MP SET 2024: State Eligibility Test, Mathematical Sciences
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JTfi
(SPACE FOR ROUGH WORK)
S/13/2024/A
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^I(Part I)
■51. ^ f„{x) = x". ^ %} : y^(x) = x", then sequence :
_ (A) [0, 1] "qr "0^ W WT ^ (A) converges pointwise to a continuous
t" I function on [0, 1].
(B)
"•i ■qr t i (B) is uniformly convergent on
(C) ?■ ■qr 37f^TORt t I (C) is uniformly convergent on
?■
(D) [0, 1] -qq TT^Tw^ t (D) is uniformly convergent on [0, 1],
52. ITHT : 52. Let ;
0, whenA; = -^;-^^,(n = I,2,3,...)
/W =
0, 7iqf^;c = -^;-^i^,(«
rt + 1 • « ^
= l,2,3,...)' 1, elsewhere
n+I n ^ '
1, 37^
then which of the following statement is
• 37^ %arT t : false ? .
(A) Upper Riemann integral
r 2
(A) "Sqft ttiRT -iHMIcbd /(x) li*: = 2. -2
f f(x)cix = 2.
'0
(B) / 37^^ [0, 2] SiqRpHd (B) f has infinite number of
■qq 37^OT ^ . 1 discontinuity points in the interval
[0, 2].
(C) / 37TtRId
°'l ■qr TtTTT (C) / is Riemann integrable over
WqTFHN t I
interval
»•!
(D) / [0, 2] -qq M ^qi^einlq (D) / is not Riemann integrable over
■q^* t I
interval [0, 2].
(2)S/13/2024/A P.T.O.
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53. a > 0 ^ pHHRrlRsid "4* ^ 53. For a > 0, which, of the following'
3TO t ? improper integrals is convergent ?
radx CO ox
fco dx raax pco ox
(A) Jo3 (B) J„3 (A) Jo;^ (B) J,7
rack r^ck rack r<x>ck
(C) Jo3 (d)It (c) Jo3 It
54. Pi+^RnPaa. ^Nh 54. . Which of the following statement is not
correct" ?
(A) 3R^ -qf^ -qiRq (A) Every bounded monotonic function
^ ^t ! is a function of bounded variation.
(B) "51?^ TOT OTR "qft^ 1^^ ^ (B) Every continuous function is of
f I boimded variation.
(C) ^ ^ OTT^* ^ "qtn (C) The sum of two functions of
^ "qf^ ^ WT ^ bounded variation is a function of
bounded variation.
(D) "qf^ fWwT qiOT (D) A flmction of bounded variation is
^ ^ "qft^ ^ t I necessarily bounded.
55. OTT "qi^ / [0, 1] "qr 'w 3OTT 55. Let /be a function defined over interval
■ -qf^nte t : [0, 1] such that :
/W =
'Jl, ^jcsnft^f {1, if X is irrational A
[0, ■ -qfqxqftMt 0, if X is rational
then on [0, l],/is :
■cR [0, 1] "qr WT/:
(A) Lebesgue integrable
(A) ^HIchdHl-M t
(B) Riemann integrable
(B) tWf TRTOTTft^ f
(C) Lebesgue and Riemann Integrable
(C) crl^'l cT®?T Tt^ TTOTiOTt^ "{lil "I" both
(D) ^ TTOTJR^ t ^ ^ M (b) Neither Lebesgue integrable nor
Riemann integrable
TTOTi^T^ t
(2)3/13/2024/A
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58. n 37^^ 1
58. The minimal polynomial of the « x «
square matrix, each of whose elements is
t, ^ 37fcq^ t:
1, is :
(A) x"-l (A) x"-\ ■
(B) (;c-w)(x + rt) ■ (B) {x-n)(x + n)
(C) x{x-n) (C) ;c(x-«)
(D)
(D) (x-l)(x-«)
59. Let :
59. 1TFTT % :
1 -1
"1 -1' B=
B= 2 -1
2 -1
R2 hR^iRo tRa<^i %, eft "RfI is a linear transformation defined on R^,
^ 37^ f 7 then which statement is false ?
(A) + 1, B ^ 37B7PTT^fe fI (A) + 1 is a characteristic polynomial
(B) B ^ 371f^ "RH R ^ of B;
t I
(B) B has no eigen value in R.
(C) B ^^ 3TTI^ RV"^* "^*
t I (C) B has no eigen vectors in R^. ^
(D) B 37T^ t i (D) B is a digonalizable matrix.
60. x^ + 2xy-\-2)P' +2xz+ 2yz-¥kz^ 60. x^ +2xy+3y^ +2xz+2yz+ kz^ is ^
^-iirH«t) RiR-cjo fs"maT "t", : positive definite quadratic form, if :
(A) k>\ (B) ^>-1 (A) k>\ (B) k>-\
(C) k> (D) k>0 (C) k> - (D) it > 0
(2)S/13/2024/A
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^II(Part II)
61. (-80 ^ PiHiT+d t : 61. One of the cubic roots of(-80 is :
, (A) (B) /^+i (A) 72-/ (B) ,-^^ + 1
(C) V3-/ (D) V3+/ .
(C) 73-/ (D) ^^+/
62. W S -n , ^ ^ W' 62. The domain of convergence of the power
2 +1
00
(^W^) PHMlTdid t :
series X Tn T is :
n=0 2''+l
(A) \z-2\<2 (B) |z-l|<2
" (A) \z~2\<2 (B) (z-l|<2
(C) \z\>2 (D) \z\<2 (C) |z|>2 (D) lz|<2
63. f{z)^x^"y^-2y+i(2x-2xy) 63. If f(z)= x^-y^-2y +i(2x-2xy),
z = x + /y,^ z^ /(z), where z = x + iy, then /(z), in terms
PiHi'Rha : of z, is :
(A) p• - + 2;z (B) 2^ + 2/z (A) z2^2/z (B) z2+2/z
(Q z + 2/z^ (^) Z-2/Z (*^) z + 2/r^ (B) Z-2/Z
64. z ^ ^ 4nni 64. A value of z for which the function
z = sinh M cos V+/cosh M sin V z= sinhMcosv +/coshMsinv ceases to
Hitri'i "t, Pi^i'Ptid ^ : be analytic is :
(A) z^\ (A)z=l
(B) z = 0 (B) z = 0
(C) z^--i (C) z.= -/
,(D) z = -1 (D)z = -1
(2)S/13/2024/A ! P.T.O.
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65. The harmonic conjugate v of the function
65. M= ^ ^IhTPh^
V PlMlPchci c
u = —logg|x^+>'^j is (c being a ,real
cilWpcicb STER t): constant) :
(A) v = tan[ —l + c (A) v = tan[-]+c
(B) v = cot[ —l+c (B) v = cot[-j+c
(C) v = tan 'l^^l+c (C) V = tan
(D) -cot-'(j^ +c
(D) v = cot '(jj-fc
66. Zj Z2 66. If Z] and Z2 both satisfy the relation
z+z = 2|z-l| arg {^\~^2)-~^,' z+z = 2|z-l| and arg =
^ cit Z| + Z2 ^>)iQrMpich then the imaginary part of Zj + Z2 is :
^ PiMiPtici "^Wt :.
(A) 0 (B) -1
(A) 0 • (B) -1 .
(C) 2 (D) 1 (C) 2 (D) 1
67. ^TfriTaT Zj, Z2 sik Z3 "O;^ 67. If complex numbers Zj, Z2 aitd Z3 are the
wrfNl ^ t, 3k Z2, imW vertices of an isosceles triangle, right-
^ zf+2z2+23 ^ PHHlPchd angled at the vertex Z2, then the value of
sImi : zf+2z2+Z3 is equal to :
(A) 1 (B) 223(2,+Z2) (A) 1 (B) 2z3(zi+Z2)
(C) 222(2,4-23) (D) 0 (C) 2z2(zi+Z3) (D) 0
(2)S/13/2024/A
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68. 1, /, 14- / ^ 68. , The equation of the circle which passes
^ pHHlf^d f : through the points 1, i, 1 + / is :
(A) 2zz+(/-1)2-(/+1)z = 0 (A) 2zz+(i-l)z-(/ + l)r = 0
(B) zz+(/+l)z+(/-l)z = 0 (B) zz + (/ +1) z + (; -1) z = 0
(C) 2zz-.(/ +l)z-(/+l)z = 0 (C) 2zz - (/■ +1) z - (/■ +1) z = 0
(D) zz +(i -1)z+(/'-1)z = 0 (D) zz+(/-l)z +(/-l)z = 0
69. /(z)= M +/v, z= x + iy 69. If /(z) = w + /v is an analytic fimction of
■ . 41(TH 3^ z = x + iy M-v = e;'^(cosy-sin^),-
M-v = e^(cos;/-sin;') ^z then /(z), in terms of z, is (c being a
■q^* f(z) rHMlRtid t c constant) :
3T^ Tlf^ "t) :
(A) +c
(A) +c
(B) Z-2 +Z + C
(B) +Z + C
(^) z^ - 2z + c
(Q z^-2z+_c
(^) ze^'-^c (^) ze^ +c
70. 41(Til w = e'*^(A:cos3^ —^'sin^y)' W^iRT 70. Function M = e*(xcosy-ysiny)
"^RtTT "i" I "^TO satisfies Laplace equation. Then the
4nni /(z) = w + /v, z "q^ corresponding analytic function
' PiHiPtiq i (^5t^ c 37^ qif^ /(z) = w + /v, in terms of z, is (c being
a constant) :
■ t) :
(A) ze^ +c (A) ze^ +c
(B) e^ + z + c (B) e +2 +c
(Q e^-z + c (Q e^-z + c
(D) ze^ + Z + C (D) ze +Z + C
{2)S/13/2024/A P.T.O.
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71. "^Tf^ I ^ C MT ^ WT 71. If /(z) is continuous on a contour C of
/(z) HcRT C yrA-^i 1^ z length /, and |/(z)|<M for every point
^ |/(z)|<M ^: z on C, then :
(A) J /(2) <M/
(A) J /(2)i3!z <Ml
(B) J/(z)i/z >M/ (B) J /(^)dz >Ml
(C)
\f{z)dz<Ul J/(z)fife < M/
(C)
(D)
\f{z)dz>Ul (D)
J /(z)fife>M/
72. -^Tf^ C,z = a 3lk z = 6 ^ ^ 72. If C is any rectifiable arc joining z = a
and z = b, then :
(A)
J dz-a-h I dz = a~b
(A)
C
(B)
J t/z = 0 I t/z =0
(B)
C
(C)
I dz= b-a I dz-b-a
(C)
(D,
0
{2)S/13/2024/A 8
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73. ^ C ^ sfk- 73. If a function /(z) is regular within and
-qr'-q^.WT /(z)"Wm on a closed contour C, and if § be a
C ^ eft : point within C, then :
1 r/(^)dz
(A) -' 2ni j^z+^
^17/ J r 4- (A) 2ra^z+^
(B)
c
(B) "m)=~l^dz
27i^z-4
Jir •> T —r
1 r/(z)t/z 1 r/(2)cfe
(C) .27t^z+4 (C) 27t^Z+5
(D) (D) 2ni%z-i,
1
74. /W = t: 74. /(2)= has ;
z'*+2z^+l z''+2z^+l
(A) ^ (A) Single pole
(B) f|3v - (B) One double pole
I
(C) ^ WT P (C) Two single poles
(D) ^ ^ (D) Two double poles
75. r^tRslcb "sft f^3;ft* zj = CO. 75. The bilinear transformation that maps
Z2 = / srk Z3 = 0 ^ W] = 0, points Zj = CO, Z2 = i, and Z3 = 0 into the
W2 = / ^ W3 = CO stiH^;i: ylaPqBia points Wi = Oy W2 = i, and ii'3 = .00,
WT t ': respectively, is :
(A) iv =-
z
(B) w=-z (A) h'=7 (B) w=-z
(C) w = z (D) >v=-- (C) w=z (D) >v =--
(2)S/13/2024/A 9 P.T.O.
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76. /(z) ^^ z ^ 76. A zero of an analytic function /(z) is a
^ ttpt t : value of z for which :
(A) fiz)^^ (B) /(z)= 0 (A) /(z) = «) (B) /(z) = 0
(C) /(z)= 2 (D) /(z)= l (C) /(z) = 2 (D) /(z) =I
1
77. z = a/ "qR: ^ 3T^ t : 77. The residue of at z = a/, is
(A) (B) (A) (B)
4a' 4a' 4cr* 4a'
(C) (D) (D)
2a' 2a' 2a'
78. C |z|= 3 i\, ^ 78. • If C is the circle |z| = 3, then by
f j Cauchy's integral formula
■ ^ ^ J(z-l)(z-2) . ^ : r .
