MH SET 2021: 26 September, 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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SEP - 30221 (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 : 190
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SEP - 30221/II—A 2
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SEP - 30221/II—A
Mathematical Science
Paper II
Time Allowed : 120 Minutes] [Maximum Marks : 200
Note : This Paper contains One Hundred Ninety (190) 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 10, Section II : Q. Nos. 11 to 100,
Section III : Q. Nos. 101 to 190.
SECTION I 2. Let f : R Q be a non-constant
increasing function. Then :
1. Let ( xn ) n N be a convergent
(A) f is constant on some interval
sequence of positive real valued
[a, b]
numbers. Which of the following
statements is true ? (B) f is one-one
(C) f is onto
(A) Limit of the sequence is positive
(D) f is not constant on any interval
(B) Limit of the sequence may be
negative 3. Which of the following is true for a
complex number z x iy ?
(C) There is a subsequence of the
(A) | x| | y| | z|
sequence which has positive
limit (B) | x| | y| 2 | z|
(D) The limit of every subsequence (C) | x| | y| | z|
of the sequence is non-negative (D) | x| | y| | z|
3 [P.T.O.
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4. The function f : C C defined by 7. Suppose P(x, y) denotes the joint
f ( z) cos z is : probability mass function of two
(A) one-one random variables X and Y where :
P(0, 0) = 0.4, P(0, 1) = 0.2,
(B) onto
P(1, 0) = 0.1, P(1, 1) = 0.3.
(C) bijective
What is the conditional probability
(D) bounded of X = 0, given that Y = 1 ?
5. Let Rn [x] denote the vector space 1
of all polynomials over R having (A)
5
degree atmost n. Then the
3
dimension of Rn [x] is : (B)
5
(A) n 2
2
(C)
(B) n + 1 5
(C) n (D) 0
(D) 8. If 3 balls are randomly drawn from
a bowl containing 6 white and 5
6. Let V and V' be finite dimensional
black balls, what is the probability
vector spaces over K and let
that one of the balls is white and
{ 1, ....., n } be a basis of V. If
other two black ?
T L (V, V ') and T is injective, then
which of the following statements is 3
(A)
11
FALSE ?
4
(A) T 1, .....,T n are linearly (B)
11
independent
8
(B) dim V dim V' (C)
11
(C) dim V = dim V'
2
(D)
(D) Ker T = {0} 11
4
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SECTION II
9. Which of the following sets are
11. Consider the set X of all sequences
convex sets ?
(xn) of natural numbers.
(i) {( x, y)| y x} Let Y = {(xn) X : xn = 0 for infinitely
many n},
(ii) {( x, y)| y | x |}
Z = {(xn) X : xn = 0 for infinitely
(iii) {( x, y)| y x2 } many n},
W = {(xn) X : xn 0 for finitely
(A) (i) only many n}.
(B) (i) and (ii) only Then which of the following sets is
countable ?
(C) (i) and (iii) only (A) Y
(D) (ii) and (iii) only (B) Z
(C) W
10. Which of the following statement(s)
(D) Complement of Z in X
is/are true for an LP problem ?
12. Which of the following real numbers
(I) The set of all feasible solutions x, the inequality (1 x)n 1 nx
holds for all n N ?
to an LP problem is a convex
(A) x R
set.
(B) x [ 4, 4]
(II) Every hyperplane is a convex (C) x ( 1, )
set. (D) x ( , 4] [0, )
(A) Both are true 13. How many solutions are there for the
equation cos x e x 0 ?
(B) Only (I) is true (A) No solution
(C) Only (II) is true (B) Finitely many solutions
(C) Countably infinite solutions
(D) Neither (I) nor (II) is true
(D) Uncountably many solutions
5 [P.T.O.
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14. Which of the following statements 16. Which of the following Mobius
holds for a function f : [0, 1] R ? transformations maps the unit disc
(A) If f is of bounded variation, then onto the right half plane ?
f is Riemann integrable z i
(A) f ( z)
z i
(B) If f is of bounded variation, then
z 1
it is uniformly continuous (B) f ( z)
z 1
(C) If f is bounded, then it is of
1 z
(C) f ( z)
bounded variation 1 z
(D) If f is uniformly continuous, 1 z
(D) f ( z) i
then it is of bounded variation
1 z
17. Let S be the set of zeros of a non-
15. Consider the following statements :
zero entire function. Then the set
(a) Every non-empty open subset of
S is :
R is uncountable.
(A) non-empty open
(b) Every connected subset of R (B) infinite and bounded
containing at least two elements (C) closed
is uncountable. (D) uncountable
Then : 18. If f u iv is analytic on C, then
which of the following is FALSE ?
(A) Only (a) is true
(A) u, v are harmonic
(B) Only (b) is true
(B) u2, v2 are harmonic
(C) Both (a) and (b) are true (C) u + v is harmonic
(D) Both (a) and (b) are false (D) uv is harmonic
6
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19. Let f be analytic on C {0} and let 21. Let G be a finite group and H, H'
1, 2 : [0, 1] C {0} be defined
be two subgroups of G. Then :
by 1 (t) e2 it and 2 (t) 10 e2 it .
Suppose I1 f ( z) dz and (A) H H' #H #H'
1
I2 f ( z) dz (B) H H' [G : H] [G : H ']
.
2
(C) H H' aH bH' for some
Then :
(A) I1 = I2 a, b G
(B) I1 = 10I2
(D) H H' both H and H' are
1
(C) I1 I
10 2 normal in G
(D) There is no relation between
22. Let I be the ideal in the polynomial
I1 and I2
ring Z 4 [X] generated by the
cot z
20. Consider the function f ( z) . polynomial 1 + 2X. Then :
z
Then :
(A) I is a prime ideal
(A) f has a simple pole at z = 0
(B) I is a maximal ideal
(B) f has a removable singularity at
z = 0
(C) I is a proper ideal which is not
(C) f has uncountably many
a prime ideal
singularities
(D) the residue of f at z = 0 is 1 (D) I is a unit ideal
7 [P.T.O.
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25. Which of the following is an integral
23. In the group S3, there are exactly :
domain but not a field ?
(A) 2 elements satisfying x3 = e and (A) Z 10
(B) Z 31
3 elements satisfying x2 = e
(C) R[x]
(B) 3 elements satisfying x3 = e and (D) C(x)
1 1
4 elements satisfying x2 = e 26. A35 if A = is :
0 1
(C) 2 elements satisfying x3 = e and 1 70
(A)
0 1
4 elements satisfying x2 = e
1 35
(B)
0 1
(D) 3 elements satisfying x3 = e and
2 35
3 elements satisfying x2 = e (C)
0 2
24. Let G be a cyclic group of order 20. 35 35
(D)
0 35
Then the number of elements of G
27. Let W be the vector space of real
of order 10 is : skew symmetric n × n matrices.
Then the dimension of W =
(A) 10
(A) n 2
(B) n2 – n
(B) 4
n2 n
(C)
(C) 6 2
n2 n
(D) 2 (D)
2
8
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28. If A is a real 3 × 3 matrix, which 30. Let W be the subspace of R4 given
by :
of the following statements can not
be true about A ? x1
x2 x1 2 x2 3 x3 4 x4 0
(A) Minimum polynomial = W
x3 x1 x2 x3 0
characteristic polynomial x4 !
(B) A has at least one complex eigen Then W " is generated by :
value
1 0 0
(C) Minimum polynomial of 0 1 0
(A) 0 , 0 and 1
A = x2 – 1
0 0 0
(D) Minimum polynomial of
A = x2 + 1 1 1
2 1
1 0 0 (B) 3 and 1
29. Matrix A 0 1 1 is : 4 0
0 1 1
1 1
(A) Diagonalizable over R 2 1
(C) 3 and 1
(B) Triangulable over R but not
4 0
diagonalizable over R
0 0
(C) Triangulable over C but not
0 0
diagonalizable over C (D) 1 and 0
0 1
(D) Diagonalizable over C
9 [P.T.O.
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31. The partial differential equation
33. If $ 1 and $ 2 are two linearly
obtained by eliminating arbitrary
independent solutions of y " ay 0,
constants from
where a is a constant, on an interval
z ae bx sin by
I. Then which of the following
is : statements about the Wronskin
#2 z #z W($1 $ 2 ) ( x) is true ?
