MH SET 2020: 27 December, Paper II, Mathematical Sciences
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Test Booklet Code & Serial No.
A
MATHEMATICAL SCIENCE
Signature and Name of Invigilator Seat No.
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JUN - 30220 (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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JUN - 30220/II—A 2
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JUN - 30220/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. Define a function f : [0, 1] R as
1. Let (X n ) be a sequence of real f ( x) x2 ex 2. Then :
numbers and x R. Then which of (A) f(x) = 0 has no solution in
the following is true ? [0, 1]
(B) f (x) = 0 has a solution in
(A) If lim xn x, then xn < x for
n
[0, 1]
all n
(C) f is increasing
(B) If lim xn x, then there
n
(D) f is not one-one
exist m and n such that
3. Let z, w be two complex numbers
xm < x < xn
such that z w R and z + w R.
(C) If lim xn does not exist, then
n Then :
(x n ) has no subsequence (A) z = w
converging to x
(B) z w
(D) If lim xn x, then xn < x for
n (C) z, w R
infinitely many numbers n
(D) z w or z, w R
3 [P.T.O.
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1 7. Suppose X is a continuous random
4. Define f : C – {0} C by f(z) = .
z variable having the following
Then the power series expansion
probability density function :
of f(z) about z = 1 in the disc
{z : |z – 1| < 1} is : f(x) = C(4x – 2x2), 0 < x < 2
(A) 1 – (z – 1) + (z – 1)2 – (z – 1)3
= 0, otherwise
+ ...........
(B) 1 + (z – 1) + (z – 1)2 + (z – 1)3 What is the value of P[X > 1] ?
+ ...........
2
(C) 1 – z + z2 – z3 + ........... (A)
3
(D) 1 + z + z2 + z3 + ...........
C
5. If T : Rn Rm is a linear map, (B)
2
then it is represented by a matrix
of order : 1
(C)
2
(A) m × n
1
(B) m × m (D)
4
(C) n × n
8. The moment generating function of
(D) n × m
a random variable X is given by
6. Let V be a finite dimensional vector
space over R and T L(V). If is Mx(t) = exp[2et – 2]. Then E[X] =
an eigenvalue of T, then which of
(A) 0
the following statements is false ?
(A) ker(T – I) {0} (B) 1
(B) ker(T – I) = {0} (C) 2
(C) rank(T – I) < dim V
(D)
(D) det(T – I) = 0
4
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9. Consider the LP problem : SECTION II
11. Let the middle number of the three
Max Z = 10x1 + 6x2
distinct real numbers a, b and c be
Subject to the constraints : mid{a, b, c}. Then mid{a, b, c} =
(A) min{max{a, b}, min{b, c}
5x1 + 3x2 < 30;
min{a, c}}
x1 + 2x2 < 18 and (B) min{max{a, b}, max{b, c}
x1, x2 > 0. max{a, c}}
(C) max{max{a, b}, max{b, c}
Then the LP has :
min{a, c}}
(A) No feasible solution (D) max{max{a, b}, max{b, c}
max{a, c}}
(B) Multi-feasible solutions
12. Let (x n ) be a sequence of real
(C) Unique feasible solution numbers. Then(x n ) has a
subsequence which is :
(D) Cannot say anything about
(A) monotone
solution
(B) bounded
10. Which of the following statement/s
(C) cauchy
is/are true for an LP problem ?
(D) convergent
(I) Every feasible solution is a basic
13. Consider the following statements :
feasible solution.
(a) There is a continuous map from
(II) If there is a feasible solution, [0, 1] onto (0, 1)
then there is a basic feasible (b) There is a continuous map from
solution. (0, 1) onto [0, 1]
Then :
(A) only (I) is true
(A) only (a) is true
(B) only (II) is true
(B) only (b) is true
(C) both are true (C) both (a) and (b) are true
(D) neither (I) nor (II) is true (D) both (a) and (b) are false
5 [P.T.O.
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14. Consider the function 17. Which of the following sets is
mapped onto the right half plane
1
x sin , if 0 x 1 under the map f(z) = sin z ?
f ( x) x
0 , if x 0
(A) z : 0 Re( z)
Then which of the following 2
statements is true ?
(A) f is Riemann integrable (B) z : Im( z)
2 2
(B) f is not continuous at x = 0
(C) f is of bounded variation z : 0 Im( z)
(C)
(D) f is differentiable at x = 0 2
15. Consider R with the metric defined
by : (D) z : Re( z)
2 2
1, if x y
d(x, y) = 18. Consider the following statements :
0, if x y
(a) Every non-negative harmonic
Which of the following statements
is true for (R, d) ? function on C is constant.
(A) (0, 1) is not closed (b) Every harmonic function on
(B) (0, ) is unbounded C – {0} has a harmonic
(C) [0, 1] is not compact conjugate
(D) {1, 2, ......, 100} is not open
Then :
16. For a complex number z, the
inequality |z – i| < |z + i| is : (A) only (a) is true
(A) never true
(B) only (b) is true
(B) always true
(C) true if and only if Im(z) > 0 (C) both (a) and (b) are true
(D) true if and only if Re(z) > 0 (D) both (a) and (b) are false
6
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19. The value of the integral 22. Let n be a square free natural
1 1 number. Then the quotient ring
z3 cos dz
2 i | z| 1 z Q[X]/<X2 – n> is :
(A) not a field
is :
(B) an integral domain which is not
(A) 0
a field
(B) 1
(C) not an integral domain
1
(C) (D) an integral domain
6
23. Every finite group is isomorphic to
1
(D) a subgroup of :
24
(A) a cyclic group
20. Consider the function
(B) Zn for some n > 1
1 .
f ( z) 2 (C) Sn for some n > 1
z sin z
(D) (R, +)
Then :
24. Let be a permutation in Sn. Let
(A) f has a pole of order 1 at
z = 0 = 1, 2 ......... k, where 1, 2,
(B) f has a pole of order 2 at ....., k are disjoint cycles of lengths
z = 0 l1, l2, ......., lk respectively. Then the
1 order of is equal to :
(C) the residue of f at z = 0 is (A) l1 + l2 + ......... + lk
3
(B) l1 . l2 ......... lk
1
(D) the residue of f at z = 0 is (C) gcd(l1, ........., lk)
6
21. Let : G G be a surjective group (D) lcm(l1, ........., lk)
homomorphism. Then : 25. Which of the following is a principal
(A) If G is cyclic, then G is cyclic ideal of Z[x] ?
(B) If G is cyclic, then G is cyclic (A) (2, x)
(C) G is cyclic if and only if G is (B) (3, x)
cyclic
(C) (x, x2 – x, x + 4)
(D) If G is not cyclic, then G is not
(D) (2x, 3x, x2)
cyclic
7 [P.T.O.
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26. T 1 , T 2 and T 3 are three maps 29. Let T be a linear operator on R3,
defined on R3 as : given by :
x x 1 x x 2x 2y z
! ! ! xy ! !
T1 y ! y ! , T2 y ! T y! 2y z !
!
"z# ! z !#
! ! ! " z# "2x 3y
" z# " z # " z#
Then the matrix representation of
x x y T with respect to standard basis of
! ! R3 is :
T3 y ! y z!
. 2 2 1
! ! !
" z# " z x# 0 2 1!
Which of these maps are linear ? (A)
(A) T1, T2, T3 "2 3 1 !#
(B) T1, T2, 2 3 1
(C) T 3 !
0 2 1!
(D) T1, T3 (B)
"2 2 1 !#
27. Let T be a linear operator on R4
having the minimal polynomial 2 0 2
(x + 1)3. Then which of the following !
2 2 3!
statements is true ? (C)
"1 1 1 !#
(A) T is diagonalizable over R
(B) T is nilpotent 1 1 1
(C) T is triangulable over R !
2 0 2!
(D) characteristic polynomial of T is (D)
!
