MAHA SET 2018 Question Paper 2 Mathematical Science

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A
MATHEMATICAL SCIENCE
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2

Mathematical Science
Paper II
Time Allowed : 75 Minutes] [Maximum Marks : 100
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 16, Section II : Q. Nos. 17 to 50,
Section III : Q. Nos. 51 to 84.
Section I 3. The modulus and argument of

6 1
1. The sequence a n = (–1) n + 1 i are :
n2
has : 1 5
(A) ,
2 4
(A) no limit point
1 3
(B) one limit point (B) ,
2 4
(C) two limit points
(C) 2,
(D) more than two limit points 4

2. The series : 1 7
(D) ,
2 4
x3 x5 x7
x ......., | x| 1 dz
3 5 7
4. is :
z2 1
represents the function :
(A) 2 i
(A) tan–1 x

(B) tan x (B) 0

(C) sin–1 x (C) 4 i

(D) log (1 + x) (D) i
3 [P.T.O.

5. If 6. If V denotes the vector space of

n × n real skew symmetric matrices,

1 1 1 then dim V =

1
A = 1 2
, (A) n2 – n
3
2 4
1 (B) n – 1

n(n 1)
(C)
2
where = e2 i/3, then A2 =
n(n 1)
(D)
2
(A) I
7. If

1 2 3
(B) A
A= 0 3 2 ,

0 6 4
1 0 0

(C) 0 0 1 then the rank of the matrix AAt

is :
0 1 0

(A) 1

0 1 0 (B) 2

(D) 1 0 0 (C) 3

0 0 1 (D) 0

4

9. Let X be a r.v. with the following
8. Matrix
F(x),

1 3 2 0 ; x 0
x
A= 0 4 2 ; 0 x 2
F x 4
3
0 3 1 ; 2 x 3
4
1 ; x 3
is similar to :
Which of the following statements
is correct ?
1 1 0
1 2
(A) P[X = 2] = , P[X = 3] =
0 1 0 3 3
(A)
1
0 0 2 (B) f(x) = ; 0 x < 3
3
1
(C) P[X = 2] = P[X = 3] =
1 0 0 2
(D) f(x) = 1; 0 < x < 1
0 1 0 10. Let X be a discrete r.v. with the
(B)
following pmf :
0 0 2
k; x 0, j; j 1, 2, ..... n
P[X = x] =
1 0 0 0 ; otherwise
Let Y = X2, then, P[Y = 4] is :
0 2 0
(C) 1
(A) 2n 1
0 0 2
2
(B) n
1
1 0 0
n
0 2 1 (C) 2n
1
(D)
2
0 0 2 (D) 2n 1

5 [P.T.O.

11. Let X1 and X2 be jointly distributed
12. Let X be a r.v. such that variance
with pdf f(x1, x2) given by :

1
of X is . Then, an upper bound for
f(x1, x2) = 2

P[|X – FX| > 1] as given by the
1
1 x1 x2 x12 x22 ; | x1 | 1 and | x2 | 1
4
0 ; otherwise
Chebychev’s inequality is :

Then, the characteristic function of

X1 is : 1
(A)
4

sin t
(A)
t
1
(B)
2
cos t
(B)
t

(C) 1
sin t
(C)
t2

3
t (D)
(D) 4
sin t

6

13. Let X be a normal random variable
14. The problem :
with mean 1 and variance 1. Define

the events :
Max. : Z = 3x1 + 2x2

A = {–2 < X < 1}, Subject to :

x1 – x2 1,
B = {–1 < X < 1},

x1 + x2 3

C = {0 < X < 2}.
x1, x2 0

Which of the following statements
has :
is correct ?

(A) Feasible solution

(A) P(C) < P(B) < P(A)
(B) Optimum solution

(B) P(A) = P(B) < P(C)
(C) Feasible but not optimum

(C) P(A) = P(B) = P(C) solution

(D) Unbounded solution
(D) P(B) < P(A) < P(C)

7 [P.T.O.

15. The set Section II

S = x1 , x2 : 3 x12 2 x22 6 17. Let :

is a : f(x) = e–1/x if x > 0

= 0 if x 0,
(A) Concave

(A) f(x) is not continuous
(B) Not concave

(B) f(x) is continuous but not C1
(C) Convex

(C) f(x) is C1 but not C
(D) Not convex

(D) f(x) is C but not analytic
16. If the set of feasible solutions of the

system AX = B, X 0, is a convex z
18. The Taylor series for 2 around
z 1
polyhedron, then at least one of the
0 is :

extreme points gives a/an :
(A) z + z3 + z5 + .......