(A) 0
# y2 #x
(A) W($1 , $ 2 ) ( x) is a constant
#2 z #2 z function
(B) 0
# y2 #x 2
(B) W($1 $ 2 ) ( x) is a monotonically
#2 z #2 z increasing function
(C) x 0
#y 2 #x 2 (C) W($1 $ 2 ) ( x) is a monotonically
#2 z #z decreasing function
(D) x 0
#y 2 #x
(D) W($1 $ 2 ) ( x) 0 for all x I
32. The solution of the linear partial 34. A solution of the differential
differential equation
equation
#2 z #2 z #2 z xy ' 2y
2 0
#x 2 #y#x #y 2
passing through (1, 2), also passes
is :
through :
(A) z $1 ( x y) $ 2 ( x y)
(A) (2, 1)
(B) z $1 ( x y) x$2 ( x y)
(B) (2, 4)
(C) z $1 ( x y) x$2 ( x y)
(C) (2, 8)
(D) z $1 ( x y) x2$ 2 ( x y) (D) (2, 16)
10
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35. The differential equation obtained 37. A company management and the
by eliminating arbitrary constants labour unions are negotiating a new
from the expression y aebx is : three year settlement. Each of these
2 has 4 strategies mentioned below :
d2 y dy
(A) y 2 0 (I) : Hard and aggressive bargaining.
dx dx
(II) : Reasoning and logical approach.
d2 y dy (III) : Legalistic strategy.
(B) y x 0
dx2 dx (IV) : Conciliatory approach.
The cost to the company are given
2
d2 y dy for every pair of strategy choice.
(C) xy 0
dx2 dx
Company Strategies
d2 y dy I II III IV
(D) 2
xy 0
dx dx
Labour I 20 15 12 35
36. Assign the programmers U, V, W to Union II 25 14 8 10
the programmes I, II, III in such way Strategies III 40 2 10 5
that the total computer time is
IV –5 4 11 0
minimum. The estimated computer
time required by the programmers Then the value of the game is :
to execute the programmes is given (A) 12
in the table. (B) –5
(C) 0
Programmers (D) 2
U V W 38. Consider the following LP problem.
I 120 100 80 Max Z = 6x1 – 4x2
Programmes II 80 90 110 Subject to the constraints :
III 110 140 120 2x1 + 4x2 4
4x1 + 8x2 16
The optimal assignment is : x1, x2 0.
(A) I W , II V , III U Then L.P. has :
(B) I V , II U , III W (A) Infeasible solution
(C) I U , II W , III V (B) Unbounded solution
(D) Optimal assignment cannot be (C) Basic feasible solution
done (D) Feasible solution
11 [P.T.O.
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39. Consider the LP problem : 41. Consider the function
Max Z = x1 – x2 + 3x3
Subject to the constraints : f : [0, 1] % [0, 1] R
x1 + x2 + x3 10 defined as :
2x1 – x2 – x3 2
2x1 – 2x2 – 3x3 6 and f ( x, y) 1 if either x or y is
x1, x2, x3 0. irrational
Then the dual of the above LP 0 otherwise
problem is :
Min W : py1 + qy2 + ry3 Then f is :
Subject to the constraints : [0,1] % [0,1]
ay1 + by2 + cy3 1 (A) 0
y1 – y2 – 2y3 –1
y1 – y2 – 3y3 3 and (B) 1
y1, y2, y3 0
(C) Undefined
where the values of p, q, r, a, b, c
are : (D) 2
(A) p = 10, q = 2, r = – 6, a = –1,
42. Consider the function f : R 2 R
b = 2, c = –2
(B) p = 10, q = 2, r = 6, a = 1, given by f ( x, y) x2 y2 4y .
b = 2, c = 2 Then in a neighbourhood of (0, 0) :
(C) p = 10, q = –2, r = –6, a = –1,
(A) x variable can be expressed as
b = –2, c = 2
a function of y
(D) p = –10, q = 2, r = 6, a = –1,
b = –2, c = 2 (B) y variable can be expressed as
40. The number of basic variables in an a function of x
(m × n) transportation table are :
(C) none of x and y be expressed as
(A) m + n – 1 a function of other variable
(B) m + n + 1
(D) both x and y can be expressed
(C) mn as a function of the other
(D) m + n variable
12
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43. Consider the function f : R 2 R2 46. Let f : C C be a non-constant
defined as f ( x, y) ( x2 y2 , 2 xy) . analytic such that | f ( z)| | z| for
1 | z| 2019 . Then which of the
Then the image of the ball B(P, ) ,
2 following is false ?
where P = (1, 0) is the centre and (A) f maps a line onto a circle
1 (B) f is one-one
is radius, is :
2 (C) f is onto
(A) a closed set (D) f '( z) 0 for all z C
(B) an open set 47. Let L|K be an algebraic field
(C) neither open nor closed set extension and ' : L L be a
(D) an unbounded set K-algebra homomorphism. Then :
44. Let f be a non-constant entire (A) ' is bijective
function. Then which of the (B) ' is injective, but not surjective
following is impossible ? (C) ' is surjective, but not injective
(A) f (C) C (D) ' is neither injective nor
(B) f (C) surjective
C {0}
48. The number of linear factors of the
(C) f (C) C
polynomial X 16 1 in R [X] is :
(D) f (C) & R
(A) 0
45. Consider the following statements : (B) 1
(a) | esin z
| does not assume its (C) 2
maximum value in C. (D) 4
(b) |sin(e z )| does not assume its 49. The number of irreducible quadratic
minimum value in C. factors of the polynomial X8 + 1 in
Then : R [X] is :
(A) Only (a) is true (A) 4
(B) Only (b) is true (B) 3
(C) Both (a) and (b) are true (C) 2
(D) Both (a) and (b) are false (D) 1
13 [P.T.O.
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SEP - 30221/II—A
50. Let X be a metric space and 53. Let G be a group of order n, p a
Y, Z ( X such that Y ) Z $ . Then : prime such that p|n. Then :
(A) d(Y, Z) > 0 (A) every Sylow p-subgroup of G is
(B) d(Y, Z) > 0 if Y and Z are closed normal in G
(C) d(Y, Z) > 0 if Y or Z is compact
(B) every Sylow p-subgroup of G is
(D) d(Y, Z) > 0 if one of Y or Z is
abelian
compact and the other is closed
in X (C) the number of Sylow p-sub-
51. Let f , g : R R be defined by groups of G is equal to the index
1 1 of N(P) in G, where N(P) is the
f ( x) sin and g ( x) x sin . normaliser of any Sylow
x x
Then : p-subgroup P of G
(A) f and g are of bounded variation (D) the number of Sylow p-sub-
(B) f is of bounded variation, but groups of G is equal to the index
g is not of bounded variation of P in G, where P is any Sylow
(C) g is of bounded variation, but p-subgroup of G
f is not of bounded variation 54. The number of elements of order
(D) neither f nor g is of bounded 3 in the direct product of groups
variation
Z9 × Z3 is :
52. Let X, Y be metric spaces and
(A) 6
f :X Y be a surjective continuous
map. Then : (B) 8
(A) f maps open sets in X to open (C) 9
sets in Y (D) 18
(B) f maps closed sets in X to closed 55. The field Q ( 2, 3, 5) is not equal
sets in Y to :
(C) f maps dense subsets of X to
(A) Q ( 30)
dense subsets of Y
(B) Q ( 2 3 5)
(D) f maps nowhere dense subsets
of X to nowhere dense subsets (C) Q ( 2 3, 5)
of Y (D) Q ( 2 3 – 5)
14
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56. Let X and Y be Banach spaces with 58. Let X be the space of all continuous
. 1 and . 2 respectively denoting
their norms. Let T : X Y be a complex valued functions on [0, 1]
bijective bounded linear map. If
with inner product
x 3 Tx 2 and y 4 x 1,
where Tx y , for all x X and 1
y Y . Then : + f, g * f ( t) g(t) dt .
