following statements
transformations maps the unit disk is false for an entire function f ?
onto itself ? (A) If f ( x) 0 x [ 1, 1] , then
2z i f 0
(A) f ( z)
2 zi
(B) If A is an uncountable set in
i z
(B) f ( z) C and f ( z) 0 z A , then
i z
f 0
2z 1
(C) f ( z)
2 z (C) If { an } is a sequence of distinct
iz 1 complex numbers and f (an ) 0
(D) f ( z)
z i
for all n, then f 0
26. Let f be analytic on
(D) f (1 / n) 0 n 1 , then f 0
D {z C /| z| 3} such that :
28. Suppose f ( z) u iv is an entire
1 2n 1
f 1 , n 1, 2, ........ function. Which of the following is
n n 1
true ?
Then f(z) =
(A) If u 0 , then f is unbounded
2z 1
(A)
z on C
z (B) If | f ( z)| | z| for all z, then
(B)
z 1
f is constant
z 1
(C) (C) If u is bounded, then v is
z
bounded
z
(D)
z 1
(D) If u 0 , then v 0
8
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29. Suppose f : C C is an analytic 32. The value of the integral
one-one function. Then which of the
z3
following statements is false ? dz , where (t) 2eit ,
( z 3)3
(A) f is a polynomial of degree one
t [0, 2 ] is :
(B) f is an open map (A) 18
(C) f is an onto map (B) 18 i
(D) f '( z) 0 for some z C (C) 6 i
1
(D) –2 i
30. Let f ( z) e z and g( z) e z and
1 ez 1
A z / 0 | z| 33. The residue of f ( z) at
. z2
2
z = 0 is :
Then, on A :
(A) 0
(A) f is one-one but g is not one-one
(B) f is not one-one but g is one-one 1
(B)
2
(C) Both f and g are one-one
(C) 1
(D) Neither f nor g is one-one
(D) 2
31. If all roots of the polynomial
1 z z2 ..... z10 are in the disk 34. The image the circle
B(0, R), then :
c {z C :| z 1| 1}
1 2 z 3 z2 ..... 10 z9 1
dz
1 z z2 ..... z10 under the map f ( z) is :
|z| R z
(A) 0 (A) a circle
(B) 10 (B) a line
(C) 10 i (C) a parabola
(D) 20 i (D) an ellipse
9 [P.T.O.
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APR - 30224/II—A
35. Which of the following polynomial 38. Consider the following statements :
is reducible over the rationals ?
(I) 3 is prime in the integral
(A) x11 – 11x – 11 domain Z [ i 5]
(B) x10 – 7 (II) 2 is a prime element in Z / 10 Z
(C) x5 – 5 but not irreducible in Z / 10 Z
(D) x9 + 5x5 + x4 + 5
Which is correct ?
36. Which is correct ?
(A) Only (I) is true
(A) The polynomial ring K[x] over
(B) Only (II) is true
a field K is local
(C) Both (I) and (II) are true
(B) The additive group of rationals
has a maximal subgroup (D) Neither (I) nor (II) is true
(C) Any ring (may not have unity) 39. Which of the following is false ?
has a maximal ideal (A) In a commutative ring with
(D) Any finitely generated ideal of unity, every prime ideal is
a Boolean ring is principal maximal
37. Consider the following statements : (B) In a commutative ring with
(I) A p-sylow subgroup of the unity, every maximal ideal is
underlying additive group of a prime
finite commutative ring R with (C) In a Boolean ring with unity,
unity is an ideal of R. every prime ideal is maximal
(II) If M is a maximal ideal of a
(D) In a Boolean ring with unity,
commutative ring R and R* be
every maximal ideal is prime
the group of units of R. Such
that R is a disjoint union of M 40. Let G be a finite abelian group. If
and R*, then R has a maximal n N and |G| are relatively prime,
ideal other than M. then the function Q : G G defined
Which is true ? by Q( a) an is :
(A) Only (I) is true (A) homomorphism but not 1 – 1
(B) Only (II) is true (B) homomorphism but not onto
(C) Both (I) and (II) are true (C) isomorphism
(D) Neither (I) nor (II) is true (D) not a homomorphism
10
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APR - 30224/II—A
41. Let G be a group and a map 43. Consider the following statements :
Q:G G defined by Q(a) = a–1. (I) If n 5 , then all cycles of
Then which of the following is length 3 are conjugates in An.
false ?
(II) Two elements of An that are
(A) Q is bijective
conjugates in S n are always
(B) Q is a homomorphism if G is
conjugates in An also.
abelian
Which is correct ?
(C) Q is a homomorphism only if G
(A) Only (I) is true
is abelian
(B) Only (II) is true
(D) Q is a homomorphism, if G is
any permutation group (C) Both (I) and (II) are true
42. Consider the following statements : (D) Neither (I) nor (II) is true
(I) If G is a finite group that has 44. Consider the following statements :
only 2 conjugate classes, then
(I) Every finite group is isomorphic
|G| = 2.
to a subgroup of the alternating
(II) If G is a group and a Z(G) , group An for some n > 1.
the center of G, then the
(II) The symmetric group S n is
conjugacy class of a is {e, a},
nilpotent for n 3.
where e is the identity of G
Which is correct ?
Which of the following is true ?
(A) Only (I) is correct
(A) Only (I) is true
(B) Only (II) is true (B) Only (II) is correct
(C) Both (I) and (II) are true (C) Both (I) and (II) are correct
(D) Neither (I) nor (II) is true (D) Neither (I) nor (II) is true
11 [P.T.O.
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APR - 30224/II—A
45. The number of words of three 48. If F is a finite field of characteristic
distinct letters formed from the p, a prime, then for some integer
letters of the word “PNTU” is : n 1 :
(A) F $ Z pn
(A) 24
(B) F $ Fp % Fp % ..... % Fp (n times)
(B) 24 where F p is the field of p
(C) 42 elements
(D) 4 6 (C) F is the splitting field of
n
xp x Fp [ x ]
46. Suppose that for any group of n
(D) F is the splitting field of
people has the property that at least n
xp 1 Fp [ x ]
two of them have birthdays that
49. The closed unit ball centered at
occur on the same day of the week.
origin is compact in :
Then which one of the following
(A) Finite dimensional normed
must be the value of n ? linear space
(A) 10 (B) (C[0, 1], )
(B) 7 (C) (L1 [0, 1], 1 )
(C) 6 (D) (L2 [0, 1], 2 )
50. Let S1 {( x, y) R 2 / x2 y2 1} be
(D) 5
the unit circle and
47. Let Fq be a finite field with q odd
D {( x, y) R 2 /| x| 1,| y| 1}
and ! a and " # a.
a Fq\{0}
Then :
a Fq
(A) S1 is homeomorphic to R
Then :
(B) S1 is not homeomorphic to D
(A) 0 and " 1
(C) S 1 is homeomorphic to an
(B) 1 and " 1
interval in R
(C) 0 and " 1
(D) D is homeomorphic to an
(D) 1 and " 1
interval in R
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APR - 30224/II—A
51. Let X A B where A and B 54. Which of the following is true ?
are connected sets such that
(A) A metric space is always
A B = . Let f : X f ( x) & R be
compact
a homeomorphism. Then :
(A) f(X) is always a finite subset (B) A metric space is always
of R Hausdorff
(B) f(X) is a singleton set
(C) A metric space is always
(C) f(X) is an interval in R
connected
(D) f(X) is not connected
(D) A metric space is always
52. Which of the following sets are
homeomorphic ? complete
(A) Square and Parabola 55. Let A and B be connected sets.