(x + 1)3 "2 2 3#
28. Let V be a finite dimensional vector 30. Let A and B be real n × n matrices.
space over K and W be a subspace Which of the following statements
of V. Then which of the following is false ?
statements is true ? (A) If A–1 and B–1 are congruent,
W then so are A and B
(A) dim V = dim W + dim
V (B) If At and Bt are congruent, then
V so are A and B
(B) dim V = dim W – dim
W (C) If both A and B are congruent
V to the identity matrix, then they
(C) dim V = dim W + dim are congruent
W
V (D) If both A and B have the same
(D) dim V = dim rank, then they are congruent
W
8
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31. The partial differential equation 35. The order and degree of the
ut + ux = u2 differential equation
2
can be classified as :
dy 3 d3 y
(A) Linear 1 3 4
" dx #! dx3
(B) Non-linear is :
(C) Semilinear (A) (3, 2) (B) (3, 4)
(D) None of the above (C) (3, 3) (D) (2, 3)
32. The complete integral of the partial 36. A dairy firm has three plants P1, P2,
differential equation P3 located in a state. They daily milk
production at each plant (in
f(p, q) = 0 thousand liters) is P1 6, P2 1,
is : P3 10. Each day, the firm must
(A) z = (a) x + y + b fulfil the needs of its four
(B) z = (a) x2 + ay + b distribution centers D1, D2, D3, D4.
The minimum requirement at each
(C) z = (a) x + ay + b center (in thousand liters) is
(D) z = x2 + ay + b D1 7, D2 5, D3 3, D4 2.
33. Consider the following two Cost in hundred of rupees of
shipping one thousand liters from
statements :
each plant to each distribution center
(I) The set {x, |x|} is linearly is given below :
independent on (– , ) Distribution Center
(II) The Wronskian W(x, |x|) = 0 D1 D2 D3 D4
on (– , )
P1 2 3 11 7
Then :
Plant P2 1 0 6 1
(A) Both (I) and (II) are true
(B) Neither (I) nor (II) is true P3 5 8 15 9
(C) Only (I) is true Then the initial basic feasible by
(D) Only (II) is true North-West Corner method is :
34. Solution of the differential (A) Non-degenerate and cost
associated with this solution is
equation
` 11,600
xdy – ydx = 0, (B) degenerate and cost associated
represents a : with this solution is ` 12,000
(A) straight line (C) non-degenerate and cost
associated with this solution is
(B) hyperbola
` 12,000
(C) parabola (D) degenerate and cost associated
(D) circle with this solution is ` 11,600
9 [P.T.O.
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37. Consider the following LP 39. Consider the LP problem :
program : Max Z = 4x1 + 2x2
Subject to the constraints :
Max
– x1 – x 2 < – 3
Z = –3x1 + 2x2
– x1 + x2 > – 2 and
Subject to the constraints : x1, x2 > 0.
x1 < 3 Then the dual of the LP problem
is :
x1 – x2 < 0
Min. W = ay1 + by2
x 1 , x2 > 0
–py1 + qy2 > 4
Then the above LP problem has : –y1 – y2 > 2 and y1, y2 > 0
(A) no feasible solution where the values of a, b, p, q are :
(A) a = –3, b = –2, p = –1, q = –1
(B) feasible solution
(B) a = 3, b = 2, p = –1, q = –1
(C) unbounded solution
(C) a = –3, b = 2, p = 1, q = 1
(D) none of the above (D) a = 3, b = 2, p = –1, q = 1
38. Consider the following statements : 40. For an LP problem which of the
following statements is false ?
(I) Dual of the dual is primal.
(A) If the feasible region is a
(II) Every solution of an LP
polyhedron, then at least one
problem is a basic feasible basic feasible solution is optimal
solution. (B) If an optimal solution exists,
Then which of the following is then the feasible region is
bounded
true ?
(C) If the half spaces corresponding
(A) Only (II) is true to the constraints do not
(B) Only (I) is true intersect, the problem has no
feasible solution
(C) Both are true
(D) The set of all feasible solutions
(D) Neither (I) nor (II) is true to the problem is a convex set
10
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41. Let f : D R be defined as 43. The vector normal to the gradient
of the function f : R3 R defined
f(x, y, z) = xyz; where D = {(x, y, z) by
R3/x2 + y2 + z2 < 1}. Let P denote f(x, y, z) = xy + y2 + z
at the point P = (1, –1, 1) is :
the plane {(x, y, z)/2x + 3y + 5z = 0}.
(A) (–3, –3, 0)
Then f dx dy dz is : (B) (3, 0, 3)
D$P
(C) (–3, 0, 3)
(A) Undefined (D) (1, 0, 0)
44. Let f be a non-constant analytic
(B) 0
function on the unit disc.
(C) Then which of the following is
(D) 2 impossible ?
(A) f(G) % {z : Re(z) > 1}
1 2 (B) f(G) = {z : |z| < 2}
42. Let A = 3 6 and f : R2 R3 be (C) f(G) = {z : |z| > 1}
1 2 (D) f(G) = {z : |z| = 1}
45. Let a, b C and D be a disc in C
defined as f(x) = Ax. Then which of and f(z) = aez + be–z for all z C.
the following statements is If f(z) = 0 for all z D, then :
false ? (A) a = 0 and b = 0
(B) a = 0 and b = 1
(A) f is continuous function
(C) a = 1 and b = 0
(B) f is differentiable function
(D) a = 1 and b = 1
(C) f maps open neighbourhood of 46. Let f : C C be a non-constant
origin to an open neighbour- analytic such that |f (z)| < f(z)| for
all z C. Then :
hood
1
(D) the directional derivative of f in (A) is not analytic
f
1 (B) f is a polynomial
direction of (1, 0) is 3 (C) f (z) = 0 for some z
1 (D) f = cf for some constant c
11 [P.T.O.
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47. Let R[X] be the polynomial ring in 50. For x, y R, let d(x, y) = |x – y|.
one indeterminate over the field of Then :
real numbers. Then in the quotient (A) (–1, 1) is complete in d
ring R[X]/<X 8 + 1> number of (B) (–1, 1) is not complete in d, but
maximal ideals is :
there is a complete metric on
(A) 1 (–1, 1) inducing the same
(B) 2 topology on (–1, 1) as of d
(C) 3
(C) every metric on (–1, 1) inducing
(D) 4 the same topology as that of d
48. Let p be an odd prime number and is complete
Z" p# {a b p }| a, b Z} (D) no metric on (–1, 1) inducing
the same topology as that of d
(% C). Then the ring Z " p # is :
is complete
(A) not a Euclidean domain, but is 51. Let X = {(x, y) R2 : |x| |y| = 1}
a PID (Principal ideal domain) and Y = {(x, y) R2 : |x| + |y|
(B) Not a PID, but is an UFD = 1}. Then :
(C) Neither Euclidean domain, nor (A) X and Y are compact
a PID, but is an UFD (B) X is compact, but Y is not
(D) not an UFD compact
49. Let & % C be a region and H(&) be (C) Y is compact, but X is not
the set of all analytic functions on compact
&, then : (D) neither X nor Y is compact
(A) H(&) is an integral domain 52. Let E be a measurable subset of R
which does not satisfy ascending such that m(k) = 0 for all compact
chain condition on ideals subsets k of E. Then :
(B) H(&) is a unique factorisation (A) E is bounded
domain (B) For ' > 0 there is an open set
(C) H(&) is a finite dimensional U in R such that E % U and
vector space over C m(U) < '
(D) H(&) has only finitely many (C) E i s a G (-set
maximal ideals (D) E is an F -set
12
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53. Let G be a group of prime power 56. Let X and Y be Banach spaces and
T:X Y be a bijective continuous
order. Then :
linear map. If )) . ))1 and )) . ))2 denote
(A) G is abelian
the norms on X and Y respectively,
(B) G is simple define )) x ))3 = )) Tx ))2 and )) y ))4 =
(C) G is solvable )) x ))1, where Tx = y, for all x X
and y Y. Then :
(D) G has a Sylow subgroup which
(A) )) . ))1 and )) . ))3 are equivalent,
is abelian
but )) . ))2 and )) . ))4 are not always
54. Which of the following rings is not
equivalent
noetherian ? (B) )) . ))2 and )) . ))4 are equivalent,
(A) Q[x1, x2, x3, ........] but )) . ))1 and )) . ))3 are not always
(B) Z[x] equivalent
(C) )) . ))1 is equivalent to )) . ))3 and
(C) Z[ 2 ]
)) . ))2 is equivalent to )) . ))4
(D) R[x, y]/(x2 + y2 + 1)
(D) neither )) . )) 1 is always
55. Let K/F be a Galois extension of equivalent to )) . ))3, nor )) . ))2 is
fields. If Gal(K/F) is isomoprhic to S3, always equivalent to )) . ))4
the number of fields L such that 57. Let T : l 2 l 2 be defined by
Ten = 0 if n is odd and Ten = en /2
F % L % K is :
if n is even. Then :
(A) 6
(A) T is normal
(B) 4
(B) T is self-adjoint
(C) 10 (C) T is unitary
(D) 8 (D) T*T = I = the identity map
13 [P.T.O.
