MH SET 2023: 26 March, Paper II, Mathematical Sciences
Text of page 1
Test Booklet Code & Serial No.
A
MATHEMATICAL SCIENCE
Signature and Name of Invigilator Seat No.
1. (Signature) ......................................... (In figures as in Admit Card)
(Name) ................................................ Seat No. ..............................................................
2. (Signature) ......................................... (In words)
(Name) ................................................ OMR Sheet No.
MAR - 30223 (To be filled by the Candidate)
Time Allowed : 2 Hours] [Maximum Marks : 200
Number of Pages in this Booklet : 48 Number of Questions in this Booklet : 180
Instructions for the Candidates
1. Write your Seat No. and OMR Sheet No. in the space provided 1.
on the top of this page.
2. This paper consists of 180 objective type questions. Each question
will carry two marks. Candidates should attempt all questions 2.
either from sections I & II or from sections I & III only. I II I III
3. At the commencement of examination, the question booklet
will be given to the student. In the first 5 minutes, you are
requested to open the booklet and compulsorily examine it as 3.
follows :
(i) To have access to the Question Booklet, tear off the
paper seal on the edge of this cover page. Do not accept
(i)
a booklet without sticker-seal or open booklet.
(ii) Tally the number of pages and number of questions in
the booklet with the information printed on the cover (ii)
page. Faulty booklets due to missing pages/questions
or questions repeated or not in serial order or any
other discrepancy should not be accepted and correct
booklet should be obtained from the invigilator within
the period of 5 minutes. Afterwards, neither the Question
Booklet will be replaced nor any extra time will be
given. The same may please be noted.
(iii) After this verification is over, the OMR Sheet Number
should be entered on this Test Booklet. (iii)
4. Each question has four alternative responses marked (A), (B),
(C) and (D). You have to darken the circle as indicated below on 4. (A), (B), (C) (D)
the correct response against each item.
Example : where (C) is the correct response.
A B D (C)
5. Your responses to the items are to be indicated in the OMR
A B D
Sheet given inside the Booklet only. If you mark at any place
other than in the circle in the OMR Sheet, it will not be evaluated. 5.
6. Read instructions given inside carefully.
7. Rough Work is to be done at the end of this booklet.
6.
8. If you write your Name, Seat Number, Phone Number or put
any mark on any part of the OMR Sheet, except for the space 7.
allotted for the relevant entries, which may disclose your 8.
identity, or use abusive language or employ any other unfair
means, you will render yourself liable to disqualification.
9. You have to return original OMR Sheet to the invigilator at the
end of the examination compulsorily and must not carry it with 9.
you outside the Examination Hall. You are, however, allowed
to carry the Test Booklet and duplicate copy of OMR Sheet on
conclusion of examination.
10. Use only Blue/Black Ball point pen. 10.
11. Use of any calculator or log table, etc., is prohibited. 11.
12. There is no negative marking for incorrect answers. 12.
Text of page 2
MAR - 30223/II—A 2
Text of page 3
MAR - 30223/II—A
Mathematical Science
Paper II
Time Allowed : 120 Minutes] [Maximum Marks : 200
Note : This Paper contains One Hundred Eighty (180) multiple choice questions
in THREE (3) sections, each question carrying TWO (2) marks. Attempt
all questions either from Sections I & II only or from Sections I & III
only. The OMR sheets with questions attempted from both the Sections viz.
II & III, will not be assessed.
Number of questions, sectionwise :
Section I : Q. Nos. 1 to 20, Section II : Q. Nos. 21 to 100,
Section III : Q. Nos. 101 to 180.
SECTION I xs 1
3. The improper integral 1 x is
1. Consider the sets given by 0
X {2m 1 / m Z } and convergent if and only if :
Y {2 n 3 / n Z} . (A) 0 s 1
Then :
(B) 0 < s 1
(A) X is a proper subset of Y
(C) 0 s < 1
(B) Y is a proper subset of X
(C) X = Y (D) 0 < s < 1
(D) Either X is finite or Y is finite 4. Let X and Y be metric spaces and
2. The series f, g : X Y be continuous. Then
1 1 1 1 1 { x | f ( x) g( x)} is :
1 .........
2 3 4 5 6
(A) Open
(A) is divergent
(B) Closed
(B) converges to zero
(C) converges to loge2 (C) Connected
(D) converges to loge1 (D) Compact
3 [P.T.O.
Text of page 4
MAR - 30223/II—A
5. Which of the following pair X, Y of 7. Let F be the class of subsets of R
sets do not have same cardinality, defined by F = {A|either A is
i.e. |X| = |Y| ? countable or Ac is countable}. Then,
(A) X = [0, 1], Y = [0, 1] × [0, 1] which of the following is not true ?
(B) X = the set of irrationals, (A) R F
Y = the set of algebraic
numbers
(B) Bn F , if B F , for all n
n
n 1
(C) X = R, Y = C
(D) X = a bounded interval [x, y] (C) [0, ) F
with x < y in R, Y = (– , ) = R
(D) Q c F , where Q is the set of
6. Suppose that {An} is a sequence of all rational numbers
sets such that :
8. A function f is defined on R by :
{ n, n 1, n 2} if n is even
An
{ n, n 1, n 2,.....} if n is odd x2 if x 3
f ( x)
a x b if x 3
Then, which of the following is/are
true ? Then, for which values of a and b,
f ' (3) exists ?
(A) nlim A n does not exist
(A) a = 6, b = –9
(B) nlim A n = {1, 2, 3, .......}
(B) a = 0, b = 9
(C) nlim A n = {3, 5, 7, .......}
(C) a = 3, b = 0
(D) nlim A n =
(D) a = 18, b = –9
4
Text of page 5
MAR - 30223/II—A
9. Consider the function g defined 11. Which of the following sets of vectors
by : are linearly independent ?
1 cos(2 x) 2 3 1
if x 0
g( x ) x2 I.
0 , 2 , 2
u if x 0 1 1 0
The function g is continuous at 0, 2 2 4
(A) if u = 2 0 , 1 , 1
II.
0 0 8
(B) if u = 1
(C) if u = 0 1 2
III. 2 , 1
(D) such a value u does not exist
1 1
10. Let f be a function from R2 to R
(A) I and II only
given by :
(B) II and III only
x if x y
f ( x, y) (C) I and III only
1 if x y
(D) None of the sets are linearly
Then, which of the following
independent
statements is true ?
12. The dimension of the vector space
S1 : Both partial derivatives of f at
consisting of all 10 × 10 real
(0, 0) exist and are equal
symmetric matrices with determinant
S2 : f is differentiable at (0, 0)
zero is :
(A) Both S1 and S2
(A) 55
(B) Only S2 (B) 54
(C) Neither S1 nor S2 (C) 99
(D) Only S1 (D) 45
5 [P.T.O.
Text of page 6
MAR - 30223/II—A
13. Let T be a linear transformation 15. Let A be a square matrix and let
from R2 to R2 such that :
B = A – 0.2 I, where I is the identity
1 1 1 2
T , T . matrix. Which of the following is
0 1 2 1
always true ?
3
Then, T :
4
(A) A and B have the same rank
3
(A) is
2 (B) B–1 exists
4 (C) A and B have the same eigen
(B) is
5
vectors
5
(C) is
3 (D) A and B have the same eigen
(D) Cannot be determined from the values
given information
14. If A and B are two m × n non-null 16. Suppose the eigen values of a 3 × 3
matrices such that AB = 0, then,
matrix A are 0, 1 and 2. Then, the
which of the following is always
true ? determinant of (I + A2)–1.