(A) Unbounded solution
z3 z5
(B) z – + – .......
3! 5!
(B) Bounded but not optimal

(C) z – z3 + z5 – .......
(C) Optimal solution

z3 z5
(D) Infeasible solution (D) z – + – .......
3 5
8

19. The matrix : 21. Which of the following statements

is not true ?
1 1
0
2 2
(A) Every closed and bounded
0 1 0
subset of a matric space is
1 1
0 compact
2 2

(B) Every compact subset of a
represents :
metric space is closed and
(A) rotation by angle /4 around
bounded
x-axis

(B) rotation by angle /2 around (C) A closed subset of a compact set
y-axis is compact
(C) rotation by angle /4 around
(D) Cartesian product of compact
y-axis
sets is compact
(D) rotation by angle /8 around
z-axis 22. The function f(x) = |x|3, x R
20. Let V denote vector space of n × n is :
real matrices, having zero trace.
Then dim V = (A) continuous but not

(A) n – 1 differentiable

(B) n2 – 1 (B) differentiable but not c1

(C) n2 – n
(C) of class c2
n(n 1)
(D) (D) of class c3
2
9 [P.T.O.

23. f(x) = sin x, x R
26. The function f(x) defined as :
(A) f ( x ) is continuous but not
uniformly continuous f(x) = 0, if x is irrational

(B) f(x) is uniformly continuous but = 1, if x is rational,
not Lipschitz
is :
(C) f(x) is Lipschitz and uniformly
continuous (A) Continuous on R
(D) f(x ) is neither Lipschitz nor
(B) Continuous at rational points
uniformly continuous

24. Total variation of the function : (C) Continuous at irrational points

x3 (D) Discontinuous at all points of R
f(x) = – x, x [2, 3]
3
27. sin (x + iy) is equal to :
is :
(A) 16/3 (A) sin x sin y + i cos x cos y
(B) 17/3
(B) sin x cosh y + i cos x sinh y
(C) 0
(D) 13/3 (C) sin x cos y + i cos x sin y

log x (D) sin x sinh y + i cos x cosh y
25. The maximum value of ,
x
x > 0 is : 28. Which of the following complex
(A) e numbers are collinear ?
1
(B) (A) 1 + 2i, 2 + 5i, 4 + 11i
e

1 (B) 1 – i, 2 + i, 1 + i
(C) e +
e
(C) i, –1 + 2i, 3 + 4i
1
(D) + e (D) 0, –3 + i, 7 + 8i
e
10

29. Let f(z) be an entire function, then 31. Let f be continuous in an open set

f(z) is also bounded if and only if :
D, then f z dz 0 for each
C
(A) f(z) is a polynomial function piecewise differentiable closed curve

(B) f ( z ) is the reciprocal of a C in D if and only if :

polynomial function
(A) f is identically zero

(C) f(z) is a polynomial in sin z and
(B) f is a polynomial function on D
cos (z)

(C) f is a periodic function on D
(D) f(z) is a constant

(D) f is an analytic function on D
30. Which of the following complex
sin z
functions has a pole at z = 0 ? 32. The value of dz is :
z
|z| 1

3
(A) f(z) = e1/z (A) 2 i

1
(B) f(z) = sin (B) 0
z

1 z 2 z3 (C) 2
(C) f(z) = 4 7
z z

(D) f(z) = z3 + 7z + 1 (D) –2 i

11 [P.T.O.

33. Which of the following rings need 35. Let a be an element of a group G

such that an = identity, then which
not have identity ?
of the following statements is
(A) The ring of homomorphisms of true ?

a vector space to itself (A) The order of G is finite

(B) The ring of automorphisms of (B) The order of the element a

is n
a vector space to itself
(C) Every subgroup of G of order
(C) The ring of n × n upper
n contains a
triangular matrices over R
(D) The order of a is finite and

(D) The ring of polynomials P0[R
R] divides n

vanishing at origin 36. Let G be a group of order n and

H = {a G|a = a–1}, then which
34. Let m be an even positive integer.
of the following is true ?
What is the product of all mth roots
(A) If the cardinality of H is odd
of unity in C ?
then n is odd

(A) 1 (B) The cardinality of H divides n

(B) –1 (C) If the cardinality of H is even

then n is odd
(C) i
(D) The cardinality of H is a power
(D) –i of two

12

37. Let G be a non-abelian group with 40. If characteristic equation of matrix
n elements, then which of the A is same as minimum polynomial
following values of n is possible : which reads as (x – 1)3 (x – 4), then
the Jordan canonical form of A
(A) n = 13
is :
(B) n = 9
1 1 0 0
(C) n = 10
0 1 0 0
(D) n = 15
(A)
38. Let F be a field with 128 elements, 0 0 1 0
then which of the following is
0 0 0 4
true ?
(A) F has a subfield with 8 1 1 0 0
elements
0 1 0 0
(B) F has no proper non-prime (B)
0 0 4 0
subfield