0
(A) . 3 and . 4 are norms and
complete Let T:X X be defined by
(B) . 3 is a complete norm and T( f )(t) t eit f (t) . Then :
. 4 is not a norm
(A) T is self-adjoint, but not unitary
(C) . 4 is a complete norm and
. 3 is not a complete norm
(B) T is unitary, but not self-adjoint
(D) neither . 3 is a complete norm
nor . 4 is a complete norm (C) T is normal, but neither self-
adjoint nor unitary
57. Let Vk {(an ) l2 : an 0 for all
(D) T is not normal
n * k} . Then :
(A) Vk is finite dimensional for all 59. Let Xi denote the set X with topology
k and l2 is a direct sum of ,i and Yi denote the set Y with
{Vk | k N}
topology - i for i 1, 2 . If f : X1 Y1
(B) Vk is finite dimensional for all is continuous, then f : X 2 Y2 is
k and l2 is a sum of {Vk | k N}
continuous provided :
and is not a direct sum
(C) Vk is infinite dimensional for all (A) -1 ( - 2 and ,1 ( ,2
k and l2 is a sum of {Vk | k N} (B) - 2 ( -1 and ,1 ( ,2
(D) Vk is finite dimensional for
(C) -1 ( - 2 and ,2 ( ,1
all k and l2 is not a sum of
{Vk | k N} (D) - 2 ( -1 and ,2 ( ,1
15 [P.T.O.
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60. Let X {( x, y) R 2 : ( x n) 2 y2 1, 63. Let G be a simple graph having no
isolated vertex and having no vertex
n Z } be the subspace of the
induced subgraph with exactly two
Euclidean space R2. Then : edges. Then :
(A) X is compact and connected (A) G is planar
(B) G is self-complementary
(B) X is connected, but not compact
(C) G is complete
(C) X is compact, but not connected (D) G is a tree
(D) X is neither connected nor 64. The coefficient of x8 in
compact ( x 2 x3 x4 x5 )5 is :
(A) 13
61. Let X be a separable first countable (B) 10
space, then X × X in the product (C) 20
topology is : (D) 0
65. Let u1 and u2 be solutions of the
(A) first countable and separable
equation :
(B) first countable, but may not .u = 0 in D
always be separable #u
g(s) on #D
(C) separable and may not always #n
be first countable x2 y2
where D {( x, y) R2 / 1}
4 9
(D) neither always first countable
is an ellipse with boundary
nor always separable
x2 y2
62. The subgroup lattice of a finite cyclic #D {( x, y) R2 / 1} . If
4 9
group is : u1 (0, ½) u2 (0, ½) 1 . Then :
(A) finite direct product of chains (A) u 1 and u 2 are dependent
solutions
(B) modular but not distributive
(B) u1 / u2 1 in D
(C) not modular
(C) u1 u2 changes sign in D
(D) not distributive (D) u1 / u2 in D
16
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66. The equation 70. Modulo the prime 101, the number
1 (50 !)2 is congruent to :
xuxx yuyy (u uy ) 0 (A) –1
2 x
is hyperbolic in the maximal (B) 1
domain : (C) 0
(A) {( x, y) R 2 / x * 0, y * 0} (D) 2
71. Which of the following statements
(B) {( x, y) R 2 / x + 0, y * 0}
is correct ?
(C) {( x, y) R 2 / xy * 0}
(A) A system of n particles can have
R 2 / x + 0, y + 0}
(D) {( x, y)
infinite degrees of freedom
67. The partial differential equation
(B) A system of n particles with
2 2 #z #z 3 2
(x z ) xy z x y is : k holonomic constraints has
#x #y
3n – k degrees of freedom
(A) linear
(C) A system of n particles with
(B) semi-linear
k constraints has n – k degrees
(C) quasi-linear
of freedom
(D) non-linear
(D) A system of n particles has
68. Every natural number is :
(A) a sum of distinct powers of 2 6 degrees of freedom
(B) a sum of distinct powers of 3 72. The Lagrangian of a dynamical
(C) a sum of distinct factorials system is given by :
(D) a sum of distinct triangular m 2
L(q1 , q2 , q1,q2 ) (q q22 )
numbers 2 1
69. Which of the following is a solution a(q1q2 q2 q1 )
to the system of congruences : where m and a are constants. Then
x / 2 (mod 5) canonical momenta of the system
x / 3 (mod 4)
are given by :
x / 1 (mod 3)
(A) p1 mq1, p2 mq2
(A) 22
(B) p1 mq1 aq2 , p2 mq2 aq1
(B) 27
(C) 47 (C) p1 aq2 , p2 aq1
(D) 67 (D) p1 mq1 aq2 , p2 mq2 aq1
17 [P.T.O.
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73. If the Hamiltonian of the dynamical 76. u ax by , v cx dy where a, b,
system is c, d are non-zero constants are
H( q1 , q2 , p1 , p2 ) p12 p22 k q12 , velocity components of a possible
then the Lagrangian is given by :
incompressible fluid motion only, if :
q12 q22
(A) L = k q12 (A) a = d
2 2
(B) a + d = 0
(B) L = q12 q22 k q12
(C) b = c
q12 q22 (D) b + c = 0
(C) L = k q12
4 4
77. If the velocity potential of the fluid
q2 q22
(D) L = 1 k q12 is given by :
2 2
74. Which of the following statements xy3 x3 y
$ x2 y2 6,
is correct ? 3 3
(A) Finite rotation can be then the velocity components of the
represented by a single vector fluid flow are :
(B) Infinitesimal transformation y3
(A) u x2 y 2 x ,
can be represented by a single 3
vector
x3
(C) The matrix representing v 2y xy2
3
infinitesimal transformation is
idempotent matrix x3
(B) u 2y xy2 ,
(D) Addition of finite rotations is 3
commutative y3
v 2x x2 y
75. Identify the velocity field (u, v) 3
which satisfies conservation of mass
x3
for incompressible plane flow : (C) u 2y xy2 ,
3
(A) u 3 x 2 y2 , v x3 4 xy2
y3
(B) u x2 , v y2 v 2x x2 y
3
(C) u 2 xy x2 , v 2 xy y2
(D) u 2x , v 2y
(D) u x2 y , v xy2
18
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79. Let
78. In two-dimensional fluid flow Edu22Fdudv Gdv2 and
Ldu2 2Mdudv Ndv2 be the first
consider a source of strength m at
and second fundamental forms
z = f, where f is real, outside the respectively of a surface patch in R3.
circular cylinder of radius a whose E F L M
If X = and Y = ,
F G M N
centre is at the origin. The image
then :
of the source in a cylinder is :
(A) X–1Y is non-singular
a2 (B) X–1Y may not be non-singular,
(A) a sink of strength m at z
f but symmetric
and a source of strength m at
(C) X–1Y may not be symmetric, but
the origin
have real eigen values
a2 (D) X–1Y is non-singular, but not
(B) a source of strength m at z
f
symmetric
and a sink of strength m at the
80. For any unit speed curve in R3 with
origin
tangent t , normal n , binormal b ,
a2 curvature k and torsion -, - t ' k b'
(C) a sink of strength m at z
f
is in :
and a sink of strength m at the
(A) the linear span of t and b
origin
(B) the linear span of t and n
a2
(D) a source of strength m at z (C) the linear span of b and n
f
and a source of strength m at (D) none of the linear spans of any
the origin two vectors in { t , n, b }
19 [P.T.O.