(A) The column space of B is
contained in the null space (A) Cannot be found from the given
of A
information
(B) The column space of A is
contained in the null space (B) is 1/10
of B
(C) Either A = 0 or B = 0 (C) 0
(D) Either B or A can be non-
singular (D) 1/8
6
Text of page 7
MAR - 30223/II—A
17. Suppose the two matrices A and B 19. Let V be an inner product space, W
have same characteristic polynomial. be a subspace of V and
W {v V/ v, w 0 for all w W}
Then :
Then :
(A) A and B have same minimal (A) dim W dim W
polynomial (B) W W
(C) W ! W
(B) A = Bt
(D) V W"W
(C) A and B are similar 20. The Jordan canonical form of a real
symmetric matrix with characteristic
(D) A is invertible if and only if B
polynomial (x – 1)4 is :
is invertible
1 0 0 0
18. Which of the following is not a 0 1 0 0
(A) 0 0 1 0
nilpotent operator ?
0 0 0 1
(A) An injective operator on R4
1 1 0 0
(B) The identity operator on R4 0 1 0 0
(B) 0 0 1 1
(C) The differential operator on the 0 0 0 1
vector space of polynomials of 1 1 0 0
degree upto 3 over R 0 1 1 0
(C) 0 0 1 1
(D) T : R4 R4 defined by : 0 0 0 1
x1 x1
1 0 0 1
x 2 x2
T 2 0 1 0 0
x3 3 x3
(D) 0 0 1 0
x4 4 x4
0 0 0 1
7 [P.T.O.
Text of page 8
MAR - 30223/II—A
SECTION II 24. Let D {z C :| z| 1} and let
21. One of the values of i(1 i) is : f :D D be a bijective analytic
# 1
(A) i e 2 function such that f 0.
2
#i Then :
(B) e 2
#i 2z 1
(C) (A) f ( z)
e2 2 z
(D) i e # / 2
2z 1
(B) f ( z) ei$ for some real
22. If f ( z) u iv is analytic on a 2 z
domain and v y 2 xy , then u = number $
(A) x 2 xy c
2z 1
(C) f ( z)
(B) x x 2
y 2
c 2 z
(C) y2 x2 y c 1
z
(D) f ( z) 2
(D) x2 y2 c 1 z
23. The radius of convergence of the
25. Suppose f is a Möbius transformation
series
on C that has as its only fixed
1 – 4z2 + 16z4 – 64z6 + 256z8 – .....
point. Then :
is :
(A) f is a rotation
(A)
(B) 1 (B) f is a dilation
(C) 2
(C) f is a translation
1
(D)
2 (D) f is an inversion
8
Text of page 9
MAR - 30223/II—A
26. A function f analytic on a disk D is 29. The radius of convergence of the
necessarily constant if f(D) is n ! zn
contained in :
power series % nn
is :
n 1
(A) a disk
(A) 1
(B) a half plane
(C) a circle (B) e
(D) C (C)
27. Let A denote the set of all complex
1
numbers lying on and within the (D)
e
square having vertices 0, 1, #i and
#i + 1. Let f(z) = ez, z C . Then 30. Let G be a region and suppose
f(A) = fn : G C is analytic for each
(A) {w / 1 | w| e} n & 1 . Suppose { fn } converges
(B) {w / | w| e} uniformly to a function f : G C.
(C) {w / 1 | w| e, 0 arg w #} Then :
#
(D) {w / 1 | w| e, 0 arg w } (A) f is continuous but not
2
1 differentiable
z4 sin dz
28. z
|z| 1
(B) f is differentiable but not
(A) 12 #i
analytic
#i
(B)
60 (C) f ' is analytic
(C) 60 #i
(D) f is analytic but f ' is not
#i
(D)
30 analytic
9 [P.T.O.
Text of page 10
MAR - 30223/II—A
31. Let G { z C :| z 1| 1} and let 33. The residue of f ( z) cot z at z #
f :G C be an analytic function is :
such that :
2n 1 (A) 0
f (1 1 / n)
n 1
(B) #
for n = 2, 3, .........
Then : (C) 1
2z 1
(A) f ( z)
z 1 (D) 2#
z 1
(B) f ( z) 34. Let G = {z : 0 < |z| < 0.1} and
2z 1
1 z f(z) = e1/z, z G. Then :
(C) f ( z)
2 z
(A) f(G) = G
2 z
(D) f ( z)
1 z
(B) f(G) = C
32. Consider the following two
statements :
(C) f(G) = {z : 0 < |z| < e10}
(i) An entire function f with
Re f Im f is constant. (D) f(G) is dense in C
(ii) There exists an analytic
function f from the unit disk ' 35. What is the degree of the splitting
1 field x3 – 2 over Q ?
into ' such that f (0) and
2
3 (A) 6
f '(0) .
4
Then : (B) 4
(A) Only (i) is true
(B) Only (ii) is true (C) 3
(C) Both (i) and (ii) are true
(D) 2
(D) Both (i) and (ii) are false
10
Text of page 11
MAR - 30223/II—A
36. Let F be a finite extension of R. 39. Suppose the ideal A = (p(x)) in F[x]
Then the degree [F : R] is : F-field is a maximal ideal. Then the
(A) a power of 2 polynomial p(x) :
(B) even (A) has no root in F
(C) a power of a prime
(B) is monic
(D) prime
(C) is irreducible
37. Which of the following statements
(D) has all roots in F
is not true ?
40. Which is FALSE ?
(A) There is a field with 4 elements
(A) The polynomial ring K[x] over
(B) There is a Boolean ring with
8 elements a field K is a Euclidean
domain
(C) Every group of order 12 is cyclic
(D) Every group of order 15 is (B) Every principal ideal domain is
abelian a Euclidean domain
38. The number of non-constant (C) The polynomial ring K[x],
irreducible polynomials of degree where K is a field, is a
3 in Z2[x] is : Euclidean domain if and only
(A) 5 if it is a principal ideal domain
(B) 4 (D) The polynomial ring K[x, y] is
(C) 6 not a principal ideal domain,
(D) 3 where K is a field
11 [P.T.O.
Text of page 12
MAR - 30223/II—A
41. Consider the following statements : 43. Consider the following statements :
(I) The polynomial ring Z[x] is a (I) Let G be a finite group. Then
principal ideal domain. every element is of finite
(II) The polynomial ring Z[x] is a order.
Euclidean domain.
(II) Let G be a group such that
Which is correct ?
every element is of finite order.
(A) Only (I) is true
Then G is a finite group.
(B) Only (II) is true
Which is true ?
(C) Both (I) and (II) are true
(A) Only (I) is true
(D) Neither (I) nor (II) is true
42. Consider the following statements : (B) Only (II) is true
(I) If R is a commutative ring with (C) Both (I) and (II) are true
unity and f : R R be a ring
(D) Neither (I) nor (II) is true
homomorphism defined by
44. Which of the following is true ?
f (a) a2 , then 1 + 1 = 0.
(A) Let G be a finite group such that
(II) If R is a commutative ring with
|G| = pk, where p is prime and
unity and f : R R be a ring
homomorphism defined by
k N . Then G has a non-trivial
center
f ( a) a3 , then 1 + 1 + 1 = 0.
Which of the following is true ? (B) The group of order p2 (p-prime)
(A) Only (I) is true is simple
(B) Only (II) is true (C) The group of order pq, where
(C) Both (I) and (II) are true p and q are primes is simple
(D) Neither (I) nor (II) is true (D) The group of order 45 is simple
12
Text of page 13
MAR - 30223/II—A
45. Consider the following statements : 47. How many distinct terms are there
(I) Let H and K be subgroups of in the expansion of (w x y z)10 ?
an abelian group G. Suppose
(A) 228
|H| = m and |K| = n. Let
d = lcm (m, n). Then G has a (B) 224
subgroup of order d. (C) 281
(II) The group of order 15 is abelian.
(D) 286
Which of the following is true ?