(B) Square and Ellipse Then :
(C) Ellipse and Parabola (A) A B is connected if A B
(D) Parabola and Hyperbola
(B) A B is connected if A B
53. A topological space X is regular
if : (C) A \B is connected if B & A
(A) Singleton sets are closed sets (D) A \B is connected if B ( A
(B) For each pair x X and an 56. Let C[0, 1] denote the space of
open set B in X with x ' B there
continuous functions on [0, 1] with
exists an open set U such that
x U and u B sup norm and G denotes the
(C) For each pair x X and a closed set of all constant functions on
set B in X with x ' B there [0, 1]. Then :
exists an open set U such that
(A) G is bounded subset of C[0, 1]
x U and u B
(D) For each pair x X and a closed (B) G is dense in C[0, 1]
set B in X with x ' B there exist (C) G is closed subset of C[0, 1]
an open sets U, V such that
x U , B & V and U V (D) G is compact subset of C[0, 1]
13 [P.T.O.
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APR - 30224/II—A
57. Let X be a topological space with a 59. Let A {( x, y) / y2 x2 y2 1} and
subspace Y. Which of the following f :A R be a continuous map.
statements is false ?
Which of the following is false ?
(A) A set F is closed in Y iff
(A) f(A) is an interval
F F1 Y where F1 is closed
(B) f(A) is a bounded set
in X
(B) A set B is open in Y iff (C) f(A) is a closed set
B B1 Y where B1 is open (D) f(A) is a union of two disjoint
in X intervals
(C) A set C is connected in Y
60. Let X = [0, 1) × [0, 1), Y = [0, 1]
iff C C1 Y where C 1 is
× (0, 1) and Z = [0, 1) × (0, 1] be
connected in X
subspaces of the Euclidean space
(D) A set K is compact in Y iff
R2. Then :
K K1 Y where K1 is
(A) X is homeomorphic to both Y
compact in X
and Z
58. Consider the set :
A {( x, y) / xy 0} (B) X is homeomorphic to Y but not
Which of the following is false ? homeomorphic to Z
(A) A is closed in R2 (C) X is homeomorphic to Z but not
(B) A is path connected homeomorphic to Y
(C) A is connected (D) X is not homeomorphic to Y and
(D) A is not connected not homeomorphic to Z
14
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61. If c1 c2 ln x is the general solution 64. Which of the following is false :
of differential equation :
(A) If 1, 2 are linearly independent
d2 y dy
x2 2 Kx y 0, x > 0 functions on I, they are linearly
dx dx
then K equals : independent on any interval J
(A) 2 contained in I
(B) 3
(B) If 1, 2 are linearly dependent
(C) –1
(D) –3 functions on I, they are linearly
62. The initial value problem : dependent on any interval J
dy contained in I
| y|, y(0) 0,
dx
(A) has no solution (C) If 1, 2 are linearly independent
(B) has unique solution solutions of a second order
(C) has non-zero solution
differential equation L(Y) = 0 on
(D) has more than one solution
63. The differential equation : an interval I, they are linearly
y " ex y 0 independent on any interval J
has a series solution y( x) ! ax
k 0
k
k contained in I
which satisfies x(0) 1 and x '(0) 0. (D) If 1, 2 are linearly dependent
Then the values of the coefficients a0, solutions of a second order
a1, a2 are :
differential equation L(Y) = 0 on
(A) a0 = 1, a1 = 0, a2 = – ½!
an interval I, they are linearly
(B) a0 = 1, a1 = 1, a2 = 2!
dependent on any interval J
(C) a0 = 0, a1 = 1, a2 = 2!
(D) a0 = 1!, a1 = 2!, a2 = 3! contained in I
15 [P.T.O.
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65. Consider the following statements : 67. The initial value problem
(I) f ( x, y) xy2 satisfies Lipschitz y' 2 y , y(0) a
condition on the rectangle has :
| x| 1 , | y| 1 . (A) a unique solution if a < 0
(B) no solution if a > 0
(II) f ( x, y) xy2 satisfies Lipschitz
(C) infinitely many solutions if a = 0
condition on the strip | x| 1 ,
(D) a unique solution if a 0
| y| .
68. Which one of the following is not the
Then : general solution of the partial
(A) Only (I) is true differential equation :
(B) Only (II) is true z( xp yq) y2 x2
(A) x2 y2 z2 f ( xy)
(C) Both (I) and (II) are true
(B) x 2 y2 z2 f (( x y)2 z2 )
(D) Both (I) and (II) are false
(C) x 2 y2 z2 f ( y / x)
66. The Wronskian of the solutions of
(D) ( x y)2 z2 f ( xy)
the differential equation
69. A partial differential equation which
y" 6y" 12 y ' 8y 0 represents a surface of revolution
is : obtained by revolving a plane curve
f ( x, y) 0 about x-axis is :
(A) 2e6x
(A) yp – xq = 0
(B) 3e3x
(B) xp – yq = 0
(C) 4e2x (C) y + zq = 0
(D) 2e2x (D) x + zp = 0
16
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70. Consider the following two first order 72. The solution of the first order
partial differential equations
partial differential equation
xp – yq = x ................. (1)
f ( x, y, z, p, q) 0 of the form
and x2p + q = xz ................ (2)
F(u, v) 0 , where u and v are
Then :
(A) they have common solutions functions of x, y, z is called :
(B) every solution of (1) is also the
(A) complete integral
solution of (2)
(C) every solution of (2) is also the (B) general integral
solution of (1)
(C) particular integral
(D) they do not have any common
solutions (D) singular integral
71. The characteristic curve and the
73. The region in which the partial
envelope of the one parameter family
of surfaces differential equation
( x a)2 ( y 2a)2 z2 1 (1 x 2 ) uxx uyy 0
are :
is an ellipse if :
(A) the great circle of the sphere and
( y 2 x)2 5 z2 5 respectively
(A) x > 1
(B) any circle on the sphere and
( y 2 x)2 2 z2 2 respectively (B) x < 1
(C) exactly the same great circles
(C) |x| > 1
(D) a circle on the sphere and
x2 y2 1, z 2a respectively (D) |x| < 1
17 [P.T.O.