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83. If the integrand f does not depend
81. The Gaussian curvature of the
on y, the Euler Lagrange’s equation
elliptic cylinder of the functional
x2 y2 x2
{( x, y, z) R3 : 1} I[ y( x)] f ( x, y ', y ") dx
2 3
x1
is : under the conditions that both y and
y' are prescribed at the end points
(A) always > 0
has the first integral as :
(B) always < 0 #f d #f
(A) constant
#y " dx #y "
(C) always = 0
#f d #f
(D) > 0 at some points and < 0 at (B) constant
#y ' dx #y "
some points
#f d #f
(C) constant
82. The extremal problem #y ' dx #y '
#f d #f
J[ y( x)] ( y '2 y2 ) dx, y(0) 1, y( ) 0 (D) constant
#y " dx #y '
0
84. An extremum of the functional
has :
1
(A) a unique extremal if 0 = 1 ( y '2 12 xy) dx, y(0) 0, y(1) 1
0
(B) infinitely many extremals if
can occur only along the curve :
0 = 1
(A) y( x) x3
(C) a unique extremal if 0 = –1
(B) y( x ) x3 2x
(D) infinitely many extremals if
(C) y( x ) x3 2x
0 = –1 (D) y( x) x2 x3
20
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85. The curve connecting two given 87. The initial value problem
points A and B that do not lie on y" y 0 , y(0) y '(0) 0
a vertical line such that a particle is equivalent to the integral
sliding down this curve under equation :
gravity from the point A reaches t
point B in shortest time is : (A) y(t) (t s) y( s) ds
0
(A) a straight line connecting A
t
and B
(B) y( t) t (t s) y( s) ds
(B) a parabola passing through A 0
and B t
(C) there is no curve that (C) y(t) (s t) y(s) ds
0
minimizes time
(D) the cycloid of which one arc t
contains both the points A (D) y(t) t ( s t) y( s) ds
0
and B
88. If K(t, s) is symmetric kernel of the
86. Let 01 0 2 be the characteristic
Fredholm integral equation. Define
numbers and $1 , $ 2 be the
the operator K by :
corresponding eigen functions for the
b
homogeneous Fredholm integral
K$ K(t, s) $( s) ds
equation with separable kernel : a
1 and consider the following
u(t) 0 k(t, s) u( s) ds .
statements :
0
(I) K is self-adjoint operator.
Then which of the following is not
(II) Inner product + K$, $ * is
true ?
always real.
(A) Fredholm determinant D(0) = 0
Then :
1
$1 (t) $ 2 (t) dt 0 (A) Only (I) is true
(B)
0 (B) Only (II) is true
(C) $1 , $ 2 are linearly dependent (C) Both (I) and (II) are true
(D) $1 , $ 2 are linearly independent (D) Both (I) and (II) are false
21 [P.T.O.
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89. The third divided difference of the 92. If
function f ( x) 1 x with arguments
p, q, r, s is : e xt g(t), t * 0
f ( t)
0 , t + 0.
1
(A)
pq Then which of the following is true ?
1
(A) L{ f (t)} e sF{ g (t)}
(B) (B) F{ f (t )} sL{ g(t)}
pqr
(C) F{ f (t)} L{ g(t)}
1
(C) (D) L{ f (t)} F{ g(t)}
pqrs
93. If f (s) is the Fourier transform of
s
(D) f(t), then Fourier transform of
pqr
f (t a) is :
90. Using Newton’s forward interpolation
(A) eisa f ( s a)
formula, the cubic polynomial which
takes the following value (B) e sa f ( s a)
(C) e sa f ( s)
x: 0 1 2 3
f ( x) : 1 2 1 10 (D) eisa f ( s)
94. Solution of the Fredholm integral
is :
equation
(A) 2 x3 7 x2 6 x 1
(B) 2 x3 7 x2 6 x 1 cos 0t u(t) dt e 0
(C) 2 x3 x2 2 x 1 0
(D) 2 x3 2 x 2 3 x 1 is :
91. For a given initial value problem 2 1
(A)
y' y x, y(0) 2 (1 t 2 )
the value of y(0.1) by Runge-Kutta 2 1
(B)
second order method is : (1 t 2 )
(A) 2.2100 2 1
(C)
(B) 2.0050 (1 t)2
(C) 2.2050 2 1
(D) 2.1900 (D)
(1 t)2
22
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95. Consider the following auxiliary LP 97. Which of the following statements
problem for two-phase simplex
is true for a graph G ?
method.
m (A) Every graph G has a perfect
Max Z * 1 ( 1) A i ,
i 1 matching
where Ai’s are artificial variables,
subject to (B) Every regular graph G with
n
even degree 1 has a 2-factor
1 aij x j Ai bi , i 1, 2,....., m
i 1
(C) Every bipartite graph G having
and x j , A i 0.
If Max Z* = 0 and at least one no pendant vertex has a 2-factor
artificial variable is present in the (D) Line graph of a regular graph
basis with positive values then the
G has a 1-factor
original L.P. problem has :
98. Consider the signed measure v on
(A) no feasible solution
[0, 2] defined by (A) f ( x) dx
(B) a feasible solution
A
(C) degenerate feasible solution
f ( x) 1 if x [0,1)
where .
(D) cannot say anything about the 1 if x [1,2]
existence of solution 1 3
Then , is :
96. There exists a graph with vertex- 2 2
connectivity a, edge-connectivity (A) equal to the Lebesgue measure
b and the minimum degree c, if : of (0, 1)
(A) a = 4, b = 2, c = 3 (B) 1
(B) a = 4, b = 3, c = 4 (C) 0
(C) a = 2, b = 3, c = 4
1
(D)
(D) a = 2, b = 4, c = 3 2
23 [P.T.O.
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99. Let (X, M, µ) be a measurable space SECTION III
101. In a class of 100 students, 40
and E M be a measurable set.
students got 60 marks and 60
Let f be a non-negative measurable
students got 40 marks out of 100.
function with support contained
Hence the mean, median and mode
in E. Define 0(A) f dµ . of the class marks are respectively.
A )E
(A) 48, 40 and 60
Which of the following statements (B) 48, 40 and 40
is false ? (C) 24, 60 and 40
(A) 0 is a measure on M (D) 24, 40 and 60
(B) 0 is a measure on 102. The coefficient of variation of a
X
M(E) {A ) E / A M} random variable X is 1.2. If Y = ,
4
(C) 0 is absolutely continuous with the coefficient of variation of Y is :
respect to µ on M(E) (A) 0.3
(D) 0 is singular with respect to µ (B) 0.6
on M(E) (C) 2.4
(D) 1.2
100. Let f : [0, 1] R be a continuous
function. Then f is absolutely 103. The random variable X follows
continuous, if : normal distribution with mean 5 and
standard deviation 6. Hence :
(A) f is uniformly continuous
(A) P[X 5] = P[X 6]
(B) f is of bounded variation
(B) P[X 5] > P[X 6]
(C) f is Riemann integrable
(C) P[X 5] < P[X 6]
(D) f' exists and is continuous (D) P[X 5] > P[X 6]
24
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104. Suppose P(A/B) = 0.4 and P(B) = 0.8. 107. Which of the following is a
Which of the following is a possible characteristic function of a random
value for P(A) ? variable ?
(A) 0.3
(A) (t) = cos(t2)
(B) 0.5
(C) 0.6 (B) (t) = cos t 2
(D) 0.1 2
105. Let X be a random variable with (C) (t) =
1 2t
E [|X|3] = 3
t
Then : (D) (t) =
1 t2
3
(A) P[|X| 2] 108. Suppose E, F and G are 3 events
8
such that P(E ) F ) G) = P(E).P(F).
3
(B) P[|X| 2] P(G) and P(E ) F ) G) = P(E).
5
P(F ) G) .
(C) One cannot say anything about
P[|X| 2] from the given Consider the statements :
information (I) F and G are independent events
5 (II) E and FC G C are independent
(D) P[|X| 2]
8 events
106. Let the joint distribution of (X, Y)
(III) EC and F ) G are independent
be given by :
events
X (IV) E, F and G are mutually
0 1 2 3
independent events
Y 0 1 1 1 1
10 20 20 20
3 1 3 5 (A) Only statements I, II and III are
1 20 5 20 20
correct
Then P[X 1] is :
(B) Only statements II, III and IV
(A) 3
20 are correct
(B) 1
20 (C) Only statements II and III are
(C) 1
2 correct
(D) 1 (D) Only statement I is correct
5
25 [P.T.O.
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109. Suppose X1, X2, X3 are independent 112. Suppose X 1, X 2 , X 3 and X 4 are
and identically distributed random independent and identically
variables each having exponential distributed tri-variate random
distribution with mean 2. Then the
vectors having normal distribution.
probability density function of
Then the distribution of a1X1 + a2X2
sample range is given by,
+ a3X3 + a4X4, where a1, a2, a3, a4
(A) h(r) = e–r, r > 0
(B) h(r) = re–r, r 3 are real numbers, is :
(C) h(r) = (1 – e–r)e–r, r 3 (A) Univariate normal
(D) h(r) = 2(1 – e–r)e–r, r > 0
(B) Four variate normal
110. The distribution with moment
1 (C) Trivariate normal
generating function t is :
3 – 2e
(D) Not normal
(A) Exponential distribution
(B) Cauchy distribution 113. Let X1, X2, ......., Xn be independently
(C) Geometric distribution and identically distributed random
(D) Logarithmic series distribution variables with N(µ, 1) distribution.
