MAHA SET 2018 Question Paper 3 Mathematical Science

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A
MATHEMATICAL SCIENCE
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2

Mathematical Science
Paper III
Time Allowed : 2½ Hours] [Maximum Marks : 150
Note : Attempt all questions either from Sections I & II or from Sections
I & III only. The OMR sheets with questions attempted from both
the Sections viz. II & III, will not be assessed.
Section I : Q. Nos. 1 to 5, Section II : Q. Nos. 6 to 75,
Section III : Q. Nos. 76 to 145.

Section I
2. Let f(x, y) = x2 + 5xy2, then the

1. Let
directional derivative of f at the
x2 y
f(x, y) = , (x, y) (0, 0)
2 2
x y
point (–2, 1) in the direction of vector
= 0, (x, y) = (0, 0).

v = (12, 5) is :
Then :

f f
(A) and exist at (0, 0) (A) 88/13
x y

(B) f(x, y) is continuous at (0, 0)
(B) –88/13

(C) f(x, y) is differentiable at (0, 0)
(C) 78/7
(D) f(x, y) is continuously differ-

(D) –88
entiable at (0, 0)

3 [P.T.O.

4. Let R 4[t] be the vector space of
3. Let
polynomials with degree 4. Find

the matrix of the linear operator
f(x) = x, if x is rational L(f(t)) = (f(t) – f(0))/t with respect to
the standard basis {1, t, t2, t3, t4}.

= 0, if x is irrational 0 0 0 0 0
1 0 0 0 0
(A) 0 1 0 0 0
then : 0 0 1 0 0
0 0 0 1 0

(A) f is continuous on R
0 1 0 0 0
0 0 1 0 0
(B) 0 0 0 1 0
(B) f is continuous on R except at
0 0 0 0 1
0 0 0 0 0
origin
1 0 0 0 0
0 0 1 0 0
(C) f is discontinuous at all points (C) 0 1 0 0 0
0 0 0 1 0
0 0 0 0 1
except at origin

0 0 0 1 0
(D) f is discontinuous at all points 0 0 1 0 0
(D) 0 0 0 0 1
0 1 0 0 0
of R 1 0 0 0 0

4

Section II
5. Suppose that the columns of an
6. Suppose f ( x , y ) is of class c 2 ,

then under the coordinate trans-
n × n -matrix M over R are formation x = r cos , y = r sin , the
2 2
f f
expression 2 transforms to :
2
x y
orthonormal. Then which of the
2 2
f f
(A) 2 2
r
following is not true ?
2 2
f 1 f
(B)
r2 r2 2

2 2
(A) For every x Rn, Mx = x f f 1 f
(C) 2 2 2
r r r

2 2
f 1 f 1 f
(D) 2 2 2
(B) For every x, y Rn, r r r r

7. For the function f(x, y) = (x – 1)

2
<Mx, My> = <x, y> (x2 – y2), the point ,0 :
3

(A) is not a critical point

(C) The rows of M are orthonormal
(B) is a local maximum

(C) is a local minimum
(D) M is symmetric
(D) is a saddle point

5 [P.T.O.

b
dx 10. Let R be a Boolean ring with
8. The improper integral n
a
x a
identity having n elements, then
converges :
which of the following values of n
(A) n
is possible ?
(B) if n > 1
(A) n = 32
(C) if n is an integer

(D) only if 0 < n < 1 (B) n = 23

9. The functions : (C) n = 27

f1(x, y, z) = x + y + z
(D) n = 15
f2(x, y, z) = xy + yz + zx
11. Let R be a ring with identity 1 ( 0)
f3(x, y, z) = x2 + y2 + z2
and m be the characteristic of R,
(A) are functionally independent

on R3/{x-axis} then which of the following is

(B) are functionally independent on true ?
the set {(x, y, z) R3/x > 0}
(A) wherever R is infinite, m = 0
(C) are functionally independent on
the set {(x, y, z) R3/x > 0, (B) wherever m 0, R is finite

y > 0, z > 0}
(C) wherever R is finite, m 0
(D) are functionally dependent

on R3 (D) wherever m 0, R is infinite

6

12. Let F be a field with at least two 14. Which of the following statements

elements. Then which of the is false for Q 2 and Z 2 ?

following statements need not be
(A) There are infinitely many units
true ?
in Z 2
(A) Either Z or Zp for some integer

p 2 is embedded in F (B) Q 2 is not a subfield of

complex numbers
(B) If n is not prime, then Zn is not

embedded in F (C) Q 2 is isomorphic to

(C) If Z is embedded in F, then the Q x x2 2

field Q is also embedded in F
(D) There are infinitely many
(D) If F is infinite, then Zp cannot
primes in Z 2
be embedded in F for any

prime p 15. The radius of convergence of the

13. Which of the following rings is a zn
power series is :
n 0 n2 1 2n
Unique Factorization domain but

not a PID ? 1
(A)
2
(A) Z
(B) 1
(B) F[x] (F a field)

(C) Z[x] (C)

(D) Z[i] (D) 2

7 [P.T.O.

16. For which of the following functions 18. The bilinear transformation :
does the function take complex
3z 4 6
values arbitrarily close to any f(z) = 5 z 6 from C – to C
5
complex number, inside any
assumes all complex values, except :
arbitrary neighbourhood of 0 ?

5
(A) f(z) = ez (A)
3

1 4
(B) f(z) = sin (B)
z 3

6
1 (C)
(C) f(z) = z3 + z2 + 5
z
3
(D) f(z) = cosh z + sinh z 2 (D)
5

17. Which of the following continuous 19. Which of the following complex

functions is not an open mapping functions has an infinite number of

from C to C ? poles ?

2 (A) ez
(A) f(z) = ez

(B) sec z
(B) f(z) = |z|
1
(C) f(z) = sin z + cos z 2 (C) cos
z

1
z2 z4 z6 (D)
(D) f(z) = 1 ....... z5 z 1
2! 4! 6!
8

20. Let 21. Which of the following subsets of R2

are not compact ?