J(z-l)(z-2)^^is:
(A) 27i/|e''-e^j (B) 27ti|e''+e
(A) 27c/ (B) 2ni\e^ + j
(C) 27ti|e^-e''j (D) 2ni(e^+e'*^ (C) 27r/|e^-e'*J (D) 2jr/|e^ + e'*j
79. "^rf^ C ^ ,|z+3/|= l A ^ 79. If G is the circle |z + 3/| = l, then by
_. 1
«HI=hQH ^ ^ h{z+ ni)'^<^ •■
C
7(
f 1
Cauchy's integral formula J z[z-k-ni) dz
IS :
(A) 1 (B) 0
(A) 1 (B) 0
(Q ni (D) -71/
(C) 71/ (D) -71/
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80. p z = 1 80. The residue of / .
(z-l)''(z-2)(z-3) (z-l) (z-2)(z-3) at
t: the pole z = 1 is : ,
53 27 53 27
(A) y (B) y (A) T (B) T
101 101
(D) 0 (D) 0
81. "INDEPENDENCE" ^ ^ ^ 81. The number of arrangements of the
letters of the word "INDEPENDENCE"
#P t: which start from P, is : '
(A) 1663200 (B) 138600 - •(A) 1663200 (B) 138600
♦
(C) 16800 (D) 13700 (C) 16800 (D) 13700
82. "DAUGHTER"^^ 3T^' ^ 8 3T^ 82. The number of different 8-letter
arrangements of the word
<=<icrl "3^ 'tioHi, f^iiA
"DAUGHTER" in which all the vowels
^ ^«T a, f :
occur together, is :
(A) 80640 . (B) 40320 . (A) 80640 (B) 40320
(C) 4320 (D) 10070 (C) 4320 (P) 10070
83. 52 3RT ^ "qiif ^ TT^ ^^ 83. What is the number of ways of selecting
^ 4 "Tdt ^ -^ 4 cards, of the same colour, out of a pack
TO ■^fro t ? of 52 playing cards ?
(A) 29900 (B) 105625 (A) 29900 . (B) 105625
(C) 270725 (D) 541450" (C) 270725 (D) 541450
• I
(2)S/13/2024/A 11 P.T.O.
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87. 8« + 5 #? 87. How many prime numbers are of the
form 8/j + 5 ?
(A) 512 (B) 181
(A) 512 (B) 181
(C) 131 (D) 00
(C) 131 (D) 00
88. 5^2 _ 312 f .
88. 5^2 _ 312 is divisible by :
(A) 5^ (B) 3^'
(A) 5 ' (B) 3
(C) 12^ (D) 13^ (C) 12 (D) 1-3
/
89. 2^ ^ f- : 89. The primitive root of 2^ is ;
(A) 3 (A) The integer 3 only
(B) lyrfe 2 -3?k 3 (B) The integers 2 and 3
(C) ^ lyfe 2- (C) The integer 2 only
(D) S sfk (D) The integer 3 and its multiples
90. cit%3: p ^ t 90. Letpbea. prime and (a.p)= I, a gS+,
{a, p) = a gZ+, ^ then which of the following statements
."t ? is correct ?
(A) :c"^a(mod;7) ^ - 1 ^ t i (A) = fl(modp) hasp- 1 solutions.
(B) ^ a(mod p) has («, p -1)
(B) jc" = a(mpdp) {n,p~l) f
solutions if =l(mod
^ a(^-'),''{«'P-') = i(modp)
P)
(C) x" = a[modp) {n,p-l) (C) x" = (mod p) has («, p -1)
t />-') ^ 1 (mod p). solutions if 4-1
(mod p).
(D) x"=(3(mod/7) ^ ^ t
(D) x"=i3(modp) has no solution if
^ = 1 (mod /?)
>-])/("'^"0^1 (modp)
(2)S/13/2024/A 13 P.T.O.
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91. G t cisn H, K 91. G be a group and H, K be its subgroups,
t, RHfansId ^ WT then which of the following statements
^t ? is not true ?
(A) HK G^ t (A) HK is subgroup of G <=> HK = KH.
oHK = KH I ■ (B) If G is an abelian, then HK is
always a subgroup of G
(B) ^ G a^nM t, eft HK ^
1 (C) H n K is the largest subgroup of G
containing both H and K.
(C) H n K ^ ^3W1F t G
^ H ^ K t t (D) H u K is always a subgroup
of G. ■
(D) HuK ^ G^ #TT I
92. Let G be a group with (a6)" = a"b" for
92. 1THT G ^t 1^1^* (aft)" =
V fl, 6 € G cI«7T K > 1 t™ tI all a, 6 e G ^d for some fixed integer
G„ = {a e G|a" = e} ^ n> Let G„ = {a G G\a" = e} and
Qn = e G}. Then which of the
G" — {a"|a € G} PiMRiRaei '^*
following statements is true ?
^t ?
(A) G" < G and G/G" = G„
(A) G« < G afk G/G" = G„
(B) G« < G G^G" = G
(B) G« < G and G^G" = G ^
(C) G« < G G„ < G and G"/G„ = G
(C) G" < Q G„ < G aSk G«/G„ = G
(D) G„ < G G" < G 3?k G/G„ s G" (D) G„ < G G" < G and G/G„ s G"
93. -(H^rHd S3 "fent wichiRctiaiy. 93. How many automorphisms are there in
the symmetric group S3 ?
^t ?
(A) 2 (B) 3
(A) 2 (B) 3
(C) 4 (D) 6
(C) 4 (D)" 6
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94. Tipn R S ^ t sk / -^ 94. Let R and S be two rings and/be an
/: R ^ S t, ^ : into homomorphism /:R -> S, then :
(A) R/Im/= S (A) Rrim/= S
(B) R/Ker/= S (B) R/Ker/= S
(C) R/Ker/= Im/ (C) R/Ker/= Im/
(D) R = S/Im/ (D) R = S/Im/
95. RnfdRsld ^ ^^ 95. Which of the following statement is not
t ?
true ?
.. • (A) Z^ Jpn^sl'^d
(A) The ring of integers Z is a UFD.
"511^ t I
(B) The polynomial ring F[x] over a
(B) ^ F FM
37feft^ -RRI t I , field F is a UFD.
(C) liUHidcril TTRT (C) Every principal ideal domain is a
TjupF^ -RRf f I UFD.
(D) T^yrMhT -^Rr (D) Commutative integral domain
R =|a+6V^|fl,Z)GZ} R =|a+/)V^|a,^? eZj is a
TTRT t" I UFD.
96. 10 + 11/ ^«TT 8 + /' ^ tTWrFT WWfe" 96. What is the g.c.d. of 10 + 11/ and 8 + / •
Z[i] ^t ? in Z[/] ?
(A) 3+2/ (B) 3-2/' (A) 3 + 2/ (B) 3 - 2/
(C) 2 + 3/ . (D) 2-3/ (C) 2 + 3/ (D) 2-3/
97. 1THT ^ G^ 108 t, 97. Let 0 be a group of order 108, then the
0 3-'3W^ #1 ? number of Sylow 3-subgroups in G is :
(A) 3 j ^(B) 5 (A) 3 (B) 5
(C) 6 , (D) 9 (C) 6 ' (D) 9
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101. : 101. For the differential equation :
(px^+y^){px+y)={p +\f (px'^ + y^){px+ y)=(p +
^ rHHPdRsld which of the following statements is
/
t-? correct ?
(A) M ^ (A) The singular solution of the given
9 9
+4(x + y)=0 t I equation is ;c 7 + 4(x +;;)= 0.
(B) M. ^ (B) The singular solution of the given
equation is x^y^ + 4{x'^ +/)=0 •
x^y^ +4(x^+y^)=0 t '
(C) M ^h1=+)<u| ^ ^ (C) The general solution of given
=c(x +j^) t,^ c equation is x^+y'^ = c(x + y),
t where c is an arbitrary constant.
(D) "3^1^ (D) All of the above
102. 31^^ -^TiteTJT 102. The solution curves of the differential
equation :
|X2x^-/) dy _ \y{2x^-y^)
tic lx(2;^^-;c^) dx [x(2;'^-x^)
are of the form :
^ ^ i^fH ^ "^f "I
3/2 . ..3/2 o_3,.3"
(A) x^''+/''=3cx'y
(A)
(B) =3c{xy)2/3
(B) +y^^^ =3c(xy) 2/3
(C) x^ +y^ = 3cxy
(C) x^^ +y^ = 3cxy .
(D) None of the above
(D) -^f ^ ^ 3^"
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103. 103. The solution of the differential equation :
{x^D^ -xD+2)y =;c log X (x^D^-xD +2)y = x log
^ ^: IS :
(A) 3;= xc,cos(logx)+ C2sin(log:3i:) (A) y = xci cos(logx)+ C2 sin(logx)
+ A:log:c + X log X
(B) y^x[Cj cos(Iogx)+ C2 sin(logx)] (B) y = x[ci cos(log x)+ C2 sin(log x)]
+ xe^ + xe^
(C) y = x[ci cos(log'x)+ C2 sin(log x)] (C) 'y = x[ci cos(log x)+ C2 Sin(log x)]
+ logx + logx
(D) y = x[q cos(log x)+ C2 sin(log x)] (D) y = x[C] cos(log x)+ C2 sin(log x)]
+ ;clogA: + xlogx
104. PiHRnRaci ^ cifi^ ^ Rm. 104. For which of the following functions
y2{x) t*, p(x) ;;i(x) and y2ix) are continuous
functions, p{x) and ^(x) can be
q(x) [-1, 1] W W
determined on interval [-1, 1] such that
1^ t and y2ix), 3;i(x) and 3^2(^) linearly
independent solutions of
y\x) + p{x)y'(x) + q(x)y(x) = 0, /(x) + p{x)yXx) + q(x)y{x) = 0,
V;c e [-1, 1]. V X G [-1, 1] ?
^ ^ ^ tfecbd: ^t ?
(A) yi(x)= xsinx; y2W = cosx
(A) yj(x)= xsinx; =
(B) y^(x)=-xe''; y2(x)=-sinx
(B) yi(x)= xe''; y2(x)~smx
■ (C) = , y2(x)= e'-i (C) y^(x)= e'-'-, y2ix)= e'~l .
(D) yi(x)= x^\ y2(x)= e"" -1 (D) = y2M = e'-\
(2)S/13/2b24/A 18
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/ \-312 105. The order and degree of the- differential
105. =0 ^ \-3l2
dx^ dx" . d'
equation = 0 are :
^ ^^R?T: f : dx' dx^
(A) ^ 1; ^ 4 (A) Order 1; Degree 4
(B) ^ 4; ^ 4 . (B) Order 4; Degree 4
(C) 1; ^ 1 - (C) Order 1; Degree 1
(D) ^ 4; ^ 1 (D) Order 4; Degree 1
106..'2Tf^ ;'^(^)=:s:cos2:c 106. If y^{x) = xQ.os2x is a particular
>'"+ay =-4sin 2x ^ solution of 3;" + = - 4 sin 2;c, then the
cR a'= '#11 I value of a is :
(A) -4 (B) -2 (A) -4 (B) -2
(C) 2 (D) 4
(C) 2 (D) 4
107. f^(x)=4,f2(x)= x^ . 107. If yi(;c) = 4,/2(x) = ;c3 and
■ :f2{x)-\+ax+bx^ ^3 (;c) = 1+ ojc + 6;c^.
If y^(x) .is orthogonal"to yj(;c) .and
/3W WT /iW /2W ^
flix) on the interval (-2, 2) then what
Wf^ (-2, 2) TR; ^ WT
is true ?
■^8^^ t :
(A) a = 0, 6 = 1 ■
(A) = 0, 6 = 1
(B) a = 0, 6 = 0
(B) a = 0,6 = 0
(C) a = 0,6=,--
(C) a = 0,6= 4
4
(D) None of the above
(D)
I
(2)8/13/2024/Ai 19 P.T.O.
Text of page 22
108. 108. The complete integral of differential
equation :
+ px = q
p^x^ +px = q
^ "'jpf eVii ;
is given by :
(A) z= a\ogx +{a^ +a)y+c
(A) z = a log x + {a^ + a)y + c
9
, . 1
CB") z= a\ogx+—y +c 1
a (B) z = log ;c + — ^^ + c
a
(C) z=a^+{a^ +a)y-^c 9
(C) z = a — + {a +a)y + c
2
(D) <iH^=Kl ^^ (D) None of the above
^a c t 1 where a and c being arbitrary constants.