111. Which of the following statements
Assume that µ 0. Let µ̂ denote
about an exponential distribution
MLE of µ. Then which of the
with mean is true ?
(A) Its variance is following statements is true ?
(B) Its moment generating function
(A) µ̂ = max X n ,0
is 1/(1 – t) exists for all t R
(B) µ̂ is unbiased for µ
(C) E(X) = e– 0x dx.
0
(C) The sample mean Xn is
(D) F(t s) F(t) F(s), 4 t, s R ,
sufficient for µ
where F( x) is the survival
function of X. (D) µ̂ may not exist
26
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114. Let X be a single observation from 117. Let X 1 , X 2 , ..........., X n be a
,e– ,x 5[0, ) ( x) , > 0. Then (X, 2X) is random sample from the density
a confidence interval for 1 with
, f(x, ) = c( ) h( x) e ( )d(x), , .
confidence coefficient :
n
(A)
1 Let T( x) = 1 d( xi ) .
2 i 1
(B) e–1 Then which of the following
1 statements is not true ?
(C) –
e 2
(A) T(X) is a complete sufficient
1
(D) –
e 2 – e –1 statistics for
115. Let X1, X2, ........, Xn be a random (B) T(X) is uniformly minimum
variance unbiased estimator of
1 1
sample from uniform ,, ,– its expected value
2 2
distribution. Consider the problem (C) T(X) is unbiased for
1 1 (D) The probability distribution
of testing H0 : , – Vs H1 : , .
2 2 depends on
Let X(1) = min X1 , X 2 ,......., X n . 118. Suppose t test is applied to test the
Consider the test, reject H 0 if equality of means of two random
X(1) > 0 and accept otherwise. Then variables X and Y, where X and Y
the power (p) and size (s) of the test
are independent. Then :
(p, s) are given by :
(A) (0, 0) (A) Distribution of both X and Y
(B) (0, 1) must be normal with equal
(C) (1, 0) variances
(D) (1, 1)
116. Which of the following theorems is (B) Distribution of both X and Y
useful for obtaining a sufficient must be normal
statistics ? (C) Distribution of X and Y may not
(A) Neyman-Pearson theorem
be normal
(B) Neyman-factorization theorem
(C) Basu’s theorem (D) Both X and Y must have the
(D) Rao-Blackwell theorem same variance
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119. X is a normal (6, '2) random variable 121. A frequency data is classified in 10
and a 95% confidence interval for 6 classes and normal N(µ, 2)
based on a random sample of size
distribution is fitted to data after
n on X is given by (15, 25]. Hence
estimating the parameters. To use
it can be interpreted that :
72 goodness of fit test, two classes
(A) The true value of µ is 20
are combined into one class. Hence
(B) The true value of µ lies in
the degrees of freedom associated
(15, 25] with probability 0.95
with 72 test are :
(C) The true value of µ is between
15 and 25 (A) 9
(D) The interval (15, 25] covers the (B) 8
true value of µ with probability
(C) 7
0.95
(D) 6
120. Suppose r12.3 is a partial correlation
coefficient between (X1, X2) and X3. 122. Let (Xi, Yi) i = 1, ......, 5 be the
Then which of the following bivariate pair of observations and
statements is not correct ? Zi = Xi – Yi i = 1, ......, 5. For a given
(A) r12.3 is a correlation coefficient sample the number of Zi > 0 is 2.
between X – X ˆ and X – X ˆ ,
1 1 2 2 Hence the p-value to test the null
where X̂1 is a line of best fit
hypothesis that H0 : median (Z) = 0
based on X3 and X̂ 2 is a line
against H1 : median (Z) < 0 is :
of best fit based on X3.
(B) – 1 r12.3 1 (A) 0.5
(C) r12.3 0 always (B) 0.25
r12 – r13 .r23 (C) 0.10
(D) r12.3 = 2 2
1 – r13 1 – r23 (D) 0.05
28
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123. Suppose that there are 1000
components in operation. The 125. A canteen has a single cash counter.
number of failed replacements at the
end of each week are as follows : A customer has to buy coupons from
W1 W2 W3 W4
50 82 128 199 the cash counter before getting the
If the cost of replacing an individual
failed component is ` 125 and the
cost of group replacement of all food items. Customers arrive at the
1000 components is ` 300, what is
the best interval between group counter according to a Poisson
replacements ?
(A) One week
process with rate of 2 per 5 minutes.
(B) Two weeks
(C) Three weeks
(D) Four weeks It takes 1.5 minutes on average to
124. A newspaper vendor buys papers
for ` 3 each and sells them for ` 5 buy coupons. What is the long-run
each. The unsold papers have no
value. The daily demand for the
probability that the cash counter is
paper has the following probability
distribution :
# of customers Probabilities free ?
200 0.2
250 0.3 1.5
300 0.3 (A) 1 –
2
350 0.1
400 0.1 1.5
If each day’s demand is independent (B) 1–
5
of the other days, the average profit
per day is : 1.5
(A) ` 75 (C) 1–
2.5
(B) ` 120
(C) ` 93
(D) 0
(D) ` 50
29 [P.T.O.
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126. In a survey of finite population, the 127. Let the variances of the regression
following method is applied to obtain estimator and the sample mean,
a sample of size n. both computed from SRSWOR of the
(i) In the 1st draw, the ith unit is same sample size, be denoted by
2 2
selected with probability VReg and VSRS respectively. Then :
N 2 2
Pi, > 0, i = 1 ............ N, 1 Pi 1 (A) VReg VSRS for a large n.
i 1 2 2
(B) VReg VSRS
1
and Pi for some i. (C) Both the estimators of the
N
population mean are unbiased
(ii) The remaining (n – 1) units are
2 2
obtained by SRSWOR method (D) VReg VSRS for any n
from remaining N – 1 units of 128. To examine whether five different
the population. skin creams A, B, C, D and E have
different effect on the human body,
Let Y denote the sample mean,
n randomly chosen persons were
then :
enrolled in an experiment. The five
(A) Y is an unbiased estimator of
creams were applied on each of one
population mean randomly chosen fingres of the same
(B) There exists an unbiased person; and the procedure is
estimator of the population repeated for each person. The
mean and it is different from the resulting design is :
(A) a completely randomized design
sample mean
with 5 treatments
(C) This is a stratified sampling
(B) a randomized block design with
procedure
5 blocks and n treatments
(D) There does not exist an (C) a randomized block design with
unbiased estimator of the n blocks and five treatments
population mean (D) a repeated measures design
30
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129. For one-way classification model the 131. Let F1 and F2 be two '-fields of
F-test is valid under the assumption subsets of 8.
that : Then which of the following is not
(A) The observations on response true :
variable are uncorrelated with (A) Both F1 F2 and F1 F2 are
common mean and variance -fields
(B) The errors are uncorrelated
(B) F1 F2 is not a -field but
with common mean and
F1 F2 is a -field
variance
(C) The errors are normally (C) Neither F1 F2 nor F1 F2 are
distributed with common mean -fields
zero and common positive (D) F1 F2 is a -field but F1 F2
variance is not a -field
(D) The errors are independently
132. Let 89 9 {1, 2, 3, 4}.
normally distributed with
Consider the two -fields :
common mean zero and
common +ve variance F1 = {$, 8,{1},{2},{1, 2},{3,4},{2,3, 4},
{1,3, 4}}
130. Under a 23 factorial design with
F2 = {$, 8,{1,2},{3,4}}
factors A, B, C arranged in a single
Let f denote a function from ( , F1)
replicate of two blocks, the containts
to ( , F2).
of block-I (in Yate’s notation) are
Then :
{1, a, ab, b}. Which of the following (A) Any f is measurable
effects is confounded with blocks ? (B) f defined by f(k) = k, k = 1, 2,
(A) AB 3, 4 is measurable
(C) f defined by f(1) = f(3) = 1 and
(B) AC
f(2) = f(4) = 4 is measurable
(C) ABC
(D) f defined by f(1) = f(2) = f(3) = 1
(D) C and f(4) = 4 is measurable
31 [P.T.O.
