MP SET 2025: State Eligibility Test, Mathematical Sciences
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(SPACE FOR ROUGH WORK) S/18/2025/A
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54. Which of the following sequences 54. RnRriRsid ^
converges uniformly ?
nx nx
• (A)
xe[0,l] (A) J^6[0, 1]
l+ n^x^
(B) nxe
-nx
X e[O, l] (B) nxe
-nx
X e[0,1]
;c X
(C) x^ [0,1] (C) [0,1]
\-\-nx' \+ nx'
(D) 1- \-x' "-^/2, >/2] (D) 1- l-:c' Xe [-y/2, ^/2'j
55. For any rational number
55. wn
r=^G(0,1),
where p and q are integers with no
common factor and for g > 0, f{r)= — p,q t,
and y(x)= 0, for all irrational
X e (0,1), then choose the correct answer
out of the following :
f(x)=0 t X G (0,1)
(A)/is not continuous at any rational .WTT t, "dl rHHR^Rsld "4* ^
number :
(B) / is continuous at every rational
number
(A)/t^ r
(C) /is not continuous at every irrational (B) / "RK w t I
number
(D)/ is continuous at every integer (C) I
number (D)/ t1
56. Let : 56. "fTFlT :
> 52 > 0 5i > 52 > 0
and J„+i +•?„-!), n > 2 5„+l — "^'^rt-l)' n>2
Then which statement is true for the
^ ai^snnl Rm, "qfhr^
given sequences ?
WT t ?
(A) {5i, 53, 55, } is an increasing
sequence.
(A) {5i, 53, 55, } 31^ t 1
(B) {^2. 54, 5g, } is a decreasing (B) {52, 54, 5^, } IRRH aq^ 11
sequence. (C) {s„} ariRTRl t i
(C) {5„} i^ a divergent sequence. (D) {5„} a^fq^ t I
(D) {5„} is a convergent sequence.
N(3)S/18/2025/A
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57. IfV = VjCK) and 57. V = V3(E) t T e A(V) ^
1 1 2 31T^. TTTTO < ej, 62, 63 >
-1 2 1
0 1 3
1 1 2
is the matrix of T s A(V) relative to the -1 2 1
0 1- 3
standard bases < 62' ®3
t, ^ T ^ 3TT??R {(1, 1, 1),(0, 1, 1),
will be the matrix of T relative to the
(0, 0, 1)} ^
basis'{(1, 1, 1),(0, 1, 1),(0, 0, 1)} ? ^?
4 3 2 4 3 2
-2 0 -1
(A) -2 0 -1
(A)
2 1 2
2 1 2
4 3 2" 4 3 2"
2. 0 -1 2 0 -1
(B) (B)
2 1 2
2 1 2
'4 3 2 4 3 2
-2 0 -1 -2
(C) 0 -1
(C)
2 ■ 1 0 2 1 0
-4 3 2
-4 3 2
-2 0 -1
-2
(D) -0 -1
(D)
2 ■ 1 0 2 1 0
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58. Let F be the real field and let V be the 58. 1TFTT F ^ t cia?T "RHT V,
set of all polynomials, in variable a:, over . ^ a: 2 W ^^"511^*^
field F of degree < 2. If in V we define %, F tI V 3f?R
inner product : "ipFT "3^ MR'Hirqd t :
(pix), q(x))= p{x)q(x)dx. (p(a:), 9W)= p{x)q{x)dx,
where p(x)y q(x)e V, then the W % pixlqix)eW, 3imR
orthonormal basis of V corresponding to {l,x,x'^} ^ M^IHM dirndl
the basis {\, x, x^} is : t:
m {;g'I"- (A)
(B) |l, X, (B) |l,
(c <-^'1
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59. Let A = be n-rowed square matrix 59. TTHT % A =[a,y] ^ t
and Ij2 be the matrix obtained by ?TS!Tr Ii2 n-'^^ ^
interchanging the first row and second
row of «-rowed identity matrix. Then AIi2 ^
AIj2 is such that its first :
(A) row is the same as its second row ■ (A) tidk ^ wm 11
(B) A^ ^
(B) row is the same as the second row
tI
of A
(C) A ^ ^ ■HRPT
(C) column is the same as the second
t I
column of A
(D) ^ ^t I
(D) row is all zero
00
60. If If{x)dx converges and 60. J /(x)tic srfq^ t ^27T
1 1
lim /(x)= L, then : lim f(x) = L t, W :
(A) L>0 . (A) L>0
(B) L<0 (B) L<0
(C) L = 0 (C) L = 0
(D) L=1 (D) L = 1
N(3)S/18/2025/A P.T.O.
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Part II(m II)
z-i z— i
61. If = 2, then the loci of the 61. =2 "rff z
Z+7 z+ 7
complex points z is a/an :
(A) Circle (B) Parabola (A) ^ (B) "TOOT
(C) Ellipse (D) Hyperbola (C) (D) a^f^TTOOT
62. The solution of complex equation 62. |z|-z=1+27 ^ ^
|z|-z = 1+27 is : t:
(A) 1-2/ (A) 1-27
(B) 2-1/ (B) 2--/
(C) 2+2.: (C) 2+ 2/
(D) 2-2/ (D) 2-2/
63. If Zj and Z2 be complex numbers such 63. Zi ^ Z2 t
that Zi +IZ2 =0 and arg (zj Z2)- tu then, fe, Z]+/Z2 =0 3lk (Z]Z2)= 71
arg (z,) is : cfl =bl"H'=h (Zj) t :
(A) J (A) J
(B)f (B)f
¥ (C) ^
¥ ¥
IM(3)S/18/2025/A
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64. Which of the following equations is 64. PHHRrlRsId "4' ^
true ? % 7
(A) sin ^ z ~\\ogliz+^ll-z^ (A) sin ^ z = ilogj/z + Vl-z^
(B) cos ■^z = ylog|/z + 's/l + z -I. 1
(B) cos • z = Tlogjzz + 's/l + z^
(C) sinh"' 2 = Iog{z-V2^ -1 (C) sinh ^z^logjz-Vz^-l
(D) cosh ' z = Iog|z + Vl + z^ (D) cosh ^z = logjz + Vl + z^
/i \n
65. The domain of convergence of the power 65.
l~Z
n=\
\n K ^ /
00
1.3.5....(2^-1) fl-z'
series ^ is :
W=1 \n. I^
(A) Inside the circle of radius — and (A) y2 srk ^ 4 ^ ^ ^
center at —
(B) Inside the circle of radius — and (B) ^ ^ ^ ^
center ~
(C) Inside the circle of radius — and (C) j sik ^ ^^^
•31^ ^ ^
center
(D) Inside the circle of radius — and (D) r. 4 N
^ j^^
V /
37^ «TFT
center at ~"j
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CO / I
66. The radius of convergence of the power 66. W M y 1+- 2"
CO / j
n=l ^
series ^ 1+- n
z" is :
t:
n=]
(A) e (A) e
® 7
(C) 1 (C) 1
(D) (D)
67. If function /(z)= secz, then : 67. /(2)= secz ^:
(A) function /(z) is analytic for every (A) /(z), z ^ "RHf ^
value of z
(B) Function /(z) is analytic at (B) HJtni /(z), z=^
n
•"^Wr
^=2
(C) Function /(z) is not analytic at (C) /(z), z = 7t
z = 7t 5V11
(D) Function /(z) is not analytic at (D) /(z), z=-^
n
5V11
'=2
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68. If f{z)= uix,y)-\-iy{x,y) is an analytic 68. ''if^ ./(z)= M(x,_y)+/v(x,>')
function and z = re'®, then polar form
of Cauchy-Riemann equations is ; ^ ^t :
du _ dv du ^ \ dv du ^ dv du ^ 1 dv
du \ dv du dv du 1 dv du dv
dr T-ae' 50' '' dr dr rdQ' m ^ dr
du \ dv du \ dv du \ dv ^ du _l dv
(C)
dr rdQ'S0 r dr dr m~rW
du _ dv _ du _ I dv du _ dv du _ I dv
'dr~''''W' m~7dr
69. If w(x,>')=jc^-3:^ is a harmonic function, 69. u(x, y)= x^ - 3x^
then the value offiinction/(z) is : t, f{z) "HH :
(A) iz^ + ic (A) iz^ + ic
'T,
(B) + ic (B) z^ + ic
(C) 3z2 + c (C) 3^2+ c
(D) None of the above (D) ^ ■?T^*
70. If a+;p = then the value of 70. ^ a+/p = /^''""'P^ ^ a^ + p2 ^
a^+ p2 is - ^ I
(A) ^-(4«+i)7tP . (A) g-(4n+l)jtP
(B) g(4«+l):tp • (B) g(4n+l)np
(C) g(2«+l)7iP (C) e(2«+l)7iP
(D) g-(2/>+i)7tP (D) g-(2«+l)7ip
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i L
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71. If a function f{z) is analytic and one- 71. ^ "^3;^ hR1.<2II C ^ 3T^ 3IR
valued inside and on a closed contour C, ^ /(z) srk
then :
(A) J[,/(zVz5'0 (B) |,/(z)rfz=0 (A) j^/(z)dz:>tO (B) J^/(z)rfz =0
(C) I,/(zWz = l (D) J[,/(zyz =i (C) I,/(z)& = l (D) ]^A^)dz=j
72. If/(z) is analytic in a ring-shaped region 72. /(z) ^ Ci aSk C2 ^
bounded by two circles C| and C2 and Cleric ^ 31^ "k'
a is a point in the region between Cj and it Cj 3ik C2 ^ ^ k* a
C2 where Cj is the outer circle then : i\, C| qq :
1 .1 m 1 .jm
(A)
c, c,
_L'r/(£)dz J_ [JM dz
27r/ J z-a 271/ J z-a
(B) (B) =
J_ r/W dz 1 rm dz
f
27:/ J z-a 2uii J z-a
(C)
C, c,
J_ ^/z
J_ r/(£)dz
271/ J Z-Cf 27t/ J z-a
(D)
C,
J_ f M dz
0-
2ni 1
J z-ha 271/ J z+ a
Cn
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1 I
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73. Let /(z) be analytic within and on the 73. TTHT /(z) D ^
boundaiy C of a simply connected region
^ C ^ ^ sfR C ^
mR^IhI ^ t sfR ttptt a,
D and let a be any point within C. Then
derivatives of all orders are analytic and
^ f fFT
given by : • ^t :
/(«)(„)=^\^^dz
(A) 27ti J (z+ «)"+' (A) 2t' J (z+ «)"+'
/(")(„)= 1 f_JM. dz /(z) •i/z
(B) ^ 2ni
2ro JJ (z-a)"""
^\n (B) (z-a)'
(C) ^ 2?:? J (z . -N«+l
+ a)' (C)
/(«)(a)=^f (fe
21' J (z+ a)"^'
(D, (D)
(z-a) 21'J (z-a)"+'
74. If /(z) be analytic within and on a 74. ^ /(2), |z - ZqI ~ r ?[Rr sR^iRia
circle C defined by \z - Zq\ = r and if ^ C ^ ^ ak C m ^
|/(z)| < M on C, then : afk ^ C tR- |/(z)| < M ^:
(A) \f^"\zo)\^n\— (A) |/"\zo)|^"!—
,M
(B) |/^"^(zo) <n\—
(B)
(C) <n\—
r'' (C) [/'"'(zq) <«!—
r"
,M
(D) (D) |/^"'(zo) >n\—
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r
•%
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75. A function which has no singularity in 75. ^ ^ WT ^
the finite part of the plane and has a pole WTT t 31% tR ^ ^
of order n at infinity is a polynomial of ^ 'W ^ 13^
degree #11 I
(A) 0 • (A) 0
(B) 1 . (B) I
(C)? (c) I
(D) n ■(D) r,
76. A function/(2) is a polynomial of degree 76. W^/(z) W n ^ #n t
n if and only iff(z) has no singularities 37k f(z) ^ "qkfkfi
in the finite part of the plane and has w ^ t 37k 37#
a of order n at infinity. /? f I
(A) Pole (A)- W
(B) Singularity (B)
(C) Zero (C) ^
(D) None of the above (D) k* ^
77. All the roots oi - Sz^ + 12 = 0 lie .77. _ 5^3 + 12 = 0 ^
between the circles \z\ = 1 and |z| = 1 3^' t I
(A) lz|=3 (B) |z|=2 (A) |z|=3 ' (B) U|=2
(c) i.i4 ■(D) |.| = 2 (c) 1.14 (D) |z| = 2
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78. If function f{z) is regular in the whole 78. z-m^ ^ 37^ 1337 f(z)
2-plane, and if |/(z)|<M for all values 2^ irf ^
of z (where M is some constant), then
/(z) must be a t) il, 'm f(z'j ^ -Qqr :
(A) Constant (A)
(B) Variable (B) ^
(C) Point (C) ^
(D) None of the above (D) -4* ^ ^
79. A of a function f{z) is a 79. WT f{z) m ,'qF t fel
point at which the function ceases to be ^t I
analytic.