A= x, y R2 x 0, 1 y 1
(i) a circle

1 1 (ii) a parabola
B= x, y R2 0 ! x ,y sin
" x
1
(iii) S = x, y R2 y sin ,
x
and X = A # B. Then :
0! x 1

(iv) T = x, y R 2 | x| | y| 4
(A) X is connected and path

(A) (i) and (ii)
connected

(B) (ii) and (iii)

(B) X is connected but not path (C) (iii) and (iv)

connected (D) (ii), (iii) and (iv)

22. A metric space is compact if and only

(C) X is not connected but path if it is complete and :

connected (A) bounded

(B) totally bounded

(D) X is neither connected nor path
(C) closed

connected
(D) countable

9 [P.T.O.

24. Consider the functions
23. Consider the following statements :
f, g : [0, 1] $ R defined by :
I. A function f on [a, b ] is of
1 if x is rational
f x
0 if x is irrational
bounded variation if, and only
and
if, f is the difference of finite-
1 / x if 0 ! x 1
g x
0 if x 0
valued monotone increasing

then :
functions on [a, b].

(A) both f and g are not Lebesgue
II. If the function f is of bounded

integrable
variation on [a, b], then f is

(B) f is Riemann integrable and
measurable.

g is Lebesgue integrable
(A) Only I is true

(C) f is not Riemann integrable and
(B) Only II is true
g is Lebesgue integrable

(C) Both I and II are not true
(D) f is Lebesgue integrable and

(D) Both I and II are true g is not Lebesgue integrable

10

25. The Lebesgue integral of the
27. Which of the following groups is
f u n ct i on f : [0, 1] $ R defined

by :
not solvable ?

%1 3 x if 0 ! x 1
f x
%0 if x 0 (A) Z 150

is :
(B) S4
3
(A)
4
(C) S5
2
(B)
3
(D) a non-abelian group of order 27
3
(C)
2
28. Which of the following rings is not
4
(D)
3
Noetherian ?
26. For which of the following values

of n is every group of order n (A) R[x1, x2, ........]
abelian ?

(A) n = 100 (B) Z[i]

(B) n = 121
(C) Z
(C) n = 24

(D) Z205[x]
(D) n = 120

11 [P.T.O.

29. The degree of the splitting field of 31. Let S be a subset of a normed space

the polynomial x3 – 2 Q[x] is : X and [S] be the span of S in X. Then

which of the following need not be
(A) 3
true ?

(B) 6
(A) [S] is a subspace of X

(C) 5
(B) [S] is a closed subspace of X

(D) 4 (C) [S] is a subspace of X

30. Which of the following constructions (D) [S] is a closed subset of X

is not possible by ruler and 32. Let X be a normed space such that

compass ? every closed and bounded set in X

is compact. Then which of the
(A) Trisection of an angle of 90°
following is true ?
(B) Construction of a regular
(A) X has a countably infinite basis
polygon of 7 sides
(B) Every bounded subset of X is
(C) Construction of length equal to
complete

1 + 2 + 42
(C) X is finite dimensional

(D) Construction of the complex (D) Every proper subspace of X is

number e"i/40 bounded

12

33. Let X be a separable inner product 35. Denote the dual space of the normed

linear space X by X . Then which
space and Y be any orthonormal set
of the following is true ?
in X. Then :
(A) X is separable & X is separable

(A) Y is linearly dependent (B) X is separable & X is separable

(B) Y is countably infinite (C) X ' X

(D) X is always separable
(C) Y is a basis (Hamel) of X
36. Let f : [0, 1] $ [0, 1] be a continuous
(D) Y is countable
function. Then which of the

34. An infinite dimensional, separable following is true ?

(A) For all x, y [0, 1],
complex Hilbert space is congruent
x y & f(x) f(y)
to :
(B) ( x [0, 1] such that f(x) = x
(A) l
(C) If for some x, y [0, 1] with

(B) l2 x y, f(x) f(y) then for all

x y, f(x) f(y)
(C) l1
(D) If there is x0 [0, 1], f(x0) = x0

(D) l3 then f(x) = x for all x X

13 [P.T.O.

37. Which of the following spaces need
39. Let X be a locally connected
not be Hausdorff ?

(A) X is a topological space such topological space. Then which of

that ) = {x × x/x X} is closed
the following is true ?
in X × X

(B) X is a metric space (A) X is connected

(C) X is an Indiscrete Topological
(B) X is locally path connected
space

(D) X is the product of two Hausdorff (C) Every component of an open set
spaces
of X is open in X
38. Let X be a non-empty subset of R
such that for every positive integer
(D) Every component of X is also a
M there is x X such that
|x| M. Then which of the following path component of X
statements is true ?
40. What is the coefficient of x2y2z3 in
(A) X is bounded with respect to
some metric which induces the (x + y + z)7 ?
same topology as induced by the
standard metric on R (A) 630

(B) X is unbounded with respect to
(B) 210
every metric on R

(C) The set X contains rationals as (C) 420
well as irrationals

(D) X is countably infinite (D) 179

14

41. The dual statement of p & q * r 43. Find a 12 if an2 1 5an2 , where

an > 0 for n 0 and a0 = 2 :
is :

(A) 31250
(A) (p * q) + (p * r)

(B) 31000
(B) (~p) + (q + r)

(C) 30350
(C) p + (~q) + (~r)

(D) 30250
(D) ((~p) * q) + ((~p) * r)
44. Eigenvalues corresponding to the

42. Find the number of integers between
following Sturm-Liouville problem :

1 and 10,000 both inclusive which d2X
+ ,X = 0, X(0) = 0, X-(1) = 0
dt2
are divisible by none of 5, 6 or 8 :
are ,n, n N, where ,n =

(A) 3666 2
(2n 1)"
(A)
2
(B) 5000
(B) (2n")2

2
(C) 6250 n"
(C)
2

(D) 6000 (D) (n + 1)"

15 [P.T.O.