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58. Let C [0, 1] = {f : [0, 1] R : f is 60. Let X = {(x, y) R2 : x2 + y2 < 1}
differentiable on [0, 1] and f is + {(x, y) R2 : (x – 2)2 + y2 < 1}
continuous} and C[0, 1] = {f : [0, 1] and Y = {(x, y) R2 : x2 + y2 < 1}
U {(x, y) R2 : (x – 3)2 + y2 < 1}
: f is continuous on [0, 1] with
and Z = {(x, y) R2 : x2 + y2 < 1,
)) f )) = sup{|f (t)| : t [0, 1]}. Let
x 0} be subspaces of the Euclidean
D : C1[0, 1] C[0, 1] be defined as
space R2. Then :
Df = f and I : C[0, 1] C[0, 1] be
(A) X and Y are homeomorphic
x
defined as I(f)(x) = f (t) dt. Then : to Z
0
(B) X is homeomorphic to Y, but not
(A) D and I are continuous linear to Z
maps
(C) X is homeomorphic to Z, but not
(B) D is continuous, but I is not to Y
(C) I is continuous, but D is not (D) none of X or Y or Z is
(D) both D and I are not continuous homeomorphic to the other
59. Let X and Y be topological spaces two
and f : X Y be continuous. 61. Let {f : R R : f is continuous such
(1) f (A) % f (A) , (2) f (Å) % f (Å) and that f(t) = 0 for all t in a subfield
(3) f (*A) % *f(A). For every subset of R} = X. Then X contains :
A of X : (A) uncountable number of
(A) (1), (2) and (3) are true elements
(B) (1) and (2) are true, but not (3) (B) countably infinite number of
(C) (1) and (3) are true but not (2) points
(D) (1) is true and (2) and (3) are (C) finitely many elements > 2
not (D) only one element
14
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62. Consider the following statements 64. Given a group of n married women
about a lattice L : and their n husbands at least how
(i) Every ideal of L is a prime ideal.
many people be chosen from this
(ii) L is a chain.
group of 2n people to guarantee the
Then which of the following is
set contains a married couple ?
correct ?
(A) n – 1
(A) (I) implies (II) but not
conversely (B) n
(B) (II) implies (I) but not (C) n + 1
conversely (D) n + 2
(C) (I) if and only if (II) 65. Let u1 and u2 be solution of the
(D) Neither (I) implies (II) nor (II)
equation :
implies (I)
,u = 1 in D
63. Let G be a connected graph with at
least 3 vertices such that removal u = 0 on *D
of any edge of G disconnects the where D = {(x, y) R2/(x – 2)2 +
graph. Then which of the following (y – 1)2 < 1/4}
is not true ? and *D = {(x, y) R2/(x – 2)2 +
(A) G has no circuit
(y – 1) 2 = 1/4} is the boundary
(B) There is a unique path in G
of D.
between any pair of distinct
If u1(2, 1) = u2(2, 1); then :
vertices
(C) G has at least two pendant (A) u1 is a constant function
vertices (B) u2 is a constant function
(D) G has at least two spanning (C) u1 - u2 - a constant function
trees
(D) u1 - u2 in D
15 [P.T.O.
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66. Consider the function 68. The linear diophantine equation
7x 8 y 5 has :
f(x) = |x| –1 < x < 1
(A) exactly one integer solution
Then the Fourier series for f is :
(B) exactly two integer solutions
1 4 cos x cos 3 x
(A) 2
.... (C) infinitely many integer
2 12 32
solutions and the difference
1 4 sin x sin 3 x
(B) 2
.... between any two values of x in
2 12 32
the solutions is divisible by 8.
1 4 cos x cos 3 x
(C) .... (D) infinitely many integer
2
2 12 32
solutions and the difference
4 sin x sin 3 x between any two values of x in
2
....
12 32
the solutions is divisible by 7
1 4 cos x cos 3 x 69. Which of the following is a quadratic
(D) 2
....
2 12 32
residue modulo 43 ?
4 sin x sin 3 x (A) –1
2
....
12 32
(B) 40
67. The equation 3uxx – x2uy + 5u = 0 (C) 2
(D) 3
transforms under the change of
70. If µ is the Mobius function and n
variables to polar coordinates is : is a positive integer, then µ(n)
µ(n + 1) µ(n + 2) µ(n + 3) is equal
(A) a non-linear equation
to :
(B) a hyperbolic equation (A) 4
(B) 0
(C) an elliptic equation
(C) 1
(D) a parabolic equation (D) 6
16
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71. A dynamical system consists of three 73. Let the Hamiltonian of a system be
particles m1, m2, m3 in motion in p2 k2 q2
H= . If C1, C2 denote
2 2
space. The distance between m1 and
arbitrary constants the general
m2, m2 and m3, m1 and m3 remain
solution of q is :
constant throughout the motion. The (A) q(t) = C1 cos kt + C2 sin kt
motion will be governed by n Euler (B) q(t) = kt + C1
Lagrange equations. Then : (C) q(t) = C1ekt + C2e–kt
(A) n = 7 (D) q(t) = C1 cos k2t + C2 sin k2t
(B) n = 6 74. Consider a single operator equation
(C) n = 9 L = IW representing the relation
between angular momentum vector
(D) n = 3
and angular velocity :
72. A particle of mass m is moving in
(A) The operator equation represents
a two-dimensional space. Its kinetic
linear transformation
energy in terms of polar co-ordinates
(B) The vectors L and W have the
is given by :
same dimensions
1
(A) m(r 2 r 2 .2 ) (C) The vectors L and W are two
2
physically different vectors
1
(B) m(r 2 r. )
2
2 having different dimensions
1 (D) The operator I acting upon the
(C) m(r 2 r 2 .2 )
2
vector W represents orthogonal
1
(D) m(r 2 r .2 ) rotation
2
17 [P.T.O.
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75. If velocity components are time 78. In a uniform stream with complex
independent, then :
potential Uz, a two-dimensional
(A) Only stream lines and path
source of strength m is situated at
lines are coincident
the point (a, 0). The stagnation point
(B) Only stream lines and streak
lines are coincident is at :
(C) Only path lines and streak lines a
(A) z m
are coincident U
(D) Stream lines, path lines and (B) z = 0
streak lines are coincident U
(C) z a
76. Velocity potential function for a m
steady flow exists, if : m
(D) z a
(A) flow is only incompressible U
(B) flow is inviscid, incompressible 79. Let the first and second fundamental
and irrotational forms associated with a surface patch
(C) flow is inviscid, incompressible be respectively Edu2 + 2F dudv +
and rotational
Gdv2 and Ldu2 + 2M dudv + Ndv2.
(D) flow is only inviscid
E F L M
77. If = (x t) (y t) represents If S = ! and T = !,
"F G# "M N#
the velocity potential of an
then :
incompressible two-dimensional
fluid then the stream lines at time (A) S and T are non-singular
t are the curves : (B) S is non-singular and T may not
(A) (x – t)2 – (y – t)2 = constant be
(B) (x – t)2 + (y – t)2 = constant (C) T is not-singular and S may not
(C) x2 – y2 = t2 be
(D) x2 + y2 = t2 (D) S and T are singular
18
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80. Let U be an open subset of R2 and /4
82. The functional ( y2 y 2 ) dx
S be a smooth surface in R3. If 0
:U S is a regular surface patch satisfying y(0) = 0 attains its
u. u u. v
extremal on :
such that det 1
u. v v. v
(A) y = x
Then :
(B) y = 0
(A) is an isometry
(B) is conformal, but not an (C) y = sin x
isometry (D) y = cos x
(C) preserves area, but need not
1
be conformal 83. Let J( y) ( y ey xy2 ) dx be a
1
(D) preserves area and conformal
functional defined on C 1[–1, 1].