(A) W
(A) Pole
(B) ^
(B) Zero
(C)
(C) Singularity
(D) None of the above (D) T? ^ ^
1 1
80. The residue of , 2 2x2 at z = ai 80. z = ai 71 71" ^ 37q^
{z +a ) (z +£7 )
IS : t:
-t -I
(A) (B) (B)
4^ 4a'
-I -I
(C) (D) (C) (D)
2a' 2a' 2a' 2a'
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81. What is the number of ways of choosing 81. "TO ^ 52 ^ ^^
4 cards from a pack of 52 playing cards,
so that all four cards are of the same suit "*7% ij] (fs, ""TH)
(diamond, club, spade, heart) ?
(A) 715 (B) 1860 (A) 715 (B) I860
(C) 2860 (D) 270725 (C) 2860 (D) 270725
82. If A = {a, b, c} and S = P(A), (power 82. A= {a, b, c} ^8TT S = P(A), (AW
set of A), then what is the number of wr ii, s
elements in S which do not contain any W P = {c} W
element of P = {c} : ^^ il :
(A) 2 (B) 4 (A) 2 (B) 4
(C) 6 (D) 8 (C) 6 ■ (D) 8
83. The number of integers in between 333 83. 333 ^ 999 ^ lyife f,
and 999 which are divisible by 2 or 3, is : 2 "ZIT 3 ^ f ?
(A) 111 (B) 223 (A) 111 (B) 223
(C) 333 (D) 445 (C) 333 (D) 445
84. Find the remainder when the sum of 84. ^S = l!+2!+3!+ + 10000!
5 = 1! + 2! + 3! + lOOOQ! is ^^^ 4 eft ^
divided by 4. wn 01 =b1f^y. I
(A) 0 (B) 1 / (A) 0 (B) 1
(C) 2 (D) 3 (C) 2 (D) 3
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Which are the three mutually incongruent 85. 6x = 15 (mod 21)
solutions modulo 21 of the congruence ■ "^-chlH^ 3 21
6x= 15 (mod 21) ? f ?
(A) -1, 6, 15 (B) -1, 6, 20 (A) -1, 6, 15 (B) -1, 6, 20
(C) -8, 13, 20 (D) 6, 13, 20 (C) -8, 13, 20 (D) 6, 13, 20
For ri > 1, the sum of the positive 86. M > 1 ^ ^
integers less than n and relatively prime
to « is : a, t :
(A)_^«<!)(«) (A)
(B) ««) (B) <1.(17)
(C) ni/{n) (C) /7(1)(«)
r.. i\
(n^\ « -l
(D) (D) '1'
V 2,
87. If the integer a has order, modulo n and 87. Tfe.a, k n W Vm t,
h > 0, then what is the order of ? h> 0 cK ^^ ?
(A) Idgcd (h^ Jc) modulo n (A) kigcd {h, k) modulo n
(B) gcd {h, modulo n (B) gcd {h, k) modulo n
(C) k.gcd {hy k) modulo n (C) kgcd Qiy k) modulo n
gcd(h,k) gcd(h,k)'
(D) modulo n (D) modulo n
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88. If A: > 3, then how many primitive roots 88. ^ > 3 t; ^ 2^
does 2^ have ? 7
(A) 20 (A) 20
(B) 2' (B) 21
(C) 2'^-' (C) 2^-1
(D) No primitive roots (D) T#n
89. How many number of non-negative
89. l^cFTt WIT A'
integer-triplets (x, y,z) will be such that
(x, 2) x+_v + 2=17
X +_y + 2= 17 when 0 <x < 3, 0 <y <2
^-0<x<3,0<>'<2 ^ z> 6 ?
and 2 > 6 ?
(A) 11 (B) 12
(A) 11 (B) 12
(C) 17 (D) Infinite (C) 17 (D)
90. What is the all roots of the congruence 90. xl^+6xl'^ +2x^ + 1 =0(mod 5)
xl7 + 6x^^^ + 7x^ + 1 = 0 (mod 5) ? ^^ ™t?
(A) X = 1 (mod 5) (A) X = 1 (mod 5)
(B) X = 1 (mod 5) and x = 2(mod 5) (B) X = 1 (mod 5)^X = 2(mod 5)
(C) X ^ 3 (mod 5) (C) X = 3 (mod 5)
(D) X = 1, 2, 3, 4(mod 5) (D) X = 1, 2, 3, 4(mod 5)
91. If/: S3 S3 is an automorphism 91. S3 -> S3 ^^IcblRdl (3iT<iMlflJh^^)
such that /(I, 2, 3) = (3, -2, 1), then t ^ /(I, 2, 3) = (3, 2, 1); ^
/(I, 3, 2)= ? /(1,3,2)=?
(A) (i; 2, 3) (B) (1, 2) (A) (1, 2, 3) (B) (1, 2)
(C) (1, 3) (D) (2, 3) ,. (C) (1, 3) (D) (2, 3)
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92. If G is a group of order 108, then it has 92. G 108 ^ -gq f,
a normal subgroup of: "qfife ^ I
(A) Order 27 and Order 6 (A) 3#^ 27 ?T81T 3#^ 6
(B) Order 27 or Order 9 (B) 371^ 27 ^ 3;n^ 9
(C) Only Order 27 or Order 6 (C) 371^ 27 "qi 3fi^ 6
(D) Only Order 9 or Order 6 (D) 3;#^ 9 "qr 6
93. If 0 is a group such that : 93. "qfq G "0;^? "gq t :
G = CixC2X xC„ G = CixC2X xC„
where each Cy is cyclic group and t; iq t 3^ ic^-i = 2,
'|C^-| = 2, then the order of G is : , " cK G "qq 371^ ^ :
(A) 2"-l (B) 2"-' (A) 2"-l (B) 2"-l
(C) 2"-2 • (D) 2" (C) 2"-2 (D) 2«
94. If G is a group which has no proper 94. "qR G t iqw ^ mqr
t
normal subgroup, then G can be : ■ qfife "^iqgq t, qq G ^ t:
(A) S3 (A) S3
(B) GL2(R) (B) GL2(R)
(C) Z2 X Z3 . (C) Z2 X Z3
(D) None of the above (D) "^qjqq ^ qj^ q^'
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95. The smallest value of n for which 95. n ^ ^
• Z2 X Z2 X Z2 is isomorphic to a Z2 X Z2 X Z2 mM ^ s„ ^
subgroup of the symmetric group is : i\ ■^TTTT, t ?