109. 3Tff^ mtwn : 109. The solution of partial differential
(d^ -DD'-6D'^)z ="cos(2x + y) equation (PDE) :
(d^ -DD' - 6D'^) z = cos(2x+y)
is :
(A) z=(t)i(>'-3x)+(t)2(y +2x)
(A) 2 = it)i(;^-3x) + (|)2(y + 2x)
1
+ — cos(2x +;^)
+ —cos(2x+>')
4
(B) z =(t)i(;'+3:j:)+(j)2(y +2x) ■
(B)' 2 = (t)i(:F + 3x) + (|)2(:v + 2x)
- cos(2x + y) 1
cos(2x +y)
4
(C) Z = i/i(y-3x)+if2(y-^x) (C) 2 = (t)i(y-3x) + (|)2(y-2x)
. + ■icos(2x + 3') + ■icos(2x+y)
(D) z = ^i{y + 3x) + ^2(y~^^)
(D) z = f (y + 3x) + (j)2 (y - 2x)
1
-■—cos (2x + y)
■ - —cos(2x + 3') 4
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Text of page 23
110. : 110. The solution of PDE (Partial Differential
dh 2 5^2 Equation) :
--a ~^= x
dx} ■ dy^ dz 2
—:r~-a —T- = X
dx dy
(A) z= ^^{y+ax)+i^2iy-^) is given by :
(A) z = ^^(y + ax) + ^2(y-ax)
+ {xy^)l(2a')
+ {xy^Wa")
(B) ^=:^\{y■^^)+^2{y-ax)
(B) - z = ^x{y-^ax) + i^2iy-ax)
- ixy'ycia')
- {xy'yila')
(C) z = ^i(y + ax) + ^2(.y~''^)-^y^ (C) 2 = (j)](;^ + ax) + (t)2(;'-czx)-xy^
(D) z = <^iiy + ax) + ^2(y-'^) (D) 2 = (l)i(y+'as:) + (|>2(y-ax)
~ {xy)/2a - {xy)l2a
111. ^ arfn^TT^ ^ ^ 111. The rate of convergence of Newton-
Raphson's method is : •
(A) tHach (B) t^cpffq (A) Linear (B) Quadratic
(C) n^Wldl^ (D) ■ (C) Third degree (D) Fourth degree
112. ^ /(-I) = - 13, /(O) = - 7, 112. If/(-I) = - 13,/(O) = - 7,/(l) = - 1,
/(I) = - l,/(2) = 11,/(3) = 35, ^ /(4) /(2) = 11,/(3) = 35, then find the value
^ "HH : of/(4): ■
(A) 0 (B) 27 (A) 0 (B) 27
(C) 37 (D) 77 (C) 37 (D) 77
113. ^ '1*^111 ^ ^ 113. How many digits are to be taken in
#TT #11 ^ 0.1% ^ H i\ ? computing so that the error does
not exceed 0.1% ?
(A) « = 2 (B) « = 3 .
(A) « = 2- (B) n = 3
(C) n<4 (D) n>4
(C) n < 4 (D) n>4
(2)S/13/2024/A 21 P.T.O.
Text of page 24
114. /(Aro,ro,*o) ^ ^ t : 114. The value of /(xq.xq.Xo) is :
(A) /(xo) (A) /(xq)
(B) /'(*o) (B) /'(*o)
(C) /"(xo) (C) /"(xo)
/"(xo) /"(Xo)
(D) • (D) Cz
Cz
115. PiHRrlfed MW 115. Whieh of the following is the formula of
^ IJ? t ? Newton-Raphson's method ?
' /(x„)(x„-x„_i)
(A) ^n+\ ~ (A) x„., x„
/(x„)-/(x„+i)
_ ■ /'(^n)(^n ^n-0 /(Xn)(Xn X„_|)
(B) '^-'-^""'/(;c„)-/(x„+l) (B) *n+l ^ /(x„)-/(x„+l)
/(xj
,
(C) *..1 - « .
/fa) (C) ■'^n+l ~ *n ~ /'(X„)
■ /fa)
(D)
. _V +1 /fa)
(D) +
116. Alogx ^ • 116. The value of A log :c is
h' h
(A) log 1+- (A) log 1+ -
h
(B) log (B) log
kX \X/
(C) log^l--
h
(C) log^i--J
(D) (D) None of the above
(2)S/13/2024/A 22
Text of page 25
117. ^ 7 ^ 7^' ■ 117. By taking the step length — and using
i •
1 'f 3
TFW J x^dx ^1 HH Simpson's rrd rule the value of nJ ^
3
0
: will be :
(A) 8g
A
(B) 4 (A) ^ (B) 4
(C)
1
(D) g (C) ^ (D)
8
118. OT,« Yfe t, ^ 1¥=#ffeT 118. If w, n are positive integers then which
^ ^t ? one of the following is true ?
(A) A'"A"f{x)= A"^f{x) (A) a'"aVW = a'"VW ."
(B) A'"A"f{x)= A'"^"f(x) (B) A'"AVW = A'"^"/W
(C) E =l-A (C) E =l-A
(D) £=.-"^ +1 ■ (D). E =e-^°+l .
119. x^ f ;- 119. The value of X is .:
^e; \ Xj J
(A) 6xh (B) 4xh ' (A) exh (B) 4xh
(C) 6xh'^ (D) 9xh (C) (ixl?- (D) 9xh
120. ^ /(6) 120. By Newton's divided difference formula
HM ^10 =hlRiy. 1 "R'41 "i" : find the value of/(6). Given that;
X : 5 9 11 13 21 X : 5 9 ■ 11 13 21
fix): 150 392 1452 2366 9702 ' fix) : 150 392 1452 2366 9702
. (A) 111 (B) 252 ■ (A) 272 (B) 252
(C) 262 (D)^^ (C) 262 (D) None of these
(2)S/13/2024/A 23 P.T.O.
J ^
Text of page 26
121. l[y(x)] ^ 121. The variation of a differential funetional
f=FT ^TRIT "t : I[y(x)] is given by the formula :
(A) 5I = j|^l[/W +a5;,][^^ (A) 8I =[Ai[yw +a5y]}^^^
(B) 5I =[-lhW +a5.]|^^^ 8I =
| — +
a=0
(C) 51 = —ifyW
da
+ aaj']
ja=o
• (C) 5I =(^l[Xx)+a5;.L
(D) "dH^=k1 "^* =t)l^ (D) None of the above
122. "Qi^ 4icni^ : 122. If a functional :
i[y(x)]= ll y{x)cix IbW]=|^ y(x)d>:
^ C.[0, 1] i\, ^ is defined on class C [0, 1] then the
I [sin Tuc] ^ eVii : value of I[sin tlc] will be :
2 7t 7U
(A) r
7C
(B) 2 (A) -
TT
(B)
2
(C) 71 (D) -t: (C) 71 (D) -71
123. XO)= 1. XI)= 2, ira 123. If XO)^ 1> XI) 2, then extremum of
the functional :
I[jvW]=]^ (xy+ y^ -2y^y')dx
I ly(x)]=\\{xy+ y^-2y^y')dx
^ 11F t :
IS :
1
(A) -J (A)
2
(B) 2 (B) 2 .
(C) -2 (C) - 2 .
(D) None of the above
(D) ^^
(2)S/13/2024/A 24
A .
J
Text of page 27
124. 124. The solution of Euler's equation is
t: called:
(A) ^ (A) Non-stationary curves
(B) (B) Extremal
(C) W{\^ (C) Definite integral
(D) "3^1^ . (D) None of the above
125. 125. A necessary condition for functional
•b
I[X^)]=f F(:c,;.,/,y')^
••fl
I[X*)] = f Kx,y, y',y")dx
'a
^ WT ^ 3im^ t ; to be extremum is :
^•9F'\ f dF\
+■ =0
(A)^ +■ =0 (A)
5x ^ K^'y dx' dx dx dx' W)
^ dF^
=0 +• =0
(B) ^ tie +■ (B) ^ dx .dy') dx ^dy j
dx
5F d ^ dF^ dF d ^dF^
=0 +• =0
(C) + (C) dy dx\dy'j
^ dx[dy\ dx' _dy") dx'
(D) ^ ^ -=1^' (D) None of the above
126. 3TcRI^ [fl, 6] W WT /(O ^ 126. If for a continuous" function f{t) on
"^fcRr Ti(0 ^Hl=hd interval [a, b\ the integral
i=fV(OTiw</'=o i= |V(OTi(Orf' = o
•>0
■ ^
^ [a, 6] "TT f(f) ^ 5V11 : for all continuous functions t|(0, then the
value of /(0 on [o, 6] will be ;
(A) 0
(A) 0
(B) 1 .
(B) 1
(C) 2
(C) 2 ■ . ■ '
(D) ^4^Jck1 "^f ^ (D) None of the above
{2)S/13/2024/A 25 P.T.O.
I
Text of page 28
n
127. y(0) = 0, = 127. If 3/(0) = 0, y I = 1> then the
extremum of functional :
^[y(x)] = \^^ {y''^-y^)dx
"FH PlH <i'=hni
t: can be attained only on :
(A) y = cosx (A) y = cosx
(B) ^= sin:c (B) 3/ = sinx
(C) y = tanx (C) 3' = tanx
(D) ^^ , (D) None of the above
128. Trf?!^* XI) = Xe) = 1 ^ 128. Extremum of the functional
4ieHc*i
lbW]= J,^ (xe^ - ye'')dx
lbW]= Jj^ (.xe>'-ye'')dx under the condition XO UX^) = 1 is :
"TO t" :
(A) +
(A) y = x +\ogx
(B) 3; = x^ + logx
, (B) y = x^ +\ogx ■ »t
(C) ;;=;c-21ogx (C) 3' = x-21ogx
■(D) ^^^ (D) None of the above
129. I[X^)] = Jq (^sin^ + cosX^^^; 129. If I[X^)] = jp (xsin;/ + cosX^»
7t
XO) = 0,X1) = -J WT^ 7C
XO) = 0,X1) = "^ be a functional, then
^ "5IM extremum can be found on the
t : ■ ■ curve :
(A) ;' = cot"^x -1
(B) y = sec"'x (A) 3/ = cot 'x (B) 3' = sec-1'x
(C) y = cosec"^^: (D) y = tan"' x (C) y = cosec 'x (D) >» = tan-1' x
(2)S/13/202'4/A 26
I
Text of page 29
r,
130. ^^ ^ MiW* 130. The curve on which the functional
jJ[(yf+i2A5']&,>'(0)=o,:»'(i)=i
^ can be extremised, mil be :
(A) y =^ {^)y = =^,
(B) ■ . (B) y--^ : '
. (€)>-=> • \
V (C) :v =* .
(P)y=.x/> (D) y = x^ ■ ■ ■ . \
I .. . •
131. sm^H 131. *Leibnitz' rule for differentiation under
'yi^l^' 15m t : integral signs is given by.:
(A) ^[j;K(x,^)«®rf4] (A) i[jr
=r
Ja dx
=f
ox
(B) ^[j;K(x,5)«©rf5] (B).f[j;K(A:,^)»(4)rf4] ,
=f ^«(§)rf5tc
ox .
= r ^uQ,)d^+C
•'« OX
(C) f[C
=r ¥«ft^+K(*.*) = r^u(.Qd^+K(x,x)
•'a Cbc
(D). iK (D) itt
r '
;
=r §««)rf5+K(*,*)«(*) =P ^M(5y^+K(x,*)w(x)
fa ^
¥C
car
(2)S/13/2024/A 27 P.T.O.
✓ ^
Text of page 30
132. TTR y"+ xy= 1, >(0)= 0 132. Volterra integral equation related to
= j;'(0) ^ <=llr^<l initial value problem y" + xy = 1,
t: ■ X0)= 0=/(0)is : . ' .
■ (A) y(x)-^+j^ (x+^).y(^)d^ ■ (A) yix)=^+ (x+^).y(^)dl^
(B) y(x) (B) y(x)=^-jjx+^).y(^)d^
(C) y(x)=^ ■ (x-^)^.y(i,)d^ ' (C) y(x)=^-j'^ (x-^)^.y(^)d^
(D) y(x)=Y+!o (D) =
133. "Sff^ Y^fniz) m D 133. If Yfni^) an infinite series of single
4icriil ^ 37% t" "Q^* valued functions in bounded and closed
. \f„iz)\<M„,^ ^ domain D and |y),(z)|<M^, where
% ^ t n^TT ■ is a series of positive constants
i, fe: "^TT and is convergent then
t: Weierstrass M-test proves :
(A) S/«W ^ (A) Only uniform convergence of
37fnWT I X/«w-
(B) Only absolute convergence of
(B) ^ ^
srfqmuT I E/«w-
(C) Uniform and absolute convergence
(C) Xi/rtW ^ W™ 31^
of S/„w.
aifeOT I
(D) Only conditional convergence of
(D) ^ ^
I
(2)S/13/2024/A 28
Text of page 31
134. ^ 134. Second kind - of 'Volterra integral
equation' is given by :
(A) r* Yi(x,^)u(^)d^+/(a:)=0 . -gRT
Ja
(A) f K(x,%)ui^)d^ + fix) = 0
Ja
(B) f^ Kix,Qu(Qd^+f(x)=0 OT (B) JjK(x,5M^y^ + /«= 0
(C) r K(x,Qu(^)d^+fix)=uix)-sm
Ja
(C) f Y.ix,QuiQd^ + f(x) = uix)
Ja
(D) K{x,^)uiQd^+fix)= u(x) OT (D) f^Kix,QuiQd^ + fix) = uix)
135. Solution of integral equation
135. 4HHlchei w(x)= 1+ J^ "(4)^^
^ f: m(x) = 1+ J|^' u(Qd^ is :
(A) w(x) = 1 (B) w(x) = X (A) m(x) = 1 (B) w(x) = X
(C) u(x) = (D) w(x) = cos x (C) w(x) = (D) m(x) = cos x
136. ■^FfteTJT 136. Integral equation
•6
f'' Kix,^)uiQd^ + f(x) = u(x)
Ja
f Kix.^)u(^)d^ + f(x) = uix)
is a.:
(A) ^ (A) "Volterra Integral Equation" of first
kind.