(C) Such a field F does not exist 0 0 0 4

(D) F has a subfield with 32 1 1 0 0
elements
0 1 1 0
39. Let T be a linear operator defined (C)
on vector space V. If there exists 0 0 1 0
v V such that v, Tv, ......., Tn – 1v
0 0 0 4
are linear independent vectors and
dim V = n, then : 1 1 0 0
(A) T is invertible
0 1 0 0
(B) T is nilpotent (D)
0 0 1 1
(C) T is diagonalizable

(D) for T 0 0 0 0 4

13 [P.T.O.

41. Let T be a linear operator defined 43. If P is a 3 × 3 matrix of rank three

on a vector space V; T : V | V. and :

If dim ker T > 0, then : 1 2 3

(A) 0 is an eigenvalue of T A= 4 5 6 ,

(B) T is invertible 7 8 9

(C) T is nilpotent then the rank of PA is :

(D) Im T = V (A) 1

42. If T(x1, x2) = (–x2, x1), then matrix (B) 2
representation of T in the ordered
(C) 3
basis {(1, 1)t; (1, –1)t} is :
(D) Depends upon matrix P
0 1
(A) 44. If A is a 2 × 2 real matrix with
1 0

trace zero and determinant 1, then
0 1
(B) the eigenvalues of A are :
1 0

(A) real and distinct
0 1
(C)
1 0 (B) real and repeated

1 0 (C) complex with non-zero real part
(D)
0 1 (D) purely imaginary

14

47. The differential equation obtained
45. The partial differential equation :
by eliminating the arbitrary
constants A and from :
x2uxx – 2xyuxy + y2uyy + xux + yuy = 0
y = Ae– t sin ( t + )
is :
is :

(A) is hyperbolic if x > 0, y > 0 d2 y dy
(A) 2 +( 2+ 2)y = 0
dt2 dt
(B) is hyperbolic on R2
d2 y dy
(B) 2t + ( 2 – 2)y = 0
dt2 dt
(C) is elliptic on R2
d2 y dy
(C) 2 t +( 2+ 2)y = 0
(D) is parabolic on R2 dt2 dt

d2 y dy
46. The Pfaffian differential equation : (D) 2 t + ( 2– 2)y = 0
dt2 dt

X · dr = P(x, y, z) dx + Q(x, y, z) dy 48. Let 1, 2 be linearly independent
solutions of the linear differential
+ R(x, y, z) dz = 0 equation :

y + a1y + a2y = 0,
is integrable if :
where a1, a2 are constants, then the

(A) X · X Wronskian W( 1, 2) is constant if
0
and only if :

(B) X · curl X 0 (A) a1 = 0

(B) a1 0
(C) X · curl X 0
(C) a2 = 0
(D) X · X 0 (D) a2 0

15 [P.T.O.

49. The differential equation : Section III

51. The mean height of 10000 children
(x3 + xy4) dx + 2y3 dy = 0
of age 6 years is 41.26" and the
standard deviation is 2.24". Then the
will become exact on multiplication
odds against the possibility that the
by : mean of a random sample of 100 is
greater than 41.7" is :
2
(A) ex
(A) 1 : 39

(B) ex (B) 39 : 1
(C) 1 : 40
(C) e–x
(D) 40 : 1
2
(D) ex + x 52. Which of the following statements
i s correct about a regression model ?
50. The differential equation : (A) Residual sum of squares reduces
with every new term added in
Y(4) + sin xy = 0
the model.

(A) has exactly 4 linearly (B) Residual sum of squares reduces
with new term added in the
independent solutions model provided the response
variable is dependent on the
(B) has at most 4 linearly new term.
independent solutions (C) Residual sum of squares
increases with the new term
(C) has less than 4 linearly added in the model.

independent solutions (D) Residual sum of squares
increases with the new term
(D) has more than 4 linearly added in the model provided the
response variable is correlated
independent solutions with the new term.
16

53. Let ( , F, P) be a probability space. 55. L et {X n } be a sequence of
Let {An} be a sequence of events
independent random variables.
such that P(An) = 1 for each n 1.
Define the -field :
Then P An
n 1 F= Xn, Xn 1 , ........ .
n 1
(A) Zero

(B) One Suppose A fe. Then, which of the

(C) Infinity following statements is more

(D) Not defined appropriate ?