(A) 4 (A) 4
(B) 5 (B) 5
(C) 6 (C) 6
(D) 7 (D) 7
96. Let G be a group and H be a non-empty 96. Hiii G "I H
subset of G Then H is a subgroup of0 t I^H (G
if and only if : ^ 'sSk tiqoi :
(A) a, h G H G H (A) a, b e U => ab e U
(B) a, 6 G H ab~^ g H (B) a, G H => GH
(C) a G H => G H (C) a G H => GH
(D) None of the above (D) ^
97. An integral domain D is said to be 97. iiim D ^
of finite characteristic if there exists a ^'iicHch "tjyife
positive integer m such that : w ^ t :
(A) ma = 0, for all a g D (A) wa = 0, aGD ^
(B) ma = 1, for all a g D (B) ma = 1, fl G D
(C) m^ = 1, for all a G D (C) = 1, aGD ^
(D) None of the above ■ (D) ^ ^ -^T^'
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98. Choose the correct option from the 98. :
following : (A) Q[x] Tiffed PID t I
(A) Q[x] is a field and also PID. (B) Q[x] PID t=> Q[x]
(B) Q[x] is PID Q[x] is a field. - tI
(C) Q[x] is neither a field nor PID. (C) Q[x] ^ ^ t ^ ^ PID t I
/
(D) Q[x] is PID, but not a field. (D) Q[x] PID t iI
99. Let R be a ring and U is a proper ideal 99. - ^R 13^1^ t ciaiT U WR
t |-d^R/Uq:^t^tl^R
of it. The RAJ is a ring known as :
1^* I
(A) Homomorphic image of R (A) ^Hichifl (Homomorphic
Image)
(B) Isomorphic image of R
(B) (Isomorphic Image)
(G) Both (A) and (B)
(C) (A) (B)
(D) None of the above (D) ^^
100. Consider the following polynomials : 100. Pi fen Had nitnlniPi^en ^ 1eT^ aiin-yi
(i) x^ + 3x + 2e Z/(7) M ■?Tc^ t ?
(i) x3 + 3x + 2 G Z/(7) [x]
(ii) + 8 e Q[x]
(ii) x4 + 8 G Q[x]
Then :
(A) t I
(A) Both the polynomials are irreducible. >
(B) MletlnlfHacn (i) ^Is^J^qen "I*
(B) Polynomial (i) is irreducible and
MldldlPH^d (ii) t I
polynomial (ii) is reducible.
(C). t I
(C) Both the polynomials are reducible.
(D)
(D) None of the above
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101. The initial value problem 101. "RPT
^=n/H. K0)= 6 ■ f = ^M,X0) = 0
has : :■
(A) No solution (A) ^ ^ t
(B) A unique solution (B) ^ f
(C) Two solutions (C) t
(D) Infinite solutions (D) t
102. The singular solution of differential 102. y^(\ + p^)==\ ^ "^ff^
»
equation y^(l+ p^)= l is (where
^ p =^ ) :
ax
-■
(A) X =±\ (B) y =±l
(A) X = ± 1 (B) > = ± 1
(C) X =±y iC) X =±y (D) X = 0
(D) X = 0
103. The ^Green's function for 103. /-2y-3 = sinx ^
y"~2y'-'i = s\nx is.: f :
(A) i[, ] . (A) i[..Kx-t) _ -(x-t)
(B) i[.,(^-0 _ ] ■ (B) J
• (C) +£-('-')]
(D) 4[e(*-0 ^ Q-ix-t) ] (D) 4[e(^-0 + e-(A:-0 ]
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104. The non-trivial solution of the Sturm- 104. :
Liouville problem :
+ Xy = 0, y(0) = 0,y(n) = 0
dx^
+^y = o, xo)= =0
<h?
^ ^ ^biT,
exists if:
(A) X = 0
(A) X. = 0
(B) X ^ TRsqr ^ I
(B) X is a prime number
(C) X
(C) X is a non-negative perfect square
number
^ I
(D) X is a non-negative odd number (D) X ^ I
105. Solution of the differential equation 105. 31^^
{y+xy-¥y )dx-vix-^ 2y)dy =0 (y+^+ y )dx+{x + 2y)dy = 0
is : ^ "t : -
(A) e^(;g;+2/)=C (A) e''{xy +2y^)= C
(B) xy+2y^-Q.e^ (B) xy+ 2y^=.C.e''
(C) xy-\-y^ = C.e^ (C) xy+ y^=C.e'
(D) e\xy^y^)= C (D) e^{xy+ y'^)= C
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Text of page 24
106. If and y2(x) are complementary 106. yi(x) Tim y2{x)
functions of
2
ao(.x)^ +a^(x)^+a2(x)y = F{x) "" ^+'>i(x)^+a2(x)y = F(x)
ax , ^
then by using the method of variation of ^ ^<ni eft Mi-qtn
parameters the complete solution is given ^ ^EF^yf
by
y(x) = v^(x)y^(x) + V2ix)y2{x\
yW = v^(x)yi(x) + V2(x)y2ix),
where Vj(x) and V2(x) are defined as
^ Vi(x) ^2TT V2(x) "51^
(W is Wronskian of and ^^2) • (W, y^ ^8IT ^2 ^ t) :
(A)
(A) =
(B)
(C) ^iW \aoMWix) ' <•:)
V (r^ = f ^(X)y2(.x)
(D) ^IW
OoixWix) <D) '■'''—
_ f FW3^i(x)
_ r FWziW dx
V2W=J ao(x)W(^)
V2W= J flo(x)WW dx
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Text of page 25
107. The solution ofpartial differential equation 107. ^qq>cn
z{xp~yq)= y^-:p- z(xp~yq) = y2 ~~x 2
IS : ^ ^ ^ :
(A) x^+y^ +2^)=0 (A) f{x + y, x^ +y^ +z^) = 0
(B) f{x + y,xyz)=0 (B) ;!^z) = 0
/(^> x^ + y^ +z^)=0 f(.xy,x^+y^ + z^) = G
(D) f(xy,x + y+z)= 0 (D) fixy,x+y + z) = 0
108. The particular integral of partial 108.
differential equation dp
dp
■^+p-q = z + xy is
^ t p=^
dx
^arr
(where, p = -^ and 9 = ^).: dz .
dy
(A) 2= \ + X + y + xy
(A) z - \ + X + y + xy
(B) z = I + X ~ y - xy
(B) z = I + X - y ~ xy
(C) z = I - X - y + xy
(C) z = l-x-^ + ;ii^
(D) z = \ - X - y - xy (D) z = \ - X - y - xy
109. Using the method of separation of 109. ^ ^ Mn ^ TPiW ^
du ^ du
variables, the solution of ^ = 2 ^+ w;
ox ot
■^-2—+«; u{x,Qi)-6e ^ ^
dx dt
M(*,0) = 6e"^'' is : #n :
(A) n = (B) u = (A) (B) „ = 6e-3^-2/
I
(C) M = 6e-3'+"2 (D) u = ee-^'-"- (C) (D) =
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L
Text of page 26
110. Which of the-following represents two- 110. rHHPdRsld ^ ^ f|3TPTTTft
dimensional heat equation ?
d^u
= C'
(A) (A)
dp-
d\ 9^w du du d^u d\ d'^u du du d^u
(B) 3/2 = C' ax'^'^ dx'dy'^ ^2^ (B)
dp
=C
d^u d\ d^u d^u
(C) (C) &2
d^u du du d^u 9m p2 5w du d\
(D) 37=^ (D) 3*2
111. The double root of the equation 111. ^ ^
f{x)=i?-x^~x +\ =0 f(x)= x^-x^-x+l=0
' using Newton-Raphson method is ^ im ^:
(A) 1.0091 (B) 1.0001 (A) 1.0091 (B) ,1.0001
(C) 1.0010 (D) 1.0032
(C) 1.0010 (D) 1.0032
112. Using Euler's method for the solution of 112. ^=^^+>'>XO)= i ^
the differential equation
f=^2+;;,X0)=l,
then the value of y (0.05) is (taking
;^(0.05)^ TM ^(-^ A = 0.05) : ■
h = 0.05):
(A) 1.051 (B) 1.025
(A) 1.051 (B) 1.025
(C) 1.075 (D) 1.0
(C) 1.075 (D) 1.0
N(3)S/18/2025/A 24
Text of page 27
■ <iy X
113. Using Picard's method for the solution of 113. ~
the differential equation
..2
^ y(0.25)
dy
y(0) = 0
■ dX
the value of y(0.25) is :
(A) 0.70 (B) 0.005
(A) 0.70 (B) 0.005
(C) 0.421 -(D) 0.321 (C) 0.421 (D) 0.321
114. The error in the Trapezoidal rule is of the 114. ^
order : ^t :
(A) h (B) (A) h (B)
(C) (D) (C) (D)
115. Simpson's one-third rule of integration is 115. ^ ^
exact for all polynomials of degree not ^ Pn, ■'TOST t "To
exceeding the following : rHMPcifeld ^ H i[ :
(A) 5 ' (B) 4 (A) 5 (B) 4
(O- 3 (D) 2 (C) 3 (D) 2
116. The function whose fust difference is e*, 116. "JTSTTT t,
is : ■ #TT : '
(A) (B) (A) (B)
£*+1 £"-1 e'^ +l
x+h
(C) e
x+A
(D) ex-h (C) e (D) ex-h
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Text of page 28
117. The function y - sinx is tabulated 117. = sin^: ^ f:
below :
X y =sinx
X ;V = sinx
0 '0
0- 0
n
7t
0.70711
0.70711 T
4 71
Tt
1.0
1.0 2
2
Using Lagrange's interpolation formula ^^ WIFt ^ sin
the value of sin — is : ^t :
(A) 0.51743 (A) 0.51743
(B) 0.51375 (B) 0.51375
(C) 0.54175 (C) 0.54175
(D) 0.57011 (D) 0.57011
118, The rate of convergence of Regula-Falsi 118. ^ ^
method is : . t:
(A) Linear (A)
(B) Quadratic (B) Urohr
(C) Cubic .(C) fWN
(D) Cannot be determined (D) W ^ -^mnTT
N(3)S/18/2025/A ■ 26
4 k
Text of page 29
119. TTie approximate value of the integral 119.. ^ ^
1= Jof -^dx
l+ x .+x
•etc
using Simpson's rule is : ^ W\Wl "TO :
(A) 0.69351 (A) 0.69351
(B). 0.63915 (B) 0.63915
(C) 0.96444 (C) 0.96444
(D) 0.69444 (D) 0.69444
120. If interval of difference is unity, then : 120. SRR ^ ^
■ 1 ^ =9
1 ^ =9
./wj ,/WJ
Af(x) A/W
/W/(x +l) (A) /W/(x+l)
A/(;c+l) A/(;c+l)
nx+\)f(x) (B) 7o?Ti)7w
Af{x) . A/W
/W/(x+l) (C) /w/(x+l)
Af(x +\) A/(;c+l)
(P) -7(iTi)7w (D) f(x +l)f(x)
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I
Text of page 31
123. A stationary function for an integral 123. ^
functional is one for which the variation "I", "qit
of that integral is : ^t :
(A) Zero (B) Positive (A) ^ (B)
(C) Negative (D) One (C) (D)
124. The extremal of the functional : 124. chiqlcHcti
ji/2 Ji/2
I [y2-/+2^]dt J [y2_/+2^]A
0 0
that satisfies the boundary conditions ■qiT ?T?ff ' xo)- = 0;
/• \
n
y{0) = 0; y =0 is : =0 ^ :
Jy
,(A) y=-^sinx (A) ;' = ^sinA:
(B) y = x~smx (B) smx
7t .