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74. The partial differential equation 76. Let E be the shift operator and µ
)2 z )2 z be the averaging operator. Then :
y 0, y > 0
)x 2 )y 2
is : (A) µ E½ E ½
(A) hyperbolic
E½ E ½
(B) µ
2
(B) parabolic
(C) elliptic E½ E ½
(C) µ
2
(D) Laplacian
E½ E ½
75. The partial differential equations (D) µ
3
f ( x, y, p, q) 0 and g( x, y, p, q) 0 dy
77. Let 1 xy with y(0) 2.
dx
are compatible if : Using Picard’s method, the first
)( f , g) )( f , g) approximation is :
(A) )( x, z) 0
)( y, z)
(A) y1 x x2
)( f , g) )( f , g)
(B) )( x, z) 0
)( y, z)
(B) y1 2 x x2
)( f , g) )( f , g)
(C) )( x, p) 0
)( y, q)
(C) y1 20 x x2
)( f , g) )( f , g)
(D) )( x, p) 0
)( y, q) (D) y1 2 x
18
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1 80. A n × n matrix A is diagonally
78. If f ( x) , then the first divided
x2 dominant of :
difference [a, b] is :
(A) The absolute value of each
(a b) leading diagonal element is
(A) a2 b2 greater than or equal to the
sum of the absolute values of the
(a b) remaining elements in that row
(B)
a2 b2
(B) The value of the diagonal
a b element is maximum in that
(C)
ab row
(C) The value of the diagonal
a b
(D) element is greater than or equal
ab
to the sum of the values of
dy
79. Let f ( x, y) be an initial value
dx remaining elements in that row
problem with y( x0 ) y0 . Then in
(D) All diagonal elements are zero
Runge-Kutta method of order 4,
the value of K 3 to find first 81. The value of
approximation is :
*10 [(1 x) (1 2 x2 ) (1 3 x 3 ) (1 4 x4 )]
(A) K 3 h( f ( x0 , y0 ))
is :
(B) K 3 h( f ( x0 h, y0 h))
(A) 24 × 10 !
h k2
(C) K 3 h f x0 , y (B) 24 × 8 !
2 0 2
(C) 12 × 10 !
h k1
(D) K 3 h f x0 ,y
2 0 k2 (D) 12 × 8 !
19 [P.T.O.
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82. The curve for which the area of 84. The first integral of Euler-Lagrange’s
surface of revolution is minimum differential equation of the functional
when revolved about x-axis is the
x2
solution of the equation : I ( y( x)) f ( x, y ', y ") dx
x1
(A) ay ' y2 a2 , a is a constant is :
)f )f
2 2 (A) f y' y" constant
(B) y a y' a )y ' )y "
)f d )f
(C) x2 a2 y ' a (B) constant
)y ' dx )y "
(D) ay ' x2 a2 )f d )f
(C) constant
)y ' dx )y "
83. The Euler-Lagrange’s differential
equation for extremization of the )f d )f
(D) constant
functional )y dx )y '
1
I ( y( x)) ( y '2 y2 ) dx 85. The extremal of the functional
0 1
y2
subject to the condition that I( y( x)) dx , when x(0) 0,
0
x3
ydx 1 x(1) 1 is :
0
is : (A) y = x
(A) y" – y = 0
(B) y = c
(B) 2y" + 2y = +
(C) y = x4
(C) y" + y = 0
(D) y" – y = + (D) y = 0
20
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86. The solution of the functional 89. The nontrivial solution of the
x2
integral equation
I( y( x)) f ( x, y, y ") dx is a :
x1
1
(A) one parameter family of curves x( t ) + et s x(s) ds
0
(B) two parameter family of curves
(C) three parameter family of is :
curves
(A) t
(D) four parameter family of curves
87. Let B be a Banach space. Then (B) t 2
f :B R is said to be coercive
(C) et2
if :
(A) f ( x) as x 0 (D) et
(B) f ( x) 0 as x 90. The Volterra integral equation
(C) f ( x) as x
t
(D) f ( x) as x x( t ) t (s t) x(s) ds
0
88. The following integral equation
2
is equivalent to the initial value
x( t ) sin t + sin(t s) x(s) ds problem :
0
(A) x "( t) x( t ) 0, x(0) 0,
is : x '(0) 1
(A) Fredholm integral equation of
second kind (B) x "(t) x( t ) 0, x(0) 1,
x '(0) 0
(B) Fredholm integral equation of
first kind (C) x "(t) x '(t) 0, x(0) 1,
(C) Volterra integral equation of x '(0) 0
second kind
(D) x "(t) x '(t) 0, x(0) 0,
(D) Volterra integral equation of
x '(0) 1
first kind
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91. For a homogeneous Fredholm 93. Eigen values of the homogeneous
integral equation with separable
kernel : Fredholm integral equation
b 2
x( t ) + k(t, s) x(s) ds , x( t ) + sin (t s) x(s) ds
a 0
consider the following statements :
are :
(I) If Fredholm determinant
D( +) 0 , then integral equation (A) 1, –1
has only trivial solution i.e.
(B) , –
x(t) = 0.
(II) If Fredholm determinant (C) 1,
D( +) 0 , then integral equation
(D) 1/ , –1/
has infinitely many solutions.
Then : 94. A particle is thrown horizontally
(A) Only (I) is true from the top of a building of height
(B) Only (II) is true h with initial velocity u. Neglecting
(C) Both (I) and (II) are true
all other forces except the gravity
(D) Both (I) and (II) are false
and the air resistance which is
92. The Neumann series solution of the
proportional to the velocity of the
Volterra integral equation
particle, then the equations of motion
t
x( t ) 1 t (t s) x(s) ds are given by :
0
is : (A) x 0, y g 0
(A) x(t) log t (B) mx kx 0 , my ky mg 0
(B) x(t) cos t
(C) mx kx mg 0 , my ky 0
(C) x(t) sin t
(D) x(t) et (D) mx kx 0 , my ky mg 0
22
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95. The Lagrangian of a particle of mass 97. Let pj and p 'j are the components
m moving on the xy-plane which is of the canonical momenta corres-
rotating about z-axis with angular ponding to the Lagrangians L and
velocity , is given by : L' respectively, where
1 dF
L m[( x ,y)2 ( y ,x)2 ] V( x, y) L' L and F F ( q j , 1) .
2 dt
Then : If H and H' are the Hamiltonians
corresponding to L and L', then :
(A) Linear momenta px and py are
conserved (A) H' = H
(B) Hamiltonian H is conserved )F
(B) H' H
)t
(C) The total energy E is conserved
)F
(D) Hamiltonian H represents the (C) H' H
)q j
total energy
96. Let L be a Lagrangian of a particle )F
(D) H' H
and p j and q j are respectively )t
the conjugate momentum and 98. A particle of mass m is attached to
generalised velocity. Then which
one end of the string and the other
one of the following represents the
Lagrange’s equation of motion ? end at a distance r is fixed in space.