46. The differential equation by
45. Consider the boundary value
eliminating arbitrary constant a
problem : from :

y = a(x – a)2
ut(x, t) = uxx(x, t), 0 < x < c, t > 0,
is :

with conditions : 3
dy dy
(A) 4 xy 8y 0
dx dx
ux(0, t) = 0 = ux(c, t), t > 0,
3
dy dy
(B) 4 xy 8 y2 0
dx dx
u(x, 0) = f(x), 0 < x < c.
2
dy dy
(C) 4 xy 8 y2 0
Substituting u(x, t) = X(x) T(t) in the dx dx

2
dy dy
above equation we generate the (D) 4 xy 8y 0
dx dx

following Strüm-Liouville problem 47. The general solution of the partial

differential equation :
for X(x) :
z z
xzp – yzq = y2 – x2, p, q
x y
(A) X--(x) + ,X(x) = 0, X(0) = X(c) = 0
is :
(B) X--(x) + ,X(x) = 0, X-(0) = X-(c) = 0
(A) .(xy, x2 + y2 + z2) = 0

(B) .(x2 – y2, x2 + y2 + z2) = 0
(C) X--(x) + ,X(x) = 0, X(0) = 1, X(c) = 0
(C) .(xyz, x2 – y2) = 0
(D) X--(x) + ,X(x) = 0, X-(0) = 0, X(c) = 0
(D) .(xyz, x2 + y2) = 0

16

48. The following partial differential
50. For which of the following primes
equation :
p , is the quadratic congruence
z z
x2 z2 xy z3 x y2
x y x2 / 2(mod p) solvable ?

is :
(A) p = 5

(A) linear
(B) p = 13

(B) non-linear
(C) p = 19

(C) semi-linear
(D) p = 23

(D) Quasi-linear
51. Let 0(n) denote Euler’s function.

49. Which of the following natural
Then :
numbers cannot be written as a
0 d
d|24
sum of 3 squares of non-negative

is equal to :
integers ?

(A) 24
(A) 2013

(B) 2015 (B) 25

(C) 2017 (C) 23

(D) 2019 (D) 20

17 [P.T.O.

52. The degrees of freedom of a rigid
54. If the Lagrangian L x, x
body are :

(A) 1 corresponding to Atwood’s machine

(B) 2
is given as :

(C) 3
1
L x, x m m2 x2 m1 gx
2 1
(D) 6

53. If a particle of mass m moves in a m2 g l x,
plane under the influence of

gravitational force of magnitude
where m1, m2, l and g are constants.
m
directed towards origin. Then
r2
Then the equation of motion is :
Lagrangian of the system is :

m m m1 m2
(A) r2 r2 2 (A) x g
2 r m1 m2

m m m1 m2
(B) r 2
r 2 2 (B) x g
2 r m1 m2

m m m1 m2
(C) r2 r2 2 (C) x g
2 r2 m1 m2

m m m1 m2
(D) r2 r 2 (D) x m1 m2
g
2 r
18

55. If Lagrangian of a system is : 1
57. Let 1 : ,2 $ R3 be defined by
2
1
L , m l2 2 gl 2 , 1(t) = (t, –t, t3). Then the torsion of
2
1 at t = 1 is :
where g, l are constants. Then the

Hamiltonian of the system is : (A) 1

(B) –1
P2 1
(A) mgl 2
2ml 2 2 (C) 0

P2 1 (D) 2
(B) 2
mgl 2
2ml 2
58. The right cylinder over the circle
2
P
(C) mgl 2 x2 + y2 = 1 has the parametrization
2
ml
x : U $ R 3 , where :
P2
(D) 2
mgl 2 U= u, v R 2 |0 ! u ! 2", !v! ,
2ml

56. Let 1 : (0, 1) $ R 3 given by x u, v cos u, sin u, v

1(s) = (s, s + 1, s2) be a curve
Then the coefficients E, F, G in the
parametrised by arc length s. Then
first fundamental form of the
the curvature of 1 at s is :
cylinder are given by :

(A) 2
(A) E = 1, F = 1, G = 1

(B) 2s (B) E = 1, F = 0, G = 1

(C) 2 4 s2 (C) E = sin2 u, F = 0, G = 1

(D) s (D) E = cos u, F = 0, G = 1

19 [P.T.O.

59. Consider the functional : 60. The extremal of the functional :

2
I u x, y = F x, y, u, ux , uy dx dy x3
I y x 2
dx
G 1
y-

subject to the conditions y(1) = 0,
over the region of integration G,
y(2) = 3 is :
where u is continuous and has (A) y = (x2 – 1)x

continuous derivatives upto second (B) y = (x2 – 1)

(C) y = (x – 1)x
order. Further u takes prescribed
(D) y = (x – 1)
values on the boundary of G. Then 61. The extremal of the isoperimetric
problem :
the differential equation for the
4
2
extremization of the functional is : I y x y- dx
1

F F F subject to the conditions :
(A) 0
u x ux x uy
4
y dx 36
F F F 1
(B) 0
u x uy y uy
and y(1) = 3, y(4) = 24 is :

F F F (A) a parabola
(C) 0
u x ux y ux
(B) an ellipse

F F F (C) a hyperbola
(D) 0
u x ux y uy
(D) a circle
20

62. The integral equation : 64. The solution of the singular integral
1 equation :
t
e 2 et s
x s 0
0 t
1
"t x s ds
is a : t s
0
(A) linear Fredholm integral
equation of second kind is :

(B) linear Volterra integral
(A) 2"
equation of second kind
(C) linear Fredholm integral (B) 2" x
equation of first kind
(C) 2 x
(D) linear Volterra integral
equation of first kind (D) " x
63. The initial value problem
corresponding to the integral )2
65. The value of E x 2 (taking
equation :

x h = 1) is :
x t x cos x x t x t dt
0
(A) –1

is : (B) 0
(A) x--(t) + x(t) = sin t, x(0) = 1,
(C) 1
x-(0) = 0
(B) x --( t ) + x -( t ) + x ( t ) = sin t , (D) 2
x(0) = 1, x-(0) = 0
where :
(C) x--(t) – x(t) = cos t, x(0) = –1,
x-(0) = 1 ) — forward difference operator
(D) x --( t ) – x -( t ) – x ( t ) = cos t ,
x(0) = –1, x-(0) = 1 E — shift operator.