(C) j/= A:+-2SinA: (C) ^ = a:+—smo:
(D) y = x (D) y = x
125. The Euler's equation for the variational 125. TPTRT ^ ^
problem
Minimize :
1
1
l[y{x)]=\C2.x-xy-y)ydx I W] =j(2x-xy- y')y'dx
0
IS : t :
.(A) 2f^y^2 (B) 2/+x=2 (A) 2/-7 = 2 (B) 2/ + y = 2
(C) /+2j^=:0 (D) 2/-y =0 (C) / + 2:i^ = 0 (D) 2/-;;= 0
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Text of page 32
126. Consider the functional 126.
Xi '2x1/2
VbMl-J' '' X V[yW]=}^
*2 n J.,.'2x1/2
dx
Then
/ «d/2
F=
[uyf a:
F=
{'*']
is valid if: ■qF=q t :
d¥ _ ^ .
(A) ^— Constant (^) 'dy' ~
i: ^ .
(B) Constant (B)
(C) ^ ^
(C) Extremal of the given functional is
a cycloid.
(D) Hidich ^
(D) Extremal of the given functional is
a straight line.
127. The extremum of the functional 127. I[X^)]= jy'^^dx, y> 0 ^
0
a
jy y > 0 and a > 0 a>0 "SIM f :
0
attains :
(A)
(A) Strong maxima
(B)
(B) Weak maxima
(C) Strong minima (C) J^lRklJ^lldl
(D) Weak minima (D)
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Text of page 33
128. If JM= J(y^+2/y+y°)A, 128. ^ J(X= j(y^+2yV + y®)^^,
;;(0)= then extremal is : ■ y(0')= e^, cR RIT f :
1 1^
(A) i=y 4x+-y (A) I = X
y e^J e /
^ 1 ^
(B) i=y^W+\ \ 5 /
(B) i=y 4x^ +—
e^J
1 ' 1^
(c) i=y 4x+-^ (c) i=y 4x +-^
V e^J V J
/
•5 1
(D)'I= J^ P) i=r 4x+-^
\ ^j
129. If F[X:t)]= J"/[:*:,X^XyW]'3^> then 129.^ F[X^)]= J/[x,X^),yW]iic, ^
a a
F is called : F ^^vildl t :
(A) Functional (B) Range (A) (B)
(C) Arc length (D) Curvature (C) (D)
130. In the Brochitiscrone problem, particle is 130. 'HRfTO
given an initial velocity Vq = 0, the path ^^ ^ WT ^ "qsiT ^
of minimiim time is : t:
(A) Parabola (B) Cycloid (A) "TOOT (B) -qsh^
(C) Catenary (D) Ellipse (C) (D)
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Text of page 34
131. The integral equation 131.
(|)(x).w(x)= f(x)+ X, K.ixd)u(t)dt,
a<x,t<b a<Xyt<b
ctieeiicii ^ :
is called :
(A) Volterra linear integral equation (A)
(B) Volterra non-linear integral equation
(B)
(C) Fredholm linear integral equation
(C)
(D) Fredholm non-linear integral
equation (D)
132. The'solution of {x-t)u{t)dt is : 132. (x-t)u(t)dt ;
(A) (B) ^u{t)dt (A) u(x) (B) ^u(t)dt
(C) lu{x)dt (D) (C) lu(,x)dt - (D) ^ix-t)dl
133. The solution of Volterra integral equation 133.
,"W =i+f"CV "W =l+f
is : ^ f:
(A) u{x)= e'' (A) M(x)= e^
(B) u(x)= e -X
(B) M(x)= e~^
(C) «(x)= 0 (C) uix)=0
(D) m(x)= oo (D) w(x)= co
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Text of page 35
134. The solution of Fredholm integral 134.
/
equation
u{x)=^x-^+]^^{x+t)u{t)dt
u(x)=^x-'^+^ ^(x+t)uit)dl
^ t' :
is :
(A) u{x) - — X (B) u(x) = X
(A) u{x) ~ ~ X (B) w(x) = X
(C) u(x) = X + t (D) u(x) = X - t (C) w(x) = X + r (D) m(x) = X - t
135. The initial value problem corresponding 135.
to the integral equation ;
y(x)=\+^ y(t)dt
y{x)=\+[y{t)dt
rx
^ #TcI TM t:
IS :
(A) y-y = o,yo)= i .
(A) y'~y = 0,y(0)= l
(B) y+>'= o, xo)= o
■(B) y'+y=^0,yi0) = 0
(C) y'-y = 0,y(0)=0 (Q y->'=o, yo)=o
, (D) y'+ y = 0,y(0)= \ (D) y' + y = 0,y(0) =l
136. The values of X for which the integral 136. X 1TR iFtoq
equation
yx)=x J^(6x-/)x/)i//
3T-"5^ T^sRTT "I, fiHpQinsia
has a non-trivial solution, are given by
OT 1^ :
the roots of the following equation :
(A) (3X-l)(2+^)-X^=0 (A) (3X-\)(2 + X)'X^=0
(B) (33,-l)(2+ X,)+2= 0 (B) (3A,-l)(2 + A,) + 2 = 0
(C) (33,-l)(2+2,)-4X,^=0 (C) (3A--1)(2 + X.)-4X^ = 0
(D) (3X-1)(2+ X)+3.^=0 (D) (3X-V){2 + X) + X^^0
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Text of page 36
137. The eigenvalues of the integral equation 137.
rl
_y(x)=^^ ~^x^)y{t)dt y(x)='k^(2xt-4x^)y(t)dt
are :
(A) 0, 3 (B) 3,3 (A) 0, 3 (B) 3,3
(C) -3,.3 (D) -3,-3 (C) -3,3 (D) -3,-3
138. The solution of the integral equation 138.
rl
IS : t:
-X -X
(A) £ (B) (A) e (B)
2x -2x 2x -2x
(C) e (D) e (C) e (P) e
139. The solution of the integral equation 139. ^Hl^d
rl M 2
g(^)=.y+|j su^g{u)du g{s)=s+ ^ su g(u)du
IS : ^^ f :
(A) g(0 =j (A) g(0=j
(B) git)=j (B) g(0^|-
(C) g(0=|- (C) g(0=f
|
(D) g(0= |
(D) g(0=
N(3)S/18/2025/A 34
Text of page 38
142. Lagrange's equation for standard system 142. q, ^ X(^,9)
S with generalized coordinates q, kinetic {Qj} ^ "HTar
energy T(^,^) and generalized forces
t:
d ax. ax d ^ ax^ ax
(A) Tt. dq: , dqj = 0 (1 < 7 < «)
(A) di a^^-j dqj
=0 (1<;<«)
\ V J
ar
(B) Tt dqjj+-^=Q; i\<j<n) d ^ax^ ax
dqj .(B) -dt
d d ^ax^ ax
\
\ !
V /
-Ik-Oj (C) Tt -=Qy (l<y<«)
oT ax
dqj) dqj-=Qj i\<j<n)
ax ax
(D) (D)
dr dV a?^-
143. Xhe motion of any mechanical system 143. ^ ^ ■qfd ^
which performs small motions near a
position of stable equilibrium is called : qieoiiol "I :
(A) Stable oscillations (A)
(B) Simple oscillations (B)
(C) Small oscillations (C) ^
(D) Simple harmonic motions (D)
t
144. Find the degree of freedom of a ball 144. ^ Its; ts;^ qft
rolling on a rough table : "t :
(A) 1 (B) 2 (A) 1 (B) 2
^ (C) 4 . (D) Not defined (C) 4 (D).
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Text of page 39
145. Which of the following is not true ? ■ 145. PHHRrlRsld ^^ f ?
(A) Each compact regular topological (A)
space is normal. ^t I
(B) Each compact Hausdorff space is (B) mrmPT
normal. t I
(C) A real line is a connected space. (C) ^ ^ ^
(D) The product of two second t I
countable spaces is not necessarily (D) ^ wTteif ^ "3^
second countable. wfhr #TT 11
146. A topological space whose topology has 146.
a countable base, is : f, ^ t :
(A). Dense (A) WT
(B) Separable (B)
(C) Compact (C) ^
(D) Connected (D)
147. Each compact subset of a Hausdorff . 147. ^ ^
space is : ^t :
(A) Open (A)
(B) Closed (B) ^
(C) Dense in itself and open (C) WT
(D) None of the above (D) "5^1^ ^^
N(3)S/18/2025/A 37 P.T.O.
Text of page 40
148. According to Heine-Borel theorem a 148.
closed interval of a real line is : ^ ^t :
(A) Connected (B) Compact (A) (B)
(C) Normal (D) Metrizable (C) (D)
149. If X and Y are two topological spaces, 149.^ X cfSTT Y ^ "HTTfeff t,
then X X Y will be a compact space if cil Xx Y ^ ^ 3k
and only if: "^Tf^ :
(A) Only X is compact (A) X ^
(B) Only Y is compact ' (B) Y ^
(G) Both.X and Y are compact (c) X ^8iT Y."#ff a
(D) None of the above (D) ^^
150. Consider the following statements : 150. :
I. Every second countable space is
I.
first countable space.
"TOh ^t I
II. A non-empty subset of the real line
II. qi'Kiiqcti i<2ii ^ 3tR^
is always connected.
"I 1
Which of the above statements is/are
true ?
(A) Only I is true. (A) I WT t I
(B) Only II is true. (B) ciiciQi II f I
(C) Both I and II are true. (C) I II ^t I
(D) Neither I nor II is true.
