MH SET 2018: 28 January, Paper III, Mathematical Sciences
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Test Booklet Code & Serial No.
A
MATHEMATICAL SCIENCE
Signature and Name of Invigilator Seat No.
1. (Signature) .........................................
(In figures as in Admit Card)
(Name) ................................................
Seat No. ..............................................................
2. (Signature) ......................................... (In words)
(Name) ................................................ OMR Sheet No.
JAN - 30318 (To be filled by the Candidate)
Time Allowed : 2½ Hours] [Maximum Marks : 150
Number of Pages in this Booklet : 48 Number of Questions in this Booklet : 145
Instructions for the Candidates
1. Write your Seat No. and OMR Sheet No. in the space provided 1.
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2. (a) This paper consists of One hundred forty five (145) multiple 2. (a)
choice questions, each question carrying Two (2) marks.
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JAN - 30318/III—A 2
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JAN - 30318/III—A
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.
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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
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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 (C)
f f 1 f
2 r 2 2
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.
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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
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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.
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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
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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.
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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
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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.
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29. The degree of the splitting field of 31. Let S be a subset of a normed space
the polynomial x3 – 2 X and [S] be the span of S in X. Then
Q[x] is :
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
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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.
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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
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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.
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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
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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.
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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
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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.
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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
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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.
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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
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70. The solution of the integral
72. Let 3* be an outer measure on H(R).
equation :
t Then E H(R) is 3*-measurable
x t e t sin t 2 x 2 d2
0
if :
by using Laplace transform method
is :
(A) for each A H(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)
–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)
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)
(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 where H(R)-hereditary 5-ring.
2
23 [P.T.O.
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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
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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.
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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
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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.
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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
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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.
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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
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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.
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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
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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
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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
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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 $
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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)
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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