48. What is the last digit in 355 ?
(A) Only (I) is true
(A) 3
(B) Only (II) is true
(B) 5
(C) Both (I) and (II) are true
(D) Neither (I) nor (II) is true (C) 1
46. Let S3 be the symmetric group, (D) 7
I n (S 3 ) be the group of inner 49. Let f : R2 R3 be a linear map
automorphisms of S3 and Z(S3) be and
the center of S3. Then which is
A = {(x, y) R2 / |x| )1, |y| ) 1}.
FALSE ?
Then :
(A) Z(S 3 ) = {e}, where e is the
(A) f(A) is a connected subset of R3
identity of S3
(B) In(S3) = S3 (B) f(A) is dense in R3
(C) In (S 3) = {e}, where e is the (C) f(A) is an open subset of R3
identity of S3 (D) f(A) is an unbounded subset
(D) I n (S3 ) ( S3 / Z(S3 ) of R3
13 [P.T.O.
Text of page 14
MAR - 30223/II—A
50. If 52. Let A and B be proper dense subsets
A = {(x, y) / –1 )x < 1, –1 |y| 1} of a metric space. Then :
and (A) A * B is dense
B = {(x, y) / (x – 1)2 + y2 1/2} (B) A + B is dense
then : (C) A\B is dense
(A) A\B is compact in R2
(D) Either A or B is a closed set
(B) B\A is compact in R2
53. Which of the following statements
(C) A * B is compact in R2 is true ?
(D) A + B is compact in R2
(A) Every connected subset of Rn
51. R is not homeomorphic to R 2 is convex
because :
(B) Every convex set in R n is
(A) (0, 1) is open in R but not open
compact
as a subset of R2
(C) Every convex subset of Rn is
(B) (0, 1) is open in R but closed
as a subset of R2 connected
(C) R\(0, 1) is not connected in R (D) The set :
and is also not connected in R2
{(x, y) R2 / x &) 0, y &) 0} +
(D) [0, 1] is closed in R but not {(x, y) R2 / x ) 0, y ) 0}
closed as a subset of R2 is convex
14
Text of page 15
MAR - 30223/II—A
54. Let f : [0, 1] S1 be defined as 56. Consider the space R with the
f ( x) e2#ix . topology induced by the open base :
If Q denotes the set of rational
{u / u = (a, b) or (a, b) \ F}
numbers and S1 is unit circle,
where F { yn / n N} .
then :
The space R with this topology is :
(A) f ([0, 1] * Q) is closed subset
of S1 (A) not T1
(B) f ([0, 1] * Q) is an open subset (B) regular
of S1
(C) both T2 and regular
(C) f ([0, 1] * Q) is dense in S1
(D) T 2
(D) f ([0, 1] * Q) is connected in S1
55. Which of the following collection is 57. Let 1 p q . Then :
not a base for Z × R, where Z (A) lp is a closed subspace of lq and
denotes the set of integers.
lq is a closed subspace of lp
(A) {{k} × I / I is an open interval
(B) lp is a closed subspace of lq and
in R, k Z}
lq is not a closed subspace of lp
(B) {Z × I / I is an open interval
in R} (C) lp is not a closed subspace of lq
and lq is a closed subspace of lp
(C) {{k, –k} × I / I is closed interval
in R, k N, the set of natural (D) neither lp is a closed subspace
numbers}
of lq nor lq is a closed subspace
(D) {{k} × (k – 1, k + 1) / k Z} of lp
15 [P.T.O.
Text of page 16
MAR - 30223/II—A
58. For x, y R , define 60. An equivalence relation ~ defined on
X = [0, 1] by s ~ t iff either s = t
d1 ( x, y) || x| | y|| or s, t {0, 1} . Then quotient space
X/~ (if [0, 1] is given the metric
and d2 ( x, y) | x2 y2 |. Then :
topology induced by modulus) is
(A) d1 and d2 are metrics on R homeomorphic to :
(A) [0, 1]
(B) d1 is a metric on R and d2 is
(B) [0, 1)
not a metric on R
(C) (0, 1)
(C) d2 is a metric on R and d1 is
(D) s1 {( x, y) R 2 : x2 y2 1}
not a metric on R
61. The initial value problem :
(D) neither d1 nor d2 is a metric
y1 f ( x, y), y( x0 ) y0
on R
where (x, y) belongs to a domain D,
has a unique solution :
59. Let A {( x, y) R 2 : x, y are integers}
(A) If f ( x, y) is continuous in D
and B {( x, y) R2 : y x2 } . Then :
(B) If f ( x, y) is continuous and
(A) R2 – A is closed
bounded in D
(B) R2 – A is connected (C) If f ( x, y) is continuous and
satisfies Lipschitz condition for
(C) R2 – B is closed
all points in D
(D) R2 – B is connected (D) For all functions f ( x, y)
16
Text of page 17
MAR - 30223/II—A
62. Let y1(x) and y2(x) be two solutions 64. Let f ( x) x 2 and g( x) x 2 log x on
of : | x| 0 . Then :
( r( x) y ')' p( x) y 0 (A) f, g are linearly dependent
such that y1(x) and y2(x) have a (B) f, g are linearly independent but
common zero in [a, b]. Then :
cannot be solutions of an
(A) y1, y2 are linearly independent ordinary differential equation
on [a, b]
(C) f, g can be solutions of an
(B) W( y1 , y2 ) 0
ordinary differential equation
(C) W( y1 , y2 ) 0
(D) Wronskian of f and g is x2
(D) None of the above
65. For the initial value problem
63. The differential equation of family
y1 y2 cos2 x , y(0) 0
of circles of radius r whose center
lie on x-axis is : the interval of existence of solution
in rectangle
dy 2 2
(A) y y r
dx R :{( x, y)|0 x a,| y| b,
dy
2 a ½, b 0}
(B) y 1 r2
dx
is :
dy (A) [0, 1]
2
(C) y 1 r2
dx
(B) [0, 1 / 2]
2
2 dy (C) [0, ½]
(D) y 1 r2
dx
(D) [0, 2]
17 [P.T.O.
Text of page 18
MAR - 30223/II—A
66. Let y be solution of y1 y | x|, 69. A given partial differential equation
x R and y( 1) 0 , then y(1) can have :
equals : (I) no integral surface passing
(A) 2/e through a given curve
(II) unique integral surface passing
(B) 2e
through a given curve
(C) 2/e2
(III) infinitely many integral
(D) 2e2
surfaces passing through a
67. Let , be a curve which passes given curve
through (0, 1) and intersects each Then :
of the curve of the family y cx 2 (A) (I) and (II) are true
orthogonally. Then , also passes (B) (II) and (III) are true
through : (C) (I) and (III) are true
(A) (0, 2) (D) (I), (II), (III) are true
(B) (1, 1) 70. The partial differential equation
(C) ( 2, 0) yp xq 0 , has :
(D) ( 1, 1) (A) unique integral surface passing
68. The locus of points of intersection through the curve x2 = z, y = 0
of two surfaces represented by (B) finitely many integral surfaces
the partial differential equations passing through the curve
xdx ydy zdz 0 and yp xq 0 x2 = z, y = 0
is a : (C) infinitely many integral
(A) parabola surfaces passing through the
(B) circle curve x2 = z, y = 0
(C) straight line (D) no integral surface passing
(D) helix through the curve x2 = z, y = 0
18
Text of page 19
MAR - 30223/II—A
71. The partial differential equation : 73. The two partial differential
f (x, y, z, p, q) = 0 and equations f ( x, y, z, p, q) 0 and
g (x, y, z, p, q) = 0
g( x, y, z, p, q) 0 are said to be
are compatible on a domain D if :
-( f , g) compatible if :
(A) 0 and the equation
-( x, y)
(A) every solution of f (x, y, z,
dz = pdx + qdy is integrable
p, q) = 0 is also a solution of
-( f , g)
(B) 0 and the equation g (x, y, z, p, q) = 0
-( p, q)
dz = pdx + qdy is integrable (B) every solution of g (x, y, z,
-( f , g) p, q) = 0 is also a solution of
(C) 0 and the equation
-( p, x)
f (x, y, z, p, q) = 0
dz = pdx + qdy is integrable
(C) they have common solution
-( f , g)
(D) 0 and the equation
-( q, y)
(D) they do not have common
dz = pdx + qdy is integrable
72. Consider the following two solution
statements :
74. The normals to surfaces represented
(I) Every surface generated by a
one parameter family of by the Pfaffian differential equation
characteristics is an integral
Pdx + Qdy + Rdz = 0 and the
surface
(II) Every integral surface is Lagrange’s equation Pp + Qq = R :
generated by the characteristic
(A) are collinear
curves
Then :
(B) are orthogonal
(A) (I) is true and not (II)
(B) (II) is true and not (I) (C) are parallel
(C) (I) and (II) both are true
(D) (I) and (II) both are false (D) intersect
19 [P.T.O.