(B) tfcfh ITOT ^ (B) "Volterra Integral Equation" of
I second kind.
(C) ^ ^Hlctid (C) "Fredholm Integral Equation" of
I ■ first kind.
(D) fefH 31^ ^ ^M14)d (D) "Fredholm Integral Equation" of
second kind.
(2)S/13/2024/A 29 P.T.O.
Text of page 32
137, ^Hlchd 137. The relation between 'Resolvent Kernel'
R(x, 5; X) and 'iterated kernel' K„(x, 5)
U{x)=f{x)+ X\^¥.ix,%)u(i,)d%
for a 'Fredholm Integral Equation'
^ 'R'Jinri'Ti R(x, ^) cWT
K„(x, ^^
Kx)=fix)+\f^K(x,^M^)d^
t: is :
(A) = (A) R(^,4;X)=2r^"K„(*,4)
(B) R(x,4;X)=2:rrt„(*,0 . (B) R(;c,4;X)= X;°X"-'K„('.?)
(C) R(x,4;X)=2;;°X"K„-i(*.4) ' (C) R(*.§:>.)=2r>-X-i(*.5)
(D) R(a:,^;X)= 2:U"-'K„.,(;c,5) (D) R(x,4;X)= X|,^""'K„-i(^.^)
138. • 138. Integral equation
rK(;c,4)a®rf^+/(x)=0
••a /
f K(;f.4)"Ky^+/W =0
•a
t, : is a :
(A) WR ^ (A) "Fredholm Integral Equation" of
first kind
(B) UW (B)."Fredholm Integral Equation" of
second kind
(C) '$r«iR ^ "qft^ (C) "Volterra Integral Equation" of first
kind
(D) fefb? TOR ^ 0) "Volterra Integral Equation" of
second kind.
(2IS/13/2024/A 30
Text of page 33
139. ^ 139. For the 'Fredholm Integral.Equation' .
y(x)^Xj^K(x,^)y(^)di
y(x)^-Xj^ K(x,^)y(,Qd^
with syminetric kernel eigen values are
^ ttpt ^^ f : always : ^
(A) ■ . (A) real
(B) chicTHpiqi (B) imaginary
(C) "RTW «RTTcq^ (C) only positive integer
(D) "TO (D) only negative integer
140.
140. Solution of the integral equation :
*=|g COS(X-0u(4y4 x =fg cos{x-'Qu{%)d%
^"I : is :
(A) 1+y ■ ' . (B) \~ (A) Uy (B) 1-y
(C) 1 + X (D) 1-a: (C) 1+x (D) 1-x
141, WUX= {a, b,c,(i} a% ilHT T ={X,([), 141. Let X = {a, b, c, d} and let T = {X, ij),
W., 6}, {c, {a, c, 6?}, {a}, {6}, {a, b], {c, d], {a, c, d},
{6, c, d)}y X ^ "^nfFTf^ f I ^ {b, c,d}} be a topology for X. Then the
W- T^ tr^-aiiqR t I collection is a basis for T.
. (A) B(x) = {{a}, {c, ^0} (A) BW - {{fl}, {c,d)}
(B) B(x)={{fl}, {6}, {c,^/}} (B) BW = {{a}, {b}, {c, d}}
I
(C) B(x) = {{6}. {c, rf}} (C) B(a;) = {{6}, {c, rf}}
I
(D) k^)={{^/}, {6}} (D) BW={{a}, {6}}
(2)S/13/2024/A 31 P.T.O.
Text of page 36
146. ^ "f^ 146. If a generalised co-ordinate has the
X^sRTI %,^ WlFlt^ M-" i^ dimensions of momentum, then the
vkm I
generalised velocity, will have the
dimensions of.
(A) ^ (B) "SFTSTf
(A) Force (B) Energy
(C) (D) 7# (b)■ Speed
(C) Inertia
147. fSFnl S\<=4HH ^ T?i^ ^^ TITO 7I#IcT 147. A particle of unit mass is projected so
%ziT ^ t f¥ 1^ ^ ^ T?T ■ that its total energy is E in field it is
"t "31^ E ^ "l^ moving in and the potential energy is
1%^ ^ r ^ f^8Tf^ -gwrf V(r) t, V(r) at a distance r from the origin, then
cit f^«&id for such a particle by principle of least
^ -qsT ^ ^4l4)<"l t : action, the differential equation of the
path is :
(A) r^+f-'''"
KdQJ J
=/[E+VW]
(A) = r''[E + V(r)]
Ue,
(B) r2- =/[E+ V(r)]
Ue; j =/[E+ V(r)]
(B)
/ , \2
(C) r^- = /-'^[E-V(r)]
U0 (C) r^-\ =/[E-V(r)]
mq).
(D) = r''[E-V(r)] (D) = r''[E-V(r)]
Ue ■ Ue
(2)S/13/2024/A 34
Text of page 37
148. A' ^ ^ 148. In variational principle the line integral
^ ^HIcbcH ofsome function between two end points
t: is ;
(A)^ (A) Zero
(B)
(B) Infinite
(C) (C) One
(D)^
(D) Extremum
149. H ^ ."CT t|' 149. Hamiltonian H is defined as;
I
I
-mi t I
(A) The difference in energy of the
(A) "f^OTT ^ "37^ 3ffT^ system.
(B) feR ^^ (B) The total energy of the system
(C) PichiM ^ "31^ ^ ^pHWT (C) The"product of energy of the system
(D) Pro 37^ (D) Constant energy
150. "zrfk ■ ^ WT ^ "Pr^ 150. If the Lagrangian does not depened on
f; cR : time explicitly, then ;
(A) f (A) Hamiltonian is not constant.
(B) tPr 3rjrf f (B) The kinetic energy is constant.
(C) tfeetPm i (C) The Hamiltonian is constant.
(D) ftsrpR 37^ t (D) The potential energy is constant.
(2)S/13/2024/A 35 P.T.O.
Text of page 38
qm III (Part III)
.61. ^ E„E2, ,E„ wfk 61. If Ei,E2, 3„ is a countable
sequence of events such that E,- 3E^^j
Ey=)E,^,; / =!, 2, ., ^ : for / = 1, 2, then :
(A) LimP(E„)= 0
V' rt->co
(A)
^
LimP(E„)
«->Q0
=0
(B) LimP(E„)= «) (B) LimP(E„)= co
^' n->®
(C) LimP(E„)= l (C) LimP(E„)= l
^ U-^00
A' n->co
(£)•) LimP(E„)= 0.5 (£)•) LimP(E„)= 0.5
^'rt->00
62. 4- ^6 t*, 62. A man draws 2 balls from a bag
i{ o?Tf^ 2 t 1 containing 4 red and 6 green balls. If he
^ M ^ 45 is to receive Rs. 45 for every red ball
t ^85[T ^ ^ 22.50 which he draws and Rs. 22.50 for every
green ball, then his expectation is :
(B) 15^ (A) Rs. 42 (B) Rs. 15
(A) 42^
(D) 63 (C) Rs. 40 (D) Rs. 63
(C) 40"^
63. 1^ 131T '^MPHd ^ t : 63. Given the symmetrical distribution :
;-a<x<a
311^ (114) ^ the fourth central moment (114) will be
(4-7t) a^(4-7t)
(A) (B) (A) (B)
71
71
(C) {\-2a)n (D) + (C) (1-2^)71 (D) a''ll+^ Stc
(2)S/13/2024/A 36
Text of page 39
64. arafqp ^ ^ 64. Using Chebyshev's inequality, how many
31^Tfe^' f«=Mi Rhdnl ^ times a fair coin must be tossed in order
^ 3^T^ ^ "^TlsJTTSlt' that the probability will be at least 0.90
"d^n -atSKnl ^ WsZTT S^^qirT 0.4 that the ratio of the observed number of
0.6 ^ 7?^ ^ yfRjctidl ^ ^ ^ heads to the number of tossed will lie
between 0.4 and 0.6 ?
0.90 ^ ?
(A) 120 (B) 80 (A) 120 (B) 80
(G) 250 (D) 375 (C) 250 (P) 375
65. "RHT Xi,X2,....,X„ mn 65. Let Xp X2,....,X^ be iid Poisson variates
^ (iid) P(l)f ^«TT P(l)and S„ =Xj+X2 +...r.+X^. Then :
S„=X,+X2+ + X„ i\, :
is equal to :
(A) 0.5 (B) 1 (A) 0.5 (B) 1
(C) e-" (D) e-i (C) e-« (D) e-l
66. X Y ^ la^T 66. Given joint distribution of X and Y as'
BVN (3, 4, 16, 25, 0.8) it, it BVN (3, .4, 16, 25, 0.8), then
P(5<X<9|Y = 6) ^ TTH ^ : P(5<X<9|Y = 6) will be :
t: Given :
z. 0.3 1.0 1.97 3.0 z 0.3 1.0 1.97 3.0
(t)(z) 0.11791 0.34134 0.47558, 0.49865 <|)(z) 0.11791 0.34134 0.47558 0.49865
. (A) 0.35767 (B) 0.48767 (A) 0.35767 -(B) 0.48767
(C) 0.59349 (D) 0.29674 (C) 0.59349 (D) 0.29674
(2)S/13/2024/A 37 P.T.O.
Text of page 40
67. ^ ^ laiT 313^5^ I 67. Given the sequence {X„} of independent
{x„} -qfwiftct t : random variables defined by :
P{X.=±2'}= 2-'"^'',
p{Xt=±2*}= 2-(^''''^
P(Xi=0)=l-2-".
P(Xt =0)=l-2''^^
B„-Var EX,
^ n ^
^ ^ lim%
n->~ fl
If B„=Var|^XX.j,then
Vi=l J
is equal to :
^ ilH ^ :
(A) 2"
(A) 2"
(B) 1
(B) 1
(C) CO
(C) 00
(D) 0 (D) 0
68
X^ 68. The two-dimensional random variables X
and Y have the joint probability density
function :
\kx{x-~y); 0<x<2,^x<y<x fjb:(:c-y); 0^x^2,-x<y<x
f{x. yh 0; Q. otherwise
■ERHT A : {X : X > 1} The-probability of the event
#Tt : A: {X:x> 1}
is :
2
(A) 3
(A) 3
h (B)
15
(C) 15
16
(C)
(D) i
(D) g
(2)S/13/2024/A 38
Text of page 41
69. ^ ^ X^ 69. The probability density function oT a
WT t : continuous random variable X is given
by:
1 •
•-e x>0, >.>0
/W= A, 1 --
0; 37^ —e^; x>0^X>0
/W= A
0; otherwise
Y = VX ^ Mll^ctidl wm
The p.d.f. of Y = Jx will be :
1
(A) ~e^
(A)
(B) ^e'>-
(B)
(C)
(C).
(D) 2y
(D) ^
70. T3^ "W^ arrpft ^ TT^ 70. A cyclist covers his first 3 miles at an
3 8 %f "Rf^ WEi ^ aRT^
average speed of 8 m.p.h., another
2 3 -Jfe -Rf^ WT ^ 7Tf% 3^
2 miles at 3 m.p.h. and the last 2 miles
37i^ 2 2 1% "Rf^ ^ Tjf^
at 2 m.p.h. The average speed for the
^^ fI Tjjft -qr^ ^
3ltetT "Rf^ : entire journey is :
(A) 4.33 (A) 4.33 m.p.h.
(B) 3.43 41d/Nu<;i (B) 3.43 m.p.h.
(C) 4.86 (C) 4.86 m.p.h.
(D) 3.33 ite/Wr (D) 3.33 m.p.h.
(2)S/13/2024/A 39 P.T;o.
Text of page 42
71. R; X„ OT (0 71. If R/ is number of return to state (/) by
^^ f sfk P/,; m chain X„ and P/,/ is the probability of
(i) ^ (0 ■^* wr return to state (/) from state (/) within
^^ ^ yfedl t ^ (HHldfed
finite time. Then, which of the following
■^' ^ MI4)T=i
is not true for transient state (/) of a
{x„}^^o ^ if^TTm HiP^ctidi sn^
Markov chain with transition
P 1^ 3T^^ (0 ^ ^
^ "=1^ t ? probability matrix P ?