21 [P.T.O.

66. The cubic polynomial which takes
the following values : 68. If u0 = 1, u1 = 5, u2 = 8, u3 = 3,

x f (x )
u4 = 7, u5 = 0, then the value of
0 1
1 2 )5u0 is :
3 1
3 10 (A) 24
by Newton’s forward interpolation
formula is : (B) –60
3 2
(A) 4x – 2x + 5x + 1
(B) 2x3 – 7x + 6x + 1 (C) 60
3 2
(C) 2x – x + 8x + 1
(D) 4x3 – 3x2 + 6x + 1 (D) –61
67. A curve passing through the points
as given in the table : where ) forward difference operator.
x y (x )
1 0.2 69. If L–1{F(s)} = f(t), then
2 0.7
3 1 L–1{F(n)(s)} = is :
4 1.3
5 1.5 (A) tn f(t)
6 1.7
7 1.9 (B) (–1)n tn f(t)
8 2.1
9 2.3 (C) (–1)n f(t)
the area bounded by the curve, the
x-axis, x = 1 and x = 9 is : (D) (–1)n f (n)(t)
(A) 11.5 sq. unit
(B) 12.0 sq. unit where L–1 be the inverse Laplace
(C) 10.5 sq. unit
(D) 10.2 sq. unit transform

22

70. The solution of the integral
72. Let 3* be an outer measure on H(R
R).
equation :

t Then E R) is 3*-measurable
H(R
x t e t sin t 2 x 2 d2
0
if :
by using Laplace transform method

is :
(A) for each A H(R
R)
(A) x(t) = e–2t + 2t – 1

(B) x(t) = 2e–2t + t – 1 3*(A) = 3*(A 4 E) + 3*(A 4 EC)

(C) x(t) = e–t + 2t – 1
(B) for some A H(R
R)
–t
(D) x(t) = 2e + t – 1

71. If F s is the Fourier transform 3*(A) = 3*(A 4 E) + 3*(A 4 EC)
of f(t), then the Fourier transform
of : (C) for each A H(R
R)

f(t) cos at

is : 3*(A) 3*(A 4 E) + 3*(A 4 EC)

(A) F s a
(D) for some A H(R
R)
(B) F s a

1 3*(A) 3*(A 4 E) + 3*(A 4 EC)
(C) F s a F s a
2

1
(D) F s a F s a R)-hereditary 5-ring.
where H(R
2

23 [P.T.O.

74. Let E and F be measurable sets,
73. Consider the following statements : f L(E, 3) and 3(E ) F) = 0,
then :

(A) f L(F, 3) and
I. A measure 3 on a ring R is
f d3 = f d3
complete if E R, F 6 E and E F

(B) f L(F, 3) and
3(E) = 0 then F R.
f d3 < f d3
E F
II. A measure 3 on a ring R is
(C) f L(F, 3) and

5-finite if, for every set E R,
f d3 > f d3
E F

E= En for some sequence
(D) f 7 L(F, 3)
n 1

75. Consider the following statements :
{En} such that En R.
I. If v1, v2 and 3 are measures
and v 1 8 3, v 2 8 3, then
(A) Only I is true v1 + v2 8 3.

II. If v and 3 are measures such
that v << 3 and v 8 3, then v
(B) Only II is true is identically zero.

(A) Only I is true

(C) Both I and II are true (B) Only II is true

(C) Both I and II are not true
(D) Both I and II are not true (D) Both I and II are true

24

Section III
77. Consider the following LPP :
76. Consider the following statements :

1. Sensitivity analysis provides a Max. : Z = 2 x 1 + x2
single value within which a

parameter may change without
Subject to :
affecting optimality.

2. While performing sensitivity
3x1 + 4x2 6
analysis, the upper bound

infinity on the right hand side
6x1 + x2 3
of a constraint means that the

constraint is redundant.
x1 0, x2 0
3. When an additional constraint

is added in the LP models,
What is the solution of this LPP ?
the existing optimal solution

can further be improved if
(A) (1/3, 2/3)
zi – cj 0.

Which of the above are correct ?
(B) (3/11, 14/11)
(A) Only 1 is correct

(B) Only 2 is correct (C) (5/7, 5/7)

(C) Only 3 is correct

(D) All are wrong (D) (2/7, 9/7)

25 [P.T.O.

78. Consider the following statements : 79. A quadratic programming problem

is given as follows :
1. The use of cutting-plane

method reduces the number Max. : Zx = 2x1 + 3x2 – 2 x12

of constraints in the given Subject to :

problem. x1 + 4x2 4

2. In a Branch and Bound x1 + x2 2

minimization tree, the lower x1, x2 0.

bounds on objective function
If we apply Wolfe’s method to solve,
value do not decrease in value.
then the correct Kuhn-Tucker

3. The 0-1 integer programming condition will be :

problem requires the decision (A) 4x1 = ,1 + ,2 + 31 and

variables to have values
x2 = 3 – 4,1 – ,2 + 32
between zero and one.
(B) 4x1 = 2 – ,1 – ,2 – 31 and

Which of the above are correct ?
4x2 = 3 + ,1 + ,2 – 32

(A) Only 1 and 2 are correct
(C) 4x1 = –2 – ,1 + ,2 – 31 and

(B) Only 1 and 3 are correct x2 = 3 – 4,1 – ,2 + 32

(C) Only 2 and 3 are correct (D) 4x1 = 2 – ,1 – ,2 + 31 and

(D) All are correct 3 + 32 = 4,1 + ,2

26

80. Let X and Y be two random variables 81. Let Y be a Bernoulli random

defined on the same probability variable with :

space such that for each y > 0, the
P(Y = 0) = P(Y = 1) = 1/2.

conditional density function of X
Let Xn = (1 – Y/n1)n, 1 0,
given Y = y is :

then, which of the following
y yx2 2
f x/ y e , y 9 0, x R.
2"
statements are correct ?
Let
d
(i) Xn ::$ Y when 0 1 < 1
1 y2
g y e , y 9 0.
2"y
d
(ii) Xn ::$ e–Y when 1 = 1
Which of the following statements
d
(iii) Xn ::$ Y when 1 > 1
is more appropriate ?