Text of page 20
MAR - 30223/II—A
75. The partial differential equation 78. The relation to solve x e x using
-2 z -2 z Newton-Raphson method is :
x2 y2 xyz
-x 2 -y 2
(1 xn ) e xn
(A) xn 1
is : 1 e xn
(A) hyperbolic (1 xn ) e xn
(B) xn 1
(B) parabolic 1 e xn
(C) elliptic (1 xn ) e xn
(C) xn 1
1 e xn
(D) Laplacian
xn (e xn )
76. Let (D) xn 1
1 e xn
( n)
x x( x h) ( x 2h)...( x (n 2)h)
79. Everett’s formula truncated after
( x (n 1)h) ,
second differences is equivalent to
where h is the step size. Then which
...................... truncated after third
of the following is correct ?
differences.
(A) ' n x( n) n ! h n and ' n 1 x(n) 0
(A) String’s formula
(B) ' x n ( n) n 1 ( n)
n ! and ' x 0
(B) Bessel’s formula
(C) ' x n ( n)
n ! h and
(C) Stirling’s formula
n 1 (n )
' x ( n 1)! (D) Lagrange’s formula
(D) ' n x( n) n ! and 80. The value of
' n 1 x (n ) ( n 1)! ' 5 [(1 x) (1 x 2 ) (1 x 2 )]
77. Let f(x) = ex, then 'n ex = ......... is :
(A) e x (eh 1)n (A) 5!
(B) e x (eh 1)n (B) 0
(C) ex (C) 1
(D) e–x (D) 5
20
Text of page 21
MAR - 30223/II—A
81. The missing term in the following 83. The shape of a curve in the plane
table with fixed perimeter that encloses
x f(x)
maximum area is :
0 1
(A) a square
1 3
2 9 (B) an ellipse
3 – (C) a rectangle
4 81
(D) a circle
using difference operator method
is : 84. The extremal of the functional
(A) 28 4
I y( x) xy ' ( y ')2 dx
(B) 29 0
(C) 30
determined by the boundary
(D) 31
conditions y(0) = 0 and y(4) = 3
82. A curve of quickest descent in the
is :
vertical plane, where y is vertical
and x is horizontal, is the solution x2
(A) y
4
of the equation :
(A) y(1 y '2 ) c x2
(B) y 3x
2
(B) y2 c2 y' c
x2
(C) y x 1
(C) y ' a2 x2 x 4
(D) 4 y x( x 1)
(D) y ' a2 x2 a
21 [P.T.O.
Text of page 22
MAR - 30223/II—A
85. The Euler’s equation for the 87. The Euler Lagrange equation
extremization of the functional :
associated with the functional
#/2
I y( x) 2 xy x '2 y '2 dt ,
1
0 I(w) |Dw|2 dx f w dx
2u
x' dx / dt u
is :
where u is an open set is :
(A) x " y 0
(B) y " x 0 (A) 'w w in u
d4 y
(C) y 0 (B) 'w f in u
dt 4
(D) x " y 0
(C) 'w w in u
86. The Euler-Lagrange’s differential
equation for the extremization of the (D) 'w f in u
functional :
x2
88. The solution of the homogeneous
I y( x) f ( x, y, y ') dx
x1 integral equation
is :
1
d2 y x(t) . t s x (s) ds ,
(A) f y ' y ' 2
( fy' x fy) 0
dx 0
d2 y dy
(B) f y ' y ' 2
fy' y for . / 0 , is :
dx dx
( f y' x fy) 0 (A) t 2
d2 y dy
(C) f y ' x 2
f y' y (B) 1 + t2
dx dx
( f y' y f y) 0
(C) t
2
d y
(D) f y ' x ( f y' x fy) 0
dx 2 (D) 1 + t
22
Text of page 23
MAR - 30223/II—A
89. The kernel of the linear integral 91. For a homogeneous Fredholm
equation integral equation with symmetric
kernel :
1
x( t ) . (t 2 s2 ) x(s) ds b
0 x(t) . k(t, s) x( s) ds
a
is :
consider the following statements.
(A) Separable
(I) The eigen values are not real.
(B) Symmetric
(II) The eigen functions corres-
(C) Convolution
ponding to distinct eigen values
(D) Singular are orthogonal on [a, b]. Then :
90. The initial value problem (A) Only (I) is true
x "(t) x( t ) 0 , x(0) 1 , x '(0) 1 (B) Only (II) is true
is equivalent to the integral (C) Both (I) and (II) are true
equation :
(D) Both (I) and (II) are false
t
92. The solution of the integral equation
(A) x(t) 1 (t s)2 x(s) ds
0 t
x( t ) t (s t) x(s) ds
t
0
(B) x(t) t (t s) x( s) ds
0 with the help of resolvent kernel
t is :
2
(C) x(t) 1 (t s) x(s) ds
(A) x(t) cos t
0
(B) x(t) sin t
t
(D) x(t) 1 t (t s) x(s) ds (C) x(t) et
0
(D) x(t) 1
23 [P.T.O.
Text of page 24
MAR - 30223/II—A
93. Eigen values of the homogeneous 95. For a particle the kinetic energy and
Fredholm integral equation the potential energy is given by
1 1 1 r2
x(t) . (1 3ts) x(s) ds T mr 2 , U 1 ,
2 r c2
0
where c is a constant. Then the
are :
Hamiltonian H of the particle is :
(A) 1, 2
(A) H = T + U
(B) 2, –2
2 1 1 r2
(B) H Pr r m r2 1
(C) 2, –1
2 r c2
(D) 1, –2 1 pr2r c2 1
(C) H 2 (mrc2 2) r
94. Let a particle of mass m be projected
from the ground with initial velocity 1 pr2r c2 1
(D) H
2 (mrc2 2) r
u making an angle $ with the
96. A Lagrangian of a particle of mass
horizontal. Then the total energy of
m is given by
the particle is :
1 1 r2
1 L mr 2 1 2 ,
(A) p2 p2y 2 r c
2m x
c is a constant. Then :
1
(B) m px2 p2y mgy
2 (A) the Hamiltonian H represents
the total energy E
1
(C) px2 p2y mgy (B) H is conserved
2m
(C) E is conserved
1
(D) m x2 y 2
mgy
2 (D) Both H and E are conserved
24
Text of page 25
MAR - 30223/II—A
97. Let a particle of mass m be projected 99. A particle of mass m falling
with initial velocity u making an vertically under gravity. Then the
angle $ with the x-axis. Then its Lagrangian L of the particle is
velocity after time t is given by : given by :
(A) v2 u2 g 2t 2 1
(A) L mz2 mgz
2
(B) v2 u2 g2 t2 2 gut cos $
1
(C) v2 u2 g 2t2 2ugt sin $ (B) L mz2 mgz
2
(D) v2 u2 2ugt sin $ 1
(C) L m ( x2 y2 ) mgy
2
98. A particle has Lagrangian
1
L
1
f ($)$2 2
2 g($) w$ w h($) V (D) L m ( x2 y2 ) mgy
2 2
1
Then f ($)$2 w2h($) V 100. A rigid body with one point fixed
2
represents : has :
(A) The Hamiltonian H
(A) Three degrees of freedom
(B) The total energy E
(B) Six degrees of freedom
(C) Both Hamiltonian H and the
total energy E
(C) Two degrees of freedom
(D) Neither the Hamiltonian H nor
the total energy E (D) One degree of freedom
25 [P.T.O.