(D) ^ ift I ^ ^ II ^ t I
N(3)S/18/2025/A 38
Text of page 43
65. For independent random variables 65. Xy, i = 1, 2, n^
X,-, / = 1, 2, «
^Xi+X2+. x„(0 txi+X2+ :x„(0
is equal to : #IT :
(A) + + + <t'x„(0 (A) •I'Xi(0+^2(0+ +'1>X„(0
' (B) "^XiW-^^x^W ^x«(0 (B) 'l>Xi(0.<l>X2W.."-'l>X„(0
(C) -t-Xi-Xj x„(0 (C) 't'x,.X2 x„(0
(D) Ail of the above (D)
66. If probability density function of a 66. X ^ yiRlchdl
random variable X is
0<x<l X, • 0 < X ^1
/w= 2-x, l<x<2 /w= 2-x, 'l<x<2
0. otherwise 0,
then its cumulative distribution function t, ^ x = i.S-'qK "sw ^ WT
F(x) at X = 1.5 is : ^ F(x) #TT :
)
.(A) 0.75 (A) 0.75
(B).l (B) r
(C) 0.875 (C) 0.875
(D) 0.732 (D) 0.732
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Text of page 44
67. Let X be distributed with probability 67. ^ X yiPlchdl WT
density function
[0, otherwise
f{x)=['
^ [o, SPOT
Then which of the following statements
is true ? t?
(A) p. (A) P x-i
^ 2
(B) P
^-2 <2J- <0.75 (B) P
^-2 <2J- <0.75
^-2 >/i2jUo.75
(C) ?• (C) P •<
^-2
(D) P <2J- >0.75 (D) P
^-2 <2J— >0.75
68. In Bayes theorem which is accurate ? 68. "^f t ?
(A) P(E^-) is priori probabilities (A) P(E,.) "RTfedT^ t
(B) P A/ is likelihood (B) t
v/E,v
(C) p(%)is posterior probabilities (C) hIwIR^k yiP^chdnt t
(D) All of the above (D)
ISI(3)S/18/2025/A 42
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Text of page 45
69. A coin is tossed until a head appears. 69. ^^ t
What is the expectation of the number of -m w: (Head) ^ I
tosses required ? ^ WIT ^ TO -RcTOT i\Jj\ ?
(A) 1 (A) 1
(B) 2 (B) 2
(C) 3 (C) 3
(D) 4 ■ , . (D) 4
70. Central limit theorem for independently, 70. TWH ^ ^
identically distributed random variables
^ t:
is :
(A) 3?^
(A) De-Moivre Theorem
(B) 31^
(B) Liapumoff Theorem
(C) Lindberg-Levy Theorem
(C) 31^
(D) Khinchine-Theorem (D) 31^
71. Which of the following statements is true 71. ^^ %i; PHMRrir^d ^
for the binomial distribution ? W f ?
(A) Mean - Variance (A) = 31WT
(B) Mean > Variance (B) > 31^OT
(C) Mean < Variance (C) < 3IWT
(D) Mean = 2 x Variance (D) = 2 X -qwT
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72. If X and Y are independent Poisson 72. X^Y ^ t"^
variates, then conditional distribution of X/(X + Y) ^ (Conditional)
X/(X + Y) is : ^t :
(A) Poisson (B) Geometric (A) (B)
(C) Normal (D) Binomial (C) (D)
73, The moment generating function of a 73. ^
geometric distribution is : f:
(A)
'*> rv \-qe
\-qe^
(B) ■ (B)
(C) p[l-ge'] m
(C) p[\-qe']
1 1
(D) i(l-pe') (D) i(l-y)
74. Which of the following distribution has 74. RHRrlfisid ^ ^ ^ "RT®? tI8?I
equal mean and variance ? w"! ^t ?
(A) Binomial (A)
J
(B) Poisson (B) WRTT
•(C) Geometric (C)
(D) Negative Binomial (D)
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Text of page 47
75. Coefficient of skewness pj for Gamma 75. "TOT ^ "f^ [q^Hcii Pi ?kTT
distribution is : t:
(A) 3.^ (B) 6+^ . (A) 3+- (B) 6.^
(D) i
76. If a random variable has an exponential 76. ^ ^ ^^
distribution with parameter 0 > 0, then 0> 0 ^ jmn
its variance is : 3TWT t :
(A) 0 (B) 02 (A) 0 (B) 02
(C) ^ (D) 9 (C) ^ (D) 0
77. The characteristic function of standard
Cauchy- distribution is :
(A) e-' (B) eH'l (A) e-' (B) e-l'l
(C) e' (D) j'l (C) e' (D) J'l
78. A Markov chain is said to be 78. ^ TOFT f
homogeneous if transition probability is : TOTO 3#T^ i\ :
(A) More than j (A) ^
(B) Less than — (B) ^ ^ TO
(C) 2 2
(D) Independent of trial number (D) (^I^) ."^F^ ^ TOF^
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Text of page 48
79. Standard deviation of the sampling 79. "Sfrr™ ^ ^
distribution is called : f:
(A) Standard enor (A) ife
(B) Probable error (B)
(C) Sampling error (C) ife
(D) Experimental error (D) wftto ife
80. If Xj, .X2, be a random sample 80. :ci,;C2, ,:c„
from a normal distribution with mean p. TTT?2T p mm a2
and variance then the sampling t;^ S" ^
distribution of sample mean 3c is : "I :
(A) Normal with mean p and (A) p cT«TT TROT ^ "^n^T
variance
(B) Normal with mean p and
(B) "mSR p cT^TT TROT —
n
.2
a
vanance
TOTTH
n
(C) Normal with mean and (C) - tTSIT TROT CT^ ^ RHSil
vanance a' TOTTFT
(D) Normal with mean — and
(D) - -dSIT TROT —
n n
a
vanance TRFOT -
N(3)S/18/2025/A 46
Text of page 49
81. If 0 is the . maximum likelihood 81.- 0 ^ 0
estimator of 0 then for any function t(0), cfl Ftrtil HiQli t(0) HiQH
the maximum likelihood estimator of
t(0) t:(0)
t(0) is x(0). Then the name of the
I ipr ^ ^:
property is;
(A) mm
(A) Consistence
(B)
(B) Unbiasedness
(C) WTPRt
(C) Asymptotic Normality
(D) Invariance (D) Rif-didl
82. For likelihood ratio test 82.
®^p0e©. /0(^b^2 ^^^^^P0e0o/e(^1^^2 ^«)
^ sup0^0/g(:vj,^2 SUP0e0/0(^l,^2 ^n) ■
which of the following is true ? ^ ^ "t :
(A) X>C (B) X<C (A) X>C (B) X<C '
(C) 0<A,<oo (D) 0<X,'< 1 (C) 0 < X < 00 (P) 0'<X< I
83. Let Xj, X2 X„ be a random 83. Xy, X2,.,.., X„'
sample from the density /(x, 0). Let f f(x, 0) t I 'W
T=/(Xj,X2,.--, be a statistic for which T= X2,..., X„) ^^
Pg(T(0) < T)= y. Then T is called : P0(t(0) <T)=y^T^W :
(A) One-sided lower confidence limit (A) 13^^
(B) One-sided upper confidence limit (B) sfdTO
(C) Both (A) and (B) (C) (A) (B)
(D) Neither (A) nor (B) (D) ^ -dl (A) 3ik (B)
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Text of page 50
84. A testy* ofHq :0= Oq versus Hj:0 = 0| 84. Hq ; 0 = Oq Hi ; 0 = 01 ^ "^1%^
is defined to be most powerful test ofsize y* 3TTTO a ^
a if: iWt, :
(A) 0 < a < 1 (A) 0 < a < 1
(B) 7r^*(0o)= a (B) 7ry*(0Q)= a
(C) (01)^ Tty(01) for any other y for (C) 7Cy»(0i)>7Cy(0i) ^
which ny(0o)^cx y^ 7iy(0o)<a
(D) All of the above (D)
85. For Cauchy distribution which is 85. cfM" si'ii 3iicfi(ril
consistent estimator of population mean ? ?
(A) Sample median (A)
(B) Sample mean (B)
(C) Sample mode (C)
(D) None of the above (D) "3^1^ ^^
I £
O<;c<0
O<;c<0 86. fM)= 0'
86. If = 0'
otherwise 0,
0,
Then for testing Hq : 0 = 1 against Hq : 0 = 1 Hi ; 0 = 2 ^ 1^4?
Hi :0 = 2, by mean of a single observed
value of X what would be the size of the 0.5 < A- ^
type one error for critical region 0.5 <x ? ■^Wr ?
■ (A) 0.25 (B) 0.5 (A) 0.25 (B) 0.5
(C) 0.75 (D) 0;875 (C) 0.75 (D) 0.875
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Text of page 51
87. If X - N (p., CT^), then the value of 87. ^ X~N(p, a2) ^ PP|>3) ^
P(|2|>3) is: ^i :
(A) 0.6512 (A) 0.6512
(B) 0.95 (B) 0.95
(C) 0.0027 (C) 0.0027
(D) 0.9999 (D) 0.9999
88. The oldest method of finding point
estimator is : t:
(A) Method of maximum likelihood (A) STfqWT
(B) Method of moments (B)
(C) Bayes estimation (C) ^
(D) Invariant estimation (D) 3PTfM%r
89. 'For large samples from N(p, 95% 89. N(p, a^) ^ ^ 311^
confidence limits for parameter p is : p ^ 95% f^^WlkrlT "#111? t :
X
3a _
I— > X H ^
3a 3a _ 3a "l
(A) (A) X I—, X + ^
v« v«. V V« V^i y
/ \
(B) *-2.58-^, 3c+2.58-^l (B) x-2.58-^, 3c"+2.58-^
V« yinj V« yjn.
(C) 3c-1.96-^, 3c +1.96-^ (C), x-1.96-^,' x.+ 1.96-^
y/n yin) V« yjn)
/ \
(D) x-1.48-^, j+1.48-^ (D) J-1.48-^, 3c+1.48-^
-Jn yjrij
N(3)S/18/2025/A 49 P.T.O,
Text of page 52
90. For Chi-square test of goodness of fit : 90. TftsPT fg :
K K * K K
(A) IO,= XE,- (A)
/=! /=!