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133. Let R denote the real line, B the 134. Let {Xn} be a sequence of random
Borel -field and m the Lebesgue variables and X a random variable
measure on R :
such that :
For A :, define µ(A) =
x2 1 P[|X n – X|* ;] < for every
n 1
exp – dm(x).
2
A
; * 0.
Consider the statements :
Consider the statements :
(I) µ is a measure on B
(I) P lim X n X 1
n
(II) The Radon-Nikodym derivative
of µ with respect to m, (II) P lim X 2n X2 1
n
dµ x2 (III) P lim X n X only if {Xn} is
( x) exp – a.e. m. n
dm 2
a sequence of independent random
(III) m < < µ, (m is absolutely
variables.
continuous with respect to µ).
Which of the above are always
Which of the above statements are
true ? correct ?
(A) All the three (A) (I) only
(B) (I) and (II) only (B) (I) and (II) only
(C) (I) and (III) only (C) (III) only
(D) (I) only (D) None of the three
32
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135. Let {Xn} be i.i.d. random variables
137. Suppose {Xn} and {Yn} are sequences
with mean zero and finite variance
2 > 0.
of random variables and X and Y are
Which of the following is false ?
random variables such that as
1
(A) (exp (X1) + exp (X2) + X3 +
n n , Xn converges in quadratic
...... + Xn) 0 in probability as
n mean to X and Yn converges in
1 quadratic mean to Y.
(B) 5/8 (exp (X1) + X2 + ....... + Xn)
n
0 in probability as n
Consider the following statements :
1
(C) 3/8(X1 + ....... + Xn) does not
n (I) The sequence (X n + Yn)
converge in probability as
n converges in quadratic mean to
1 (X + Y) as n
(D) 3/8 (X1 + ....... + Xn) converges
n
in distribution to a r.v. as (II) nlim P |X n Yn XY|> ; 0,
n
136. Suppose the sequence of random 4; * 0
variables {X n } converges in
distribution to X. (III) nlim E ei X n = E ei X
Then which of the following is not
always true ? Which of the above are true ?
(A) lim
n
E[Xn] = E[X]
(A) All the three
(B) E[|X|] lim inf E[|X |]
n n
(B) (I) and (II) only
|X n | |X|
(C) lim
n
E 1 |X n | = E 1 |X|
(C) (I) and (III) only
Xn
(D) lim
n
= 0 in probability (D) (II) and (III) only
n
33 [P.T.O.
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138. Let {X n } be a sequence of 140. Suppose a distribution function
independent random variables. F : R [0, 1] is as follows :
x + 0,
1 X k (w) converges
0, if
Let E = w|
k 1 ! 1 / 4, if 0 x + 1,
Then which of the following is F x 1 / 2, if 1 x + 2,
always correct ? 1/2 x 2 / 2, if 2 x + 3,
(A) P(E) = 0
1, if x 3.
(B) P(E) = 1
(C) 0 < P(E) < 1 Then F(x) = l 1 F 1 (x) + l 2 F 2 (x)
where :
(D) P(E) is either ‘0’ or 1 (A) l 1 = 1/4, l 2 = 3/4, F 1 is a
139. Let {Y n } be a sequence of distribution function of a
discrete random variable with
independent random variables with
support {0, 1} with respective
mean zero and variance 1. probabilities 1/4, 3/4 and F2 is
Let Fn = (Y1, ......, Yn) the -field a distribution function of
uniform U(2, 3) distribution.
generated by {Y1, Y2, ....... , Yn} (B) l 1 = 1/2, l 2 = 1/2, F 1 is a
Consider the statements : distribution function of a
(I) The sequence Xn = Y1 + Y2 discrete random variable with
support {0, 1} with respective
+ ........ + Yn is a Fn - martingale. probabilities 1/2, 1/2 and F2 is
n a distribution function of
(II) The sequence Xn = < Yk is a uniform U(2, 3) distribution.
k 1 (C) l 1 = 1/4, l 2 = 3/4, F 1 is a
Fn - martingale. distribution function of a
discrete random variable with
2
n support {0, 1} with respective
(III) The sequence Xn = 1 Yk –n probabilities 1/2, 1/2 and F2 is
a distribution function of
k 1
uniform U(2, 3) distribution.
is a Fn - martingale.
(D) l 1 = 1/4, l 2 = 3/4, F 1 is a
Which of the following is correct ? distribution function of a
(A) (I) and (III) only discrete random variable with
(B) (I) and (II) only support {1, 2} with respective
probabilities 1/4, 3/4 and F2 is
(C) (II) and (III) only a distribution function of
(D) All the three uniform U(2, 3) distribution.
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141. Which of the following statements 144. Suppose X has bivariate normal
is not correct ? The set of points of distribution with mean vector (3, 4)'
discontinuities of a distribution and
function may be : V(X1) = 2. Further, it is known that
(A) empty X1 and X2 are positively correlated.
(B) finite Which of the following can be the
(C) uncountable conditional distribution of
(D) countable (X1|X2 = 2) ?
142. Suppose X is a non-negative random
(A) N(1.5, 2.64)
variable whose mean exists. Then :
(B) N(2.6, 1.84)
(A) E(log(X)) < log(E(X))
(B) E(log(X)) log(E(X)) (C) N(3.5, 1.52)
(C) E(log(X)) = log(E(X)) (D) N(3.5, 2.84)
(D) None of (A), (B), (C) is true 145. Let X1, X2, ........, Xn be i.i.d. Bernoulli
143. Suppose ( . ) is a characteristic ( ). Let the loss function for the
function of a random variable with problem of estimation of be given
distribution function F. Then : by
(A) F(b) – F(a) =
(, – a)2
u
[exp(– iua) – exp( iub)] $(u) du
L ( , a) = . Let the pdf of
lim
1 ,(1 – ,)
u 2 iu
–u
the prior for ( ( )) be given by
(B) F(b) – F(a) =
u
( ) = 1 0 1.
[exp(– iua) – exp( iub)] $(u) du
lim Let X be the sample mean. Then :
u iu
–u
(C) F(b) – F(a) = (A) X is Bayes but not minimax
u
1 [exp(iua) – exp(iub)] $(u) du (B) X is not Bayes but it is
lim
u 2 iu
–u minimax
(D) F(b) – F(a) =
u
(C) X is both Bayes and minimax
1 [exp(– iub) – exp( iua)] $(u) du
lim
u 2 –u iu (D) X is not admissible
35 [P.T.O.
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146. Let X1, X2, ........, Xn be iid Bernoulli 148. Suppose {X1, X2, ......, Xn} is a random
( ) random variables. We wish to sample from uniform U(,9 – 1/2,
follow the Bayes procedure for
, + 1/2) distribution, , R . Then :
estimation of . Then, a conjugate
(A) Sample mean is unbiased for ,
prior for :
(A) is9 (,) = 1 0 < < 1 (B) Sample median is unbiased
for ,
(B) is (,) = , =9 – 1 (1 – ) – 1
=9 > , > > 0 (C) X(1) + 1/2 is unbiased for
e, (D) X(n) – 1/2 is unbiased for
(C) is (,) = ,
1 e
149. Suppose {X 1 , X 2 , ......, X n } are
(D) Cannot be decided since the loss
function is not given independent and identically
147. Suppose X1, ......, Xn are independent distributed random variables each
and identically distributed random having U(–,, ,) r.vs. Then the
variables each having U(– , ) maximum likelihood estimator of
distribution. Then the minimal
is :
sufficient statistic for is given by :
(A) X(n)
(A) –X(1)
(B) X(n) (B) (X(1) + X(n))/4
(C) max {–X(1), (X(n)} (C) –X(1)
(D) min {–X(1), (X(n)}
(D) max {|X1|, ...... |Xn|}
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150. A random variable X follows N(µ, 1) 152. Suppose X 1 , X 2 , ......, X n are
distribution. A conventional test independent and identically
based on sample mean is used to test distributed random variables each
the hypothesis H0 : µ = µ0 against having exponential distribution with
H1 : µ = µ1 > µ0. Consider the mean . Then the asymptotic
following statements : X
distribution of n –1 where
(i) If H0 is rejected at = 0.05, it S
n
is also rejected at = 0.10. 1
S2 = 1
n 1
X 2i – (X)2 is :
(ii) If H0 is accepted at = 0.05,
it is also accepted at = 0.01. (A) N(0, 1)
Then : (B) N(0, )
(A) (i) is correct but (ii) is wrong (C) N(1, 1)
(D) N(0, 2)
(B) (i) and (ii) both are wrong
153. Let Xn be asymptotically normally
(C) (i) and (ii) both are correct
distributed with mean µ and
(D) (i) is wrong but (ii) is correct variance ' n2 , such that ' n2 0 as
151. Suppose {X1, X2, ......, Xn} is a random n . Which of the following
sample of size n from uniform statements is not true ?