(A) (i) and (ii)
(A) E(X|Y) = 0

(B) E(E(X|Y)) = 0 (B) (ii) and (iii)

(C) E(X) = 0 (C) (i) and (iii)

(D) E(X) E(E(X|Y)) (D) Only (i)

27 [P.T.O.

82. Let {Xn} be a sequence of random
84. Let < be a countably infinite set and
variables such that :
let F consists of all subsets of <.
P(Xn = –n1/2) = P(Xn = n1/2) = 1/2n

and P(Xn = 0) = (n – 1)/n. Define :

n
0 if A is finite
Let Sn Xi. 3 A
if A is infinite
i 1

Then, which of the following Which of the following statements
statements is not true ?
are true ?
(A) E(Xn) = 0
(i) 3 is finitely additive
(B) V(Xn) = 1

d
(C) Sn n ::$ Z, Z ~ N(0, 1) (ii) 3 is countably additive

d
(D) Sn n ::$ Z, Z is not (iii) {An} is an = sequence of sets
Gaussian
with 3(A n ) = 0 n , but
83. Let X n , Fn be a martingale.
n 1
3(<) = .
Let ;n = 5(X1, X2, ......, Xn). Then,

X n , ;n is a ................ . (A) (i) and (ii)
n 1

(A) Martingale (B) (i) and (iii)

(B) Sub Martingale
(C) (ii) and (iii)
(C) Super Martingale

(D) White noise (D) (i), (ii) and (iii)

28

85. Let X and Y be two independent
87. Let X be a single observation with
zero-mean unit variance Gaussian
random variables defined on a unknown mean E(X) = 3 (– , )
common probability space. Define
U = X + Y and V = X – Y. Let and variance Var(X) = 3 2 + 1.

F = 5(X). Then, which of the
Consider the problem of estimating
following statements is not true ?

(A) E(U|F) = X a.s. 3 based on X under the squared

(B) E(V|F) = X a.s. error loss L(3, a) = (a – 3)2. Let T1
(C) E(U + V|F) = 2X a.s.
and T2 be defined as follows :
(D) E(U|F) and E(V|F) are
independent T1 = X, T2 = X/2 + 1/2.
86. Let X1 and X2 be two iid random
variables with : Which of the following statements

P(X1 = 1) = P(X1 = –1) = 1/2.
is true ?
Let Z = X1 + X2, Ai = Xi–1({1}),
(A) Risk of T1 = 32
i = 1, 2.

Which of the following is not true
(B) Risk of T2 = (1 – 3 + 32)/2
–1
on Z ({0}) ?

(A) P(A1|Z) = 1/2 (C) Risk of T2 is larger than that

(B) P(A2|Z) = 1/2
of T1
(C) P(A1 4 A2|Z) = 0

(D) P(A1|Z) P(A2|Z) = P(A1 4 A2|Z) (D) T1 is an admissible estimator

29 [P.T.O.

88. Suppose X1 ...... Xn are independent 89. Suppose two control charts have the
same in-control average run length
observations from a Poisson
(ARL). Then which of the following
distribution with probability mass is true ?

function : (A) Both the charts will perform in
e ,,x similar way
f(x/,) = , x = 0, 1, 2, .......,
x!
(B) The charts can be compared
, > 0 unkonwn.
using out-of-control ARL
Suppose the prior distribution of ,
(C) Two charts cannot have the
is desired as : same in-control ARL

(D) The charts must be only to
g(,) > ,, e–2,, , > 0.
monitor population mean
Let n = 3 and the data (X1, X2, X3)
90. Let Lq be the average number of
= (1, 1, 2). Then, suppose we estimate customers in the queue, , be the

, under squared error loss L(,, a) customer arrival rate and 3 be the
average service rate. Then average
= (, – a)2. Then, the Bayes estimator
waiting time for a customer in the
based on the given data is :
queue for all infinite source

(A) 1.5 queueing models is :

(A) Lq/3
(B) 1.65
(B) 3/,
(C) 2.25
(C) ,/3

(D) 1.2 (D) Lq/,
30

91. Let X be a random variable with 93. Let X and Y be independent
Poisson variables with E(X) = ,
distribution function :
and E(Y) = , + 1, when , > 0 is an
1
F1(t) = 1 – exp(–(,t) ), ,, 1 > 0 unknown parameter. Based on
single observations X and Y;
t 0
(XY) = (2, 3), the MLE of , will
Then : be :

(A) F1 is IFR (A) 2

(B) F1 is DFR (B) 2.5

(C) 3
(C) F1 is IFR if 0 < 1 1
(D) 3.5
(D) F1 is IFR if 1 1
94. Let X 1 and X 2 be independent
92. Suppose that in a K-unit parallel observations with X1 ~ N(3, 252) and

system each unit has life time X2 ~ N(23, 52). Suppose we define
T1 = X1 – 4X2 and T2 = 2X1 – X2.
distribution with distribution
Which of the following statements
function F. Then reliability function
is not true ?
of the system at time t is given
(A) (T1, T2) is sufficient
by :
(B) When 5 2 is known, T 2 is
(A) (F(t))K ancillary

(B) (1 – F(t))K (C) When 52 is known, T1 and T2
are independent
(C) 1 – (F(t))K
(D) When 52 is known, T1 is not
(D) 1 – (1 – F(t))K sufficient

31 [P.T.O.