If the particle starts with velocity
)L
(A) p j )q j u from its lowest position, then the
velocity of the particle at any
)H
(B) q j )q j
angular distance - is given by :
(A) v2 u2 2 gr (1 sin -)
)L
(C) p j )q j (B) v2 u2 2 gr (1 cos -)
(C) v2 u2 2 gr (1 cos -)
)L
(D) q j )p j (D) v2 (r-)2
23 [P.T.O.
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99. The first integral of equation of SECTION III
motion of the simple pendulum
101. Which of the following is not true
represents :
for an exponential distribution ?
(A) Only the total energy E
(A) It satisfies the lack of memory
(B) Only the Hamiltonian H
property
(C) Both the Hamiltonian H and
the total energy E (B) The hazard rate is constant
(D) Neither the Hamiltonian H nor (C) The distribution of nth order
the total energy E statistics is exponential
(D) The distribution of the first
cos - sin - 0
order statistics is exponential
100. If A sin - cos - 0 is the
0 0 1 102. Consider the following two
statements :
matrix of orthogonal transformation
of a rigid body with one fixed, then (1) The correlation coefficient
A–1 is given by : between two variables X and Y
is the geometric mean of two
cos - sin - 0 regression coefficients "YX and
sin - cos - 0
(A) "XY.
0 0 1
(2) The arithmetic mean of the two
0 0 1 regression coefficients "YX and
cos - sin - 0 "XY is greater than or equal to
(B)
sin - cos - 0 the correlation coefficient
between X and Y.
1 0 0 Which of the above statement(s)
(C) 0 cos - sin - is/are true ?
0 sin - cos -
(A) Only (1)
1 0 0 (B) Only (2)
(D) 0 cos - sin - (C) Both (1) and (2)
0 sin - cos - (D) None of (1) and (2)
24
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103. The ogive more than and ogive 105. Which of the following is the smallest
less than of the following grouped
field containing a non-empty subset
data
A of . ?
Class Frequency
0—8 8 (A) Trivial field
8—20 12
(B) {A, AC, , .}
20—30 18
30—35 10 (C) Power set of .
35—40 4
(D) None of the above
intersect at which of the following
points ? 106. Which of the following statements
(A) 23.00 cannot be always true ?
(B) 23.33
(A) Let X : . . * and if /j be the
(C) 23.67
0-field of subsets of .*, then
(D) 24.00
X 1 (/ j ) is also 0-field
104. Suppose a class contains 44 students.
On March 10, 25 students were
(B) 0{X 1 (/) X 1 {0(/)} where, /
present in the class and on March
16, 30 students were present in the denote the class of subsets of
class. Then the minimum number .* and X : . .*
of students who were present in
the class on March 15 as well as (C) If B C , then
March 16 is equal to : X(B) X(C) where
(A) 5
X(A) = {X(w)|w A}
(B) 9
(D) X 0 and E(X) 0, then
(C) 11
(D) 14 X = 0 a.s.
25 [P.T.O.
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107. Let . = {1, 2, 3, ...., 10}, A = {1, 2}
109. The joint probability mass function
and F = { , A, AC, .}, consider the
following functions : PX,Y of the random variables X and
X(w) 100 w . Y is as given in the following
100 if w A table :
Y(w)
400 if w 'A
Z(w) w w . PX,Y (x, y) Y PX (x)
Then which of the following is
true ? 1 2 3 4
(A) Only X and Y are F-measurable
1 0.02 0.04 0.06 0.08 0.20
r.v.s
(B) All the functions X, Y and Z are
X 2 0.03 0.06 0.09 0.12 0.30
F-measurable r.v.s.
(C) Only X is a F-measurable r.v.
3 0.05 0.10 0.15 0.20 0.50
(D) All the functions X, Y, Z are not
F-measurable PY (y) 0.10 0.20 0.30 0.40
108. Suppose that probabilities are
assigned to the subsets of Then, which of the following
. = {a, b, c, d, e, f } according to the
statements is true ?
classical definition of probability.
Which of the following statements
(A) E(Y) E(X)
is/are true for the sets A = {a, b},
B = {b, d} and C = {a, c, d} ? (B) P(X > 2, Y > 2)
(I) C and A are independent
(II) C and B are independent 1 – P (X 2, Y 3)
(III) C and A B are independent
(C) X and Y are uncorrelated
(A) Only (I) and (II) are true
(B) Only (I) and (III) are true (D) X and Y are positively
(C) Only (II) and (III) are true
correlated
(D) All the three are false
26
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110. In order to detect an error, a 112. Suppose X is an arbitrary random
computer program is being tested variable. Which of the following
independently by three different
statements is true ?
testers. The probabilities with which
the testers can detect the error are E(X 4 )
(A) P[|X| 3]
0.2, 0.4 and 0.5 respectively. Then, 81
which of the following statements is
E(X 4 )
false ? (B) P[|X| 3]
81
(A) The probability that the error
will not be detected is 0.24 E(X 4 )
(C) P[|X| 3]
81
(B) The probability that exactly one
tester will find the error is 0.46 E(X 4 )
(D) P[|X| 3]
(C) The probability that at least two 81
testers will find the error is 0.30
1
113. Suppose (t) is a
(D) All the above are false (1 t 2 )
111. If E and F are events with characteristic function of a random
P(E) P(F) and P(E F) 1 0 ,
variable X. Then E(X) and Var(X)
then :
are respectively given by :
(A) Occurrence of E 2 occurrence
of F 1 1
(A) ,
(B) Occurrence of F 2 occurrence 2 8
of E
(B) 0, 2
(C) Non-occurrence of E 2 non-
occurrence of F (C) 0, 1
(D) None of the above implications 1 1
(D) ,
hold 2 16
27 [P.T.O.
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114. Suppose the probability space 116. Suppose {X n , n 1} is a sequence
(., A, P) is defined as follows :
of random variables defined as
. [0, 1] , A is a sigma field of
subsets of . and P is a Lebesgue 1
Xn 1 X , where X is a discrete
measure. Suppose a sequence n
{X n , n 1} of random variables is random variable with support {0, 1}
defined as X n (w) ew wn , n 1
and a random variable X is defined
2
and P[X 0] . Following are
3
as X(w) ew on (., A, P) . Which
of the following is not true as three statements :
n ?
(A) Xn X almost surely (I) Xn 0 in probability
(B) Xn X in probability
(C) Xn X in distribution (II) X n 0 in law
(D) Xn X pointwise
115. Suppose {X n , n 1} is a sequence of
(III) X n 0 in quadratic mean
random variables such that :
1
P[X n 0] 1 and Which of the following options is
n3
true ?