(A) P(Ry = co|Xo = /)<l (A) P(R,=a>lXo = /)<l
N
(B) P/,;<1 (B) P;;, <1
(C)
I[P"] <00
(C) ![""] <00
«^1 /, /
(D) E(R,|Xo = i) = «' (D) E(R, I Xfl = /■) = <»
72. 133? liH<l=*iiI 33^^ (0 ^ 72. A recurrent state (/) is said to be null
w I5irai t, Ilf3 n,(/) ^ 313^ (/) recurrent if (*) that is expected time
^ (0 3?t Hri3lfilW : of return to state (/) from (z") is :
(A) H,(i) =l (A) H, (') =l
(B) l<n,(/)<co (B) 1<H,(0<" s
(C) n,(0 = <» (C) n,(0="
(D) ti,(/) = 0 (D) n,(/) = 0
(2)S/13/2024/A 40
Text of page 43
73. 4Hi Xj X2 wdsf 3^ 73. Let Xj wd X2^two independeht ahd
Iqa<"i nnt*w Minimal ideMically distributed rajtldom variables
With probability mass function :
I
1 Jt = 0,l,2
p/v -fr^-_i_ A = 0;i>2i 'R-'j-i..,. ,.u
" 2*^'" /= 1,2
f, ift X,+ X2 = 7 Trifw ^ Then the probability that Xj + X2 = 7
#ft ? is : >
(A) ^
8
(B)
7
2*
(A) ^
8
(B) ^
8 7
(G) ^ (D)
.2'
(C) J (D) J
74. ^ ftr X|,Xj X, 74. Let Xp Xi,..„....Xg be independent and
ak^^^ ftrafta N(0,1) <«5Pd4> identically normally distributed N(d, 1)
t I^ raiidbm v^ables. If
T= xf+xi+ +X^ v2
T = Xf+X^+ +X|,
t, Tft E^y]^ ^^^ : then eI yJ equals:
. 1
(A) J (A) 5
(B) I (B) g
to H: to n
(D) aHtJiTO ^ ^ (D) Ndiie of the above
I
I
(2|S/li/2024/A 41 P.T.O.
c\
r t
Text of page 44
75. yif^ctiai 75. In a Binomial distribution with
: probability mass function (pmf) •
/ -\
n n
p[x)= ■• p^{^-pr, pi'')=[Jp'i^~pT '■,
\X./
x = 0,1, 2, ,«; 0</?<1. x = 0,1, 2 0<p<l.
f I ■'if^ ch'^OM 1^3 "HWT
the third central moment \x^ is k times
M-2 ^ ^ iff eft ^ "HR ^ the variance p-j • What is the value of /? ?
?
1 • \-k
1 \-k (A) (B)
(A) (B) k +l
ifc +1 2
\ +k
l+ k 1
(C) (D) (C) (D)
.k-\
76. -Tn^ ^ 76. An absorbing state of Markov chain is :
(A) 3ik ^ t ■ (A) Periodic and Transient
(B) 3rf -^^Twrf ^ t (B) Periodic and Recurrent
(C) ^RReff sftr ^t (C) Aperiodic and Transient
(D) 3lk ^^iWf ^ t (D) Aperiodic and Recurrent
77. Xj, X2 WTPT 1^ 77. If Xj, X2 are independent standard
x^ normal variates, then the distribution of
^ v2 ■^ ^1 sVll ?
Aj Xf
X2' •
(A) "RPRT y^lHM
(A) Standard Normal
(B) ^ '^'
(B) Student's W
(C)
(C) Chi-square
(D) F (1, 1) (D) F (1, 1)
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Text of page 45
78. -m X'-F(3,9) ■ sk 78. Let X'-F(3,9) and Y~F(9,3)'. If
Y~F(9,3) I ^ P(X^4)= 0.46 P(X > 4) = 0.46 and P(Y ^ A:) = 0.46,
sfiT P(Y ^ k)= 0.46 t, W then the value of A: is :
Ml*! ?
(A) 0.025
(A) 0.025
(B) 0.25
(B) 0.25
(C) 2
(C) 2 .
(D) 4
(D) 4
79. Let X be a random variable with
79. X ^ t,
yi^Fhdl "^Fl^ . probability density function (pdf)
/(:c;0)= 0.e-®^; x>0, 0>O,i ^ /(*;e) = e.e"®-'; *>0, 0>O, then ;
(A) E(X)<V(X);O<0<1 (A) E(X)<V(X);0<e<l
(B) E(X)>V(X);O<0<1 (B) E(X)>V(X);0<e<l
(C) E(x)<y(x)ve>o (C) E(X)<V(X) V 0>O
(D) ^ (D) None of the above
80. W\ ^ Xi, Xj, X3, X4 80. Let Xi, Xj, X3, X4 be independent and
WT ^^ fttifer t, identically distributed random variables
WT ^ U (0, 1) t 1 ^ w, Xf ^ with U (0, 1) as their common
■qfeR t, ^ W ^ HTMlFflCl "RFI ^ distribution. If W is the range of X-s
#TT ? .
then expected value of W will be :
(A) 3/4
(A) 3/4
(B) 3/5
I
(B) 3/5
(C) 4/5
(C) 4/5
(D) 1/2
(D) 1/2
43 P.T.O.
(2)S/13/2024/A
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Text of page 46
81. Xi,X2,....,X„;X ^ 81. A random sample X^ X2,....,X„ drawn
^lai "i", ^ TIH 1 3^ 0 "^cIT • on X which takes the values 1 or 0 with
t, -SlfeBT 0 3fk *0-0) respective probabilities 0 and (1-9).
Xx, Xx,-1 Ix, Xx, -1
t I ^ /=!• \i=l . Then M__v/=i is an unbiased
n{n-\)
t: estimator of:
(A) 0 ^ (A)- 0
(B) 02 ^ (B) 02
(C) 0+ « ^ (C) Q + n
(D) «0 ^ (D) 730
82. Xi,X2, X„ 82. Let Xj, X2,....,X„ be a random sample
TRTFTFI ^Era ^ 1ot "^imT t, "RTS^T from normal population with mean p and
^ 3lk 3ROT f I Ml ^ variance a^. Estimates of p and g by
(T ^ ■? : using the method of moments are :
(A) X Mx-Xf/n
/=1 ^
(A) X, IX(x,-x//«
(B) «xJX(X,-X)/«- (B) «xJX(x,-X)/«
(C) xJx(x?-x^)/« (C) xJg(x?-x^)/«
(D) nXj^{x,~xf/n (D) nXjY^{X,-xf/r,
(2)S/13/2024/A 44
Text of page 47
83. -qR Ti T2. r(0) ^ 83; If T| and T2 are two unbiased estimators
01=^1 "^, "I" of. r(0), having the same variance and
p t I fTHT TIc^ 3TT^T^ p is correlation between them. Let e be
^ e t, cT^ : the efficiency of each estimator. Then :
(A) p>>/e (A) p>Ve
(B) p>e-l (B) p>e-l
(C) p>2e-\ (C) p>2e-I .
(D) p>Ve-1 (D) p>Ve^
84. 1TFTT X ^ t, ^ 84. Let the random variable X haye a
"HTTH t, f^'<H=hl 41(Til ^ uniform , "distribution with density
/(x.0)=/(x;^.a)=^ function
/U,e) = /(.;p„a) =^
■^I^T _y| yol'A sslil ®1-sl
where yj .is smallest and y„ is largest
t, -oo<p<co,. CT >0 1
p 3|k CT f : observation, where -oo<p<°o and^
CT > 0. The likelihood estimates of p and
(A) CT are :
(A)
. ■
(B)
(C) ■\{y,-y„)> -^iy,+y^)
(C) -^(a •^U+:>'i)
(D)
(D)
(2)S/13/2024/A 45 P.T.O.
Text of page 48
85. TTRT = :c>0 ^ 85. Let — for x>0. The value
0
P[0<e<fcc]=l-a of k for confidence interval
^ ^ -qpi t 0>o- t): P[O<0<fa:]=l-a is (for 0>o):
1
(A) (A)
log a log a
log(l+a) log(l+a)
-1 -1
(C) (C)
log a log a
-1 -1
log(l-a) . log(l-a)
86. ftmrr % 86. Let p be the probability that a coin vwll
Uq : p = \/2 i: ^i:P = 2/3 fall head in a single toss in order to test
371^ ^ yiRi^di p Hq :;? = 1/2 against Hj . p = 2/3. The
."t I ^ 'FIT ^ I
coin is tossed 4 times and Hg is rejected
Hq ^ ^ ."TO 2
if more than 2 heads are obtained. The
^ 3Tf?7^ aTRIT I "3^'"SRiK
probabilities of type-1 error and power of
^ ife sfk "T&OT ^ ?ife yi(^c(,cji
the test is :
t:
. (A) 5/16,33/81
(A) 5/16,33/81
(B) 5/16, 48/81
(B) 5/16, 48/81
(C) 3/16, 33/81
(C) 3/16,33/81
(D) 3/16,48/81 (D) 3/16,48/81
(2)S/13/2024/A 46
Text of page 49
87. Let Xj, X2,....,X„ be a random sample
^ ^ w f,- r^^chi pdf i from a population with pdf
/(:c; 01,02, .,0^, (H) A: 3T™ /(;c;0i,02, ,0^), where the
yi^tn ■§■ I "^THT '(h'^cki parametric space @ is k dimensional.
hR^^thhi Suppose we want to test the composite
hypothesis,"
Ho:0,=0;, 02 = 02', ,e,=0;;
r<^, ^ "t*, Ho:0,=0;, 02 = 02', ,e,=e;;
0/, 0^, 0/ WIR t I ^ r<k, where 0/, 02* 0/ are specified
MRchrr^HI ^ #lt, TR X WTFTt numbers. When Hq is true, the
^2^ Wit : asymptotically distribution of ^ is ;
(A) 21ogX,x%_,) ■ (A) 2Iog3.,x%_r)
(B) -21ogX, x^(;.) (B) -21og?.,x^(,)
(C) log2X„x2j^) (G) log2X,x^(,)
(D) -21ogX,xV.) • (D) -21ogX,x%_,)
88. TTHI X,~n(h, CT^), 1 = 2,3,4,5 T!3? 88. Let X, ~n(h, ct^) for i = 1, 2, 3, 4, 5
irfcR!?! t I Tra rHH^feltt if be a random sample. Then which of the
^ 31l+el4i MVUE t, ? following estimators is MVUE ?
X,+X,+X3 2X,+X3+X; . ^ X,+X2+X3 ^ 2X,+X3+X2
1 3 ' ^ , 2 1 3 , 2 2
j
_ 5X,+3X,+X3 X3+X3+X3 „ 5X,+3X3+X3 X3+X3 + X3
' 9 ' ' 6 ' 9 ' ' - 6
(A) T, ,(A) T,
(B) T2 ■ (B) Tj
(C) T3 (C) T3
(D) T4 ■ (D) T4
(2)S/13/2024/A ' 47 P.T.O.
Text of page 50
89. -RHT TifaJsf X|,Xj X„ 89. Let X,, X2,....,X„ be a random sample
siPraiTO ^ /(*,0)=i, O<;c<:0> from a rectangular distribution
u
e>0 ^ 'M t I ws^ /(jc;0)--, O<x<0, 0>O. Then the
consistent estimator pf the geometric
'(»-(nx,)"' ^ «'in 3jRi?i^
/n y/"
sVii I
mean K^)= Tl^n is :
^ /-I /
(A) e-'e (A) 0-»e
(B) 0e (B) Qe
(C) Be-' (C) Qe-'
(D) 0-'e-' (D) 0-'e-'
90. ^^ W mi ^^ 90. Random samples of 400 men and 600
nwici ^ ^ 400 women in a city were asked whether they
3?k 600 nleeiisilf would like to have a bus stop near their
™ t I ^ 1?^ ^ m residence. 200 men and 325 women were
^ 200 ^ 325 ®ff I 5% in favour of the proposal. What will be
1% ^iT^Icbdi "^TT IfWr the inference at 5% and 1% level of
sSk Hpediit ^ ^ significance that men and women both
t ? are in favour of the proposal ?
(A) "^l^dicll ^ ^ (A) Both are in favour at both levels of
#ff ^ bW significance
(B) ^ir«fen i> ^ (B) Both are not in favour at both
f levels of significance
(C) ^ ^ -qqfe ^ t (C) Given information is not sufficient
(D) "psfecIT ^ 1% ^ TR T^f (D) Both are in favour at 1% level of
significance but not at 5% level of
^ t significance
(2)S/13/2024/A 48
Text of page 51
91. ^ 91. Suppose we have a sample
^ F(X) Xj, X2,....,X^ from a population with
t, ^ ^ t, ^ 1^ 31WNrei=h cumulative distribution function F(X).
V^ TrfcR^ ^ WTh The non^parametric one sample test
3OT ^ ': corresponds to :
(A) MR=bcrMHI ■ F(M) = 0.5 . (A) Testing the hypothesis F(M) = 0.5
F(M)]> 0.5 ^ "q^tOT, ^ M ^ against F(M) > 0.5, where M is the
^ Tnfeq^ ^ t median of the distribution
(B) WT "sftT^ (B) Wilcoxon one-sample test
(C) "qr ^ (A)"^n (B) (C) Either (A)' or (B)
(D) ^ (A) 3fk ^ (B) (D) Neither (A) nor (B)
92. 3^yNRrl«h 92. The Kolmogorov-Smimov non-para
t: metric test is :
(A) "Q;^ (A) One-sample test
(B) 1|-3tW^ (B) Two-sample test
(C) (A)^(B) (C) Either (A) or (B)
(D) "^T cit (A) sSk H ^ (B) (D) Neither (A) nor (B)
93. : " 93. Consider the sequence :
XXXXYYXYYYXXYYYYXXYY XXXXYYXYYYXXYYYYXXYY
-cpTOi ^ frofl ^ ;■ The count of runs will be :
(A) 7 ; (A) 7
(B) 8 . (B) 8
(C) 20 (C) 20
(D) 2 . (D) 2
(2)S/13/2024/A 49 P.T.O.