Text of page 26
MAR - 30223/II—A
SECTION III 103. The waiting time of the customer
(in min) is grouped as :
101. If the probability generating
Waiting Time Frequency
function of a random variable X is
4—6 6
given by : 6—8 8
PX (t) (0.4 0.6t)10 , 8—10 12
10—12 10
then the distribution of X is :
12—14 6
(A) binomial with mean 4
The mode of waiting time is :
(B) binomial with mean 6 (A) 9.33
(C) negative binomial with mean (B) 9.67
(C) 10.33
20
(D) 10.67
3
104. In how many ways can a lady
(D) negative binomial with mean 15 having 10 dresses, 5 pairs of shoes,
102. Suppose the distribution of a and 2 hats be dressed ?
(A) 50
random variable is normal (µ, 02).
(B) 100
Then, which of the following (C) 500
statements is true ? (D) 10 ! 5 ! 2 !
(A) The coefficient of skewness 105. Which of the following statements
is correct ?
11 = 3
(A) Probability measure is countably
(B) The coefficient of kurtosis sub-additive
12 = 0 (B) Distribution function of
Binomial (n, p) distribution has
(C) 95% area under the normal
n discontinuity points
curve lies in the range of µ ± 0 (C) Power set of 2 is always a finite
(D) The mean deviation from the field
(D) If X is simple random variable,
mean is same as the mean
then its set of continuity points
deviation from the median is countable set
26
Text of page 27
MAR - 30223/II—A
106. The distribution of a random 107. Which of the following statements
is not true ?
variable X is given by :
(A) E(X) may not exist
1
(B) E(X Y)2 & EX 2 EY 2
0 if x 0 2
1 1 (C) If X is integrable, then |X| is
if 0 x
4 4 also integrable
1 1 1
if x (D) Moments of even order always
2 4 2
F( x) P[X x]
3 1 3 exist
if x
4 2 4
108. Suppose that 21 = {1, 2, 3, 4, 5, 6}
x 3 3
if x 2
5 4 and that P is a probability measure
1 if x&2 such that P({j}) = 1/6, for j = 1, 2,
3, 4, 5, 6. Let IA and IB be the
indicator functions of the subsets
1 3
Then P X is : A = {1, 3, 5} and B = {2, 3, 4, 6}
2 4
respectively. The following are two
3 statements :
(A)
4 (I) I A and I B are identically
distributed
1 (II) I A and I B are independent
(B)
2
random variables
Which of the following is true ?
1
(C)
4 (A) Both (I) and (II) are true
(B) Only (I) is true
1 (C) Neither (I) nor (II) is true
(D)
8
(D) Only (II) is true
27 [P.T.O.
Text of page 28
MAR - 30223/II—A
109. Suppose that 12 independent 111. Suppose that the conditional
observations on a random variable
distribution of X given that N = n
X are given by 2, 2, 2, 2, 4, 4, 5,
is binomial with parameters n and
5, 6, 9, 9, 10 and that Y is a random
variable with c.d.f. F12, where F12 0.25 and that the marginal
is the empirical distribution function distribution of N is geometric with
of the above data. Then :
mean 2 and variance 2. Then,
(A) E(Y) = 4.5
variance (X) is equal to :
(B) P[Y 6] = 0.75
(C) P(Y = 7) = P(Y = 6) (A) 1
(D) P(Y = 2) P(Y = 9)
(B) 10/16
110. The cumulative distribution function
F of a random variable X is given
(C) 8/16
by :
F(x) = 0.2 '0 (x) + 0.4 '1 (x) + 0.2 (D) Cannot be determined
'2 (x) + 0.2 '3 (x),
where ' a is the degenerate 112. If X is a non-negative random
distribution function degenerate at variable, which of the following
‘a’. The following are two statements :
statements is not true ?
(I) P(X = 1|X 1.5) = 2/3
(II) Variance (X) = 0 (A) E(X2) & (E(X))2
Which of the following is true ?
(B) E(X3) & (E(X))3
(A) Both (I) and (II) are true
(B) Only (I) is true
(C) E(X1/4) & (E(X))1/4
(C) Only (II) is true
(D) Neither (I) nor (II) is true (D) E(X–2) & (E(X))–2
28
Text of page 29
MAR - 30223/II—A
113. Suppose (t) exp ( | t|) is a 115. Suppose {X n , n & 1} is a sequence of
characteristic function of a random independent and identically
variable X. Then the distribution of distributed random variables each
X is : following uniform U(2, 6) distribution.
(A) Normal Which of the following is not
true ?
(B) Laplace
(A) X(n) converges to 6 in probability
(C) Student’s t with 1 degree of
freedom (B) X(1) converges to 2 in probability
(D) Gamma (C) X (n) converges in law to
exponential distribution
114. Suppose {An, n & 1} is a sequence
of events on a probability space (D) Mn converges in probability to
(2, A,P) . Which of the following is 4, where Mn is the median of
always true ? {X1, X2, ......., Xn}
(A) P(lim sup A n ) 03 116. Suppose X is a real random variable
on a probability space (2, A, P) and
% P(A n ) {A n , n & 1} is a sequence of events
n&1
in A such that :
(B) % P(A n ) 3
n&1 An 2 as n .
P(lim sup A n ) 0 Which of the following is true ?
(C) P(lim sup A n ) 1 3 P
(A) X I A n 44 0
% P(A n ) L
(B) X I A n 44 0
n&1
(D) % P(A n ) 3 P
(C) X I A n 44 X
n&1
P
P(lim sup A n ) 1 (D) X I A n 44 1
29 [P.T.O.
Text of page 30
MAR - 30223/II—A
117. {X n , n & 1} , {Yn , n & 1} and Y are 118. Let {X n , n & 1} be independent and
identically distributed random
random variables defined on the
variables with mean 0 and
same probability space. Suppose as
variance 1. Which of the following
n , Xn converges in probability
is true as n ?
to 3 and Yn converges in distribution
n
to Y. Then which of the following is (A) % X i / n converges in
i 1
? distribution to a degenerate
true as n
random variable
(A) X n Yn2 converges in probability
n n
and in distribution to 3Y2
(B) % X i / % X 2i converges in
i 1 i 1
probability to 0
(B) X n Yn2 need not converge in
n
probability nor in distribution (C) n % Xi converges in
i 1
distribution to a non-zero
(C) X n Yn2 converges in probability
random variable
but not in distribution
n
(D) X n Yn2 need not converge in
% X 2i
i 1
(D) n converges in
n % Xi
probability but converges in
i 1
distribution to 3Y2 probability to 1
30
Text of page 31
MAR - 30223/II—A
119. The following are three statements 121. Suppose fii(n) denotes the probability
related to a Markov Chain :
of the first return to i in n steps in
(I) For any state i, µi & fii
a Markov chain. The following are
(II) For a persistent state i, µi & 1
three statements :
(III) For a transient state i, µi 1
where µi denote the mean (I) For any state i, fii(n) pii(n),
recurrence time for state i and fii is n & 1.
the probability of first return to
state i. (II) For any state i, % pii(n) & 1 .
n &1
Which of the following is true ?
(III) For a persistent state i,
(A) Only (I) is true
(B) Both (I) and (II) are true 5 pii(n) & 1.
(C) Both (I) and (III) are true Which of the following is true ?