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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
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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 )
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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
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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.
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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
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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.
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JAN - 30318/III—A
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
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JAN - 30318/III—A
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.
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JAN - 30318/III—A
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
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JAN - 30318/III—A
ROUGH WORK
47 [P.T.O.
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JAN - 30318/III—A
ROUGH WORK
48
Text of page 1
MAHARASHTRA STATE ELIGIBILITY TEST - 28TH JANUARY 2018
FINAL ANSWER KEY FOR PAPER -III (Booklet Code: A)
SUBJECT CODE & NAME: (30) Mathematical Sciences
1 9 41 D 81 A 121 D
2 B 42 D 82 C 122 A
3 C 43 A 83 A 123 f
4 B 44 A 84 B 124 B
5 D 45 B 85 D 125 C
6 D 46 B 86 D 126 B
7 C 47 A 87 B 127 A
8 D 48 D 88 D 128 D
9 D 49 B 89 B 129 A
10 A 50 D 90 D 130 B
11 C 51 A 91 D 131 C
12 D 52 D 92 C 132 C
13 C 53 B 93 A 133 A
14 B 54 C 94 D 134 B
15 D 55 B 95 B 135 D
16 B 56 A 96 C 136 B
17 B 57 C 97 B 137 9
18 D 58 B 98 C 138 B
19 B 59 D 99 D 139 A
20 B 60 B 100 A 140 B
21 B 61 A 101 A 141 D
22 B 62 C 102 B 142 C
23 D 63 C 103 C 143 D
24 D 64 C 104 C 144 B
25 C 65 D 105 C 145 C
26 B 66 9 106 B
27 C 67 A 107 C
28 A 68 D 108 B
29 B 69 B 109 D
30 B 70 D 110 A
31 B 71 C 111 A
32 C 72 A 112 C
33 D 73 B 113 B
34 B 74 A 114 C
35 A 75 D 115 9
36 B 76 D 116 D
37 C 77 D 117 B
38 A 78 B 118 C
39 C 79 D 119 D
40 B 80 A 120 B
9 = No option is correct or the question is wrong, a= A and B, b = A and C, c = A and D, d = A,B and C, e = A,B
and D,f = A,C and D, g = A,B,C and D, h = B and C, i = B and D, j = B,C and D, k = C and D
Text of page 2
MAHARASHTRA STATE ELIGIBILITY TEST - 28TH JANUARY 2018
FINAL ANSWER KEY FOR PAPER -III (Booklet Code: B)
SUBJECT CODE & NAME: (30) Mathematical Sciences
1 B 41 A 81 A 121 D
2 C 42 C 82 B 122 C
3 B 43 B 83 C 123 D
4 D 44 D 84 C 124 B
5 9 45 B 85 C 125 C
6 B 46 A 86 B 126 D
7 B 47 C 87 C 127 D
8 D 48 C 88 B 128 B
9 D 49 C 89 D 129 D
10 C 50 D 90 A 130 A
11 B 51 9 91 A 131 A
12 C 52 A 92 C 132 C
13 A 53 D 93 B 133 A
14 B 54 B 94 C 134 B
15 B 55 D 95 9 135 D
16 B 56 C 96 D 136 D
17 C 57 A 97 B 137 B
18 D 58 B 98 C 138 D
19 B 59 A 99 D 139 B
20 A 60 D 100 B 140 D
21 B 61 D 101 D 141 D
22 C 62 C 102 A 142 C
23 A 63 D 103 f 143 A
24 C 64 D 104 B 144 D
25 B 65 A 105 C 145 B
26 D 66 C 106 B
27 D 67 D 107 A
28 A 68 C 108 D
29 A 69 B 109 A
30 B 70 D 110 B
31 B 71 B 111 C
32 A 72 B 112 C
33 D 73 D 113 A
34 B 74 B 114 B
35 D 75 B 115 D
36 A 76 C 116 B
37 D 77 B 117 9
38 B 78 C 118 B
39 C 79 D 119 A
40 B 80 A 120 B
9 = No option is correct or the question is wrong, a= A and B, b = A and C, c = A and D, d = A,B and C, e = A,B
and D,f = A,C and D, g = A,B,C and D, h = B and C, i = B and D, j = B,C and D, k = C and D
Text of page 3
MAHARASHTRA STATE ELIGIBILITY TEST - 28TH JANUARY 2018
FINAL ANSWER KEY FOR PAPER -III (Booklet Code: C)
SUBJECT CODE & NAME: (30) Mathematical Sciences
1 C 41 9 81 A 121 A
2 B 42 A 82 C 122 C
3 D 43 D 83 B 123 A
4 9 44 B 84 C 124 B
5 B 45 D 85 9 125 D
6 B 46 C 86 D 126 D
7 C 47 A 87 B 127 B
8 D 48 B 88 C 128 D
9 B 49 A 89 D 129 B
10 A 50 D 90 B 130 D
11 B 51 D 91 D 131 D
12 C 52 C 92 A 132 C
13 A 53 D 93 f 133 A
14 C 54 D 94 B 134 D
15 B 55 A 95 C 135 B
16 D 56 C 96 B 136 C
17 D 57 D 97 A 137 B
18 A 58 C 98 D 138 C
19 A 59 B 99 A 139 D
20 B 60 D 100 B 140 A
21 B 61 B 101 C 141 A
22 A 62 B 102 C 142 B
23 D 63 D 103 A 143 C
24 B 64 B 104 B 144 C
25 D 65 B 105 D 145 C
26 A 66 B 106 B
27 D 67 B 107 9
28 B 68 D 108 B
29 C 69 D 109 A
30 B 70 C 110 B
31 A 71 B 111 D
32 C 72 C 112 C
33 B 73 A 113 D
34 D 74 B 114 B
35 B 75 B 115 C
36 A 76 B 116 D
37 C 77 C 117 D
38 C 78 B 118 B
39 C 79 D 119 D
40 D 80 A 120 A
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 4
MAHARASHTRA STATE ELIGIBILITY TEST - 28TH JANUARY 2018
FINAL ANSWER KEY FOR PAPER -III (Booklet Code: D)
SUBJECT CODE & NAME: (30) Mathematical Sciences
1 B 41 D 81 D 121 D
2 D 42 C 82 A 122 C
3 9 43 D 83 f 123 A
4 B 44 D 84 B 124 D
5 C 45 A 85 C 125 B
6 D 46 C 86 B 126 C
7 D 47 D 87 A 127 B
8 A 48 C 88 D 128 C
9 A 49 B 89 A 129 D
10 B 50 D 90 B 130 A
11 B 51 B 91 C 131 A
12 A 52 B 92 C 132 B
13 D 53 D 93 A 133 C
14 B 54 B 94 B 134 C
15 D 55 B 95 D 135 C
16 A 56 B 96 B 136 B
17 D 57 B 97 9 137 C
18 B 58 D 98 B 138 B
19 C 59 D 99 A 139 D
20 B 60 C 100 B 140 A
21 A 61 B 101 D 141 A
22 C 62 C 102 C 142 C
23 B 63 A 103 D 143 B
24 D 64 B 104 B 144 C
25 B 65 B 105 C 145 9
26 A 66 B 106 D
27 C 67 C 107 D
28 C 68 D 108 B
29 C 69 B 109 D
30 D 70 A 110 A
31 9 71 B 111 A
32 A 72 C 112 C
33 D 73 A 113 A
34 B 74 C 114 B
35 D 75 B 115 D
36 C 76 D 116 D
37 A 77 B 117 B
38 B 78 C 118 D
39 A 79 D 119 B
40 D 80 B 120 D
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.