Text of page 53
96. (M) 96. The following is an arrangement of Men
(W) OT ^ (M) and Women (W) to purchase ticket
^ ^ 1¥=Tf^TfM ^ •^-
for a rock concert. Test randomness for
TM t :
this (Hq) at 5% level of significance :
MWWWMMMWMWMMMWWMMM
MWWWMMMWMWMMMWWMMM
MWWMWMMMWMMMWWWMWM
i-
MWWMWMMMWMMMWWWMWM
MMWMWMMMMWWM
MMWMWMMMMWWM
%q; ^l^r^=6dl ^ 5%
Given Z = 1.96 or the table value,
(Hq)
we :
Z = 1.96 ^ trh ^ ^:
(A) Reject Hq
(A) Hq ^ 3iWl=bK ^
(B) Accept Hq
(B) Hq^ ^ s
(C) Either (A) or (B)
(C) -ZTT ^ (A)^(B)
(D) None of the above
(D) "3^^ ^^
97. In calculation of the rank correlation
97. M WTO ^ ^8T X T3r^' Y ^
. between X and Y with m tied ranks, we
w^fro ^"wn
add the factor. to for each
^ ^ X sk Y ^
Hc^cfi ^ 1^ -31101 f : repeated value in both the X and Y
series.
(A) /w(w^-l)/12
(A) w(m^-l)/12
(B) /m(w-1)/12
(B) ot(w7-1)/12
(C) m[m^-\)/l2
(C) m(w^-l)/12
(D) ^^
(D) None of the above
(2)S/13/2024/A 51 P.T.O.
Text of page 54
X 1 2 ■ 3 .... n~2 fi-\ n X 1 2 3 .... 77-2 77-1 77
98. 98.
Y n n~\ 77-2 3 2 1 Y n n-1 77-2 3 2 1
X Y ^^ ^
For the above ranks in two samples
^ FThR^ ^^
of X- and Y variables with no tied
ii"n'=h ranks, the Spearman's rank correlation
f. ? is :
(A) +1 (B) 0 .
■ (A) + 1 (B) 0
(C) - 1 (D) ^^^ -^T^ (C) - 1 (D). None of these
99. "^rf^ ^ ^ ■^, : 99. If Bayes* decision rule exists, then : ■
(A) ^iqidcft 1^ mRimhi (A) Bayes' estimator is a regression
t estimator
(B) ^ ^ (B) Bayes' solution minimizes 'a
^ ''^' ^jIIRsIH ^ ^ ^TT?IT posteriori' risk for accepting a
t . - hypothesis
(C) #ff (A) sfk (B). (C) Both (A) and (B) ,
(D) "5^7^ -^f ^ (D) None of the above
= 1/0; O<:c<0
100. 1THT /(^/0) = 0; 100. Let f{x/Q) = 1/0; O<x<0
37^ = 0; otherwise
and the prior distribution of 0 is
3fk -zif^ e
/i(0) = 0.exp(-0), for 9>o, then
/i(0) = 0.exp(-0), 0>o rft' 0
(t)(0/x), the posterior distribution of 0
(|)(0/:c) "mr X ^ %T
given Xf will be : •
ql^ll :
(A) exp(0-jc) (A) exp(0-x)-
(B) exp(:c~0) . (B) exp(x-0)
(C) exp(x/0) (C) exp(x/0)
(D) <5H^=K1 ■
(D) .None of the above
(2)S/13/2024/A 52
Text of page 55
101.^ -t 3lk 101. The test statistic for testing the
SicKri'ichii ^ ^*<941 "t, ^ significance of Ho:p = 0 with V is
HqIp^O ^^ f^ ^ srfet sample correlation coefficient and 'n' is
t: the number of observations is :
r^K-,
(A) t = ,(« - 2) W Wd^dl (A) ^ = with (n - 2) degree of
\ln-2 V«-2
freedom
r^ln-2 fJyi-2
(B) /= ,(« - 2) ^TO ^d=idl (B) t = ^ ^ - with (n - 2) degree of
1-r' \-r
^ "^n^T freedom
r^n-2 r-Jn-2
(C) t = , (rt - 2) W Wd5ldl (C) ' - ^ with (« - 2) degree of
^^ H«T ■ • freedom
(D) ^^ (D) None of the above
102. ANOVA cTlfel ^ ^"^1^ 102. Consider the following ANOVA table :
^ftf^ : Source Degree Sum of Mean Test
f^'^dl ^TTfT "SPif "551 of of squares sum of statistic
m^ Wdddl AjVmid T1T«3T ^
variation freedom squares
%iw 3if^
Treatment w x y 5.22
w JC .' y 5.22
^ 12 z 25 Error 12 z 25
Wtl 14 561 Total . 14 561
ciM^qo difeiqii ^ ■'JJ'f q)<A Rim, w, The value of w, x, y and z that compute
X, ^ ^fk 9bH^\: ^ : the above table are respectively :
(A) 2, 261, 130.5, 300 (A) 2. 261, 130.5, 300
(B) 2, 300, 150, 261 (B) 2, 300, 150, 261
(C) 2,250, 125,311 (C) 2, 250, 125, 311
(0) 2,311, 155.5,250. (D) 2, 311, 155.5, 250
(2)S/13/2024/A 53 P.T.O.
Text of page 56
103. TTHT W Y ^ 103. Let Y follow multivariate normal
(0,1) ^ A distribution N„(O, I) and let A and B be
^ B nxn imto t, nxn symmetric idempotent matrices.
^ troMw Then which of the following statements
t ? is true ?
(A) Y'AY Y'BY ^ ^ (A) Y'^y Y'^Y Chi-square
f I distribution.
(B) ^ Y'(A + B)Y ^ ^ (B) If Y'(-^"''®)Y has Chi-square
t ^ Y'AY ^ Y'BY ^ distribution, then Y'AY and Y'^Y
# I are independently distributed.
(C) •'Tf^ AB = 0-1, ^ Y'AY (C) If AB = 0, then Y'AY and Y'BY
Y'^Y ^t 1 ^e independently distributed.
(D) (D) All of the above
104. X 3lk Y ^ ^ t, I^TW 104. If X and Y are two independent variates
^IWI 4 CI2TT cT^ f, eft X 3fk with variances 4 and cTy respectively,
(X-Y) ^ ^ Tpji% . then correlation coefficient between X
(A) c^y^^Jolal and pC - Y) is equal to :
(A)
(B) CTx/i^4+CTy
(C) Oy/^(jl+Gy (B) CTjj /^4 +^Y
(C) (Sy/yj(y^+c\
(D) CTxay / + CTy
(D) Uj^Uy /^a^+G\
{2)8/13/2024/A . 54
Text of page 57
105. ^ ife f^WT ^ 105. The estimate of error variance in two
t" : variable linear'model is :
(A) e'el{2-rt) (A) e'el{2-n)
(B) ee'l{2-n) (B) ee'}{2-n)
(C) e'e/(rt-2). (C) e'e/(n-2) -
(D) ee'l{n~2) (D) ee'l(n~2)
106. tfe yRlHH E(Y)= XP , 106. Consider the following linear model
Cov(Y)= a^I,^ X ^ 3WT nxp E(Y)= Xp, Cov(Y)= ct^I, where X is
t, ^ r < p ^ 13^ a^To^ f I a matrix of size ny.p having rank
PiHPdndd "qr : r<p. Consider the following
statements:
(I) wff
(I) The set of estimable linear functions
form a vector space of dimension r.
wr t I
(II) If E(C'Y)= 0 for some non-zero
(II) qg® c^
vector C, then there is a function
e(C'Y)= o t, cil %tt "qqr WT
/'P which is not estimable.
/'P t ^ t I
(HI) If all linear functions /'p are
(m)^ "qi^ /'P
estimable then r - p.
t, fit r = pt !
Then which of the above statements is
TR "3q^ ^srff %8iq ^ true ?
f ?
(A) I, n and IE
(A) I, liafklll
(B)T,n
(B) I, II
(C) II, in
(C) II, III
(D) =tiqci III (D) IH only
i ,
(2)S/13/2024-/A 55 P.T.O.
Text of page 58
107. "SFf ^ ^ ^HHdl : 107. The name of the test for testing the
equality of p means :
Ho:^i=^2= — =
^ ^ %T ^ tth Ho:Hi = ^2=--=I^p
t: of several population means :
(A) t-'^^m (A) /-test
(B) ■(B) x^-test
(C) F-^W ■ (C) F-test
(D) (D) Z-test
108.^ wi[^ 108. If an experiment involves two or more
treatments in which some treatments are
^ t 37k ^ "^TT^fW fixed and the others are of random
^ f, fit Frk -qifey, ? nature, one should choose :
/ *
(A) f^wr ^ kfe^ (A) Analysis of variance model
(B) ikf«m (B) Mixed effect model
(C) krf^ Tmr^ kfs^ .(C) Fixed effect model
(D) "57^ kf^ (D) Random effect model
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109. #T ^ Yj, Y3 109. Consider three independent random
^ a- ^ ■QR^rmr variables Yj, Y2 and Y3 having common
RHPriR^d t : variance and expectations :
E(Yi) = ^i,+^3 E(Y,) = ^,+ti3
E(Y2) = ^i+ii2 E(Yj) = fi,+H2
cf8TT E(Y3>^i2+^3 and B(Y3) = 112 + 1^3
^ |X2 1^3 ^ 37^ ^l^hQlch The best estimators of pj, P2 :
:
Y +Y
(A) A.=y„A2 = y,.A3=^^^.
■ (A) A,=YpA2=Y,,A3=^y^
(B) Ai=0,A2=Y2, A3=-^^^|^ '
(B) Al=0, A2=Y2, A3=-^^^
-(C) A.=Y„A2=0,A3=Y3
(C) A,=Y„A2=0,A3=Y3
(D) None of the above
(D) ^M^CKI '^*
110. In the Gauss-Markov model :
110. ifra-ilT^ :
e(y) = ap
e(y) = ap
D(Y) = a^I, ,
D(Y) = a^I, ,
where meaning of symbols is usual, then
■^, ^ a^af HHFT f, ^
the value of Error Sum of Square (SSE)
^^ ■2f|TI (SSE) ^ :
is : -
(A) Y'{i-A(A'A)'A'|y (A) Y'|i-A(A'A)'A'}y
(B) Y{l-A(A'A)"'AjY'
(B) y|i-A(A'A)"'A'|y'
(C) y'{i-a-'(aa')"'a-'}y
(C) Y'|i-A-'(AA')''a-'}y
(D) dM^eki "^f ^ .
(D) None of the above
(2)S/13/2024/A . 57 P.T.O.
/V-
Text of page 60
111. ^ X~N(y.,S) 111. Let X'-N(|x,£), where
x'=(XiX2X3), £'=(0 00) 3fk X'=(XiX2X3), ^'=(000) and
(l - 3 2' (1 3 2^
s= 3 4 1 t, cit X,+ 2X2- 3X3 z= 3 4 1 ,then the distribution of
2 1 2
2 1 2
^ #11 : Xj + 2X2 - 3X3 is :
(A) N(10, 29) (A) N(10, 29)
(B) N(0,29) (B) N(0, 29)
(C) N(0, 18) (C) N(0, 18)
(D) N(0, 28) (D) N(0. 28)
J \ /11 -6 2
\N
(n -6 2^'
112. X~N3 -6 10 ^ ,^ 112. If X~N3 -6 10 -A , then
2 -4 6 JJ 2 -4 6 //
2
P3.12 t: P3.12 IS :
10 10
(A) Tl (A)
47
10 10
m 37 (B) -
15
»=> i (C) -
15
15
(D) Tl (D) ^
(2)S/13/2024/A 58
Text of page 61
1
»1—
113. -RR % «! = 11 3fk «2 =12 113. Suppose /3j = 11 and «2 = 12
X I.
SiqeiWi to
^feil Xj
II
X2 observations are made on two random
Rt)i< i, torjn«i> ®nt «nnr vectors and X2 respectively, which are
t 1^ ^ WIRT ^ assumed to have bivariate normal
WT ^^M^<u| S^ t, distribution with common covariance S,
171^ ^foi |j,j afk \l2 • but possibly different mean vectors pj
i I ^nfoi 3^ k.=tif^a and P2. The sample means vectors and
fFT t : pooled covariance matrix are
fl -1
Xi = S
»"Spooled -(^
~ "H
2 Xi =
-1
, X2 = _j l> Spooled ~
\
-1 5
tit fti?R ^ tu^ Then Fisher's linear discriminant
f: function is :
(A) -|(*l-*2) (A) ~{x^-X2)
(B) -^(*1+^2) (B) -•j(*l+^2)
(C) —(^1-^2) (C) -j(*i-:>:2)
'(D) -^(^1+^2) . (D) ~(x^+X2)
114. 1TH fe X~N3(|I,S),^ 114. Let X~N3([i,Z), when p'=(lll) and
1 1 5 n 1 5^
H'=(111) 41 2^= 1 4 K 4 K 1= 1 4 K
. The value of K such
5 K 3 5 K 3
^ 1^114 X2 4l -Xi + Xj- X3 that ,X2 and -Xj + X2 - X3 are
'f : independent is :
(A) -3 (B) 1 . (A) -3 (B) 1
(C) 3 (D) 0 (C) 3 (D) 0
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115.^ (Yi Y2) ~N2(!I,2), ^ S 115. If(Yj Yj)-N2(|i,Z), where Zisnon-
t I iiFt Yj = -2Y2 singular. Let Y3 = -2Y2, then which of
(fr -pFi if ^ feOT WIPI feiw the following has singular normal
■ t ? . distribution ?