1
P[X n n] , n 1
n3
Which of the following is not true ? (A) Only (I) and (II) are true
(A) Xn converges to 0 in probability
(B) Xn converges to 0 in quadratic (B) Only (II) and (III) are true
mean
(C) Xn does not converge to 0 almost (C) Only (III) is true
surely
(D) Xn converges to 0 in law (D) All three are true
28
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117. Let {X n , n 1} and X be random 118. Let {X n , n 1} be a sequence of
variables defined on the same independent identically distributed
random variables with mean ‘0’ and
probability space. Which of the
variance ‘2’. Let :
following statements is not always
n
true as n ?
Sn !X.
i
2
i
(A) If X n converges to X in
Which of the following statements
probability and in distribution
is true ?
then eXn converges to eX almost
Sn
surely (A) converges to 0 almost surely
n
as n
(B) If Xn converges to X almost
Sn
surely then eXn converges to eX (B) converges to 2 almost surely
n
in probability as n
(C) If X n converges to X in Sn
(C) converges to 0 in
n
distribution then eXn converges
probability as n
to eX in distribution
Sn
(D) If X n converges to X in (D) n converges in distribution
2
probability then eXn converges to a standard normal random
to eX in distribution variable as n
29 [P.T.O.
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119. Following are three statements. In 121. Suppose {X n , n 0} is a Markov
a Markov chain, a persistent state
chain with state space S = {1, 2, 3}
i is a null persistent if :
( n) and the transition probability matrix
(I) lim sup pii 0
(II) mean recurrence time is P given by :
(III) lim sup pii( n) 1 0
Which of the following is true ? 1 2 3
(A) Only (I) is true 32 3 4
1 55 06
5
(B) Both (I) and (II) are true 5 6
3 5
P 25 06
(C) Only (III) is true 58 8 6
51 1 16
(D) Both (II) and (III) are true 35 6
96 3 2:
120. Suppose {X n , n 0} is a Markov
chain with state space S {1, 2, 3}
and the transition probability matrix Suppose X 0 = 3. Then realized
P given by : values of X1 and X2 corresponding
1 2 3 to random numbers 0.36 and 0.64
1 3 0.3 " 4
5 respectively, from U(0, 1) distribution
P 2 0.4 0.16
5 6
3 59 7 0.2 8 6: are :
If P is a doubly stochastic matrix,
(A) X1 = 2 and X2 = 2
the value of " :
(A) Cannot be computed from the
(B) X1 = 1 and X2 = 1
given information
(B) is < 0.4 (C) X1 = 1 and X2 = 2
(C) is > 0.4
(D) is 0.4 (D) X1 = 2 and X2 = 1
30
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122. Suppose {X n , n 0} is a Markov 124. Suppose {X(t), t 0} is a pure birth
chain with state space S = {1, 2, 3, 4} process with birth rate + i + for all
and the transition probability matrix
i S and X(0) 0 . The following
P given by :
are four statements.
1 2 3 4 (I) {X(t), t 0} is a process with
31 1 4 stationary and independent
1 52 0 2
06
5 6 increments.
1 26
2 50 0
5 3 36 (II) E(X(t)) e +t
P 52 6
1
35 0 06 (III) X(t) follows Poisson P(+t)
53 3 6
5 distribution for every fixed t.
1 36
4 50 0 6
9 4 4: (IV) X(0) cannot be 0 in a pure birth
process.
Which of the following options is
true ? Which of the following is true ?
(A) States 1 and 3 are essential
states (A) Only (I) is true
(B) States 1 and 3 are inessential (B) Only (I) and (II) are true
states
(C) States 1 and 3 are transient (C) Only (I) and (III) are true
states
(D) Only (IV) is true
(D) State 1 and 3 are null persistent
states 125. Assume that X ~ B(n, p) for some
123. Which of the following options is
true ? n 1 and 0 p 1 and Y ~ Poisson
Messages arrive on a mobile phone (+) for some +; > 0. Suppose
according to the Poisson process
E(X) = E(Y). Then :
with rate 5 messages per hour. The
probability that no message arrives (A) V(X) = V(Y)
during 10 : 00 am to 12 : 00 noon
is : (B) V(X) < V(Y)
(A) 2e–5 (C) V(X) > V(Y)
(B) e –10
(C) 1 – e–10 (D) V(X) may be larger or smaller
2
(D) e 5 than V(Y)
31 [P.T.O.
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126. Suppose X and Y are independently 128. Suppose that X1, X2, ........, Xn are
i.i.d. random variables such that X1
and uniformly distributed random
has uniform distribution over (0, -),
variables on the interval (0, 1). Then - > 0. Then, which of the following
statements is true ?
the distribution of max (X, Y) is :
(I) The conditional density f (.) of y
(A) Beta (1, 1) X(1)|X(n) = y is given by :
n –2
(B) Beta (2, 2) 1 x
<(n 1). 1 ,if 0 x y
f y ( x) y y
<
(C) Beta (1, 2) 0, otherwise
(D) Beta (2, 1) (II) The conditional distribution of
X(1)|X(n) = y does not depend
127. Suppose that X and Y are i.i.d. on -.
random variables with c.d.f. F which (A) Neither (I) nor (II) is true
is binomial (n, p). Then, (B) Both (I) and (II) are true
(C) Only (I) is true
P (X = j|X + Y = k)
(D) Only (II) is true
is equal to : 129. Suppose that F is a distribution
function with mean support [0, )
(A) j/k, for j = 0, 1, 2, ......., k
and mean µ. Then, what is the value
k of the integral,
(B) j pj qk–j, for j = 0, 1, 2, ...., k
k [F( x a) F( x)] dx ?
k 1 0
(C) j , for j = 0, 1, 2, ...., k
2 (A) µ
n n 2n (B) 0
(D) j k j / n , for j = 0, 1,
(C) a + µ
2, ...., k (D) a
32
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130. A geometric distribution with 132. Cramer-Rao lower bound of variance
support {1, 2, ....} has variance 6. for estimating the parameter - of the
Then, the height of the jump at 2 distribution with pdf :
is equal to : 1 1
f ( x, -)
1 ( x -)2
(A) cannot be determined x
where is :
(B) 8/27 2
(A)
n
(C) 2/27
1
(B)
n
(D) 6/27
2
(C)
131. A random sample X1, X2, ....., Xn is n2
drawn from a Poisson distribution 1
(D)
with parameter +. Which one of the 2n
following is an unbiased estimator 133. Based on a sample of size one from
of +2 ? Binomial (6, P), it is decided to test
2
the hypothesis Ho : P 1 / 2 Vs
n
1
(A) !X
n i1 i
Hi : P 1/4 using the critical
region X 3 . Then, the size of the
1 n 2
(B) !X
n i1 i
critical region would be :
(A) 15/64
n
1
(C) ! X (X 1)
n i1 i i
(B) 35/64
(C) 22/64
1 n
(D) ! X (X 1)
n i1 i i (D) 42/64
33 [P.T.O.