(D) Only (III) is true (A) All three are true
120. Suppose fij denotes the probability
(B) Both (I) and (II) are true
of the first visit to j from i in a
Markov chain with finite state (C) Both (I) and (III) are true
space. The following are two (D) Only (I) is true
statements :
122. The following are four statements
(I) If C is a closed communicating
class of persistent states, then about a Markov chain. A unique
for any transient state i which stationary distribution exists if the
leads to j, k C, fij = fik. Markov chain is irreducible and (I)
(II) If C is a single closed ergodic (II) non-null persistent, (III)
communicating class and if
transient and (IV) null persistent.
i j where i / C and j ) C,
then fij = 1. Which of the following is true ?
Which of the following is true ? (A) Only (I) is true
(A) Both (I) and (II) are false
(B) Only (II) is true
(B) Both (I) and (II) are true
(C) Both (I) and (II) are true
(C) (I) is true but (II) is false
(D) (I) is false but (II) is true (D) Either (III) or (IV) is true
31 [P.T.O.
Text of page 32
MAR - 30223/II—A
123. Which of the following is true ? 125. To find out the prevalence of a virus
The life time of a component of a in a metrocity with population size
machine is modeled by the 10,00,000 a blood test was carried
exponential distribution with mean out on 200 randomly selected
2 per week. Failed component is citizens. Test found 8 positive
immediately replaced by a new one.
cases. Then the distribution of
Then the probability that no
number of affected persons in a
component is replaced in two weeks
is : random sample of size 500 from the
population of the same metrocity
(A) e –2
can approximately be taken as :
(B) e –4
(A) Poisson (40)
(C) e–0.5
(D) e –1 (B) Poisson (20)
124. Which of the following is true ? (C) Poisson (8)
A linear death process is a (D) Poisson (4)
continuous time Markov chain,
126. If X and Y are independent N(0, 1)
where :
random variables, then E{max (X, Y)}
(A) all states are non-null persistent
equals :
(B) all states are transient
(A) 1
(C) 0 is a non-null persistent state
and all other states are (B) 3
transient 1
(C)
(D) 0 is a non-null persistent state #
and all other states are null 1
(D)
persistent 2#
32
Text of page 33
MAR - 30223/II—A
127. Suppose that X is a non-negative 129. Suppose that X 1 and X 2 are
integer valued random variable independent and identically
having 6 as its probability
distributed random variables such
generating function. Then, variance
that X1 has exponential distribution
(X) = 1, when :
with scale parameter 1/.. Let S1 = X1,
(A) 6( s) s / (3 2 s), 0 s 1
S2 = X1 + X2. Then, which of the
(B) 6( s) 1 / (2 s), 0 s 1
following statements is true ?
(C) 6( s) (1 s) / 2, 0 s 1
(A) E(S1|S2 = 4) = ./2
(D) 6(s) (s s3 ) / 2, 0 s 1 (B) E(S1|S2 = 4) = 2.
128. Suppose that the random vector (C) E(S1|S2 = 4) = 2/.
(X1, X2, X 3)' has a multinomial
(D) E(S1|S2 = 4) = 2
distribution with parameters n, p1,
p 2 and p 3 . Then, which of the 130. If (X, Y)' follows bivariate normal
following statements is FALSE ? distribution with mean vector
(0, 0) ' and variance-covariance
(A) X1 and X 2 are independent
1 0.5
random variables having matrix 5 , which of the
0.5 1
binomial distributions with
following does not have a
parameters n and p1, and n and
72–distribution ?
p2 respectively
(A) X2 + Y2
(B) Covariance (X1, X3) = –n p1p3
(C) Variance (X3) = n p3(1 – p3) (B) X2
(C) Y2
(D) X1 + X2 has binomial distribution
with parameters n and (1 – p3) (D) 4/3 (X2 – XY + Y2)
33 [P.T.O.
Text of page 34
MAR - 30223/II—A
131. If one observation X from Bernoulli 133. Let X be random sample of size 1
distribution with parameter from U(0, $). Consider the critical
1 2 region C { x| x 1} to test
p , is obtained, then the
3 3 H0 : $ 1 against H1 $ 2 . The
MLE of p is : size of the test is :
(A) X (A) 0
2X 1 (B) 0.01
(B)
3
(C) 0.05
X 1
(C) (D) 0.10
3
134. Suppose {X1, X2, ....., Xn}, n & 2 is
X 1
(D)
3 a random sample from Poisson
132. Let X1, X2, ......., Xn be i.i.d. Bernoulli distribution with mean $. If
n
n
(p) random variables and S % Xi . T % X i , then the MVUE of
i 1
i 1
e–$ is :
Then unbiased estimator of p3 is :
T
n
3 (A)
S n 1
(A)
n
T
n 1
S(S 1) (S 2) (B)
(B) n(n 1) (n 2) n
T
(S 1) (S 2) (S 3) (C)
(C) (n 1) (n 2) (n 3) n T
n T
(D) None of the above (D)
T
34
Text of page 35
MAR - 30223/II—A
135. Suppose that X 1 and X 2 are 137. Suppose {X1, X2, ......, Xn} is a random
independent random variables
sample from N($, 1) distribution.
having Bernoulli distribution with
mean p and 2p respectively, where Which of the following is an MVUE
0 p ½ . The following are two of $2 ?
statements :
(A) X 2n 1
(I) (X1, X2) is minimal sufficient
for p (B) X 2n
(II) 2X1 + 5X2 is minimal sufficient 2
n
for p
% Xi
(C) i 1
Which of the following is true ?
n
(A) Only (I) is true
2 1
(B) Only (II) is true (D) X n
n
(C) Both (I) and (II) are true
138. Which of the following distributions
(D) Neither (I) nor (II) is true does not belong to the Cramer
136. A pharmaceutical company claims family ?
that its weight loss drug allows
(A) Exponential distribution with
women to lose 8.5 lb after one month
location parameter $ 0 and
of treatment. If we want to conduct
scale parameter 1
an experiment to determine if the
patients are losing less weight than (B) Cauchy distribution with
advertised, which of the following location parameter $ and scale
hypothesis should be used ? parameter 1
(A) H0 : µ = 8.5, Ha : µ > 8.5
(C) N($, $) $) > 0
(B) H0 : µ = 8.5, Ha : µ < 8.5
(D) Laplace distribution with
(C) H0 : µ = 8.5, Ha : µ 8.5 location parameter 0 and scale
(D) H0 : µ > 8.5, Ha : µ 8.5 parameter $
35 [P.T.O.
Text of page 36
MAR - 30223/II—A
140. Suppose {X1, X2, ....., Xn} is a random
139. Suppose {X1, X2, ....., Xn} is a random
sample from normal N(µ, 0 2 )
distribution. Suppose Mn is the
sample from a Laplace distribution
sample median. Which of the
following is true ?
with p.d.f. f(x, $) given by :
d
(A) n (M n µ) 44 Z where
Z ~ N(0, 80 2 )
1 | x|
f ( x, $) exp , (B) n (M n
d
µ) 44 Z where
2$ $
#0 2
Z ~ N 0,
2
x R, $ 0.
d
(C) n (M n µ) 44 Z where
#
Z ~ N 0,
Which of the following estimators is 20 2
d
consistent for $) ? (D) n (M n µ) 44 Z where
02
Z ~ N 0,
n
2#
|X i |
(A) % n 141. Suppose {X1, X2, ....., Xn} is a random
i 1
sample from Cauchy C($, 1)
distribution. A test procedure for
½
1 n testing H0 : $ = 0 against the
(B) %
ni 1
X i2
alternative H1 : $ 0 is not
consistent if the test statistic is
based on :
(C) Sample mean (A) Sample median
(B) Sample mean
(C) Sample first quartile
(D) Sample median
(D) Sample third quartile
36
Text of page 37
MAR - 30223/II—A
142. The Mann-Whitney U statistic is 144. Suppose n randomly chosen persons
defined as : were enrolled to examine whether
(A) The minimum absolute different skin creams A and B have
difference between the two different effects on the human body.