U( – 1, + 1) distribution. Then (A) For µ 0, 1/Xn ~ AN (1/µ,
which of the following is not true ? ' 2n /µ2)
(A) X(n) is consistent for (B) For any µ, e X n ~ AN (e µ ,
(B) Sample median is consistent e2µ ' 2n )
for (C) For µ = 0, log |X n | n| d
(C) Sample mean is consistent log |N(0, 1)|
for (D) For µ 0, log |X n| ~ AN
(D) X(n) – 1 is consistent for (log |µ|, ' 2n /µ2 )
37 [P.T.O.
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154. Let X1, X2, ......, Xn be iid U(0, ). 156. Let (X, Y) ~N (0, 1) where
Consider T1 = sample median (= Mn)
1 @
n 1 , @ 0 and
1 @ 1
and T2 = 2 X , where X =
ni 1
Xi .1
X Y 1 @
Then, ARE (T1, T2) is : W .
X Y 1 @
(A) 3/4 Then which of the following
(B) 2/3 statements is not correct ?
2
(C) 1 (A) W ~ F1,1
(D) 4/3 (B) 1/W 2 ~ F1,1
155. Let X1, X2, ......, Xn be iid observations (C) (X – Y) and (X + Y) are
from a continuous distribution F. independently distributed
Define : (D) W ~ 712
0 x + X (1) 157. Let X ~ N n (µ, ' 2I) and A be a
Fn ( x) k / n X ( k) x + X ( k 1) n × n matrix. Then Y AX are
1 x X ( n)
uncorrelated normally distributed if
Then for any fixed x, and only if which of the following
d
n (Fn ( x) F( x)) ?? Z , where Z conditions hold ?
is a normal r.v. with mean 0 and (A) A is any non-singular matrix
variance :
(B) A is an idempotent matrix with
(A) F2 ( x) / 2 rank < n
(B) (1 – F( x))2 (C) A is an orthogonal matrix
(C) F2 ( x) (D) All elements of A are equal
(D) F( x) (1 F( x)) to 1
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158. If S1, S2, ......, S5 are the corrected 161. y1, y2, y3, y4 are four uncorrelated
sum of squares and sum of products random variables with common
matrices based on five independent variance '2 and E(y1) = E(y2) = ,1
samples of sizes 100 each coming + 2,2 + ,3, E(y3) = E(y4) = ,1 + ,3.
from a four variate normal
Hence :
population. The distribution of
(A) ,2 is not estimable
5
1 Si . (B) 2,2 + 3,3 is not estimable
i 1
(C) ,1 + ,3 is not estimable
(A) is Wishart with k = 496
(D) ,1 +9 ,2 +9 ,3 is not estimable
(B) is Wishart with k = 495
162. In a simple linear regression set up
(C) is Wishart with k = 499
y > ae>, X e;, the prediction of
(D) cannot be decided on the basis
of given information response variable y :
159. Let M ~ W p (n, 1) and CC' 1 (A) does not depend on regressor X
where C is a non-singular matrix. (B) will have same variation
Then C –1 M C –1 is distributed as : irrespective of X
(C) will have larger variation for
(A) W p (n,C)
larger values of X
(B) Wp (n, 1 –1 )
(D) will have larger variation if the
(C) W p (n, I p ) distance between X and mean
(D) Wp (n,C–1 ) of regressors increases
160. The characteristic function of a 163. In a multiple linear regression set
vector X ~ N p (µ, 1) is given by : up Y X > e ; under the assumption
1 that the random errors ;1.....; n are
(A) eµ ' t t'1t
2
uncorrelated and homoscedastic the
1 residuals e1.....en are :
(B) ei µ ' t t '1t
2
(A) correlated and homoscedastic
i
(C) eµ ' t t'1t
2 (B) correlated and heteroscedastic
i
(C) uncorrelated and homoscedastic
(D) e iµ't t '1t
2 (D) uncorrelated and heteroscedastic
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164. In a multiple linear regression 166. Let N = 4. Two units are selected
model Y X > e; with E(;) 0, for inclusion in the sample based on
Var(;) ' 2I : the following design. The i-th unit
(A) the regression sum of squares
and error sum of squares are is selected with probability pi where
independent p1 = 1/6, p2 = 1/6, p3 = 2/3. The
(B) the regression sum of squares
and error sum of squares are second unit is selected randomly
positively correlated from among the remaining three
(C) the regression sum of squares
and error sum of squares are units in the population. Then, the
uncorrelated first unit is included in the sample
(D) the regression sum of squares
and error sum of squares are with probability :
negatively correlated (A) 2/9
165. Suppose Yi >1X i1 >2 X i2 ;i
(B) 4/9
2
where E(;i ) 0, Var(;i ) ' (C) 6/9
i 1,2,3 and X i1 i 4 , i 1,2,3
(D) 7/9
X i2 i 2 , i 1, 2 and X i2 i for
167. The regression estimator of the
i 3 . A least-squares fit is carried
sample mean based on a SRSWOR :
out and sum of squares are
calculated then : (A) is always better than the
(A) the sum of squares due to >1
and >2 are uncorrelated with sample mean
each other
(B) and the sample mean have the
(B) the sum of squares due to >1
and >2 are independent of each same efficiency
other
(C) the sum of squares due to >1 (C) is worse than the sample mean
and >2 are correlated with each
(D) is better than the sample mean
other
(D) the sum of squares due to >1 if n, the sample size, is
and >2 are uncorrelated with
total sum of squares sufficiently large
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168. Let p be the observed proportion of 170. For a BIBD with parameters
units with the attribute A in a (v, b, r, k, 0) to exist, the condition
SRSWOR of size n from a finite
b v is :
population size N. Let P be the
(A) neither necessary nor sufficient
population proportion and Q = 1 – P.
(B) not necessary but sufficient
Then, Var(p) equals :
(C) necessary but not sufficient
PQ N n
(A)
n N 1
(D) necessary and sufficient
PQ
(B) 171. A row-column design with 7
n
treatments arranged in 3 rows and
PQ 1
(C)
n N 1 7 columns is given below :
PQ N n 1 2 3 4 5 6 7
(D)
N N 1
2 3 4 5 6 7 1
169. A block design with v treatments
and b blocks is connected. Hence the 4 5 6 7 1 2 3
rank of c matrix is : Hence the design is :
(A) v – 1 (A) Latin Square Design
(B) v (B) Youden Square Design
(C) b – 1 (C) Quasi-Latin Square Design
(D) b (D) Split-Plot design
41 [P.T.O.
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172. In a 23 factorial design with three 174. Given two time series,
treatments A, B, C each at two levels,
blocks of 4 plots are available. The Xt 0.3 X t 1 0.2 X t 2 Zt
design is replicated twice and the
Yt 0.4 Yt 1 Zt 1.2 Z t 1
allocation of treatment combination
is : where {Zt} iid WN(0, '2), which of
Replication 1 Replication 2
the following statements is true ?
Block I II Block I II
1 b 1 a (A) {Yt} is not causal, but invertible
a c b c
bc ab ac ab (B) {Xt} is not causal
abc ac abc bc (C) {Yt} is causal, but not invertible
Hence the two confounded treatment
combinations in replication 1 and 2 (D) {Xt} is not invertible, but causal
respectively are :
175. Let {Xt} be a stationary AR(1) model
(A) AB and ABC
(B) AB and AC with Xt µ $ Xt 1 Zt , where
(C) BC and ABC Z t ~ WN(0, ' 2 ) . Then, which of the
(D) BC and AC
following statements is true ?