;=1 /=1
(B) Total frequency should be greater (B) ^ 50 ^ i\ -
than 50
(C) ^ 5 ^^
(C) No theoretic cell frequency be less
than 5
/
(D) All of the above (D)
91. Degree of freedom for Chi-square 91. 4x5
statistic in case of a contingency table of
order 4 x 5 is :
(A) 9 (B) 20
(A) 9 (B) 20
(C) 12 (D) 16 (C) 12 (D) 16
92. Let 15.4, 16.4, 17.3, 18.2, 19.2, 19, 22.7, 92. 1TH #1^ 15.4, 16.4, 17.3, 18.2, 19.2,
33.6, 24.5 are the observations for testing
19, 22.7, 33.6, 24.5 Hq : M = 20
Hq : M = 20 V5. :M 20, where M
Hi ; M 20 ^ t,
is the population median. The value of
M t I
the Wilcoxon signed rank statistic T^
T" ^ "HH #TT :
IS :
(A) 21 (B) 25.5 (A) 21 (B) 25.5
(C) 20 (D) 20.5 (C) 20 (D) 20.5
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93. LetXi,X2, ,Xi2andYi,Y2, , Yg 93. -qpT Xi, X2, , X12 srk
be two free samples from the two
Yi,Y2, ,Yg^TOT: Fi F2
populations Fj and F2 respectively. The
^^ 1 "^PTFTHT
mean of Wald-Wolfowitz run test statistic
Riy, qirs-qltrMAfqeyi (Wald-Wolfowitz)
for identicalness of population is :
(A) 7
(A) 7
(B) 9
(B) 9
(C) 8 (C) 8
(D) 24 (D) 24
94. Which of the following statements, is ,94. PclcrlchTcHid
correct in the context of"^Icoxon signed PHHRrinsld 1^* ^ ^
rank test ? t ?
(i) ^
(i) Used with two independent
^t 1
samples.
(ii) Used in matched pairs samples.
^t I
(iii) Used with multiple independent
(iii) ^ "^27 "SFlW
samples.
^sTIcTT f I
Codes :
W'
(A) Only (i) (A) (i)
/
(B) Only (ii) (B) chcici (ii)
(C) (i) and (ii) (C) (i)3ftT(ii)
(D) (ii) and (iii) (D) (ii) sik (iii) .
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Text of page 54
95. The following table represents mortality 95. ^ iFlt dlPd^hl
data for two groups of patients receiving "sra xtfrql A 3^
different treatments A and B : B ^ ^ ^t :
Outcome
Dead Alive -allfqa
A 20 10 A 20 10
C3
U
B 15 15 B 15 15
The value of the Chi-square test statistic
for testing independence of treatment and •qfwr ^ ^d'ddi ^ ^
outcome based on above table is : t :
(A)'% (B) (A) 30^^ ^ W 150/
--/35
(C) 30 (D) 150 (C) 30 (D) 150
96. Which of the following is not true 96. (-C) ^ rnHrdPsid
regarding Kendall's tau (t) ? WT f ?
(A) |t|<1 (A) |t1 < 1
(B) If T =7 0, then two random variables (B) ^ T = 0, "dt ^
are independent t
(C) T = - 1 indicates the perfect (C) T = - 1 lyt ^ ^^rfriT
disconcordance ^ I
(D) Asymptotic distribution of estimated (D) T ^ WTFft ^
T is Chi-square 1
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97. If for large n the value of the Kendall's 97. ^ ^ (t) -r-
o
tau (t) is -T-, then Spearman's rank
o
correlation coefficient (R) is (R) w]m ^ itm I -m
approximately where n is sample size. n 311TO t I
(A) I <B, f (A) I (B) f
(D) ^ i
98. Formula for rank correlation coefficient 98. WTPT ^ ^
between.two paired random variables
with usual notation is : IpifsF % I
^cif
/=1
(A) R = l-. (A) R = l-. /=!
?7(«^-l)
6Zd,'
(B) R =l-J=l— (B) R = 1 j=L
n{n +lf «(« +!)■
6E'^/
(C) R=I (C) R = 1
-I)
(^di
(D) R =l—^ (D) R = l- <•=1
/7(rt + l) 72(77 + 1)
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Text of page 56
:M.'.s
99. Let X - B(«, P) and 7t(P) = 1 be the prior 99. TIPT X - B(m, P) 7i(P) = 1,
l^X^ SIcIchUh
distribution of P. If observed value of X
ttft 1 "i", ^ ^ "5^^ "?Tf^ (squared error
is I, then under squared error loss, what
loss) P "f^
will be the Bayes estimator for P ?
■ 2 2
(A) in+ 2) (A) (« +2)
1 1
(B) (« +2) (B) (77+ 2)
2n 2n
(C) (n +2) (C) (77 +2)
2n^ 277^
(D) (n + 2) (D) (77 +2)
100. Let X ~ N (p., 1) and the prior 100. "HH X N (p, 1) 3Sk p ^ ^
\
distribution of p, be N(0, 1). Based on ^ N(0, 1). t I WT "qr Tjyf
single observation the Bayes estimator of (absolute) "^Tf^ ^ Wd p
p under absolute loss function is : i:
(A) ^ (A) X
(B) (B) jc2
I (C) f
(D) ^ (D) iL
N(3)S/18/2025/A 54
Text of page 57
: '.s
101. In regression model Y = pg + pjX + s 101. MPdJm ifer Y = Po + p,x + s (t^
■ (notations have their usual meaning), if M ^ WIFT t) i{
.2 .2
a
variance of Pi —,then"the variance Pi ^ ■=— ^ pn ^
^xx ^XX
A
of pQ will_be : 5V11 :
2
—+•
2 X^ ^
(A) —+
n SXX (A) n SX X
/ \
—n \
1 X^ ^ 1 X'
(B) n SXX (B) n SXX
—'i' \
2 X' 2 X^
(C) n SXX (C) n SXX
—1 N
1 X' 1 X
-+ —+
(D) n SXX (D) n SXX
102. If ordinary least square estimator of.p is 102. P ^
p =(X'X)v-l X'Y, then the variance of- P = (X'X)"'X'Y t,
the estimate A.'P of the linear parametric Vp ^ ;^,'p ^ "SROT
function A.'p is : t :
(A) ah'iX'Xy^ (A) CT-^V(X'X)-^ .
(B) a^X'(X'X)"^X, (B) a^X'iX'Xy^X
(C) a^(X'X)"'A, (C) CT'^(X'X)-^X.
(D) a\XX'X)X (D) CT^VCX'X)^
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mm-
Text of page 58
103. The 100(1 - a)% confidence interval for 103. yPdOHH TTT^ Y = pQ + p,X + s
the regression coefficient pj in linear ^ yRl'lHH Pi 100(1 -a)%
regression model Y = Pq + PiX + 8 is : t:
(A) Pi-/c(/2,w-2S.E.(Pi)^Pi ^Pl + (A) Pi-/ct/2,n-2S-E.(Pi)<Pi<Pi +
'a/2,n'-2S-E-(Pl) ^a/2,M-2S-E.(Pi)
(B) Pi -/a/2,n-I S.E.(p]) < Pi < Pi + (B) Pi-r^2;.-lS.E.(Pi)<Pi^Pi +
W,h-iS.E.(P])
. (C) Pi-ra/2,«S.E.(Pi)^Pl^Pl + (C) Pi-ra/2,«S.E.(Pi)<Pi^p,+
'a/2,w®-^-(Pl) W,nS.E.(Pi)
(D) Pl-^a/2,rt-2S-E.(Pi)<Pi-l<Pi + (D) Pi-/ct/2,/i-2S.E.(Pi)<Pi-l<Pi +
W,/7-2S.E.(Pi) 'a/2,n-2S.E.(Pi)
104. In the linear model =|iy + e^, i =1,2,... n 104. y,. = ii.+ e-,
ofanalysis ofvariance, is true value and i =1, 2,... n jiy f TT^TT
is the error for the characteristic under eft
study, the is due to : ■pFT ^ t :
(A) Random causes (A)
(B) Mixed causes (B) Mm
(C) Assignable causes (C)
(D).Fixed causes (D) RRjm
N(3)S/18/2025/A 56
Text of page 59
: u'/.s
105. In the linear model = p. + a,- + e^j, 105. yy = \i + ai + ey,
. z = 1, 2, tj= I, 2,.... r, tty being i = 1, 2, tj= 1, 2,.... r, a,- ^
normal with mean zero and variance TJcf (y2 ^ Tfpfe, f I
y-j has the same expectation p.
2 2 2
and variance . If the ctJ = + Og i I
observations are not statistically
independent, then the model is known
nT-stri ^ ^HdT t" :
as :
(A)
(A) Mixed Effect Model
.(B) PtRf-cia ■JITO hT-sqi
(B) Fixed Effect Model
(C)
(C) Linear Effect Model
(D) Random Effect Model (D)
106. In the simple linear regression model 106. ^ yPdhMH y,- = Pq + Pi^/
x,--x
Vi = Po PF/ ' + e., / - 1, 2,......, >z H w w -
'xr
x.-x
C/ =-=—,then the estimate p| of pj
■f, rft pj ^ ST^iTH p] ®M=W PtiMl
may be expressed as : ^1^ t :
n ■ C- n p
(A) Z (B) Z (A) Z (B)
»=1 /=! i=\ /=1 '
n p " r.
(C) Z*^/^/
/=1
(c) 'Z '^.y,
i=\
(D)^ Z^
/=1
N(3)S/18/2025/A 57 P.T.O.
Text of page 60
107. In the simple linear regression model 107. WT yPdJ WH yi=Pg +P iX,- + Cy,
yt = Po + Pi^i ^ 2, , n with / = 1,2, ,n "STWT a^ t, ^
variance cr^,'if SSE is the residual sum SSE
of squares, then the estimate of is ; t-:
SSE SSE SSE SSE
in-2) (B) («-l) (A) M (B) („-i)
SSE SSE SSE SSE
(C) (n +2) (C)
n « in+2)
108. Which ofthe following measures is for the 108. iHMidfed ^ RR "RSR
goodness offit of the regression model ? ^ ^ f ?