empirical distribution functions Cream A was applied to one of the
(B) The maximum absolute randomly chosen arm of each
difference between the two person, cream B to the other arm.
empirical distribution functions Response is measured as a
continuous variable. Which of the
(C) The number of times a Y
following tests is most appropriate
precedes X in the combined
to examine the difference ?
ordered arrangement of two
independent random samples on (A) Two-sample Kolmogorov-
X’s and Y’s Smirnov test
(B) Two-sample t-test if normality
(D) The sum of ranks of Y’s in the
can be assumed
combined ordered arrangement
(C) Paired t-test if normality can be
of two independent random
samples on X’s and Y’s assumed
(D) Mann-Whitney test
143. Consider the following statements :
145. Suppose given $, the random
(I) The Fisher information I($) variable X follows a normal
associated with a logistic distribution with mean $) and
distribution with location variance 1. Let the prior density of
parameter $) and scale $) be Bernoulli with parameter ½.
parameter 1 is 1/3. Then the posterior distribution of
(II) The Fisher information I($) $ is :
2
associated with a Cauchy (A) 2 e 1/ 2 ( x $)
distribution with location
1 1/2 ( x $)2
parameter $) and scale (B) e
2
parameter 1 is 1/2.
$2 / 2
(A) Only (I) is true e x$
(C) 1/ 2
(B) Only (II) is true 1 ex
(C) Both (I) and (II) are true 2
x$ $2 / 2
ex
(D) 1/ 2
(D) Both (I) and (II) are not true 1 ex
37 [P.T.O.
Text of page 38
MAR - 30223/II—A
146. Suppose we have 10 observations 148. In a linear model with 6 observations,
y ( y1, y2 ,....., y10 ) from Poisson (.)
E(Y1) = Q1 + Q2, E(Y2) = E(Y4) –
and we assume a Gamma (a, b) prior
distribution for .. Q1 + Q3, E(Y3) = Q2 – Q3, E(Y5) =
1 2E(Y1) and E(Y6) = 3E(Y2).
p(.) (a) ba e b. . a 1, a 0, b 0
Then which of the following
Then the posterior predictive
statements is true ?
distribution p( ynew / y) of a new
observation ynew is given by : (A) Q1, Q2, Q3 are all estimable
10
(B) Q1 + Q2 + Q3 is estimable
(A) Gamma a % yi , b 10
i 1
(C) Q1 + Q2 + 2Q3 is estimable
(B) Negative Binomial
(D) Q1 + 2Q2 – Q3 is estimable
10
a % yi , b 10 149. For a given data, the two-way
i 1
ANOVA table is :
10
(C) Gamma b 10, a % yi Source df Ss Mss F ratio
i 1
Treat 5 6 — c
10
(D) Poisson a % yi Block 3 4.5 — —
i 1
147. Consider the linear model Error a 2.0 —
yn 1 X $ p 1 9 n 1 where E(9) 0, Total 23 12.5
2
cov(9) 0 I n . The assumption that
Hence the values of a, b and c are :
9 is multivariate Normal is needed
for : (A) a = 15 b = 6.5 c = 1
(A) estimating the parameter $
(B) a = 15 b = 4.5 c = 2.25
(B) estimating the parameter 02
(C) splitting the total sum of squares (C) a = 15 b = 6.0 c = 3
into two orthogonal parts
(D) using F test to test the model fit (D) a = 15 b = 6.0 c = 9
38
Text of page 39
MAR - 30223/II—A
150. In case of a one-way ANOVA, the 152. In case of a multiple linear
data strongly supports the null
regression set up Y r B : with
hypothesis that all the treatments
have the same effect. Hence the covariance matrix of : being 02 tn,
p-value corresponding to the F 02 > 0, which of the following
ratio is :
statements is true ?
(A) much larger than 0.05
(A) mean regression sum of squares
(B) between 0.01 and 0.05
(C) much smaller than 0.01 (ss) and mean error ss are
(D) nothing can be said about the unbiased estimators of 02, mean
p-value total ss is a biased estimator
151. In simple linear regression with
of 02
regression X and response variate Y
the least squares regression line : (B) all three mean ss are unbiased
(A) may be used to predict a value estimators of 02
of Y given the value of X
(C) mean error ss is an unbiased
(B) implies a cause-effect
relationship between X and Y estimator of 02, mean regression
(C) can be determined only if a ss and mean total ss are biased
linear relationship exists estimators of 02
between Y and X
(D) all three mean ss are biased
(D) all the above statements are
true estimators of 02
39 [P.T.O.
Text of page 40
MAR - 30223/II—A
153. In case of regression of Yi on Xi 155. Suppose X 3;1 ~ MNormal3
with fitted values ˆyi , i 1,....., n we
n n
1 0.5 0
get % yi % ˆyi . 0 0.5 4 0
.
i 1 i 1
0 0 1
(A) always
Hence var (2X2|X1 = x) is equal
(B) only if the regression equation
to :
is y 10 11 X :
(A) 16
(C) If the regression equation is
(B) 3.75
y 10 11 X :
(C) 8
(D) if and only if the regression
(D) 15
equation is y 10 11 X :
156. Suppose X 2;1 ~ Bivariate Normal
154. For the logistic regression model, the
response variables u is : 4 2 1
,
1 1 5 .
(A) binary
Then the distribution of
(B) discrete with finite sample space
3Y 3X1 4X 2 is :
consisting of more than two
(A) Normal (–1, 26)
values
(C) discrete with countably infinite (B) Normal (8, 62)
sample space (C) Normal (8, 98)
(D) continuous (D) Normal (8, 122)
40
Text of page 41
MAR - 30223/II—A
157. Based on n iid observations from 159. Suppose X 3;1 ~ multivariable
multivariate normal distribution Normal3 (µ, 5) and let
with mean µ and covariance matrix
Y 2X 12 X 22 X 32 .
5. The likelihood ratio test statistics
Then the probability distribution of
to test whether µ = µ0 follows :
Y is :
(A) F distribution with one-sided
critical region (A) 72 with 2 degrees of freedom
(B) 72 distribution with one-sided (B) 72 with 3 degrees of freedom
critical region (C) 72 with 6 degrees of freedom
(C) t distribution with one-sided
(D) not 72
critical region
160. Given three random variables X1,
(D) t distribution with two-sided
X2, X3 with <12 = 0.6, <13 = 0.8 and
critical region
< 23 = 0.24. Hence the partial
158. The quadrate form
correlation coefficient between X2
9X12 4X 22 X 32 4X1 X 3 2X 2 X 3 and X3 after removing the effect of
is : X1 is :
(A) Positive semidefinite (A) –0.24
(B) Positive definite (B) +0.24
(C) Negative semidefinite (C) –0.5
(D) Negative definite (D) +0.5
41 [P.T.O.