81. L et S = {(x, y, f(x, y) R3|(x, y) U}
Then the variation of the integral
where U is open in R 2 and
J[y] is :
f: U R is smooth. If f admits a
1
local maximum at p and a local
(A) ( y ey ey y 2 xyy ) dx
minimum at q, then the Gaussian 1
curvatures k1 and k2 to S at the 1
points (p, f(p)) and (q, f(q)) (B) ( y 2 ey y2 2 xyy ) dx
1
respectively satisfy :
1
(A) k1, k2 > 0 [[ y e y 2 xy] (y e y (y ] dx
(C)
1
(B) k1 > 0, k2 < 0
1
(C) k1 < 0, k2 > 0
(D) [[ y e y 2 xy y 2 ] (y e y (y ] dx
(D) k1, k2 < 0 1
19 [P.T.O.
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84. Consider the functional 86. For the Fredholm integral
b equation
J[ y] F(x, y, y / dx
b
a
u(t) k(t, s) u( s) ds
2
1 y a
where F(x, y, y / 2
y with separable kernel, Fredholm
for admissible function y(x). Which determinant D( ) 0. Then the
of the following are extremals integral equation has :
for J ?
(A) unique zero solution
(A) y(x) = A sin x where A is
(B) unique non-zero solution
constant
(C) infinitely many solutions
(B) y(x) = A sinh x + B cosh x,
(D) no solution
A, B are constants
87. Eigen value and eigen function of the
(C) y(x) = sinh (Ax + B) where
homogeneous integral equation
A, B are constants
1
(D) y(x) = A sin x + B cos x, where u(t) et s u( s) ds
0
A, B are constants
is :
85. The shortest distance between the
parabola y = x2 and the straight line 2 2
(A) 2
, u(t) et
e 1
x – y = 5 is :
2
(A) 0 (B) 2
, u(t) et
e 1
(B) 5 e2 1
(C) , u(t) et
19 2 2
(C)
8 e2 1 2
(D) , u(t) et
(D) 5 2
20
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88. The solution of the Volterra integral 91. Let y = f(x) takes the values y0, y1,
equation of first kind ........, yn, for a = x0, x1 = x0 + h, ........,
t xn = x0 + nh = b. Then the value
f (t) et s u(s) ds,
b
0
of f ( x) dx by Trapezoidal rule
where f(t) is known function, is : a
is :
(A) tf (t) f (t )
h
(B) f (t) f ( t) (A) [2( y0 y1 ) ( y2 ....... yn )]
2
(C) f (t) tf (t) h
(B) [( y yn ) 2( y1 y2 ...... yn 1 )]
2 0
(D) t f ( t) f (t )
h
(C) [2( y0 yn ) ( y1 y2 ...... yn 1 )]
89. Which of the following is not 2
correct ? h
(D) [( y y1 ) 2( y2 ...... yn )]
2 0
(A) E = 1 + ,
92. Laplace transform of J0(a t) where
(B) , = 0(1 – 0)–1
J0 denotes the Bessel function of
(C) 0 = E–1 , zero order is :
(D) 1 + , = (E + 1)0–1 1
(A)
s2 a2
90. If u1 = 1, u3 = 17, u4 = 43 and
1
u5 = 89, then value of u2 is : (B) 2
s a2
(A) 3
1
(B) 5 (C) 2
s a2
(C) 13 1
(D) 2
s a2
(D) 15
21 [P.T.O.
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93. Fourier cosine transform of 95. Consider the non-linear prog-
2
ramming problem
f (t ) e t is :
M ax. Z = 10 x1 – x22
Subject to the constraints :
1 2
(A) e 4s x1 + x2 < 14
2
– x1 + x2 < 6 and x1, x2 > 0
1 2 Then the Lagrangian function is :
(B) e 2s
2 (A) (10 x1 x22 ) r12
1 ( x1 x2 14)
2( x1 x2 r22 6)
s2
1 2
(B) (10 x1 x22 ) 1 ( x1 x2 r1 14)
(C) e
2
2( x1 x2 r2 6)
(C) (10 x1 x22 ) 1( x1 x2 r1 14)
s2
1 4 2 ( x1 x2 r2 6)
(D) e
2 (D) (10 x1 x22 ) 1 ( x1 x2 r12 14)
2 ( x1 x2 r22 6)
94. If L{f(t)} = F(s) and L{g(t)} = G(s), 96. The number of positive integers, not
then L–1{(F(s)G(s)} is : greater than 100 which are not
divisible by 2, 3 or 5, is :
t
(A) 20
(A) f ( x) g ( t x) dx
0 (B) 25
(C) 26
t
(D) 30
(B) f ( x) g( x) dx
0 97. L et k, k and ( be the vertex
connectivity, edge connectivity and
t minimum degree respectively of
(C) f ( x ) g (t x) dx
graph G. Then :
0
(A) ( < k
t (B) k < k
(D) f ( x t ) g( x t) dx
(C) k = k , whenever G is 3-regular
0
(D) ( = k , whenever G is regular
22
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98. Consider (R, µ) be Lebesgue SECTION III
101. F or a dat a-set of n observations the
measure space. Define
sample mean, median and mode are
f ( x) d2 where 63, 60 and 58 respectively. Hence the
1(A) =
A data are :
sin x 0 x 2 (A) Symmetric about sample
f ( x) median
0 otherwise
(B) Skewed to the left
Then :
(C) Skewed to the right
(A) 1 is a measure
(D) Bimodal
(B) 1 is a signed measure
102. The regression equation of Y on X
(C) 1 is both measure and signed
is given by y = 2 – 1.2x and that
measure
of X on Y is given by x = –1 – 0.3y.
(D) 1 is neither a measure nor a
Hence the correlation coefficient
signed measure
between X and Y is :
99. Let 1 be a signed measure on (X, m).
(A) + 0.36
Let P be a positive set and N be a
(B) + 0.6
negative set. Then :
(C) – 0.6
(A) P $ N is a positive set
(D) – 0.36
(B) P $ N is a negative set
103. The height of adult men in a
(C) P $ N is a null set
population follows normal
(D) P $ N is not measurable
distribution with mean 1.65 meters
100. Which of the following subsets of R
and standard deviation . If 40% of
has non-zero Lebesgue measure ?
the men are shorter than 1.5 meters,
1 the percentage of population which
(A) n :n N
n is taller than 1.80 meters is :
(B) The cantor set (A) 80%
(B) 60%
(C) { a b 2 : a, b Q}
(C) 40%
(D) The set of limit points of Q (D) Cannot be determined based on
in R the given information
23 [P.T.O.
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104. Suppose E[X] = 5. Which of the 107. Let the joint distribution of (X, Y)
following is always false ? be specified by :
(A) E[X2] < (E[X])2 X
1 2 3 4
(B) If E[X 2 ] = 25, then X = 5
1 3 1 5
wp 1 1
16 32 8 32
Y
(C) E[X2] > 25 3 1 5 3
2
(D) E[exp|X|] > exp(E[|X|]) 32 8 32 16
105. Let E and F be two disjoint events Then P[X = 2Y] is :
such that 0 < P(E) < 1 and 0 < P(F) < 1. (A) 0
Which of the following is always 9
(B)
true ? 32
11
(A) P(E|F) = P(E) (C)
32
(B) P(E|F) = P(F) 1
(D)
(C) P(E|F) P(E) 2
(D) P(EC|F) = 1 108. Which of the following matrices is
106. Suppose X and Y are independent a covariance matrix ?
r.v.s. each with mean 1. 1
1
2!
Let X
(t) denote the characteristic !
(A) 1 !
function of a random variable. Which 0!
"2 #
of the following is false ? 1 2
(A) (t) = (t) . (t) 3t (B) !
X+Y X Y "3 1#
(B) E[XY|X] = 1 2 2
( t) (t) . (t)3t (C) !
(C) X2 Y2 X2 Y2 " 2 3#
(D) (1 – X 2 ) and exp(Y) are 2 3
(D) !
independent " 3 2#
24
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109. If a random variable X has Cauchy 111. Suppose X is an absolutely
distribution with location µ and continuous non-negative real-valued
scale , then the distribution of random variable whose distribution
function is F(x) and probability
1
Y = is :
X density function is f(x). Suppose it
(A) Cauchy with location µ and is given that f(x)/[1 – F(x)] = k, a non-
1 negative constant. Then which of the
scale
following statements is true ?
2 (A) f(x) = 1, 0 < x <
(B) Cauchy with location
2 2 2
(B) f(x) = k e–kx, x > 0
and scale (C) f(x) = k e–kx + C, x > 0, C a
2 2 2
constant
(C) Cauchy with location µ and
0 x k,
scale (D) f(x) =
1 x 4 k.