54. Let X be a zero-mean unit variance (A) P(A) = 0
Gaussian random variable. Define
(B) P(A) = 1
the random variable Y as :
(C) P(A) = P(AC)
X if |X|
Y=
(D) P(A) = 0 or P(A) = 1
X if |X|
56. If X is a positive random variable
is some positive real number.
What is the distribution of X + Y ? with probability density function f(x),

(A) Gaussian (0, 1) then X–1 has probability density

function :
(B) Gaussian (0, 2)

(C) Not Gaussian, but having (A) 1/f(x)

a distribution which is
(B) f(1/x)
discontinuous at origin
(C) 1/x2 f(1/x)
(D) Not Gaussian, but a continuous
distribution (D) 1/x f(1/x)

17 [P.T.O.

57. Let X 1 and X 2 be iid random 59. If X is F(m, n) (F-distribution with

m and n degrees of freedom) and Y
variables with distribution function

is F(n, m), given that P[X c] = a
F(x). Then, P(X1 X2) is :
1!
then P " Y is :
$ c #%
(A) 1/3
(A) a

(B) 2/3
(B) 1

(C) 1/2 a
(C)
2

(D) 3/2 (D) 1 – a

60. If X is distributed as Binomial
58. Let X be a r.v. with U(– , ). The

(n, p), 0 < p < 1, then :
distribution of Y = X2 is :

(A) –X is distributed B(–n, p)
(A) U(0, )
(B) n – X is distributed B(n, 1 – p)

(B) U(0, 2)
(C) X – k is distributed B(n – k, p),
y½
(C) f(y) = ; 0 < y < when k is constant
2

1 1
(D) f(y) = (2 ) y–½; 0 < y < 2 (D) is distributed as B n,
X p
18

61. Suppose X and Y are independent
63. Let X1, X2, .......... Xn be a random
r.v.’s such that :
sample from Poisson distribution
e xx 1
f(x) = ; x > 0, > 0
with parameter ). Then covariance
e y y& 1
f(y) = ; y > 0, & > 0 between X and (X1 – X2) is :
&

X
The distribution of is : (A) 1/2
X+Y
(A) Beta ( , &)
(B) 1
(B) Gamma ( + &, 1)
(C) Beta ( ' ' &, ' ' &) (C) –1
(D) Gamma ( &, 1)
62. Let X1, X2, .......... Xn be iid r.v.’s (D) 0
from an exponential distribution
with mean . Then the distribution 64. Based on a random sample of size

n
n from N(*, 2), where * is known
of X1 given T = ( 1
Xi

and variance is unknown, sufficient
n 3
n 2 t x1
(A) ; 0 < x1 < t statistic for * is :
tn 2

n 1
(B)
n t x1
t n
; 0 < x1 < t (A) (X i

n 1 t x1
n 2 (B) ( X ,( X
2
i i
(C) ; 0 < x1 < t
tn 1
(C) X n
n 1
n· t x1
(D) n
, tn
; 0 < x1 < t (D) (X 2
i

19 [P.T.O.

65. Let X1, X2, .......... Xn be a random 67. Let X1, X2, .......... Xn be a random
sample of size n from Poisson sample of size n from U(0, ). Define
distribution with mean ). The log T1 = X(n) and T2 = X n . Then :
likelihood function L()|x1, x2, ..... xn)
(A) T1 is UMVUE and T2 is not
is :
unbiased estimator for
(A) constant in )
(B) T2 is UMVUE for /2
(B) monotonic function of )
n 1 n 1
(C) concave function of ) (C)
n
T1 – 2T2 and
n
T1

(D) convex function of ) are correlated

66. Let X be a Bernoulli r.v. with n 1 n 1
(D) T1 – 2T2 and T1
parameter . Suppose we wish n n
to test H 0 : = 1/3 against are uncorrelated
H1 : = 2/3, based on random 68. Let X1, X2, .........., Xn be a random
sample of size 1. Then : sample observed from exponential

(A) there exists infinitely many distribution with location parameter

most powerful tests of size 0.05 . For testing H0 : = 0 against
H1 : > 0, the UMP test rejects
(B) there exists unique non-
H0, if :
randomized most powerful test
of size 0.05 (A) X > C

(B) X(1) > C
(C) there exists unique randomized
most powerful test of size 0.05 (C) X(n) > C