173. Let {Xt} be a stationary time series
with autocovariance function r(h). (i) E(X t ) µ
Then, which of the following
statements is not true about r(h) ?
(ii) V(X t ) '2 / 1 $2
(A) A(0) Var (X t ) (iii) Corr(X t , X t s ) $s
(B) A (k) A ( k) (iv) V(X t ) 1 $2
(C) | A ( k)| A (0)
(A) (ii) and (iii)
(D) 1 1 =i= j A (|ti t j |) 0
for
i j (B) (i) and (iii)
every set of time points
(C) (i) and (ii)
{t1 , t2 , ......, tn } and real
members {=1 , = 2 , ......, = n } (D) (iii) and (iv)
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176. Consider a Markov chain on 178. Let the off-spring probability
S {1,2,3, 4,5} with its transition distributions of the three Branching
probability matrix given by :
processes be given by :
1 2 3 4 5 1 1 1
I. p0 p1 p2
1 3
4 0 14 0 0 4 4 2
2 1 0 0 12 0
2 1 1 1
P 1
3 2 0 2 0 0 1 II. p0 p1 p2
4 2 4
4 0 0 0 2 3 13
1 3
5 0 0 0 1 2 . III. p1 p2
3 3 4 4
Then : Then, the extinction probabilities of
(A) there exists a unique stationary the three Branching processes are
distribution respectively given by :
(B) there does not exist any
(A) 0, 0, 1
stationary distribution
(C) all states are transient (B) ½, 0, 1
(D) there exist infinitely many (C) ½, 1, 0
stationary distributions (D) ½, ½, ½
177. Let {X(t), t 0} be a time-homo-
179. If all states of a Markov chain are
geneous Poisson process with rate 0.
persistent non-null :
Let T be a positive continuous r.v.
distributed independently of (A) it has a unique stationary
{X(t), t 0} . Then : distribution
1 (B) it does not have a stationary
(A) Cov(X(T), T)
2
distribution
(B) Cov(X(T), T) 0 / (0 E(T)) ( x)
(C) nlim pij * 0 for all i and j
(C) Cov(X(T), T) 0 Var(T)
(D) it has at least one stationary
(D) Cov(X(T), T) + 0 distribution
43 [P.T.O.
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180. The method of age-standardisation 182. The following sampling plan is to be
of fertility rates is used to :
applied for acceptance or rejection
(A) exclude the effect of age
of large lots.
distribution in measuring
fertility “Select two items from the lot and
(B) exclude effects of differences in inspect both the items. If both are
fertility levels
good, accept the lot, if both are
(C) compare fertility in two
defective, reject the lot; otherwise,
population groups
select one more item from the lot and
(D) compare birth rates in two
accept the lot if and only if this item
populations
181. A column L x in the life table is good.”
denotes :
If the incoming lot quality is 0.8,
(A) the number of years lived by the
then the average outgoing quality
persons in the age interval
(AOQ) is equal to :
(x, x + 1)
(B) the number of individuals (A) 0.512
surviving to age x
(B) 0.896
(C) the number of individuals
surviving to age x + 1
(C) 0.800
(D) the average number of years of
life remaining after age x (D) 0.640
44
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183. Identify the correct match between 185. As the variance of service time
the terms in Column-I below with
the definitions given in Column-II. increases, the waiting time of a
Column-I customer in M|G|1 queue :
(i) AOQ
(ii) AQL (A) increases
(iii) ATI
(B) decreases
(iv) RQL
Column-II (C) remains the same
(=) Number of items per lot that will
be inspected on average (D) cannot be determined
(>) A lower bound on the incoming 186. Consider a system of two servers
quality for which a lot is to be
rejected where customers from outside the
(A) Proportion of defective items system arrive at server 1 at a Poisson
remaining in the lot after
inspection rate 4 and at server 2 at a Poisson
(B) An upper bound on the
rate 5. The service rates servers 1
incoming quality for which a lot
to be accepted and 2 are respectively 8 and 10. A
(i) (ii) (iii) (iv)
customer at server 1 is equally likely
(A) (A) (B) (=) (>)
(B) (A) (>) (=) (B) to join server 2 upon completion of
(C) (>) (B) (=) (A)
(D) (=) (B) (A) (>) service or leave the system. Similarly
184. If the demand distribution is a customer at server 2, upon
exponential with mean 1/=, then the
optimal inventory level S* is : completion of the service will go to
h p server 1 with probability 1/4. Then
(A) = log
h c the expected number of customers
1 h c in the queue waiting for service is :
(B) log
= h p
(A) 7/9
1 h p
(C) log (B) 1/4
= h c
h c (C) 1/5
(D) = log
h p (D) None of the above
45 [P.T.O.
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187. The dynamic programming approach 189. Consider the LPP :
to an optimization problem gives the
Minimize Z = 4x1 + 6x2 + 18x3
recurrence equation for optimal
solution as : Subject to : x1 + 3x2 3,
f k (bk ) min { x 2 fk 1 (bk 1 )}
0+ x b x2 + 2x3 5
for k * 1 ; and f1 (b1 ) (b2 x)2 .
and x1, x2, x3 0.
Then, f3 (15)
Then, which of the following is not
(A) 25
(B) 112.5 a constraint in the corresponding
(C) 75 dual problem ? (Assume that w1 and
(D) None of the above
w2 are the dual variables.)
188. Which of the following characteristics
of dynamic programming is not (A) w1 4
true ?
(B) w2 9
(A) The problem can be subdivided
into stages (C) 3w1 + w2 6
(B) Every stage consists of a
(D) w1 0, w2 0
number of states
(C) Linear programming problems 190. In critical path analysis, CPM is :
cannot be solved using dynamic
(A) event oriented
programming
(D) The optimal solution to the (B) deterministic in nature
problem is obtained using
(C) dynamic in nature
optimal solutions to sub-
problems (D) none of the above
46
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ROUGH WORK
47 [P.T.O.
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ROUGH WORK
48
Text of page 1
MAHARASHTRA STATE ELIGIBILITY TEST - 26th September 2021
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 B 81 C 121 D
2 A 42 B 82 D 122 A
3 C 43 B 83 B 123 C
4 B 44 D 84 B 124 A
5 B 45 A 85 D 125 C
6 C 46 A 86 C 126 B
7 C 47 A 87 C 127 A
8 B 48 A 88 C 128 C
9 C 49 A 89 C 129 D
10 A 50 D 90 A 130 D
11 C 51 D 91 C 131 d
12 C 52 C 92 C 132 B
13 C 53 C 93 D 133 A
14 A 54 B 94 A 134 B
15 C 55 A 95 A 135 B
16 C 56 A 96 C 136 A
17 C 57 D 97 B 137 A
18 B 58 C 98 C 138 B
19 A 59 B 99 D 139 B
20 9 60 B 100 D 140 B
21 C 61 A 101 B 141 C
22 D 62 A 102 D 142 B
23 B 63 C 103 D 143 A
24 B 64 D 104 B 144 B
25 C 65 B 105 D 145 C
26 B 66 C 106 C 146 9
27 C 67 C 107 B 147 C
28 D 68 A 108 A 148 A
29 D 69 D 109 D 149 D
30 B 70 A 110 C 150 C
31 B 71 B 111 D 151 A
32 C 72 D 112 C 152 A
33 A 73 C 113 A 153 A
34 C 74 B 114 D 154 D
35 A 75 C 115 C 155 D
36 A 76 B 116 B 156 D
37 A 77 A 117 C 157 C
38 A 78 B 118 A 158 B
39 B 79 C 119 D 159 C
40 A 80 A 120 C 160 B
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 - 26th September 2021
FINAL ANSWER KEY FOR PAPER -II (Booklet Code : A)
SUBJECT CODE & NAME: (30) Mathematical Sciences
QueNo Answer
161 B
162 D
163 9
164 C
165 C
166 B
167 D
168 A
169 A
170 C
171 B
172 D
173 C
174 C
175 A
176 D
177 C
178 C
179 D
180 C
181 A
182 B
183 A
184 C
185 A
186 A
187 C
188 C
189 C
190 B
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.