95. Let X1, X2, X3 be iid r.v.s with 96. Let X1, X2, .......... Xn be a random

sample from N(3, 52), where 3 and
2
U( , ); > 1. The maximum

52 are unknown. Then which of the
likelihood estimator (mle) of is :

following statements is not correct ?

2
(A) X (1) n n
(A) Xi, X12 is jointly
1 1
1/ 2
(B) X (3)
sufficient for (3, 52)

2
(C) X (1) X (3) (B) X, s2 is jointly sufficient for

1/2
(3, 52)
(D) 0.2X (1) + 0.8 X (3)

(C) X is sufficient for 3 and s2 is
where :

sufficient for 52

X(1) = Min (X1, X2, X3)
(D) If 5 2 is known then X is

X(3) = Max (X1, X2, X3). sufficient for 3

32

97. Let X be a r.v. having the pmf, 99. Let X1, X2, .......... Xn be iid r.v.s
|x|
1 |x| satisfying the following regression
P[X = x] = 1 ;
2
equation :
x = –1, 0, 1, 0 < < 1

The complete statistics for : Xi = 1zi + ei; i = 1, 2, ....... n

(A) is X where z1, z2, ....... zn are fixed and

(B) is |X| ei’s (i = 1, 2, ..... n) are iid r.v.s with

(C) is X2 N(0, 52), 52 is unknown MLE of 1

(D) Does not exist is given as :

98. Let X1, X2, .......... Xn be iid r.v.s n

with U( , + 1), then E[X(n) – X(1)], xi zi
1
(A) n
where X (n) = Max X i, and
xi2
X(1) = Min Xi, is given by : 1

n 1 n
(A) xi zi
n 1
1
(B)
1 n
(B) n 1
zi2
(C)
n 1 n
(C) n 1
xi zi
n (D)
(D) n 1
zi2

33 [P.T.O.

100. If, for a given 1, 0 < 1 < 1, non- 102. Let X1, X2, .......... Xn be iid random
randomized Neyman-Pearson and sample of size n from exponential
likelihood ratio tests of a simple distribution with mean . The MP
hypothesis against a simple
test of size 1 for testing H0 : = 0
alternative exists, then :
against H1 : = 1
< 0
, is :
(A) They are equivalent
(B) They are one and the same 2@ 22 n, 1 1
%1 ; T <
(C) They are exactly opposite (A) . x = 0
%
(D) One can’t say anything about 0 ; otherwise
it
101. A sample of size n is obtained @2 1
%1 ; T < 0 2n, 1
from a Poisson distribution with (B) . x = 2
parameter m. The most powerful %0 ; otherwise

(MP) test of less than size 1 to test
2
H0 : m m0 against H1 : m > m0 0 @ 2 n, 1
%
is given as : (C) . x = 1 ; T < 2
%0 ; otherwise
1 ; T > t0
(A) . x =
0 ; otherwise
2@ 22 n, 1
%1 ; T <
1 ; T > t0 (D) . x =
% %
0
(B) . x = ? ; T = t0 0 ; otherwise
%0 ; otherwise
n
1 ; T < t0 where T = Xi
(C) . x =
0 ; otherwise 1

103. A test .(x) is called an unbiased
1 ; T < t0
% test if :
(D) . x = ? ; T = t0
%0 ; otherwise (A) EH0 . X 1
(B) EH1 . X 1
n
(C) EH0 . X 1 and EH1 . X 1
where T = Xi
1 (D) EH0 . X 1 = EH1 . X
34

104. L et X 1, X2, .......... Xn be a random 106. Based on a random sample of size
sample of size n observed from n from Poisson distribution with
Cauchy distribution with location m ea n ,, asymptotic variance of
parameter . Define T1 = sample X e X is :
mean and T2 = sample median.
,e ,
Then : (A) , , > 0
n
(A) T 1 and T 2 are consistent ,
e 2, ,
2
(B) 1 , , 1
estimator of n

(B) T1 is asymptotically normal ,
(C) 1 , e ,, , 1
n
(C) T2 is asymptotically normal
,
(D) (T1 + T2)/2 is asymptotically (D) , , > 0
n
normal
107. Consider the problem of testing
105. Let Fn(·) be empirical cumulative H0 : = against H1 : , where
0 0
distribution function, based on a is specified value of and is
0
random of size n from a continuous the parameter of one parameter
distribution with cumulative Cramer family. If ,( x ) is the
distribution function F(·). Then : likelihood ratio statistic based on

(A) Fn is not a consistent estimator sample of size n,

for F (A) –2 log ,(x) $ @12 as n $

(B) Variance of Fn converges to a (B) –2 log ,(x) $ @ 2n as n $
positive constant (C) –2 log ,(x) $ @12 , for = , as
0
(C) Asymptotic distribution of F n n $
is normal (D) –2 log ,(x) $ N(0, 1), for = 0
(D) Asymptotic mean of Fn is 1/2 as n $

35 [P.T.O.