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134. The following are some statements : 135. In the context of testing of statistical
(I) If T is a boundedly complete hypotheses, which one of the
sufficient statistic for - and S following statements is true ?
is an ancillary statistic for -, (A) When testing a simple
then T and S are independent. hypothesis H 0 against an
(II) An ancillary statistic may or alternative simple hypothesis
may not be independent of a H1, the likelihood ratio principle
sufficient statistic. leads to the most powerful test
(III) It is possible to have variance (B) When testing a simple hypo-
smaller than that of the thesis H0 against an alternative
Cramer-Rao lower bound at simple hypothesis H1
some points of the parameter
P{rejecting H0|H0 is true} +
space.
P{accepting H0|H1 is true} = 1
(IV) The parametric function need
(C) For testing a simple hypothesis
not be differentiable for the
H 0 against an alternative
Cramer-Rao lower bound to
simple hypothesis H1, rando-
hold.
mized test is used to achieve
Which of the following options is
the desired level of the power
true ?
of the test
(A) (I), (II) and (III) only
(D) UMP tests for testing a simple
(B) (II), (III) and (IV) only
hypothesis H 0 , against an
(C) (I), (III) and (IV) only alternative composite H 1 ,
(D) (I), (II) and (IV) only always exist
34
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136. Let X1, X2, ......, Xn be independently 138. Which of the following probability
and normally distributed with
laws is not a member of an
mean 0 and variance 02. What is
the minimum variance unbiased exponential family ?
estimator of 02 ?
(A) f ( x, -) - (1 x)- 1 , 0 x 1,
n
1
(A)
n 1
!Xi 1
2
i -10
1 n (B) f ( x, -) - x (1 -)1 x , x = 0, 1,
(B) ! (X i
n 1 i1
X)2
0 - 1
1 n
(C) !
n i1
X 2i (C) N(-, -2 ) , - R
1 n (D) N(-, -) , - 1 0
(D) ! (X i
n i1
X)2
139. Suppose {X 1, X 2, ......., X n } are
137. A Statistics professor wants to see
if more than 80% of her students independent random variables,
enjoyed taking her class. At the end where Xi follows uniform U(0, i-)
of the term, she takes a random
sample of students from her large distribution for i = 1, 2, ......., n.
class and asks in an anonymous Which of the following statements
survey, if the students enjoyed
is true ?
taking her class.
Which of the following set of
2 n Xi
hypotheses should she set ? (A) !
ni1 i
is not consistent for -
(A) H0 : p = 0.80 against
H1 : p < 0.80 (B) Sample mean is consistent for -
(B) H0 : p > 0.80 against
H1 : p = 0.80 X2 X3 X
(C) max X1 , , , ......, n is
(C) H0 : p = 0.80 against 2 3 n
H1 : p > 0.80
consistent for -
(D) H0 : p = 0.80 against
H1 : p 0.80 (D) X(n) is consistent for -
35 [P.T.O.
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APR - 30224/II—A
140. Suppose {X1, X2, ......., Xn} is a 141. Suppose X ~ B (1, p) distribution,
0 < p < 1. On the basis of a random
random sample from N(µ, 0 2 )
sample of size n from the
distribution. Then the approximate distribution of X, we want to test
dispersion matrix of H0 : p = p0 against the alternative
H1 : p p0 , where p0 is a specified
constant. Then asymptotic null
2
n (X n , S )'
n (µ, 0 )'
2
distribution of which of the following
is : test statistics is N(0, 1) ?
n(X n p0 )
(A)
X n (1 X n )
02
0
(A)
0 04 Xn p0
(B)
X n (1 X n )
02 0 n (X n p0 )
(B) (C)
0 20 4 p0 (1 p0 )
Xn p0
(D) p0 (1 p0 )
1
0
02 142. If T is MLE of - and =(-) is
(C) 1
0 one-to-one function of ->; then =(T)
04
is MLE of =(-). This follows from :
(A) Invariance property of MLE
1
0 (B) Asymptotic property of MLE
02
(D) 1 (C) Consistency property of MLE
0
20 4
(D) Regularity condition of MLE
36
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APR - 30224/II—A
143. In an r × s contingency table for 145. Suppose
testing H0 : Two attributes A and n
P(Y y / -) - y (1 -)n y ,
B are independent against the y
alternative H1 : A and B are not y {0, 1, 2, ....., n} . If - ~ uniform
independent, the expected frequency (0, 1), then the posterior distribution
eij of (i – j)th cell is given by : will be :
ni j
(A) (A) Uniform (0, y)
n
(B) ni.n. j (B) Beta (y, n – y)
ni. n. j (C) Beta (y + 1, n – y + 1)
(C)
n
(D) Uniform (0, 1)
ni. n. j
(D)
n2 146. Let X ~ Poisson (+) and a single
144. A manufacturer claims that the realization of X resulted in 3. If +
median lifetime of its product is more has a prior distribution
than 500 hours. To verify the claim
P(+) q + e 3+ , + 1 0 ,
based on 20 randomly observed
then the posterior mean is given
lifetimes, which of the following tests
by :
is most appropriate ?
(A) 4/5
(A) Sign test
(B) Mann-Whitney test (B) 3/5
(C) Kolmogorov-Smirnov test (C) 5/4
(D) Chi-square test (D) 2/5
37 [P.T.O.
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APR - 30224/II—A
147. Consider the linear model 149. The values of a, b and c in the
Yn 1 X-p 1 ? with E(?) 0 following ANOVA table
Cov(?) 0 2I n . Then which of the Source d.f. Ss Mss
following statements is always Treat 4 b —
true ?
Error a 0.35 c
(A) - is estimable
Total 29 1.25 —
(B) the best linear unbiased
are :
estimate of - exists
(A) a = 15, b = 0.35 and c = 0.15
(C) l ' - is estimable for any
(B) a = 25, b = 0.35 and c = 0.015
p – 1 vector l
(C) a = 15, b = 0.90 and c = 0.14
(D) l ' - is estimable for only that l
(D) a = 25, b = 0.90 and c = 0.014
which belong to the column
150. In a one-way ANOVA model
space of X'
yij = µ + di + eij, j = 1, ..., ni,
148. Y 1 , Y 2 , Y 3 are three random
i = 1, ..., 4 with E(eij) = 0. For all
variables with E(Y1) = Q1 – Q2 + Q3,
j and i, which of the following
E(Y2) = Q1 and E(Y3) = Q3 – Q2.
statements is true ?