(A) Akaikes Information Criterion . (A) 37^^
(B) Available Information Criterion (B) hm-a's
(C) Andrews Information Criterion (C)
(D) Akokes Information Criterion (D)
109. Regardless of the form of a regression 109. p yi-q(nl "Sffdwr "qfSR
model with p parameters, the average fe M qpff qq
1^ 1 n
variance var(j))=-2]v(j),-) of the fitted arteq r^^<u| var(j))= —^v(j)y) frr^
/=!
values is given by : "JRiR "f^ «n?TT t :
.2 .2
pa m' «a
(A) (B) (A) P^ (B)
n P P
a^
(C) ^ (D) (C) ^
n n y
N(3)S/18/2025/A 58
«
Text of page 61
110. In the regression model Y= Xp + e,ifthe 110. "SlfWR Y = Xp + G ife
error term € has not constant variance ■q^ G ^ ct2 ^ t, eft
then this problem is known as : w ^ ^ ^ t :
(A) Homoscedasticity (A)
(B) Heteroscedasticity (B)
(C) Multicollinearity (C)
(D) Autocorrelation (D)
111. The maximum correlation between X,- 111. X,. wim a'X(2) •% Tm
and linear combination of a'Xt^) is : srteeFT t :
(A) Partial correlation (A) sTif^ -mw^
(B) Multiple correlation (B)
(C) Both (A) and (B) (C) (A) (B)
(D) None of the above (D) ^
112. If A and Z be partitioned as : 112. "qfq A Z "fro t :
("A,
■11 ^12^ 2ii ^12
f^Aii ■^12^ ■^11 ^12
A= A= , 2=
, 2=
.-^21 ^22 .^21 ^22. V^21 A22; V^21 ^22.
Suppose A is distributed according ■qfq A qq W(Z, n), 'ii,
to W(Z, n), the distribution of ei®i Aj j 2 — Aj j — Aj2 A22 A21 '^1 ^le'i
:
^11.2 ^ ■'^ll ~ -^12 ^22 ^21 :
(A) W(ln.2''^-P+?) (A) W(Zn.2,«-p + ^)
(B) W(Zn.2'2rt-;? + 9) (B) W(Z„.2, 2/7-/7 + ^)
(C) W(Sn.2,«-(p + ^))
(C) W(Zii.2»«-(P + ^)) '
(D) W(Sh.2,2«-(p + ^))
(D) W(Zn.2,2/2-(;?+^))
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Text of page 62
113. Maximum likelihood estimator of 113. wm m
principal component is where 2=XlI t ^ X.I t :
is;
1 _P N
1 P N (A)
^ i=l a=l
(A)
/=! a=l
p N
/=1 a=l
(B)
/=! a=l
\ P ^
1 P N
P -2 (C)
(C) /=! a=l
/=! a=I
1 P N _ _, p N
(D) (D)
/=! a=l ;=1 a=I
114. The general problem which clusters 114. WTF^ ^
addresses appears in : ^ t, fFT STTcll t :
(A) Image processing (A) "stter
(B) Archaeology (B)
(C) Medicine (C)
(D) All of the above (D)
115. A problem of symmetry is use of which 115. ^
statistic ? f^Mi >3iiai ^ ?
(A) Wishart (A)
(B) Factor Analysis (B) =t)K=t)
(C) T2 (C) T2
(D) D2 (D) D2
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116. If X is multinomial distribution with 116. ^ X ^t
mean [i and variance 2, then the variance iTT^ p 2 t, ^
of its conditional distribution is : ^^ "SIWT 'i\m :
(A) 1^22^11 (A) 2ii-2ii2222ii
(B) 222-^11^22^21 (B) 222"^11^22^21
(C) T-i I-1.12^22^21 (C) 2i|-2]222222i
(D) ^22-2j2222221 (D) ^22-2|2222221 ,
117. Multiple correlatiomcoefficient is defined 117. "ipTO ^ ^ i :
as :
ayy-Cj;^^-222Cyyy
^ii ~ ^'iij^22^iij (A)
(A)
^ii-^'iij^22^iij
^ii-^'iiJ^22^Uj (B)
(B) a.-
C;
Gijj ^a;y222^// »J
(C)
"J (C) a
"
j
'llj
CTyy;
'IIJ - CT-y222^//
CTyy,- -CTyy222ayy
(D) (D)
'llj '"J
118. If Xy ~Np(p,2), then the distribution of 118. ^ Xy~N^(p,2), ^ X ^^
' X is : sl'ii t
/ '> N
/ 'I \ >2
2
(A) Np (B) Np (A) ■ (B) N.
N;.
N
V. y
^ 9 ^
^ 9 > / \ 2^ ^
(C) Np (D) Np
(C) Np (D) > Np N' /
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Text of page 64
119. Probability density function of p-variate
normal distribution is : 4>Qii ■§■ :
7hp/2 .4(x-a)'a(x-a) -i{X-iyA(X-A)
(A) ® (A)
{2%y (271)'''^
.-kx-^)'A"'(X-A) 7R 4(^-AyA-'(X-A)
(2) (27t)J
.,p/2 ® (B)
I
1 ~{X-6yA(X-6) 1 ~(X-AyA(X-4)
(C) ffTn::^p/2 ^ (C) /TTTXTT/'/Z ^
V|a|(27i)' V|A|(27i)'
1 -:^(X-AyA-'(X-A) 1 -i(X-6yA-'(X-A)
120. The Wishart distribution is a multivariate 120. f°r?TT^
extension of which distribution ? pciwidcbiui t ?
(A) r (B) F (A) t (B) F
(C) (D) Z (C) (D) Z
121. In simple random sampling, the 121. iri^^RPT ^ "felt
probability of a specified unit, from a ^ N snro ^ ^ n
population of size N, being included into
. the sample of size n is :
(A) 1 (B) i N
(B) -
^ (D) ^
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Text of page 65
122. The all possible samples for a sample of 122. N ^ ^n ^ Wm
size n from a population of size N in
Simple Random Sampling With "^ri^cT (SRSWR)
Replacement (SRSWR) are : fr" t:
(A) Nc„ (B) N" (A) Nc„ (B). N"
(C) (D) "Cn (C) (D) "Cn ■
123. The variance of sample mean y in 123. Trftresrm yPd-d^n
Simple Random Sampling Without (SRSWOR) ^^=^1 y ^
Replacement (SRSWOR) is : •5ROT t:
N-U2 (A)
N-1s2
(A) N«
N?7
N-2 2 N-2^2
(B) (B)
Nn
«-1q2
(C) (C)
N«
N-?7ci2
(D) (D)
N« N«
124. The covariance between two sample 124. WT Tlflcr "SffcT^
(SRSWR) "5^ n ^ 3rf^
means y„ and x„, each of size under
Simple Random Sampling With ■frf x„ ^ WRTm ^
Replacement"(SRSWR) is : t :
o
a yx
(A) (B) (A) ^yx (B)
yx
rn rn
(C)
'yx
(D) ■ (C)
'yx
(D) V' •:jw
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Text of page 66
125. In stratified random sampling, unbiased 125. wRa
estimator for population mean is :
(A)
/=!
(B) ST /=!
(A) S^/=I
(B)
/=!
(C)
/=!
(D)
j=l '
(C) Eif- (D, i=\
yi'
126. Which of the following relations is true 126. '41'^r^ch ^ %IT fFT
in stratified random sampling ?
(A) < Vp,,p < Vj, (A) V,p,<Vp,,p<Vi,
(B) Vp,,p<V,p,<VR (B) Vp,,p < V,p, < Vr
(C) V,pt < Vr < Vp,,p (C) V,p, < Vr < Vp,,p
(D) VR<Vp,,p<V,p, (D) VR<Vp,,p<V,p,
127, The value of intraclass correlation 127. ^
between units of the same systeniatic 3Td:'^dRT ^ ttH ^^ "^iff
sample cannot be less than : ^ ■^TqFcir :
(A) ^ n
(A)
-1 -1
(B) (n-l) (B) («-l)
(C) - (C) -
n n
(D) («-i) (D) (.-1)
N(3)S/18/2025/A 64
Text of page 67
128. The ratio method of estimation is used 128. ^ a;i3qRT ^ ^ fen
when study variable Y and auxiliary
variable X are : ^X t:
(A) Negatively correlated (A)
(B) Positively correlated (B)
(C) Both (A) and (B) (C) (A) 3fk (B)
(D) None of the above (D) ^^
A y
129. The bias of ratio estimator R =^ m
y : 129. WT ^
X
sample random sampling is :
R =Z ^ ^f :
X
(A) Cov(R,Y)/X (A) Cov(R,x)/X
(B) Cov(R,7)/Y (B) Cov(R,y)/Y
(C) -Cov(R,Y)/X (C) -Cov(R,x)/X
(D) -Cov(R,3;)/Y (D) -Cov(R,37)/Y
130. The ■ bias of regression estimator 130. wf yPd-ci-MH
?reg in simple random yrQg=y + bQi~^)
sampling is : 'M t :
(A) Cov(jc, b) (A) Cov(^,6) -
(B) Coy(y,b) (B) Cov(y,6)
(C) -Cov(3c,6) (C) -Cov(Y,i)
(D) ~Cov(y,b) (D) -Coy{y,b)' '
I\I(3)S/18/2025/A 65 P.T.O.
Text of page 68
131. Completely Randomized Design (CRD), 131. (CRD)^
provides for the error variance : "aWT ^ t:
(A) Maximum number of degree of
(A) Tm ^
freedom
(B) -Rm ^ wn
(B) Minimum number of degree of
freedom
(C) ^ felt "fr "5^
(C) Free from degree of freedom
(D) None of the above
(D) ^^
132. In the analysis of CRD, the sum of 132. ipt fen^ ^ "^f
squares between the treatments is given ^ ^ *2it7T frrH OT fen "w
by :
S *
^ 2
(A) ^^Between ~^ (^(/ ~ ) (A) ® ^Between ~^ (^i/ ~ )
/=! /=!
(B) ® ^Between ~^ (^/.~ X.) (B) ^^Between'~^ {yi.~ X.)
j=l /=!
s n, .
■ ^ f y
(C) ^^Between ~X.) (C) ^^Between ~ ~X.)
/=iy=i
(D) None of the above (D) ^ "sfitf
N(3)S/18/2025/A 66
Text of page 69
133. Let Z,'-N(0,1), / = 1, 2,......, v, be
133. 1THT Zf-N(0,l), / = 1, 2, ,v
independent random variables and
^t PiHPdnsld
assume the following disjunctive
decomposition :
|;z? = Q,+Q2+ +Q, 2] =Qi +Q2+ + Qj
/=!