'Y1-Y2-Y3 'Y1-Y2-Y3
(A) (A)
2Y,+2Y2 J 2Y1+2Y2 j
'Y,+Y2 + Y3^ Yi + Y2 + Y3
(B) (B)
I Y,+Y2 . Y1 + Y2
r Y1 + Y2 ^ Y1+Y2 1
(C) I2Y1 + 2Y3J (C) 2Yi+2Y3j
'Y1-2Y2 "Y1-2Y2"
(D) (D) VY2-2Y3,
Iy2
Y,-2Y3
116. "RH X TTRT p X 1 116. Let X be p X 1 random vector such that
t,^ .x-Np(0,S), (S)=P X ~ Np(0,S), where rank (2)= p.
t I r-IMrdR^tl ^ W? t. ? Which of the following is true ?
(A) E^x^S~'xj=/>,'var^x^S~'xj=/) (A) E|x^I~^xj= p,var(x^2"^xj= p
(B) E^x^S~'xj= p,var^x^S"'xj=2p (B) E^x^I'^xj= p,var|x^2"'xj=2p
(C) E(x^S~'xj=2pi var(x^Z~'xj=2p ■ (C) E(x^2"^xj=2p,var(x^2"^xj=2p
(D) E|x^Z~'xj= 2p,var^x^Z~'xj= p (D) E(x^2"^xj=2p,var^x^2"'xj= p
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117. ^ eftf^TT Y„-Y2, , Y„,- 117. Let Y|, Y2, , be a random sample
WTPT 3iiqiK ri ^ of size n from p-variate normal
%, r^Mch\ TTTe^r ^ distribution with mean If and positive
tTWr^ 1^Tf^ ^mWT 2t 1 definite covariance matrix L. Which of
PjHRrlfeid ^ ^t ? the' following statements is true ?
(A) YY"^ t fm^ p (A) Y Y"^- has Wishart distribution with
■ferit ^ ."t I p degree of freedom.
(B) -(Yi-p)'^2-^(Yi-p) 1 feft (B) (Yj- p)'^I"^ (Y] - p) has Chi-square
^dddl ^ t1 distribution with 1 degree offreedom.
(C) Yi + Y2 sfk Yj -Y2 ^ (C) Y] + Y2 and Yj - Y2 are indepen
t I dently distributed. "
n
(D) "4* V feift (D) X (X/ ~ has Wishart
. y=i
^d-^di ^ "^TT^T ■ fq5(11(1 distribution with. «-degree of
t I freedom.
118, 1TPT dlPdM. X], ;C2, X3, X4 118. Let ;cj, JC2, x^, x^ be independent and
3Tk ^ ^ fTTTO identically distributed standard normal
"HHRT ^ f I ■ rdHRdRsld ■ 4* variables. Which of the following has Wsh^
4^2 feft ^ "HI^T t ? distribution with 2 degree of freedom ?'
^2 ^2-
(A) 2 2 (A) 2 2
X3 a:4 .^3 ^4.
2 2
^1+^2 V3+^2^4 Xj + X2 ;CiA:3 + X2X^
(B) (B)
^1^3+^2^4 ^3+^4 X1X2+X2X4 Xy + X^
xf2 + x|
'2 2 2^^
xi+xi xf+xi x^+xl
(C) 2 2 2 2 (C) 2 2 2 2
^1 +^2 ^3 +^4/ +^2 -^3 +^4
;cf + ;C2 0 "^Xi +X2 0
(D) (D)
0 ■ ;C3+;C4 \-
0 x^ + X4
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119. ^ Y =(Yi,Y2,....,Yp/ 119. If Y = (Yi,Y2 Yp)'^ be a random
M\k^l "I 1^RT^ IJ. sfk vector with mean ji and positive definite
2f I variance-covariance matrix 2. Then the
P
coefficient vector (/j, /2, Jp) of the first
TR "RKW XOX/ p
M El
^ V Y-J IS• :
^ (/i, /2' ,y ^ ,
(A) 2 ^^^ 3it^ tth ^ a;T3^ (A) the eigen vector corresponding to
the largest eigen value of 2
(B) 2"^ ^ 3TT^ tmI' ^ (B) the vector of all the eigen values of
2-^
(C) 2 ^ -qn ^ (C) the eigen vector corresponding to
3T3^ an^ the smallest eigen value of 2
(D) 2 ^ 3TT?^' frf (D) the vector of all eigen values of 2
120. ■3#5RT X ^ 3rf^ 3#f^ WT 120. The reaction time to a stimulus X (in
■^f) ^ 1 '^" [^«Rci seconds) is distributed normally in group
f, I^RRH TTT^I 2 aik 8 t sSk ■ 1 with mean 2 and variance 8; group 2
2 ■# 1^RR?T 4 alk Ri-4<ui 1 with mean 4 and variance 1. The two
t 1 "^RH ■jpilf k* t I groups appear in equal proportion. If ;c is
^ :c, X ^ 133? ^aicidUHl^ 1TR t ^ an observable value of X, the best
arssT Hi^rn (mqio discriminant function (in the sense of
RJt ^"^wnaff ^ ^ ^ araf k') minimizing misclassification proba
W^- •^' ci'lf-^d t I bilities) is to classify into group :
(A) 2 ^ 0 < a: < 8/3; ar^sTT (A), 2 if 0 < X < 8/3; o'therwise in
1 k* group 1
(B) I ^ 0 < X < 8/3; ^ (B) 1 if 0 < X < 8/3; otherwise in
2 k* group 2
(C) 2 ^ X > 3; arqarr ^ 1 k" (C) 2 if a: > 3; otherwise in group 1
>
(D) 1 X > 3; 31^811 ^ 2 T? (D) 1 if X > 3; otherwise in group 2
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Text of page 65
121. "Rfe ■£[ P| ("q?^ TOTO 121. What does the first principal component Pj
^ t ? represent in principal component regression ?
(A) -tfeRT "^qhnr j^pET^* (A) The linear combination that has the
second highest variance among all
it I linear combinations.
(B) R%3F fel^f Trq^ (X) ii (B) The linear combination that has the
. ^H'^knT highest variance among all linear
^tcTT t" I combinations of the columns of X.
.(C) ^ R^ft RBWr^' ^ (C) The average of all covariates in the
R^RcT 1 dataset.
(D) "gfwrr ^ r??^ (D) The least significant covariate in the
R^RRt^q^ I regression model.
122. RRt^ ^ ^ %13; 1^ rM" 122. Arrange the steps involved for drawing
oHcjR^ chlf^y. : , stratified sampling :
(i) RrNrr "RT ^ ■wd^a R'^ (H) (i) Select the stratification variable(s)
RR 'RRR I and the number of strata (H).
. (ii) <iM^^ 3^ RR RRR I (ii) Select the suitable sampling frame.
■ (iii) wrAqi RR 1 N^ RRT (iii) In" each stratum, number the
(RR A RRR^sqr ^R^tR) shHl*Ptia elements from 1 to (the
«*i1Ir"< I population size of stratum h).
■ (iv) RRRW °fit H RRt i"fRRlf^ (iv) Divide the entire population into H
qilPilM, I RRNrR it RT^R strata. Based on the classification
RRR^ it yc4ch. H RRf variables, assign each element
■^f .1^Rlf^ "3?^ RR i tRsrfi^ of population to one of the H
■Rt^. I strata.
(v) ar^qif^ -RT R^qrfe-Tfei RRi^ (v) Determine the sample size of each
■Rpr ^ ^ aiT^TR -R 3Tc% RR stratum based on proportionate or
■qJT RJR RTTO ■I disproportionate stratified sampling.
(A) (i), (iii), (ii), (v), (iv) (A) (i), (iii), (ii), (v), (iv)
(B) (ii), (i)i (iv), (iii), (v) (B) (ii), (i), (iv), (iii), (v)
(C) (ii), (i), (iii), (iv), (v) (C) (ii), (i), (iii), (iv), (V)
(D) (iv), (iii), (v), (i), (ii) (D) (iv), (iii), (V), (i), (ii)
(2)S/13/2024/A ! 63 P.T.O.
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123. PlHrdRsid ^ 123. Which of the following is not true about
®fl^ "I" ? systematic random sampling- ?
(i) ^^ ^ ^ WH (i) Every element has an equal chance
^tcit MrA^h of being selected, but each sample
^ "^T^' I does not.
(ii) A' ^ -STT^!?^ (ii) If population consists of some
f^^Tcnit a sfh ^ • periodic variation and the sampling
mm 3TRfe?TT Tfe ^ interval coincides with periodicity,
^ itht then the sample would be
^^snrnn I considered as a good representative.
(iii) TT^* ^ WT,^ ts^H ■^' ^ (iii) Elements are selected from the
■n^ WT ^sFRfeqr ^ population at a uniform interval that
"mi t I is measured in time, order or space.
(iv) Tricjfrf^ ^ WM t (iv) Representativeness may decrease if
^ I there are cyclical patterns.-
(A) (i) ^ (ii) (A) Only (i) and (ii)
(B) "^m (i), (ii) sfk (iv) (B) Only (i), (ii) and (iv)
(C) '^im (ii) (iv) (C) Only (ii) and (iv)
(D) (i), (iii) 3lk (iv) (D) Only (i), (iii) and (iv)
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Text of page 67
124. "Hfen 124. Arrange the steps involved in Sampling
Design Process :
(i) 1 (i) Determine the sampling frame.
(ii) ^ TiqRt^ ^ "^PH =t)1l^y. I
(ii) Select a sampling technique(s).
(iii) ^ i
(iii) Define the target population.
(iv) ^ "Jif^ ^ trorfe
I (iv) Execute the sampling process.
(v) ^ 3TTTO fT^lffer I (v) Determine the sample size.
(A) (ii), (iii), (i), (iv), (v) (A) (ii), (iii), (i), (iv), (v)
(B) (iii), (i), (iv), (ii). (v) (B) (iii), (i), (iv), (ii), (v)
(C) (iv), (iii), (ii), (v), (i) (C) (iv), (iii), (ii), (v), (i)
(D) (iii), (i), (ii), (v), (iv) (D) (iii), (i), (ii), (v), (iv)
125. ^ Y ^ 125. In the ratio estimator for population
mean y, the ratio estimator is defined
MR"Hif^a ^STTcTT t" : as :
V -2!« (A)
(A) >'R a:.
(B) >^R=f-X (B) Fr=^X
(C) ?R=|X (C) yR=|x'
_ X _ X
(D) :>'RfY (D) .Vr=-
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126. chMH IneiM qilRny. : 126. Match the columns :
Term Description
(1) (a) Y ^ ^^ 3Tk (i) Cross- (a) The difference between
■qf^TWT ^HHl=h<ui Validation the observed value of Y
Wqqpit -qR and the value predicted
^ "qN "qir 3TfR I /
by the regression
(ii) (b) ^ tt^ ^ -^f equation.
MPdJIHH wfteTTT (ii) Residual (b) Predictor variable enter or
■^f -31^ "f "qr leave the regression
f I equation one at a time.
(iii)^- (c) '^fim ^ f ■ (iii) Multi- (c) Test examines whether a
qqr t3^ hT-sci 13^ collinearity model holds on
comparable data not
■f^ 51<nil^ . "^IHT used in the original
^ ^ f I estimation.
(iv) TOR (d) ^ #Er (iv) Stepwise (d) A state of very high
HPdMm SicCsR? "MRPfe
Regression intercorrelation among
^ l^s# I
independent variables.
(A) (i)-(a), (ii)-(c), (iii)-(b), (iv)-(d)
(A) (i)-(a), (ii)-(c), (iii)-(b), (iv)-(d)
(B) (i)-(c), (ii)-(a), (iii)-(d), (iv)-(b)
(B) (i)-(c), (ii)-(a), (iii)-(d), (iv)-(b)
(C) (i)-(b), (ii)-(d), (iii)-(c), (iv)-(a)
(C) (i)-(b), (ii)-(d), (iii)-(c), (iv)-(a)
(D) (i)-(a), (ii)-(c), (iii)-(d), (iv)-(b)
(D) (i)-(a), (ii)-(c), (iii)-(d), (iv)-(b)
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Text of page 69
127. ^ ^t : 127. Given below are two statements :
I: ^*, 'W^ StatementI: In stratified sampling, we
^ ^1WM ^ try to secure heterogeneity within
■#^ ^HJy^Mdl ^fdRl^d ^ ^ TIW ^ subgroups and homogeneity between
t I subgroups.