(D) not Cauchy
112. Suppose Y 1 , Y 2 and Y 3 are
110. In a certain communication system,
on the average, there is 1 independent and identically
transmission error per 10 seconds. distributed p-variate random vectors
Suppose the distribution of errors is having normal distribution. Then
Poisson. The probability of at least the distribution of a1Y1 + a2Y2 +
one error in a communication of half a3Y3, where a1, a2, a3 are vectors of
a minute duration is : order 1 × p, is :
(A) 1 – 2 exp (–1) (A) univariate normal
(B) 1 – exp (–3) (B) four variate normal
(C) 1 – 4 exp (–3) (C) trivariate normal
(D) 1 – exp (–1)
(D) not normal
25 [P.T.O.
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113. Let X1, X2, ......., Xn be a random 115. Let X1, X2, ........., Xn be a random
2
sam pl e fr om N (µ, ), µ is unknown
sample from U(., . + 1). If X(1) < X(2)
and a2 > 0. Let qn, 5 be the (1 – 5)th
< .......... < X(n) are corresponding quantile of 62(n) distribution.
Consider the tests for testing
order statistics, then : 2 2
H0 : < 1 Vs. H1 : > 1 : T1 :
(A) (X(1), X(n)) is jointly sufficient Reject H0 if and only if
n
for .
8 (X i X)2 7 qn 1, 5 and T2 :
i 1
(B) X(1) is sufficient for .
Reject H0 if and only if
(C) X(n) is sufficient for . n
8 (X i X)2 7 qn, 5 then which of
(D) (X(1), X(n), + 1) is not jointly i 1
the following statements is correct ?
sufficient for .
(A) T1 is a uniformly most powerful
(UMP) level 5 test
114. Let Y(1) and Y(2) be order statistics
(B) T2 is a UMP level 5 test
based on a sample of size 2 from (C) Both T1 and T2 are level 5 tests
(D) T1 is level 5 test but T2 is not
a U(., 2.) distribution. Then
116. Let 9(.) be a parametric function and
[Y(1), Y(2)] is a confidence interval for T be an estimator of 9(.). Let V.(T)
denotes variance of T. The Cramer-
. with confidence coefficient :
Rao bound gives a :
1 (A) Lower bound for V.(T) for any
(A)
2 arbitrary estimator T of 9(.)
(B) Upper bound for V.(T) for any
3
(B) arbitrary estimator T of 9(.)
4
(C) Lower bound for V.(T) among
1 all unbiased estimators T of 9(.)
(C)
4
(D) Upper bound for V.(T) among
(D) 1 all unbiased estimators T of 9(.)
26
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117. Let {X1, ......, Xn} be a random sample 119. In the method of constructing
confidence interval for an unknown
2
from N(µ1, ) and {Y1, ......, Ym} be parameter based on pivotal quantity
2 Q, which of the following statements
a random sample from N(µ2, ). Let
is correct ?
n m n
T1 8 X 2i 8 Yi2 , T2 8 Xi, (A) The distribution of Q is known
i 1 j 1 i 1 and depends on the unknown
m n m parameters
T3 8 Yi , T4 8 X 2i , T5 8 Yi2 . (B) The distribution of Q is known
i 1 i 1 i 1
and free from the unknown
Then which of the following is a
parameters
jointly sufficient complete statistic
(C) The distribution of Q is not
for (µ1, µ2, 2) : known but is free from the
(A) T2 , T3 , T4 , T5 unknown parameters
(B) T3 , T4 , T5 (D) The distribution of Q is known
(C) T2 , T3 , T1 and normal
(D) T2 , T3 , T5 120. To test H0 : p = p0 against the
alternative H1 : p p0, where p is
118. Suppose {X1, X2, ......., Xm} is a random
the population proportion and p0 is
sample from the distribution of X with a specified value of p. Suppose Pn
mean µ1 and variance 12 , {y1, ......, denotes the sample proportion based
yn } is a random sample from the on a random sample of size n from
distribution of Y with mean µ2 and the population. The test statistic for
variance 22 . The test statistic to test testing H 0 : p = p 0 against
H0 : 12 2 against H1 : p p0 is given by :
2
2 2 is T
S2X n (Pn p0 )
H1 : 1 2 n , where S2X (A)
S2Y p0 (1 p0 )
and S2Y are sample variances. Then
the null distribution of Tn is : (B) n (Pn p0 )
(A) F if X and Y are independent
and have normal distributions Pn p0
(B) F if X and Y have normal (C) p0 (1 p0 )
distributions
(C) F if X and Y are independent Pn p0
(D) F if X and Y have normal (D)
distributions with equal mean p0 (1 p0 )
27 [P.T.O.
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121. To test the equality of r indepen- 123. There are four jobs J1, J2, J3, J4 that
need to be processed on two
dent multinomial distributions,
machines M1 and M2 in sequence.
multinomial ( k, pi1 , pik ) i = 1 .... r The processing times are given in
the null hypothesis is H0 : Pij = Pj the following table :
3 j = 1 ...... k where pj are unknown J1 J2 J3 J4
2
j = 1 ....... k, a 6 test statistic is used. M1 6 10 4 7
2
The degrees of freedom of the 6 test M2 4 8 9 2
are : The total processing time to complete
(A) k(r – 1) all the jobs on both the machines
is :
(B) r(k – 1)
(A) 30
(C) rk – 1 (B) 32
(D) (r – 1) (k – 1) (C) 33
(D) 35
122. X and Y are two continuous random 124. Re-order level for an item is
variables independent of each other. always :
The combined sample based on two (A) Less than minimum stock
(B) More than minimum stock
independent samples of size 5 and
(C) Same as the minimum stock
4 on X and Y respectively has the (D) More than the average stock
following pattern : 125. Consider the following two M/M/1
queueing systems where :
y4 < x1 < x5 < y3 < x2 < y1 < y2 <
In Q1 : arrival rate , service rate µ
x4 < x3 In Q2 : arrival rate 2 , service rate 2µ
Hence the value of Mann-Whitney- Let W 1 and W 2 be the average
waiting times of an arriving customer
Wilcoxon test statistic is : in Q1 and Q2 respectively. Then :
(A) 6 (A) W1 < W2
(B) W1 = W2
(B) 8
(C) W1 > W2
(C) 10 (D) The relation between W 1
(D) 12 and W2 cannot be decided
28
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2 be the variance of the usual
127. Let VSY
126. Consider a population of N = nk
units with k > 1. A sample of size estimator in a systematic sampling
2
n is obtained by the following design and VSRS the variance of the
method : sample mean in a SRSWOR design.
It is known that the population
(i) A unit is drawn from the set of
has a linear trend, i.e., y i is
units labelled {k, 2k, ......., N}
approximately equal to ic, for a
with probability 1/n and is
positive c, i = 1, 2, ......, N. Then :
included in the sample. 2 2
(A) VSY VSRS
(ii) A SRSWOR of size n – 1 is then
2
(B) VSY 2
VSRS
obtained from the remaining
N – 1 units in the population
2
(C) VSY 7 VSRS
2
and these units are included in (D) It is not possible to compare the
the sample. two estimators based on the
given information
Then,
128. Under one way fixed effects ANOVA
(A) The sampling method
which of the following parametric
corresponds to a systematic
functions are estimable, where the
sample
model is :
(B) There is no unbiased estimator
i 1 ......... p
yij 2 5i ij j 1 ........ ni ,
of the population total
(C) Inclusion probability is not the ij
are independent identically
same for all the population distributed with zero mean and
finite variance.
units
(A) 2 51 52
(D) The sample mean is an
(B) 51 53
unbiased estimator of the
(C) 251 52
population mean
(D) 51 5p 253
29 [P.T.O.