108. L et X 1, X2, .......... Xn be the random 110. If T2 is Hotelling T2-statistic, then
sample observed from Poisson T2 n p
the distribution of
distribution with mean ,. Then n 1 p

consistent estimator of 1 – e–, is : would be :

(A) unique (A) Non-central F-distribution

(B) not unique
(B) Central F-distribution
(C) fu n ct i on of X only
(C) Chi-square distribution
(D) not asymptotically normally
distributed (D) Student t-distribution

109. If X be a p-component random 111. If x1 , x2 , ........, xn (n p + 1) are
vector with E(X) = 0 and variance distributed independently each
covariance matrix A, positive according to Np( 3 , A), then the
definite. If X is partitioned into X (1) distribution of
of p1 components and X (2) of p2
n
1
components such that p1 + p2 = p S = x1 x x1 x-
n 1
1 1
and p1 p2. Then the square of
canonical correlation are the
is :
characteristic roots of the matrix :
(A) Wp(n – 1, A/(n – 1))
(A) A
(B) Wp(n, A/n)
(B) A111 A12

(C) A 221 A 21 (C) Np( 3 , A)

(D) A111 A12 A221 A21 (D) Np( 3 , S)

36

112. If X ~ Np( 3 , A), then the distribu- 114. Given (X, Y) ~ N2(5, 10, 1, 25, B) and

tion of N( X – 3 )- A–1( X – 3 ) would P[4 < Y < 16|X = 5] = 0.954, then

be : B is equal to :

(A) Multivariate normal distribu- (A) +0.80
tion
(B) –0.80
(B) @2 with N – p degree of freedom
(C) ±0.8
(C) @2 with p degree of freedom
(D) 0.96
(D) @2 with N degree of freedom
(It is given that
113. Let X a p-component random vector
2 1 2
(1) (2) 1 z
is partitioned into X and X e 2 dz 0.477 )
2"
where X (1) has q-components and 0

X (2) has (p – q) components and 115. In a multiple regression with 3
X ~ Np ( 3 , A). If 3 and A are regressors and 10 observations, the
also partitioned as of X , then the total variation is found to be 25.549,
variance covariance matrix of the whereas the explained variation by
conditional distribution of X (1) the regression is 24.875. What is the
given X (2) is : value of adjusted-R2 ?

(A) A11 A12 A 221 A 21 (A) 0.964

(B) A11 A12 A 221 A 21 (B) 0.962

(C) A 22 A21 A111 A12 (C) 0.974

(D) A 22 A21 A111 A12 (D) 0.982

37 [P.T.O.

116. For a standard multiple regression 118. Consider the multiple regression

model {Y, XC, 52I}, which of the
set-up {Y, XC, 52I} with p-regressors.
following statements is not true ?
The ordinary least squares estimator
(A) E ( Ŷ ) = E(HY) = XC, where

H = X(X-X)–1X- (OLSE) Ĉ is the vector which

(B) Cov( Ŷ ) = 52H minimizes E(Y – XC) (Y – XC)-.

(C) Cov( Ŷ , Y – Ŷ ) = 0 Which of the following statements

(D) Ŷ ~ N n (C, 5 2 H), under is not true under this set-up ?

normality
(A) X Ĉ is always unique
117. Under the standard multiple

regression model {Y, XC, 5 2 <}, (B) Ĉ is unique if and only if rank

which of the following statements is (X) = p

not true ?
(C) Ĉ is unique if and only if rank
(A) E( Ŷ ) = XC, E( Ĉ ) = C
(X) < p
2
(B) Cov( Ŷ ) = 5 <

(D) OLSE K- Ĉ of K-C is unique if
(C) Cov( Ĉ ) = 52(X-<–1X)–1

(D) Cov(HY, MY) = 52H<M and only if K-C is estimable

38

120. Which of the following statements
119. Which of the following statements
i s not true in the context of a
is not true in the context of the simple linear regression with one

transformation of a response predictor ?

(A) The ratio SSReg/SSTot will be
variable under a regression set-up ?
same whether Y is regressed on

(A) When E(Y) = V(Y) = C X or X is regressed on Y

(B) A value R2 = 0.02 indicates
(a constant), square root
that X and Y are not related
transformation is more
(C) When the fitted regression line
appropriate is horizontal, then SSE = SSTot
and R2 = 0
(B) When E(Y) = C and V(Y) = C2,
(D) When the fitted regression line
log transformation is more is horizontal, then Yi = X .

appropriate 121. If the regression estimator of Y is
Sxy
y b X x where b = , then
(C) Scale transformation preserve S2x

the directions of the association an exact expression for bias of
regression estimator is :
between Y and X
(A) –Cov( y , b)
(D) The Box-Cox transformation
(B) Cov( y , b)

Y $ g(Y, ,) is a discontinuous (C) Cov(b, x )

function of , (D) –Cov(b, x )

39 [P.T.O.

122. The bais in ratio estimator decreases 124. If "i and "ij are respectively the first

with : order and second order inclusion

probabilities of a sampling design in
(A) increasing the sample size n
PPSWOR, then which of the
(B) decreasing the sample size n
following relation is true ?
(C) both (A) and (B)
(A) "ij "i
(D) increase in the population size j

123. A simple random sample of (B) "ij n 1 "i
j
n clusters is selected from a
(C) "ij "i
population of N clusters each of j

size M, then cluster sampling will
(D) "ij n n 1
j
be less efficient than SRSWOR if

(B d = intra class correlation 125. PPSWR sampling reduced to SRS

coefficient between elements if the probability of proportion to

belonging to same cluster) : size i.e. pi is :

(A) M > 1 and Bd > 0 (A) 1/n

(B) M > 1 and Bd < 0 (B) 1

(C) M = 1 and Bd = 0 (C) 1/N

(D) M > 1 and Bd = 0 (D) n/N

40

126. Consider a two-factor factorial, fixed 127. Consider the following statements

effect model with main effects A and about BIBD(a, b, k, r, ,) :

B used at a and b levels respectively. (1) If a = b, the design is said to

Then the estimate of operating be symmetric

characteristic curve parameter for A (2) ,(k – 1) = r(a – 1)

is :
(3) The adjusted treatment sum

a
na 2 2i of squares is free from block
i 1
(A) 2
b5 effects.

a
nb 22i
Which of the above are correct ?
i 1
(B) 2
a5
(A) Only 1 is correct
b
na C2j
i 1 (B) Only 1 and 2 are correct
(C) 2
b5

b (C) Only 3 is correct
nb C2j
i 1
(D) 2
a5 (D) Only 2 and 3 are correct

41 [P.T.O.