Statement 11: In stratified sampling, we
^ ?:^ ^ ^ f, randomly choose several subgroups that
^ ^WHd: ^ 37«zm "^ we then typically study in depth.
In the light of the above statements,
■jiT^ ■% i{, ^M choose the correct answer from the
options given below :
(A) t I (A) • Both the statements are correct.
(B) I f 3|k II (B) StatementIis correct and statement
tl II is incorrect.
(C) I "n^ t 3;lk II ^ (C) Statement I is incorrect and
statement II is correct.
(D) ^TSH t I (D) Both the statements are incorrect.
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128. : 128. Match the columns :
Vcicim Term Description
(i) (a) "ZT^ 17TW t % (i) Durbin-Watson (a) It measures the
. HRlJlPuid ^ Statistics amount by
which regressed
TTFTif
values are away
j&om actual
t I values.
(ii) (b) "^TF "3?% (ii) Variance Inflationary (b) It measures the
3T^fW 3?tT' factor degree of
correlation
between each
residual and the
residual of the
^ ^TF^T^
immediately
^ feft 1TTW
preceding time
f I period.
(iii) ife (c). (iii) Standard error (c) Collinearity is
^^TRsqi^ ^
measured by it
for each
bfM^ ^ explanatory ,
wm t I ■ variable.
(A) (i)-(b), (ii)-(c), (iii)-(a) (A) (i)-(b), (ii)-(c), (iii)-(a)
(B) (i)-(a), (ii)-(c), (iii)-(b) (B) (i)-(a), (ii)-(c), (iii)-(b)
(C) (i)-(c), (ii)-(b), (iii)-(a) (C) (i)-(c), (ii)-(b), (iii).(a)
(D) (i)-(b), (ii)-(a), (iii)-(c) (D) (i)-(b), (ii)-(a), (iii)-(c)
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129. ^ ^ <UHirci 129. In the Midzuno-Sen sampling strategy,
Inclusion Probability Proportional to what condition must be satisfied for
Size (IPPS) "^Th^RT achieving an Inclusion Probability
chl'itil ?Tcf Proportional to Size (IPPS) sampling
■qif^k. ? plan ?
(A) " ' < Pi<- (A) " ^ < p,<-
n(N-l) ' « n(N-l) ' n
1
(B) P,=^ (B) P,=^
N-1 N-1
(C) ^ P.- ^ 1 (C) —< P, ^ 1
N-n. ■ ' N-«
n
(D) P, — (D) P.=^
130. Jllell^K ^ T3^ 130. In circular systematic sampling, after
' qi^r^cji "RR^ r ^ selecting a random start r, the skip
aTcRIR K : interval K is :
(A) ^ "^RsRT (A) Rounded to the nearest whole
t I number
(B) ^ "SR^W (B) A fixed number of every
WH ^ t I population
N (C) Calculated as the nearest integer to
(C) ^ ^ RcbidM "TTte ^ ^
n N
WTT ^ ^ t I n
(D) ^ ^ (D) Selected randomly from the
%ziT "^STRT "I I sample."
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134. 32--WWt^ ^ AB ^ 134. In a 32-factorial design, if the interaction
WR Ti'm ^ ^8T effect AB is confounded with' the block
t^ 3 371^ ^ t, ^ effect and the blocks are of size 3, how
■Htia^ «<nT«^i "I' ?. many blocks are there ?
(A) 1- (B) 2 (A) 1 _ (B) 2
(C) 3 . (D) 9 (C) 3 (D) 9
135. iJ^rilR^d ■^, 3?^' 135. In a factorial design, what are 'main
™t ? effects' ?.
(A) ^ ^ (A) The combined effect of all factors
(B) yr^cti ^ci*^ (B) The effect of each independent
■3M^ . variable individually
(C) STtRtcT ■=£R ^ "3^ (C). The effects of the dependent variable
(D) t^Rm ^ ^ TT^ (D) The effects of the control variable
136. 3x3 ^ fenfT 136. In a 3 X 3 Latin Square Design, what is
the minimum number of replications
wn TO t ? required to achieve a balanced design ?
(A) 1 ■ (B) 2 (A) 1 (B) 2
(C) 3 (D) 4 (C) 3 . (D) 4
137. 5 T3:^* 5 fi 137. Assume that there are 5 treatments and
3R7M c^Tcji 6 ■ 5 blocks. In addition, we assume that
f alR TT?^' "3W, each block contains 6 treatments and
4 ^ f (^ 5 ^ each treatment occurs 4 times in the
^ t) I "TOfT ^ design (or is replicated 5 times).
yd^=h ^ sciTcfi ■^' Rtiail Calculate the number of times each pair
^'t 1 of treatments appears in the same block.
(A) 2 ■ . (B) 3 (A) 2 (B) 3
(C) 5 - (D) 4 (C) 5 (D) 4
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138. 3 6 ^ 138. In a Randomized Block Design with 3
o^Tcti Hr^ch blocks and 6 treatments, if each
" ^ "51?^ '^* 2 ^ imm treatment is replicated 2 times in each
t/ "dt ^ %?Rt block, how many experimental units are
f ? there in total ?
(A) 18 (B) 36 (A) 18 (B) 36
(C) 72 (D) 12 (C) 72 (D) 12
139. I^T^* 139. Suppose the constant ■ failure rates for
"f^ Pi<ci< [qmcnai "I afk both the main and standby units are
0.02 -grf?! -Ef^ t w r^r^'Ji constant and are given by 0.02 per hour
T3^' ^ Mr ^ and the constant failure rate of the
0.01 ^Tf!r ^ t, 50 ^ switching and sensing unit is 0.01 per
W7 %g ^'m^.^ Mdt hour. Find the reliability of this system
t ? for an operating time of 50 hours ?
(A) 0.6574 (B) 0.6374 (A) 0.6574 (B) 0.6374
(C) 0.7158 (D) 0.6948 (C) 0.7158 (D) 0.6948
140. ^ ^ 140. An electronic device has a failure rate of
0.0025 Ri4>ddli| 1# "dsr t I 0.0025 failures per hour. What is the
^ 1M MTTF M MrT t ? MTTF for this device in days ?
(A) 10 M (B) 16 M. (A) 10 days (B) 16 days
(C) 20 M (D) 40 M (C) 20 days (D) 40 days
141. Mrf^ffer -^*^^13:^ 141. Which one of the following is not
WTT (LPP) i 7 associated With an LPP ?
(A) ^Mi^Hiddi (B) srfdfMmr (A) Proportionality (B) Uncertainty
(C) ■ (D) (C) Additivity (D) Divisibility
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142. 31^ (m < n) 'rrC Y™ 142. Given a system of 'w' simultaneous
^ ^ "W f ^ linear equations in unknowns (m < «),
^ WTT "^Ot : the number of basic variables will be :
(A) (A) m
(B) (B) «
(C) n~ m (C) n- m
(D) W + OT (P) n + m
143. -Rf^ Mr ^ ^ 143. What is. the role of artificial variables in
t ? simplex method ?
(A) oHqei4 3JFI (A) To aid in finding initial basic
■H^TOT ^ "R' feasible solution
(B) "RF^ t^ ^ ^R. (B) To start phases of simplex method
(C) 3TfFT wit wn (C) To find shadow price from the final
^ niRr R ■ simplex method
(D) "5^^ R ^ . (D) None of the above
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144. t. : 144. In queuing theory, is :
(A) "4* ■ W^* ^ ^ ^ (A) Probability of customers in
^*TO=TT ' queue
(B) 'n' ^ ^ (B) Probability of customers in
service
(C) ^^ TIT?^* ^ ^ (C) Probability of 'n' customers in
system
(D) ^ (D) None of the above
145. (M I M |1):(oo I FCFS) 3#^(W) 145. In (M I M |1) : (oo | FGFS)
■^f : model :
(A) *M' ^ t I (A) There are 'M' servers.
(B) 'N' f I (B) There are 'N' servers.
(C) ai^RT ^ t I (C) There are infinite servers.
(D) TT^ ^ t I (D) There is one server.
146. "qf^ (Dual) ^ W 13^ arot^ 146. If dual has an unbounded solution, then
W«TH t, ^ Wl^ ^ W t:/•
the primal has :
(A) srotfim W«TH (A) An unbounded solution ■
(B) W«^PT (B) A feasible solution
(C) T3^ W«TFf (C) An finite optimal solution
(D) "^RTVFT (D) An infeasible solution
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147. ^ PHMRrlRsld 147. Which of the following is not an indices
^ ^ TI^ ■'EJITO ■^* of system performance for a queuing
t ? system ?
(A) "SRT fePT (^)- % (A) Average time of customer spent in
"SdtST fePTT TFIT "^TO waiting and queue of system
(B) wittoT (B) Utilization factor
(C) ^ZRc[ afk 3T^ (C) Busy and idle periods
(D) ^ (D) None of the above
148. ?fri^FT 148. An airline is planning to open a ticket
■^f <sil(rtA ^ desk at a new shopping complex staffed
t, by one ticketing agent. Requests for
I ""^RTT lectKi sfk ^"qii tickets and information takes about 15
1^ cpq^ "q^ 15 ^ "^ITPT per hour on Poisson distribution. At
WM t I ^ ""R ^ exponential rate service time is assumed
■af^ 3?5^tw 3 1^ wrar t i fern ^ to be 3 minutes per request. Find the
(system) -bMHlM W ^tilPny, : system usage ;
(A) 0.65 (A) 0.65
(B) 0.95 (B) 0.95
(C) 0.75 (C) 0.75
(D) 0.85 (D) 0.85
t
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149. (LPP) ^ ^ 149. Given an LPP to Max Z = Xj + X2 such
Max Z = Xi + ^2 (s.t.) X| that X| + ^2 < 1; -3xj + ^2 > 3,
■ + ^2 — "•" ^2 — -^l* ^2 — ^ '
Xj, X2 > 0. Using graphical method,
we have:
TO t :
(A) No feasible solution
(A) ^ TOTTO
(B) SRTtfBcT TOT«ipT (B) Unbounded solution
(C) 37l|dt^ TOieiPT (C) Unique optimum solution
(D) 375^^ TOITO . (D) Multiple optimum solution
150. TOPI Mi ^ H^R!^ 150. At any iteration of the usual simplex
■^, ^ ^ "'R 37FR TO ^ TO method, if there is at le^t one basic
TO t TOt (z^. - Cy) > 0 t, variable in basis at zero level and all
."dt ^1-0 TPTf^TR "$Vlf : - Cj) > 0» the current solution is :
(A) aToq^ (A) Infeasible
(B) TOtfe (B) Unbounded
(C) ■^-■qfe WTO (C) Non-degenerate solution
(D) "qM TOTO (D) Degenerate solution
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(SPACE FOR ROUGH WORK)
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(SPACE FOR ROUGH WORK)
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(SPACE FOR ROUGH WORK)
irn^ WTW
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(SPACE FOR ROUGH WORK)
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INSTRUCTIONS TO THE CANDIDATES
1. 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.Part H has 90 questions in order of61 to 150. This part is for
Mathematics subject and PartHI has 90 questions in order of61 to 150.This part is for Statistics
subject. Candidates for Mathematics subject should attempt Part I and H or Candidates for
Statistics subject should attempt Part I and HI. Examinee clearly mention in the OMR Sheet
whether opted Mathematics or Statistics.
3. Each question has four options. There is only one correct answer to each question. Choose the
appropriate option and darken/blacken the corresponding circle on the Answer Sheet(OMR Sheet)
with black point pen.
4. In this examination there are two question papers, consisting of150 objective type questions and
each question carries 2 mar^. 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
any marks. There is no provisions for Negative Marking.
Paper I General paper (Q. No.1-50)
Paper H Question paper on the Subject opted by the examinee (Q. No.51-160)
5. No. of pages consisting ofthis question booklet is indicated over the cover page. Candidates are
advised to ensure that all the pages ofQuestion Booklet are properly printed and binded. Otherwise
they may demand the other Question Booklet of the same set.
6. Kindly makfi necessary entries on the Answer Sheet(OMR Sheet)only at the places indicated and
nowhere else.
7. Examinee should do all rough work on the spaces meantfor rough work on the pages given in the
Question Booklet and nowhere else, not even on the Answer Sheet(OMR Sheet).
8. If there is any sort of mistake either of printing or offactual nature in any question; then out of
the Hindi and English versions ofthe question,the Hindi version will be treated as standard one.
9. Use of any type of calculator, log table or any type of electronic devices etc. are not allowed. .
10. Use only black ball point pen to mark the answers in the Answer Sheet(OMR Sheet)only after
you are instructed to open the Question Booklet.
, 11. Candidates will leave the Examination Hall only after handing over the Answer Sheet(OMR
Sheet)to the Invigilator at the end of the examination.
12. Before answering on Answer Sheet(OMR Sheet)ensure to follow the instructions given
for that.
S/13/2024/A
Page 1 of 84
Source: Madhya Pradesh Public Service Commission. Question paper published by Madhya Pradesh Public Service Commission.