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129. Under a randomized block design 130. Under an unconfounded 23 factorial
with V treatments and r replication experiment with factors, A, B, C
which of the following is most in two replicates arranged in a
correct ? randomized block design which of
the following statements is not
(A) All pairwise contrasts 5 i 5j
correct ?
among the unknown treatment
(A) All factorial effects are
effects 5. i = 1 ...... V are
estimated with same precision
estimable and are estimated
(B) Best linear unbiased estimators
with common variance
of the 7 factorial effects are
(B) All contrasts of the type
V V V
mutually uncorrelated with
8 C i 5 i , 8 Ci 0, 8 Ci2 1 are
each other
i 1 i 1 i 1
estimable and are estimated (C) The error sum of squares has
with common variance an additional degree of freedom
(C) The best linear unbiased than that of the treatment sum
estimators 5ˆ i 5ˆ j 1 i j V of squares
are all uncorrelated with each (D) Total number of degrees of
other freedom equal twice those
(D) All the above (A), (B), (C) are carried by the treatment sum of
correct squares
30
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131. Let B be the -field of subsets of R 133. Let R be the set of real numbers,
generated by {(a, b)|a, b R}. B the Borel -field. Let ‘m’ denotes
Which of the following is not
the Lebesgue measure and 2 the
true ?
measure defined by the function
(A) Every countable subset of
F(x) = (1 – e–x) if x > 0
R B
and F(x) = 0 elsewhere.
(B) All closed intervals B
Consider the statements :
(C) All uncountable subsets of
(I) µ << m, (i.e. µ is absolutely
R B
continuous w.r.t. m)
(D) Intervals of the form [a, b) B
132. Consider the following set function (II) m < < µ
d2 d2
2 on a -field : (III) ( x) e x x > 0, ( x) 0,
dm dm
2(A) = 0 if A is empty
x 0
= 1 if A is non-empty and finite
Which of the above statements are
= if A is infinite
true ?
Then :
(A) (I) and (II) only
(A) 2 is not monotone
(B) (I) and (III) only
(B) 2 is not -additive
(C) (II) and (III) only
(C) 2 is finite
(D) 2 is a measure in : (D) All the three
31 [P.T.O.
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135. Let {Xn, n > 1} be a sequence of i.i.d.
134. Let {Xn, n > 1} be a sequence of random variables with mean 3 and
variance 11. Which of the following
independent identically distributed is not always correct ? As n .
1 n 2
random variables with mean 0 and (A) 8X
nk1 k
converges in
probability to 2
2
finite variance > 0 : 2
1 n
(B) 8X
nk1 k
converges in
X1 X2 ....... X 2
n probability to 9
Let Sn
n2
1 n
(C) 8 (X 3)2 converges in
nk1 k
Then, which of the following is
probability to ‘0’
2
false ? n Xk
(D) 8 n
converges in
k 1
S
(A) P lim Sn e n 0
!#
1 probability to ‘0’
"n
136. Let {Xn, n > 1} be a sequence of
Sn
(B) As n , Sn e converges to
random variables such that
a normal random variable with P lim X n X 1 for some r.v. X.
"n !#
2 Then which of the following is not
mean ‘0’ and variance
always correct ?
(C) As n , Sn converges to ‘0’ (A) nlim E " cos(X n ) # E[cos(X)]
in probability (B) nlim E " exp(it X n ) #
E[exp(it X)]
(D) As n , eSn converges to 1
(C) P lim exp(X n ) exp(X) ! 1
"n #
with probability 1
(D) nlim E[exp (X n )] E[exp (X)]
32
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138. Let {X n } be a sequence of
137. Let {X n } be a sequence of
independent random variables such
1
independent and uniformly bounded that E[Xn] = 0 and Var(Xn) = ,
n2
random variables, and E[Xn] = 0, n = 1, 2, ........ Let E = ; |8 X n (w)
n
1
and Var(Xn) = for each n.
(2n) converges .
Then which of the following is
Let S n = X 1 + ....... + X n and
always correct ?
2
Var(Sn ) (A) 0 < P(E) < 1
n
1
(B) P(E) =
Then which of the following is not 2
(C) P(E) = 0
true ? (D) P(E) = 1
139. Let Y1, Y2 ......... be a sequence of
Sn independent random variables with
(A) does not converge in
n mean zero and let Fn = (Y1, ..., Yn).
n
distribution Define X n 8 Yk ; W1 = Y1 and
k 1
Sn W n+1 = Wn + Y n+1 Wn2 and
(B) converges in distribution to
n n
Tn = 8 (Yk 1) . Then :
a normal random variable k 1
(A) Both {X n } and {W n } are
n Fn-martingales
1
(C) nlim 8 3
E[|X k|3 ] = 0
(B) Only {Xn} is a Fn-martingale
k 1 n
(C) Both {X n } and {T n } are
Sn Fn-martingales
(D) nlim 2
0 in probability
(D) All the three sequences are
n
Fn-martingales
33 [P.T.O.
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140. Suppose a distribution function 142. Suppose X and Y are two random
F:R [0, 1] of a random variable variables whose third order
X is as follows :
moments exist. Then (E(|X + Y|3))1/3
0, if x 2, is less than or equal to :
1/3, if 2 x 0,
(A) (E(|X|3))1/3 (E(|Y|3))1/3
F( x) 1/2, if 0 x 5,
1 / 2 (x 5)2 / 2, if 5 x 6, (B) (E(|X|3))1/3 + (E(|Y|3))1/3
1, if x 4 6. (C) (E(|X|) + E(|Y|))1/3
Then E(X) is : (D) (E|X| . E|Y|)1/3
11 143. Suppose X is an absolutely
(A)
6
continuous random variable with
1
(B)
6 probability density function f(.) and
125 characteristic function (.). Suppose
(C)
6
(.) is a Riemann integrable on R.
13
(D)
6 Then :
141. Which of the following statements 1
(A) f ( x) eiux (u) du
is correct ? 2 R
The set of points of continuities of f ( x) e iux (u)du
(B)
a distribution function is : R
(A) uncountable 1
(C) f ( x) e iux (u) du
2 R
(B) countable
(C) finite 1
(D) f ( x) e iux (u)du
2 0
(D) empty
34
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144. It is given that the distribution of 146. Let f ( x|.) e ( x .) x 4 . and let the
Y given X = x is Normal (x, x2). prior for . be given by (.) = e–.
0 < . < . Then :
Suppose X ~ U(0, 1). Then, which
(A) the posterior distribution for .
of the following statements is not
does not exist
true ?
(B) the posterior distribution for .
(A) E(Y) = 1/2
is U(0, x).
(B) Var(Y) = 5/12
(C) the posterior distribution for .
(C) Cov(X, Y) = 3/12
is exponential distribution
(D) Corr(X, Y) = 1 (D) the posterior distribution for .
145. Let X (t) be the characteristic exists but its mean is not finite
function of X ~ N 3 (2, <). Then 147. Suppose X1, ......., Xn are independent
EX 1 X 22 X 3 is given by : random variables with probability
density function
( 1)4 *4 X t
(A) 1
*t1*t22*t3 , if i. xi i.
t 0
f ( xi , .) 2i.
otherwise.
3 3 0,
( 1) * X t
(B)
*t1*t2*t3 Then the minimal sufficient statistic
t1 1, t2 2, t3 1
for . is :
( 1)3 *4 X t
(C) (A) {X(1), ....... X(n)}
*t1*t22*t3
t1 1, t2 2, t3 1 (B) max{|X1|, |X2|/2, ..... |Xn|/n}
*4 X t (C) max{|X1, X2/2, X3/3, ..... Xn/n}
(D)
*t1*t22*t3 (D) min{|X1, X2/2, X3/3, ..... Xn/n}
t1 1, t2 2, t3 1
35 [P.T.O.
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148. Su ppose {X 1, X2, ......., Xn} is a random 150. Suppose {X1, X2, ......., Xn} is a random
2
sample of size n from the N(µ, )
2 sample Poisson distribution with
distribution, µ R, > 0. Suppose
ˆ 2n and Tn denote the maximum m ean .. Then the uniformly most
likelihood estimator and the
powerful test for testing H0 : . = 1
2
UMVUE of respectivley. Which
against H1 : . > 1 :
of the following statements is
correct ? (A) rejects H0 if X < c1 or X > c2
(A) ˆ 2n has smaller variance than
for some c1, c2
that of Tn
(B) rejects H0 if X < c for some c
(B) ˆ 2n has smaller MSE than that
of Tn (C) rejects H0 if X > c for some c
(C) ˆ 2n has larger variance than (D) rejects H0 if c1 < X < c2 for some
that of Tn
c1, c2
(D) ˆ 2n and Tn have the same MSE
151. Suppose {X1, X2, ......, Xn} is a random
149. Suppose f(x, .) is a probability
density function of X for which sample of size n from the U(0, .)
differentiation under integral sign is distribution. Then :
permissible.
(A) Sample mean X n is consistent
d
Then E log f (X, .) is :
d. for .