Then which of the following
(A) µ and 1, 2, 3, 4 are all
statements is not true ?
estimable
(A) Q2 – Q3 is estimable
(B) µ is not estimable but 1, 2,
(B) all three parameters Q1, Q2, Q3
3 and 4 are all estimable
are estimable (C) any linear function of 1, 2, 3,
(C) Q1 is estimable, Q2 Q3 are not 4 is estimable
estimable (D) i – k i = k = 1, 2, 3, 4 are
(D) 2Q1 – Q2 + Q3 is estimable estimable
38
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APR - 30224/II—A
151. In the multiple linear regression 153. If e1, e2, ......, en are the residuals
setup under the assumption that the obtained on fitting a simple
random errors 81, 82, ....... 8n are linear regression y "0 "1 X 8 to
uncorrelated and homoscedastic, (Yi , X i ) i 1...n under the standard
the residual variables e1, e2, ....... en assumption, then :
are : n
(A) ! X e is always positive
i 1
i i
(A) Uncorrelated and homo-
n
scedastic (B) ! X e is always negative
i 1
i i
(B) Correlated and homoscedastic
n
(C) Uncorrelated and hetero-
(C) ! X e is always zero
i 1
i i
scedastic (D) Nothing can be said about
n
(D) Correlated and heteroscedastic ! Xe
i 1
i i
152. In a multiple linear regression
154. Consider a logistic regression model
model y X" 8 , the covariance
consisting of only one explanatory
between y and ŷ , the predicted
variable X which takes values in
value of y using least square
{ 1, 1} . Then the odds ratio on
estimator of " is :
changing X from –1 to 1 is :
(A) ! X(X ' X) 1 X '
(A) exp (2")
(B) ! (I – X (X' X) 1 X ') (B) exp (")
(C) ! (C) exp (1 + ")
(D) None of the above (D) exp (1 – ")
39 [P.T.O.
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APR - 30224/II—A
155. Suppose 157. Suppose X 3 %1 ~ M Normal3 (µ, ! 3 3 )
3 2 0.5 24 and let Y X ' AX where the matrix
X 3 %1 ~ MNormal3 0, 50.5 1 0 6
5 6 . A has all eigenvalue equal to 1.
59 2 0 4 :6
Hence the probability distribution
Then the correlation coefficient
of Y
between X1 – X3 and 2X2 is :
(A) cannot be determined based on
1
(A)
2 2 the given information
(B) 0
(B) is normal (µ ' A µ, ! ' A 8)
1
(C)
2 (C) is @2 with 1 degree of freedom
(D) 1
(D) is @2 with 3 degrees of freedom
3X4
156. Suppose 5 6 ~ Bivariate Normal
9Y: 158. Corresponding to the three variables
(µ 2%1 , A 2m ) . Then : X 1 , X 2 , X 3 the quadrate form
(A) E(Y|X x) x E(Y) X 12 2X 22 4X 32 X1X 2 X 1 X 3 is :
(B) Var(Y|X x) x 2 Var(Y) (A) Positive definite
(C) E(Y|X x) E(Y) + a linear
(B) Positive semidefinite
function of x
(C) Negative definite
(D) Var(Y|X x) Var(Y) a
quadrate function of x (D) Negative semidefinite
40
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APR - 30224/II—A
159. Suppose X ~ Normal2 (µ, 8) and 161. The Euclidean distance between the
~ 2 %1
Y 0.5X 12 0.5X 22 X 1 X 2 . Then the
3 24
probability distribution of Y is : 5 26
point X 3 1 5 6 and a cluster
59 2 6:
(A) @2 with 2 degrees of freedom
3 24
5 1 6 is equal to :
(B) @2 with 1 degree of freedom center X 3 1
5 6
59 –26:
(C) F with 1, 2 degrees of freedom
(A) 5
(D) t with 1 degree of freedom
(B) 4
160. For the three random variables
(C) 2
X1, X2, X3 let X̂ 1 and X̂ 2 be the
(D) 0
fitted values of X1 and X2
respectively after regressing on the 162. Principal component analysis is used
linear function of X3. The partial for :
correlation coefficient between X1
(A) reducing the number of
and X2 after removing the effect
observations in the data
of X 3 is the simple correlation
coefficient between : (B) reducing the variability present
in the data
(A) X1 and X̂ 2
(C) getting information on variable
(B) X̂ 1 and X2
as a function of other variables
(C) (X1 – X3) and (X2 – X3)
(D) reducing the number of
ˆ ) and (X – X
(D) (X1 – X ˆ ) variables in the data
1 2 2
41 [P.T.O.
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APR - 30224/II—A
n
1 164. Consider a population of 50 units,
163. Suppose S Y
2
n 1 r1
! (Y r Y)2
{1, 2, 3, ....., 50} and suppose that 50
N
possible samples are listed as {1},
1
s2y
N 1
! (y i y )2 {1, 2}, {1, 2, 3}, ....., {1, 2, 3, ....., 50}.
i 1 One of these samples is choosen at
random. Let i be the probability
Y is sample mean under SRSWR that unit i of the population is in
the selected sample. Then which of
y is population mean the following is necessarily true ?
(A) The expected sample size is 25.5
Y sy is sample mean under
50
systematic sampling (B) !
i 1
i 1
50
Which of the following statements
(C) !
i 1
i 25
is not true ?
(D) The expected sample size is 25
(A) Probability of selecting ith 165. Consider the following hypothetical
population of 6 units. What is the
element of the population on the estimated value of population total
1 of y-variable based on SRSWOR
rth draw is under SRSWOR sample of 2 units selected as unit
N
4 and unit 6 from the population
design for i = 1, 2, 3, ......, N and
using ratio method of estimation :
r = 1, 2, 3, ......., n Unit No. x y
1 0 1
2 1 3
(B) ES2Y s2y under SRSWOR
3 3 11
4 6 18
(C) E(Ysy ) y under systematic 5 8 29
6 10 46
sampling
(A) 27
(B) 112
N 1 2
(D) V (Y) s y under SRSWR (C) 198
n
(D) 432
42
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166. In stratified random sampling with 167. Under systematic sampling, if s2wsy
denotes the variance among the
a linear cost function units belonging to the systematic
sample then the variance of the
L
sample mean Ysy is :
C C0 !Cn, h h
h 1
(A) [(N 1) s2 2
K(n 1) swsy ]
(B) [(N 1) s2 2
K(n 1) swsy ]
the variance of estimated mean (Yst )
1
(C) 2
[(N 1) s2 2
K(n 1) swsy ]
is minimum for a specified cost C N
1
and the cost is minimum for a (D) [(N 1) s2 2
K(n 1) swsy ]
N
specified variance of Yst when : 168. In a BIBD with 7 treatments and
block size 5, what will be the number
of replications required ?
< wh sh <
(A) nh B (A) 56
< Ch <
(B) 21
(C) 8
L
<< ! wh sh << (D) 3
(B) nh B
h 1
169. In a completely randomized design
< Ch <
< < with unequal group numbers, that
is, n1 = 5, n2 = 7, n3 = 6, what is
the degree of freedom for the error
N h sh
(C) nh B term ?
sh
(A) 17
(B) 15
< n sh <
(D) nh B C (C) 120
< h<
(D) 18
43 [P.T.O.