(D) the most powerful test rejects (D) + (Xi – X(1)) > C

H0, if X = 0 is observed where C is to be chosen suitably.
20

69. If r12.3 is the correlation coefficient 70. The manager of a cyber cafe says

between the variables X1 and X2 that number of customers visiting on

after eliminating the linear effect of week days followed a Binomial

X3, then which of the following are distribution. Which one of the

correct ? following techniques can be used to

(1) –1 r12.3 1 test the hypothesis at a given level

(2) 2 + 2 + 2 – 2r r r
r12 r13 r23 1 of significance ?
12 13 23

(3) 2
r12.3 = b12.3 b21.3, where b’s are (A) Test of significance of mean

partial regression coefficients (B) Test of significance of difference

(A) (1) and (2) only of means

(B) (1) and (3) only (C) Chi-square test as a test of

(C) (2) and (3) only goodness-of-fit

(D) (1), (2) and (3) (D) Correlation analysis

21 [P.T.O.

71. In a normal population N(*, 2) 72. Consider the 2 × 2 contingency table

with 2 = 4, in order to test the on two attributes A and B :

null hypothesis * = * 0 against A1 A2

B1 10 20
H1 : * = *1, where *1 > *0 based
B2 30 40

on a random sample of size n ,

What is the value of ,2 for testing
the value of K such that X > K

the independence of the attributes
provides a critical region of size

A and B ?
= 0.05 is :

(A) *0 (A) 0.645
1.645 n

(B) *0 3.290 n (B) 0.794

(C) *0 1.96 n (C) 0.812

(D) *0 3.92 n (D) 0.853

22

73. The following is an arrangement
74. In an ANOVA for an experimental

of men (M) and women (W), lined

design involving 3 treatments and
up to purchase tickets for a rock

concert :
10 observations per treatment, if

MWMWMMMWMWMMMWWMMMMWW

SSE = 399.6, then the estimated
MWMMMWMMMWWW

What is the expected value of runs variance of error term is :

for testing the hypothesis that the
(A) 133.2
arrangement is random ?

(A) 14.92 (B) 87.8

(B) 15.88
(C) 66.6

(C) 16.76

(D) 14.8
(D) 18.20

23 [P.T.O.

75. If orders are placed with the size 77. In usual notations of queueing

determined by the economic order systems, which of the following

quantity, the re-order costs relationships is not true ?

component is : 1
(A) Ws = Wq +
*
(A) equal to the holding cost
(B) Ls = )Ws
component
(C) Lq = )Wq
(B) greater than the holding cost

(D) Ls = Lq + 1/)
component

(C) less than the holding cost 78. For any primal problem and its

component dual :

(D) either greater or less than the (A) optimal value of objective

holding cost component function is same

76. Gantt chart is used to solve the : (B) primal will have an optimal

(A) Job sequencing problems solution iff dual does

(B) Inventory problems (C) both primal and dual can not

(C) Replacement problems be infeasible

(D) All of the above (D) optimal solution does not exist

24

79. The payoff value for which each 81. If - wsy denotes the intra-class

player in a game always selects the
correlation coefficient between pairs

same strategy is known as :
of units that are in the same

(A) saddle point
systematic sample and if -wsy = 0,

(B) equilibrium point
then :

(C) both (A) and (B)
(A) Systematic sampling is as

(D) none of the above
efficient as SRSWOR

80. The number of non-negative
(B) Systematic sampling is more
variables in a basic feasible solution
efficient than SRSWOR
to a m × n transportation problem

is : (C) Systematic sampling is as

(A) m + n – 1 efficient as SRSWR but less

efficient than SRSWOR
(B) mn

(C) m + n (D) Systematic sampling is more

(D) m + n + 1 efficient than SRSWR

25 [P.T.O.

82. For an SRSWOR (N, n ), the 84. Identify the treatments x1, x2, x3

probability that a specified unit is
and x 4 from blocks 1, 2, 3, 4
included in the sample is :
respectively so that the design is
1
(A)
N
BIBD :
n
(B)
N
Block 1 : A, B, C, x1
1
(C) N
Cn
Block 2 : A, x2, C, E

1
(D) N N 1
Block 3 : A, B, D, x3

83. In a 25 factorial design in block of
Block 4 : A, x4, D, E
8 plots each, total number of

interaction confounded with blocks Block 5 : B, C, D, E

is :
(A) x1 = B, x2 = E, x3 = C, x4 = D

(A) 2
(B) x1 = D, x2 = B, x3 = E, x4 = C
(B) 7
(C) x1 = C, x2 = D, x3 = B, x4 = E
(C) 3

(D) x1 = E, x2 = C, x3 = D, x4 = B
(D) 4

26

ROUGH WORK

27 [P.T.O.

ROUGH WORK

28