128. Consider the following statements : 129. Consider the following statements :

(1) If the presence of interaction (1) In Resolution-III designs main

inflates the error mean square, effects are aliased with two-

one should use factorial designs. factor interactions.

(2) Two contrasts with coefficient (2) In Resolution-IV designs, two-

{ci} and {di} are orthogonal if factor interactions are aliased
a
ci di 0. with each other.
i 1

(3) In Resolution-V designs two-
(3) Confounding is a design

technique for arranging a factor interactions cannot

complete factorial experiment in be aliased with three-factor

blocks. interactions.

Which of the above statements are Which of the above statements are

correct ? correct ?

(A) Only 1 and 2 are correct (A) 1 and 2 are correct

(B) Only 1 and 3 are correct (B) 1 and 3 are correct

(C) Only 2 and 3 are correct (C) 2 and 3 are correct

(D) All are correct (D) All are correct

42

130. A 23 design with 4 replicates are 132. The sample autocorrelation of
under consideration. Two of the certain time series data was found
replicates are shown below :
to be significant at lags one and

five, whereas the sample partial

autocorrelations were not signifi-

cants for first 30 lags. What model

would you suggest for such a time
Identify which of the treatment series ?
combinations are partially
confounded in each replicate ? (A) ARMA(1, 5)

(A) AB and AC (B) ARMA(5, 1)
(B) AB and BC
(C) MA(5)
(C) AC and BC
(D) AR(5)
(D) A and BC

131. Consider the time series model : 133. In an AR(1) model Xt = 0.5Xt – 1 + Zt,

Xt = 0.2Xt – 2 – 0.6Xt – 1 + Zt + 1.2Zt – 1, Zt ~ iid normal (0, 1), the best linear
where Zt ~ iid Normal (0, 52). Which p r edi ct or aX1 + b X3 of X2 using
of the following statements is
(X1, X3) will have :
true ?
(A) {Xt} is stationary and invertible (A) a = b

(B) {Xt} is invertible, but not causal (B) a < b
(C) {Xt} is causal but not invertible
(C) a > b
(D) {X t } is neither causal nor
invertible (D) Cannot be determined

43 [P.T.O.

134. Let the time series {Y t } be an 136. Let {Xn} be a MC on [0, 1] with
ARIMA(2, 1, 2). Then {Yt} is :
TPM :
(A) a stationary model
1 1 1
(B) having one unit root for the AR P= ,
polynomial C 1 C

(C) having one unit root for the MA 0 < 1 < 1, 0 < C < 1.
polynomial
( n)
Then lim P11 is :
(D) having unit roots for both AR n$

and MA polynomials 1
(A)
135. Consider the MC consisting of the 1 C
three states 0, 1, 2 and having C
TPM : (B) 1 C

1 1 1C
0 (C) 1 C
2 2
1 1 1
P= 1
2 4 4 (D) 1
1 2 C
0
3 3
137. There are n units in the system at

Which of the following is correct ? time t and number of arrivals take

places during the time interval )t,
(A) All states do not communicate
the probability of the event is :
(B) The stationary distribution does
(A) Pn(t) (1 – 3)t)
not exist
(B) Pn(t) (1 – ,)t)
(2) 1
(C) P00 =
4 (C) Pn – 1(t) (1 – ,)t)

(D) MC is irreducible (D) Pn(t) (1 – ,)t) + Pn – 1(t) )t

44

138. The probability generating function 140. Consider the following statements :

of a particular random variable is (1) Changing basic objective
function coefficient cj will affect
given as : the entire 0-raw to change.
1 (2) Changing right-hand side of a
P(s) = (s + s2)
2 constraint will retain the
What is the variance of the random current basis to be optimal even
if the constraint is negative.
variable ?
(3) Changing the column of a non-
1
(A) basic variable xj will affect, the
3
coefficient of xj in row-0 is still
1 non-negative and the current
(B)
4 basis is optimal.
1 Which of the above are correct ?
(C)
2
(A) 1 and 2 are correct
2
(D) (B) 1 and 3 are correct
3
(C) 2 and 3 are correct
139. Given the following table :
Age group o f Number of Total (D) All are correct
c hild bearing female Women (’000) Births
141. Let X1, X2, ........, Xn be a random
15 — 19 16.0 260
sample of size n from Bernoulli
20 — 24 16.4 2244
25 — 29 15.8 1894 distribution with parameter . Then
30 — 34 15.2 1320 variance of conditional expectation
35 — 39 14.8 916
n
40 — 44 15.0 280 X1 X2
of given X i is :
45 — 49 14.5 145 2
i 1
What is the value of TFR ?
(A) 0
(A) 2251.75 per thousand
(B) 2135.48 per thousand (B)

(C) 2106.96 per thousand (C) (1 – )

(D) 2018.88 per thousand (D) (1 – )/n
45 [P.T.O.

142. Let X be Poisson variate with 144. Suppose (X, Y) have joint pdf f(x, y),
mean ,. Then distribution function where :
of X, evaluated at 1.5 is :
2 0! x! y!1
f x, y
(A) 0 0 otherwise

(B) less than e–, Then E(X/y) is given by :

(C) equal to (, + 1)e–, (A) 2y

(D) greater than ,(, + 1)e–, (B) y/2

143. Let X be a r.v. with cdf, F(x), where :
(C) 1

0 ifx!0 (D) y
% x/2 if0 x!1
%
F( x)
%3/4 if 1 x ! 2 145. Let X be a degenerate random variable,
%1 if 2 x
at X = c . Then characteristic

Then E(X) is given by : function of X at ‘t’ is :

(A) 3/8 (A) c

(B) 1/2 (B) 0

(C) 3/2 (C) exp(itc)

(D) 1 (D) exp(–itc)

46

ROUGH WORK

47 [P.T.O.

ROUGH WORK

48