Text of page 42
MAR - 30223/II—A
161. The Mahalanobis distance between 163. Under systematic sampling :
the two Bivariate Normal Population
(A) First unit in the sample is
2 2 2
N2 , and
3 2 4
selected by probability
1 2 2 proportionate to size (pps)
N2 ,
2 2 4
sampling and remaining (n – 1)
is :
units are selected by simple
(A) 10.5
random sampling
(B) 8.5
(C) 4.0 (B) Variance of sample mean is
(D) 2.0 always less than that of sample
162. Which of the following techniques is mean from SRSWOR
used for the dimension reduction of
(C) Each population unit has
the data ?
unequal chance of being
(A) Linear discriminant analysis
selected in the sample
(B) Quadratic discriminant analysis
(C) Principal component analysis (D) Sample mean is unbiased
(D) None of the above estimator for population mean
42
Text of page 43
MAR - 30223/II—A
164. Suppose there are five units, u1, u2, 166. Post stratification is used when :
u3, u4, u5 in a population. The
probabilities of selecting a sample of (A) Stratum sizes are equal
size two with different units are :
(B) Number of strata of the
1 1
p ({u, u2 }) , p ({u3 , u4 })
2 6 population is not known
1 1
p ({u3 , u5 }) , p ({u4 , u5 }) (C) Population size and stratum
6 6
If # i denote the probability of
sizes are not known
including ith unit in the sample,
then (#1, #2, #3, #4, #5) = (D) Stratum to which a unit belongs
1 1 1 1 1
(A) , , , , is not known before sampling
2 2 3 3 3
1 1 1 1 1 167. Double sampling is useful :
(B) , , , ,
5 5 5 5 5
(A) When complete information on
1 1 1 1 1
(C) , , , ,
2 3 2 3 3 auxiliary variable is not
1 1 1 1 1 available
(D) , , , ,
3 3 2 3 2
165. Under Midzuno’s sampling design (B) When population is
the inclusion probability of ith heterogeneous
element #i is at least :
(C) To improve the performance of
n 1
(A)
N 1 ratio estimator for population
n mean
(B)
N
n(n 1) (D) To improve the performance
(C)
N–1
of regression estimator for
n(n 1)
(D)
N population mean
43 [P.T.O.
Text of page 44
MAR - 30223/II—A
168. Consider the model E(Y1) = 11 + 12, 170. Suppose we wish to have a
E(Y2) = 11 – 12, E(Y3) = 11 + 212 confounding arrangement in 2 r
with the assumption that Yi’s are
blocks of a 2n factorial experiment.
uncorrelated and var (Yi) = 02 for
Then, how many interactions have
all i = 1, 2, 3. Then the best linear
to be confounded ?
unbiased estimator of 12 is :
(A) 2 r – 1
(A) (Y1 + Y2 + Y3) / 3
(B) (4Y1 + 8Y2 + 2Y3) / 14 (B) r – 1
(C) (Y1 – 5Y2 + 4Y3) / 14 (C) n – r + 1
(D) (4Y1 – 8Y2 + 4Y3) / 3 (D) 2r – 1
169. Which of the following properties is 171. In a completely randomized design
not true about the incidence matrix
with unequal group numbers, that
M of a BIBD (r, b, s, k, .) ?
is, n1 = 5, n2 = 7, n3 = 6, what is
(A) Every column of M contains
the degrees of freedom for the error
exactly k ‘1’s
term ?
(B) Every row of M contains exactly
(A) 17
s ‘1’s
(B) 18
(C) M is a symmetric matrix
(D) Two distinct rows of M contains (C) 15
both 1’s in exactly .) columns (D) 120
44
Text of page 45
MAR - 30223/II—A
172. When the conditional effect (also 174. If the failure rate of a random
known as sample effect) for one of variable is h(t) = 3t + 2, t & 0, then
the factors in a 22 design differ only its distribution function :
in sign, then the corresponding main
3t 2
(A) is 1 exp 2t , t & 0
effect estimate will be : 2
(A) zero (B) is 1 exp (3t 2)2 , t & 0
(B) one
(C) is exp (3t 2)2 , t & 0
(C) (a + b) / 2
(D) cannot be obtained from the
(D) (a – b) / 2 given information
173. Suppose the lifetimes of systems A 175. Suppose the lifetime of a system
and B are independent and follow follows the exponential distribution
exponential distributions with with failure rate 0.8. Then the
failure rates 3 and 2 respectively. probability that the system will
Then the probability that system A survive longer than twice its mean
fails before system B is : time to failure is closest to :
(A) 0.4 (A) 0.331
(B) 0.5 (B) 0.261
(C) 0.6 (C) 0.135
(D) 1 (D) 0.101
45 [P.T.O.
Text of page 46
MAR - 30223/II—A
176. The two LPPs P and Q are duals 178. Solution to the balanced assignment
of each other, where problem is binary due to :
P : Max ax + 2y s.t. 2x + y 3,
(A) linear formulation of the
bx + 2y c, x, y & 0, and
problem
Q : Min 3u + 4v s.t. 2u + 4v & 2,
u + 2v & 5, u, v & 0. (B) its non-empty feasible region
Then, (a, b, c) is equal to : (C) the approximate nature of the
(A) (a, b, c) = (5, 4, 4) algorithm
(B) (a, b, c) = (4, 5, 5) (D) the uni-modular property
(C) (a, b, c) = (5, 4, 4) 179. In terms of the usual notation, for
(D) (a, b, c) = (4, 5, 4) an M|M|1 queueing system, which
177. Consider the LPP : of the following equations does not
Max Z = 3x + 5y hold ?
Subject to x + 5y 10; (A) Ls = </(1 –) <)
2x + 2y 5;
(B) Wq = Ws
x &) 0, y & 0.
(C) Ws = Wq + 1/µ
Which of the following statements
is true ? (D) Lq = . Wq
S1 : There exists a unique optimal 180. For a M|M|3 queueing system with
solution to the LPP. arrival rate 18, which one of the
S2 : There exists a unique optimal statements is true ?
solution to the dual problem.
(A) The traffic density is 2
(A) Only S1 is true
(B) The traffic density is 6
(B) Only S2 is true
(C) Both S1 and S2 are true (C) The traffic density is 3
(D) Neither S1 nor S2 is true (D) The traffic density is 12
46
Text of page 47
MAR - 30223/II—A
ROUGH WORK
47 [P.T.O.
Text of page 48
MAR - 30223/II—A
ROUGH WORK
48
Text of page 1
MAHARASHTRA STATE ELIGIBILITY TEST - 26th March 2023
FINAL ANSWER KEY FOR PAPER -II (Booklet Code : A)
SUBJECT CODE & NAME: (30) Mathematical Sciences
QueNo Answer QueNo Answer QueNo Answer QueNo Answer
1 C 41 D 81 D 121 C
2 C 42 A 82 A 122 C
3 D 43 A 83 D 123 D
4 B 44 A 84 D 124 C
5 B 45 C 85 C 125 C
6 D 46 C 86 B 126 C
7 C 47 D 87 D 127 D
8 A 48 D 88 C 128 A
9 A 49 A 89 A 129 D
10 C 50 B 90 D 130 A
11 9 51 A 91 B 131 C
12 B 52 B 92 B 132 B
13 C 53 C 93 B 133 A
14 9 54 C 94 C 134 B
15 C 55 k 95 D 135 C
16 B 56 D 96 B 136 B
17 D 57 D 97 C 137 D
18 9 58 D 98 A 138 A
19 C 59 B 99 B 139 A
20 A 60 D 100 A 140 B
21 D 61 C 101 B 141 B
22 B 62 B 102 D 142 C
23 D 63 D 103 A 143 C
24 B 64 C 104 B 144 C
25 C 65 C 105 A 145 C
26 C 66 C 106 C 146 B
27 C 67 C 107 B 147 D
28 B 68 B 108 C 148 D
29 B 69 D 109 B 149 D
30 C 70 A 110 B 150 A
31 C 71 B 111 C 151 A
32 C 72 C 112 C 152 C
33 C 73 C 113 C 153 C
34 D 74 B 114 B 154 A
35 A 75 A 115 C 155 D
36 A 76 A 116 C 156 D
37 C 77 A 117 D 157 D
38 A 78 A 118 B 158 B
39 C 79 B 119 B 159 D
40 B 80 A 120 B 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 - 26th March 2023
FINAL ANSWER KEY FOR PAPER -II (Booklet Code : A)
SUBJECT CODE & NAME: (30) Mathematical Sciences
QueNo Answer
161 9
162 C
163 D
164 9
165 A
166 D
167 A
168 C
169 C
170 D
171 C
172 A
173 C
174 A
175 C
176 9
177 C
178 D
179 B
180 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 DPage 1 of 48
Source: Savitribai Phule Pune University. Question paper published by Savitribai Phule Pune University.