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APR - 30224/II—A
170. Suppose the analysis of variance 172. In a completely randomized design
suppose yi . is computed from ni
calculations have been performed
observations (ni not all equal with
for a problem where there is a single k
factor with two levels. This would
!n
i 1
i N ), then what will be
produce results equivalent to : Var ( yi . y.. ) ?
(A) 0 2 / Nni
(A) a t-test with alternate
N ni
(B) 0
2
hypothesis greater than
Nni
(B) a F-test with alternate
ni
(C) 0 (N – 1)
2
hypothesis less than
(C) a @ 2 -test with alternate 0 2 (ni 1)
(D)
Nni
hypothesis not equal to
173. Suppose the lifetimes of two systems,
(D) a t-test with alternate system 1 and system 2, follow
exponential distributions with
hypothesis not equal to
failure rates 2 and ½ respectively.
171. The ‘C’-matrix of a block design Let the probability that system ‘i’
having r treatments is : fails prior to its mean time to failure
be denoted by pi, i = 1, 2. Which of
(A) Positive definite the following is true ?
(B) Positive semidefinite (A) p1 > p2
(B) p1 = p2
(C) Asymmetric
(C) p1 < p2
(D) Having rank r (D) p1 . p2 = 1
44
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APR - 30224/II—A
174. A system consists of 4 components, 176. The final tableau of the simplex
3 of which form a parallel method of the LPP
subsystem connected in series with
Max : Z = 3x1 + 2x2
the 4th component. The components
work independently of each other. If is given below :
the reliability of each component is x1 x2 x3 x4
0.8, then the system reliability is :
3 2 0 0
(A) 0.8381
x1 3 1 0.75 0.25 0 30
(B) 0.7132
x4 0 0 2.25 –0.25 0 60
(C) 0.4096
zj – cj : 0 0.25 0.75 0 90
(D) 0.7936
If the coefficient of x1 in the objective
175. Suppose only the first 6 failure
function is changed from 3 to 1, then
times of 18 items kept on test are
observed. Suppose the failure times which of the following statements
are independent and follow an is/are true ?
exponential distribution with mean
(I) x 2 enters as a new basic
-. If the data are :
variable in place of x4.
1.5, 2, 2.5, 3, 4, 5
(II) The optimal value decreases by
the maximum likelihood estimate
33.33.
of -; C
(A) Only (I) is true
(A) 9
(B) Only (II) is true
(B) cannot be obtained
(C) is 3 (C) Both (I) and (II) are true
(D) is 13 (D) Neither (I) nor (II) is true
45 [P.T.O.
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APR - 30224/II—A
177. In graphical solution method of 179. In inventory control, which costs can
LPP, the redundant constraint is the
one : vary with order quantity ?
(A) Which forms the boundary of (A) unit cost only
the feasible region
(B) Which does not form the (B) reorder cost only
boundary of the feasible region
(C) holding cost only
(C) Which does not optimize the
objective function (D) all of the above
(D) Which optimizes the objective
180. Customer arrivals at a service
function
178. Four workers are required to be facility follows a Poisson distribution
assigned to 4 tasks. The number of with mean 10 per hour and the
hours required to complete the tasks service times have an exponential
by each of the workers is given in
the following table : distribution with average service
Workers rate of 4 minutes per customer.
20 30 40 50 Then, which of the following
Tasks 40 50 60 70 statements is false ?
70 80 90 80
(A) The probability of two or more
30 50 80 40
customers waiting in the queue
Then, which of the following is
true ? is 4/9
(A) The minimum time required to
(B) The average length of the queue
complete all the tasks is 200
hours is 4/3 customers
(B) The minimum time required to (C) The average waiting time of
complete all the tasks is 210
hours a customer in the queue is 8
(C) The basic feasible solution is minutes
non-optimum
(D) The probability that the server
(D) The basic feasible solution is
non-degenerarate is idle is 2/3
46
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ROUGH WORK
47 [P.T.O.
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APR - 30224/II—A
ROUGH WORK
48
Text of page 1
MAHARASHTRA STATE ELIGIBILITY TEST - 7th April 2024
FINAL ANSWER KEY FOR PAPER -II (Booklet Code : A)
SUBJECT CODE & NAME: (30) Mathematical Sciences
QueNo Answer QueNo Answer QueNo Answer QueNo Answer
1 D 41 D 81 A 121 A
2 A 42 A 82 A 122 A
3 B 43 A 83 B 123 B
4 C 44 A 84 C 124 C
5 C 45 A 85 C 125 B
6 C 46 A 86 D 126 D
7 C 47 A 87 C 127 D
8 C 48 C 88 A 128 B
9 D 49 A 89 D 129 D
10 A 50 B 90 A 130 D
11 A 51 D 91 C 131 C
12 A 52 B 92 D 132 A
13 D 53 D 93 D 133 D
14 D 54 B 94 D 134 A
15 C 55 B 95 B 135 A
16 B 56 C 96 A 136 C
17 D 57 C 97 A 137 C
18 A 58 D 98 B 138 C
19 a 59 D 99 C 139 C
20 A 60 C 100 A 140 B
21 D 61 9 101 C 141 C
22 c 62 D 102 C 142 A
23 C 63 B 103 B 143 C
24 C 64 A 104 C 144 A
25 A 65 A 105 B 145 C
26 C 66 9 106 C 146 C
27 C 67 C 107 A 147 D
28 C 68 C 108 A 148 B
29 D 69 C 109 C 149 D
30 B 70 A 110 D 150 D
31 D 71 A 111 D 151 D
32 9 72 B 112 C 152 A
33 C 73 C 113 B 153 C
34 B 74 C 114 D 154 A
35 D 75 C 115 C 155 A
36 D 76 B 116 D 156 C
37 A 77 B 117 A 157 D
38 B 78 A 118 B 158 A
39 A 79 C 119 B 159 B
40 C 80 A 120 D 160 D
9 = No option is correct or the question is wrong, a= A and B, b = A and C, c = A and D, d = A,B and C, e = A,B
and D,f = A,C and D, g = A,B,C and D, h = B and C, i = B and D, j = B,C and D, k = C and D
Text of page 2
MAHARASHTRA STATE ELIGIBILITY TEST - 7th April 2024
FINAL ANSWER KEY FOR PAPER -II (Booklet Code : A)
SUBJECT CODE & NAME: (30) Mathematical Sciences
QueNo Answer
161 A
162 D
163 D
164 A
165 B
166 A
167 D
168 9
169 B
170 D
171 B
172 B
173 B
174 D
175 D
176 A
177 B
178 A
179 D
180 A
9 = No option is correct or the question is wrong, a= A and B, b = A and C, c = A and D, d = A,B and C, e = A,B
and D,f = A,C and D, g = A,B,C and D, h = B and C, i = B and D, j = B,C and D, k = C and DPage 1 of 48
Source: Savitribai Phule Pune University. Question paper published by Savitribai Phule Pune University.