(A) equal to the Fisher information (B) Sample median is cosistent
(B) less than zero for .
(C) less than 1
(C) 2 X n is consistent for .
(D) equal to zero
(D) X(1) is consistent for .
36
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152. Suppose {X 1 , X 2 , ......., X n } are 154. Let X1, X2, ......., Xn be iid U(0, .)
independent and identically
distributed random variables each Define T1 = X and T2 = (X(1) +
having the probability density X(n))/2. Where X(1) = min(X1, ... Xn),
function :
. X(n) = max (X1 .... Xn). Then :
f ( x) , x 4 .
x2 (A) both T 1 and T 2 are equally
Then :
efficient when n = 2
n 4.
(A) n X ! 1 L (B) ARE(T1, T2) > 1 when n > 2
"4# 3
z ~ N(0, 16 . /27) (C) ARE(T1, T2) < 1 for all n
n 4. (D) ARE(T1, T2) 0 as n
(B) n X ! 1 L
"4# 3
155. For a sample correlation coefficient
z ~ N(0, ./3)
r, it is known that :
n 4.
(C) n X ! 1 L n r = d N(0, (1 =2 )2 ),
"4# 3
z ~ N(0, 16 .2/27) where = is the population correlation
n 4. coefficient. For what transformation
(D) n X ! 1 L
"4# 3 g(.),
z ~ N(0, 16 ./27)
153. Let X1, X2, ........, Xn be iid f(x, ., 5), n ( g(r) g(=)) d N(0, 1) ?
where :
1 =
e x / . x5 1 (A) log 1 =
f ( x; ., 5) .
(5) .5
1 1 =
Then, n (log X log 5.) d z, where (B) log
2 1 =
z is distributed as :
1
(A) N(0, 1) (C) log(1 =)
2
(B) N(0, 1/5)
1 1 =
(C) N(0, 52/.2) (D) 2 log 1 =
(D) N(0, .2/52)
37 [P.T.O.
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156. Let X ~ N4 (2, <) where eigen values 158. Let X ~ N(0, <) where < is a +ve
of < are given by 1
= 6, 2
= 4, definite matrix, and A is an
3
= 3, 4
= 2. Let Yi be the principal idempotent matrix of rank k < P. Let
component corresponding to ,
i Z = X 1 < –1 AX. Then mean and
i = 1, 2, 3, 4. Then which of
variance of z respectively equal :
the following statements is not
(A) n, 2n
correct ?
(A) Yi, i = 1, ...., 4 are mutually (B) k, 2k
independently distributed (C) k, k
(B) Y1, Y2, Y3, Y4 are identically (D) n, n
normally distributed 159. Let X ~ N3 (2, <) with
(C) The percentage of variation
1 = =2
explained by Y 1 and Y 2 is !
<= = 0! 1
greater than 60% !
"= 0 1 #
2
(D) Y4 explains the smallest portion Then the conditional distribution of
of variance (X1, X2) given X3 = x3 is N2(2*, <*)
where 2* and <* are respectively
157. Let X ~ Nn (2, <) where < is a +ve
given by :
definite matrix. Then which of the
21 1 =
following is a necessary condition for (A) !, !
" 22 # " = 1#
Y = AX to have independently
21 =2 ( x3 23 ) 1 =
distributed marginal distributions ? (B) !, !
" 22 !# " = 1 #
(A) A is any non-singular matrix
21 =2 ( x3 23 ) 1 =4 =
(B) A is an idempotent matrix (C) !, !
" 22 !# " = 1 !#
(C) A is an orthogonal matrix
21 1 =4 =
(D) Columns of A form orthonormal !, !
(D)
eigen vectors of < " 22 # " = 1 !#
38
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161. y1, y2, y3 are three uncorrelated
2
160. Let X ~ N n (0, In ) . random variables with common
2
v ar i an ce and E(y1) = .1 + .3
n E(y2) = .2 + .3, E(y3) = .1 – .3.
Let Y 8 X 2i / 2 and Qi = X A i X Hence :
i 1
(A) Best Linear Unbiased Estimator
with rank (Ai) = ri, i = 1, 2, ....., k. Let
(BLUE) of . 2 is y 2 with
k 2
variance
Y = 8 Q i . Then Q1, Q2, ...., Qk are ( y1 y3 )
i 1
(B) BLUE of . 2 is y2
2
independently distributed if and only 3 2
with variance
2
if : y1 y3
(C) BLUE of .1 is with
2
variance 2
k
y1 y2
(A) ri > 0 3 i and 8 ri n (D) BLUE of .1 is with
i 1
2
2
3
variance
k 2
(B) ri > 0 3 i and > ri n 162. In a simple linear regression
i 1
analysis the fitted line using least
squares theory will :
(C) A 1 , ....., A n are all positive (A) definitely pass through the
point ( x y ) when the intercept
semidefinite matrices and is present
n (B) definitely pass through the
8 ri n
point ( x y )
i 1
(C) never pass through the point
( x y)
(D) Each Q1, Q 2, ....., Qk have 6 2
(D) definitely pass through
n
the point ( x y ) if slope is
distribution and 8 ri n
i 1 positive
39 [P.T.O.
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163. In a multiple linear regression 165. Consider a two-way classification
model yij 2 9i ?j 'ij i = 1 ... v,
set up y XB E under the
j = 1, ........, b with E(' ij ) = 0,
2
assumption E( E ) = 0, Var( E ) = . Var('ij) = 2
, 'ij uncorrelated where
I n t h e r espon se var i abl es y1 .... yn 9i ( = 1 ..... v are fixed effects and
?j j = 1, ....., b are random effects
are : 2
with mean 0 and variance ? . Hence
(A) identical and independent the hypothesis regarding the
insignificance of random effects
(B) non-identical uncorrelated
is :
(C) non-identical and independent
(A) H0 : ?1 = ?2 = ....... = ?b
2
(D) identical and uncorrelated (B) H0 : ? 0
164. Consider a multiple linear (C) H0 : 91 = 92 = ....... = 9v
2
(D) H0 : = 0
regression model y XB E and
n 1
166. A systematic sample of size 2 is
suppose ˆyi and ei i = 1, ...., n are drawn from a finite population with
the fitted values of response variable N = 4 and k = 2. The samples are
drawn independently of each other.
y and residuals respectively. Then :
The probability that the two units
n
(A) 8 ei 0 (2, 4) are included in sample
i 1 equals :
n
(A) 0
(B) 8 ei2 n 2
i 1 1
(B)
n 2
(C) 8 ei yi 0
i 1 (C) 1
n 1
(D) 8 ei yiˆ 0 (D)
4
i 1
40
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167. The variance of the usual estimator 169. The incidence matrix of a general
in a stratified sampling design is block design is :
given by :
1 1 1
L Wh2S2h !
V2 8 nh
(1 fh )
N=
1 0 0!
0!
h 1
0 1
in the standard notations. We wish !
to obtain an allocation that "0 1 1 !#
minimizes the sample size. Then,
Hence the number of estimable
(A) such an optimal allocation does
not exist orthogonal block contrasts is :
(B) proportional allocation is (A) 2
optimal
(B) 1
(C) the optimal allocation is the one
for which nh is proportional to Sh (C) 3
(D) the optimal allocation is the one (D) 4
for which nh is proportional to 170. Con si der a BIBD with parameter
WhSh
(v, b, r, k, ). If v = b = 11 and
168. Let us consider the two sampling
strategies, both based on a sample r = 5, then :
of size n : I. SRSWOR II (A) k = 5, = 2
post-stratification. To estimate the
population total : (B) k = 5, = 1
(A) If the stratum sizes are (C) k = 2, = 5
sufficiently large for each (D) k = 11, = 1
stratum and if the error effects
in Wh can be ignored, post- 171. In a M/M/K queueing system the
stratification is better than departure rate 2n when there are
SRSWOR
n customers in the system is :
(B) SRSWOR is always better than
post-stratification (A) 2
(C) If n is sufficiently large, post-
(B) n2
stratification is better than
SRSWOR (C) n2 if n k and 0 if n > k
(D) Post-stratification is always
better than SRSWOR (D) n2 if n k and nk if n > k
41 [P.T.O.