/=l
with 5 < V. Hence the Qj, Q2, , are -4' 5 < V I 37^: Qi, Q2, ,.Q,
independent Xv.'xi» »x5 distributed
I Z S
XvXv .Xv,
random variables respectively if and only
:
if: ^
(A) V = V2+V3+V4+ + Vj
(A) V = V2+V3+V4+ + Vj
(B) v = Vi+V2 + v_y_i
(B) v = v,+V2+ + v^_,
(C) v = Vi+V2 + V3+ +v^
(C) V = V|+V2 + V3+ +v_5
(D) "3^1^ i{
(D) None of the above
134. "^f r
134. In a Randomized Block Design (RBD)if
s "t, ^ ■wadcii "felll
there are r blocks, s treatments, then the
"gKl ?
degree of freedom of error is given by :
(A) sr - \
(A) sr - 1
(B) i-r - .y - /• + 1
(B) 5/- - 5 - /- + 1 .
(C) i-/* - /• + 1
(C) j-r - r + 1
(D) .yr - 5 + 1
(D) sr - s I
N(3)S/18/2025/A 67 . P.T.O,
Text of page 70
135. Consider the case that the experimental 135. "RHT ^^
units can be grouped with respect to two
factors as in a contingency table. Hence
two block effect can be removed from
the experimental error. This design is
ife ^ fcldirMd ^ t I^
called : ^^ ?
(A) Incomplete Block Design
(A) SFjyf (IBD)
(B) Factorial Design
(B)
(C) Balance Incomplete Block Design
(C) (BIBD)
(BIBD)
(D) Latin-Square Design (LSD) (D) ^ (LSD)
136. If in the Latin Square Design (LSD) 136. ^ (LSD)
there are s treatment to be compared then
*
the degree of freedom for error effect ^ iirai t :
is :
(A) ^ - I
(A) ^-1
(B) s2 _ 1
(B) ^2 - 1
(C) i2_3j + 2
(C) 52 _ 3^ + 2
(D) ^2 + 3^ _ 2
(D) s^ + 3s~2
[\I(3)S/18/2025/A 68
Text of page 71
137, Interpretation of condition of BIBD 137. spjyf (BIBD)^
X(v-1)= Y(K-1), ®cnTct>
A,(v-I)= y(K-l), the number of pairs
f
•^nTci ^ "i" I STcI:
of plots in a block is SO that total
v2.
number of pairs of plots such that
each pair consists of plot within a block
W i\, ^:
is :
bK.(K-\)
bK(K-l) (A)
(A) 2
6(K-1)
b(K-l) (B)
(B) 2
bK-l
(C)
(C)
(D) None pf the above (D)
138. In a Latin Square Design (LSD) if there 138. ^ (LSD) ^
are 4 treatments to be compared, then the 4 ^ ^^t ^
degree of freedom of treatment and error
is given by ; t ?
(A) (3, 6) (B) (6, 3) (A) (3, 6) (B)^(6, 3)
(C) (3, 3) (D) (3, 15) (C) (3,3) (D) (3, 15)
139. If the system has only one component 139,
having exponential lifetime distribution 'Jilqicwtn ^
with reliability function R(T)= e~^^, R(T)= e"^'^, ;\,>o ^ Tn«T t, ^
X>0 then this function represents : WT R(T) t:
(A) The probability that the system will (A) ^ T Wl ^^
fail within T time units
(B) The probability that the system will (B) T ^^ ^
not fail within T time units
(C) A discrete model of reliability (C) Pdiiq^HHlqdl ^ tfer
(D) None of the above (D) ^ -^T^*
l\I(3)S/18/2025/A 69 P.T.O.
Text of page 72
140. If the hazard rates of two vectors of 140.
covariates are to be compared with
each other, which of the following PinrcriRdd
relations is valid ?
t ?
(A) X(t\x2)-'
(B) Kt\xO = e
^aiA^i) _ -(fa-^yp)
(C) X(1\X2) (C) X(1\X2)-'
(D) None of the above (D) ^
141. Which of the following is not correct ? 141. rdMRHpslcl ^ "=1^ t ?
(A) If optimal solution of an L.P.P. (A) ^ felt L.P.P. m IWT ■?RT'^FT
exists, it exists on at least one t, ^ ^ ^ ^ 13^
vertex.
(B) If optimal solution of an L.P.P. (B) ^ L.RP. ^ iwT -mm'
exists at two points, then the L.P.P. w: t, cit L.P.P.
has infinite optimal solutions. SPRT "Sira ^ I
(C) The infeasibile optimal solution (C) oiJd^l4 ^ ^ ^ ^ ^
may exist even if feasible region vmm ii
dpes not exists. t I
(D) None of the above (D) ^ ^
142. Which of the following is not an 142. fFTfcfer L.P.R ^ mrni
assumption of an L.P.P ? ■=1^ t ?
(A) Linearity (B) Multiplicativity (A) (B)
(C) Probabilistic (D) Additivity (C) (D)
143. The L.P.P. cannot have constraint as: 143. PHHrdRsid ^ L.P.P.
(A) Inequality with '>' sign
^ t ?
(A) '>' ^ sRmpmr
(B) Inequality with sign
(B) '>' ^ 3i^MHdl
(C) Inequality with '< 'sign
(C) '<' ^ "HTST
(D) An equation with *=' sign (D) '=' ^ "^TT^T
N(3)S/18/2025/A 70
Text of page 73
144. In the solution of an L.P.P., the incorrect 144. L.P.P..^ TFnVH- '4*,
statement is : ^ ^t ?
(A) Iffeasible region is bounded,then the (A) ^ ^ -qftro t, ^
optimal solution is also bounded. ^ ^I
(B) If feasible region is unbounded,
(B) ^ ^ i, cTl
then the optimal solution is
'mm ^ #tt i
unbounded.
(C) Ifoptimal solution is unbounded then
(C) ^ -mm 3TOtfiT?T t, ^
the feasible region is unbounded.
^ #1T I
(D) If feasible region does not exist, (D) ^ ^ -^T^* t,
then the optimal feasible solution eft
does not exist. sl'ii I
145. For Z = 6^1 - 2x2 145. Pi+^RnRaa ^ 3ia4a Z=6xj — 2x2
subject to the constraints : ^ 37f%T^)cR m t :
2xy- X2<2 2x X2 < 2
jcj < 4
xj < 4
^2 > 0
Xi, X2 > 0
the maximum value of Z is :
(A) 0 (B) 6 (A) 0 (B) 6
,(C) 12 (D) 24 (C) 12 (D) 24
146. Which of the following is correct ? 146. "M^rfer ^ ^t ?
(A) Each basic feasible solution is (A) "3^ MlPelch
an extreme point of the feasible
region. f I
(B) If all" basic feasible solutions are (B) ^
non-degenerate, then the number of f, ^ ^
basic feasible solutions is equal to
the number of extreme points.
wm t I
(C) If the number of basic feasible
(C) ^ ^ wn,
solutions is greater than the number
of extreme points, then the system
has degenerated solution.
(D) All of the above (D)
N(3)S/18/2025/A 71 P.T.O.
Text of page 74
147. Dual Simplex method is applicable to the 147. "3^ L.P.P.'s
L.P.P.'s which start with : "5^ ^ flH "loldl "I" :
(A) an infeasible solution (A) ^
(B) an infeasible but optimum solution (B) ^
(C) a feasible solution (C) ^
(D) a feasible and optimum solution (D) ^
148. Which of the following is correct ? 148. PiMpHRsld ^ WT f ?
(A) Duality does not play any role in the (A) L.RP. ^
post-optimal analysis of an L.P.P.
(B) It is necessary to convert the t I
inequality constraints into equality (B) 13:^ L.P.P. ^ ^ ^
constraints for writing the dual of an smra ^
L.P.P.
t I
(C) If the variables in a primal problem
are all greater than or equal to zero, (C) ^ ^ ^
then the variables in the dual problem
would be all less than or equal to zero. ^^^"RT ^^ ^
t i
(D) If dual has an unbounded solution,
then the primal has an infeasible (D) ^ 3Rftto t fit
solution. Mi^Hd RtT eldi % I
149. The role of artificial variables in simplex 149. Mr ^ t:
method is: (A)
(A) To aid in finding initial basic feasible .-mm I
solution (B) ^ ^
(B) To start phases ofsimplex method i
(C) To find shadow prices from the final (C) 3#R ^^
simplex table
(D) None of the above (D) ^
150. "rtrbr r;^*
150. For a "Poisson exponential, single server
and infinite population" queueing model, 3TO "^PRPsRl" (hTki) ^ IvlA
which of the following is not correct ? PHHRdRsid ^ ^ t?
(A) -E(m)= A,E(co) (A) E(m)= >.£(©)
(B) E(rt)= E(m)-A./^i (B) E(/?) = E(w)-
(C) E(«)= ?.E(v) (C) E{n) = XE(v)
(D) E(v)= E(co)+ \l\i (D) E(v) = E(co) + l/^i
IM{3)S/18/2025/A 72
Text of page 75
chl4
(SPACE FOR ROUGH WORK)
S/18/2025/A
Text of page 76
:w'.s
INSTRUCTIONS TO THE CANDIDATES
I.. This Question Booklet is printed in two languages—Hindi and English. Candidates can select
any one of the two languages according to their convenience.
2. Examinee whose optional subject is Mathematical Sciences, they have to answer 100 questions
in total, there are three parts in the question paper.Part I is compulsory,in which there are 10
questions - in the order of51 to 60. PartII has 90 questions in order of61 to 150. This part is for
Mathematics subject and Part III has90 questions in order of61 to 150.This part is for Statistics
subject. Candidates for Mathematics subject should attempt Part I and 11 or Candidates for
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each question carries 2 marks. Both the question papers and all the questions are compulsory.
Two(2)marks shall be awarded for each correct answer. Unanswered question will not be given
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S/18/2025/A
Page 1 of 76
Source: Madhya Pradesh Public Service Commission. Question paper published by Madhya Pradesh Public Service Commission.