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172. In a 24 factorial design with two
174. L et {X i, i = 1, 2, ........ 10} be a time
blocks of 8 plots each the treatment
combinations are allotted as series with mean zero, variance one
below :
1
Block I Block II and equicorrelation . Let
a 1 2
b ab 1 10
ac c
X 10 8 X , then variance of
10 i 1 i
bc abc
X 10 is :
d ad
(A) 0.1
abd bd
cd acd (B) 0.325
abcd bcd (C) 0.2
Hence the confounded treatment (D) 0.125
combination is :
175. Let {Xt} be a stationary time series
(A) ABD
(B) ACD model given by :
(C) BCD Xt = µ + Xt–1 + Zt, Zt ~ iid normal
(D) ABCD 2
(0, ). Then, which of the following
173. Let {Xt} be a stationary AR(1) time statements is not true if PX (X t k ) t
series given by the recursive
is the forecast of X t+k . Given
equation X t X t 1 Z t , where
2 X 1 , X 2 ..... X t ?
Z t ~ W N (0, ). Let P 3 X 4
represesents the forecast of X4 given (A) The forecast mean square error
the observations (X1, X2, X3). If C( ) of PX (X t k ) approaches to
t
is defined as : 2 2 2
µ + /(1 – )
2
E(X 4 P3 X 4 ) (B) The forecast mean square
C( ) ,
E(X 4 E(X 4 ))2
error of PX (X t k ) approaches to
t
then, which of the following is not 2
/(1 – 2
)
true ?
(C) PX t (X t k ) E(Xt) as k
(A) C(0) = 1
(B) 0 < C ( ) < 1 (D) The forecast PX t (X t k ) is a
(C) C( ) > 0 when > 0 weighted average of E(Xt) and
(D) C( ) < 0 when > 0 the latest observation Xt
42
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176. Let {X n }0 be a Markov chain with
178. Consider a Branching process with
1
P[X0 = 0] = P[X0 = 1] = and
4 µ as the mean of the off-spring
1
P[X0 = 2] = . The t.p.m. is given 1
2 distribution. Let 2 and X0 = 1.
2
by :
Then E 8 Xn equals :
0 1 2 0
1 1 1 1
0 (A) 1 (B) 1
2 3 6 2
1 2
p 1 0
3 3 (C) 2 (D)
2 1 0
1
2 2 179. Consider a Markov chain on
Then P[X1 = 1] is given by : S = {0, 1, 2, ......} with p01 = 1,
1 1 1
(A) (B) 3
3 4 pi0 = , pi(t) = 3 i = 1, 2 ........
4 4
1 1
(C) (D) then :
5 6
177. Let {X(t), t > 0} be a time (A) all the states are persistent non-
homogeneous Poisson process with
rate . Let X(T) = n > 0 and let null
T1 < .... < Tn < T be times at which
(B) all the states are persistent
these n events occur. Then,
E(T1|X(T) = n) is given by : (C) all the states are transient
T
(A) (B) T. n
n 1 (D) all the states do not
n 1
(C) T (D) communicate with each other
n 1 2T
43 [P.T.O.
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180. The denominator in infant mortality 182. For a production process, the sample
rate is : ranges are found to be 1.2, 1.5, 1.1,
(A) total number of live births in a 1.4 and 1.5. Suppose that the
given period subgroup size is 5. Then, what
(B) total number of live births and will be the process standard
still births in a given period
deviation, given that A2 = 0.577 and
(C) total number of pregnancies in
d2 = 2.326 ?
a given period
(A) 0.511
(D) total population in a given
(B) 2.463
period
(C) 2.322
181. In terms of life table functions, the
(D) 0.576
chance that a child just born to a
mother of age 31 and a father of age 183. The respective failure rate functions
33 will be alive till 40 years but will of two components are r1(t) t
be orphaned by both parents is : and r2 (t) 1 t . Then the life time
l40 l71 l31 l73 l33 distribution of a series system with
(A) l0 l0 l0
the above components is :
(B) p40q31.q33
(A) IFR
L 40 L71 L31 L73 L33
(C) (B) DFR
l0 l0 l0
(C) Neither IFR nor DFR
l40 l71 l31 l73 l33
(D) l0 l31 l33 (D) Both IFR and DFR
44
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184. In the inventory model S = Q – u.t 186. Let Dn be the waiting time of the
nth customer in G|M|1 queue.
with Q = 1000, u = 40 and t = 10,
Consider the following recurrence
the current stock level is :
equation :
(A) 400
Dn Vn U n if D n Vn Un 4 0
Dn 1 .
(B) 1200
0 otherwise
(C) 1000
Which of the following interpre-
(D) 600
tations of the variables in the above
185. Which of the following is an example
equation are valid ?
of the control of the queue
(i) U n is the interarrival time
discipline ?
between nth and (n + 1)th
(A) Customer is allowed to leave the
customer
facility
(ii) Dn + Vn is the departure time
(B) The number of servers are
of the nth customer from the
varied depending on the queue
system
length
(A) Both are not valid
(C) Separate server is allocated for
special category customers (B) Both are valid
(D) The mean arrival rate is (C) Only (i) is valid
adjusted (D) Only (ii) is valid
45 [P.T.O.
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JUN - 30220/II—A
187. Suppose that R is the shortest route 189. Which of the following is not
from city 1 to city 10 passing true regarding the Karmarkar’s
through some other cities. The
algorithm ?
principle of optimality implies
(A) The algorithm starts with an
that :
interior point of the feasible
(i) For every city j on R, the
region
subpath in R from city j to city
10 must be the shortest. (B) One of the basic conditions
(ii) For every city j on R, there must 1 1 1
is that x , , .....,
n n n
be a unique path from city j to
city 10. satisfies the equation Ax = 0.
Which of the above statements is (C) Constraints are assumed to be
correct ? non-homogeneous
(A) (i) only (D) The algorithm reduces the
(B) (ii) only number of iterations
(C) Both (i) and (ii)
190. What is the failure rate function of
(D) Neither (i) nor (ii)
the series system of two independent
188. By the principle of optimality, the
components when the failure rate
minimum value of y12 y22 s.t.
functions of the two components are
y1 y2 c, y1 4 0, y2 4 0 is :
respectively r1(t) and r2(t) ?
(A) 2c2
(A) min(r1(t), r2(t))
(B) 2c
2 (B) r1(t) + r2(t)
c
(C)
2 (C) r1(t) × r2(t)
(D) c2/2 (D) max(r1(t), r2(t))
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 - 27th December 2020
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 A 121 D
2 C 42 C 82 C 122 B
3 D 43 B 83 C 123 A
4 A 44 D 84 C 124 B
5 A 45 A 85 C 125 C
6 B 46 D 86 A 126 C
7 C 47 D 87 B 127 B
8 C 48 D 88 B 128 D
9 B 49 A 89 D 129 D
10 B 50 B 90 B 130 D
11 B 51 C 91 B 131 C
12 A 52 B 92 B 132 D
13 B 53 C 93 D 133 B
14 A 54 A 94 A 134 B
15 C 55 A 95 A 135 b
16 C 56 C 96 C 136 D
17 A 57 9 97 C 137 A
18 A 58 C 98 B 138 D
19 D 59 D 99 C 139 A
20 D 60 A 100 D 140 D
21 A 61 D 101 C 141 A
22 D 62 C 102 C 142 B
23 C 63 D 103 C 143 C
24 D 64 C 104 A 144 D
25 D 65 D 105 C 145 A
26 C 66 A 106 B 146 A
27 C 67 D 107 B 147 B
28 C 68 C 108 C 148 B
29 A 69 B 109 B 149 D
30 D 70 B 110 B 150 C
31 C 71 B 111 B 151 C
32 C 72 C 112 A 152 C
33 A 73 A 113 A 153 B
34 A 74 C 114 A 154 A
35 C 75 D 115 D 155 D
36 D 76 B 116 C 156 B
37 h 77 A 117 C 157 D
38 B 78 D 118 A 158 B
39 C 79 B 119 B 159 C
40 B 80 C 120 A 160 C
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 - 27th December 2020
FINAL ANSWER KEY FOR PAPER -II (Booklet Code : A)
SUBJECT CODE & NAME: (30) Mathematical Sciences
QueNo Answer
161 B
162 A
163 B
164 c
165 B
166 B
167 B
168 A
169 A
170 A
171 9
172 A
173 C
174 B
175 A
176 D
177 A
178 C
179 A
180 A
181 D
182 D
183 C
184 D
185 C
186 B
187 A
188 B
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.