SSB Odisha Lecturer 2018: Lecturer recruitment for non-government aided colleges, Mathematics
Text of page 1
DO NOT OPEN THIS TEST BOOKLET UNTIL YOU ARE ASKED TO DO SO
TEST BOOKLET Sl. No. - STSII
Subject Code : 18 Subject : Mathematics
LECTURERS FOR NON-GOVT. AIDED COLLEGES OF ODISHA
Allowed : 3 Hours Maximum Marks: 1
: INSTRUCTIONS TO CANDIDATES :
1 flUMEDIATELY AFTER THE COMMENCEMENT OF THE EXAMINATION, YOU SHOULD
CI{EGI(THATTI{ISTESTB@KLET CONTAINS3I PAGESANO DOESNOTHAVEAI{YUNPRINTEDOR
TORN OR MTSSING PAGES OR ITEMS ETC. IF SO, GET IT REPI.ACED BY A COMPLETE TEST
BC'OIGEI.
2. You have to enter your Roll No. on the Test Booklet
in the Box provided alongside. DO NOT write
anything else on the Test Booklet. a
rn
3. The Test Booklet contains 165 questions. Each question comprises four answers. You
have to select the correct answer which you want to mark (darken) on the Answer Sheet.
ln case, you feel that there is more than one correct answer, you should mark (darken) the f-
answer which you cbnsider the best. ln any case choose ONLY ONE answer for each
question. lf more than one answer is darkened it witl be considered as wrong.
4. You have to mark (darken) all your answers ONLY on the separate OMR Answer Sheet
provided, by using BLACK BALL POINT PEN. You have to do rough work on the space
provided in the Test Booklet only. See instruction in the Answer Sheet.
5. All questions carry equal marks, i.e. of one mark for each conect answer and each wrong
.answer will result in negative marking of 0.25 mark.
6. Before you proceed to mark (darken) in the Answer Sheet the answers to various
questions in the Test Booklet, you have to fill in some particulars in the Answer Sheet as
perthe instructions in yourAdmit Card.
7 . Afteryou have completed filling in allyour answers on theAnswer Sheet and after completion
of the examination, you should hand ovbr to the lnvigilator the Originat Answer Sheet
(OirlR Answer Sheet) issued to you. You are allowed to take with you the candidate's
copy/second page of the Answer Sheet along with the Test Booklet after completion of the
examination for your reference.
Candidate's full signature lnvigilator's signature
RS - 24fi2 ( Turn over )
> Dl8
Text of page 2
1 Which of thefotlowing'is notconect ? (C) n dobs not divide k
(A) The setQ*ofpositive rationals (D) n and k are primes
is a group under ordinary
4 The order of (123)(145) in the
multiplication.
permutation group Su is:
(B) The subset {1, - 1, i, -i} of
(A) 6
' complex numbers is a group
(B) 3
under complex multiplication.
(c) 5
(C) Theset Zn={0,1,2, .....n- 1} :
(D) e
forn2l isagroupunder
addition modulo n. 5 The group of even permutations of
n symbols is denoted by An and
(D) The set {0, 1,2,3} is a group
it is called alternating group of
under multiplication modulo 4.
degree n. For n > 1, An has order:
2. The set ,^l {O;1,2,.......,n- 1} is a (A) n!
group unUer multiplication modulo n
(n + 1)!
if and only if : (B)
2
(A) n is a prime (n - t)t
(c)
2
(B) n is even
n!
(c) n is odd (D)
2
(D) n is not a prime
6. For every integer a and every
3. Let G be a group and let a be an pnme p:
element of order n in G. tf ak = e, (A) pa mod p -.p mod a
then: (B) aP,-mod,,,.r-.P = a niod P
,- -' :r-i'!..r:r-,, I...
'(A) n divides k (C) aPmodp=pmodp
(B) k divides n (D) aPmodp=pmoda
RS - 24t12 (2) Contd
Text of page 3
7 The group of rotations of a cube is 11. The polynomial 3x5 + 15xa -20x3 +
isomorphic to : 10x + 20 is irreducible over:
(A) A4 (A) n
(B) 55 (B) R-Q .
(c) 54 (c) a
(D) As (D) Z
8. An integra.ldomain is a commutative 12. Let G be a group and a, b e G such
that o(a) = 6, o(b) =2 anda% = ba.
ring with unity and :
Then o(ab) is:
(A) Zero-divisors
(A) 6
(B) No zero-divisors
(B) 8
(C) Zero multipliers
(c) 12
(D) None of these
(D) 2
9. The' characteristic of an integral
13. The numberof subgroups of order2
domain is:
in the permutation group S, is :
(A) 0 or non-prime (A) 1
(B) 0 oreven number (B) 3
(C) 0 or odd number (c) 12
(D) 0 or prime (D) 2
10. lf F is a field of characteristics 0, then 14. lf G is an abelian group, then the
F contains a subfield isomorphic to number of conjugacy classes equal
the: to:
(A) lrationalnumbers (A) o(G)
(B) Rationatnumbers (B)l '
n'
(C) Even numbers (C) o(G)- o(Z(c) )
(D) Odd numbers (D) 2
RS -24t12 (3) (Turn over)
Text of page 4
I
15. Let G be an abelian group with the (B) Kernel f = {(123), (132), e}
identity e. Which one of the following
(C) Kernel f ={(123),(12)I
statement is true ?
(A) H ={xe G : orderofxisodd}is (D) None of these
a
a subgroup ofG
19. Let A be a 5 x 5 real matrix.
(B) H={x€G;orderof xis
Suppose 0 is one of eigenvalues of
even) v{e} is a subgroup of G
A. Which of the following statement
(C) Every subgroup of G is normal
is true ?
(D) G is cyclic ,,, t \,.,,. ,
,.r, i <_ii,!i, ii
(A) System Ax = 0 has unique
16. ln the ring Zrlxl , the element
solution
4x2+6x+3is:
(A) Anilpotent System Ax = C has unique
(B) Aunit solution forany C
(C) A idempotent (c) SystemAx = 0 has a non-trivial
(D) A non-zero divisor solution
17. Suppose that g : Zro + Zrois an (D) None of these
automorphisim such that (5) = 5, the
" ' )'r:ilr
number of possibilities for Q(1) is : 24. Which of the following subset is a
(A)4 , subspace of the vector space IR.3
(B) 1 overthe field R ?
(c) 5 (A) {(u, v, w) e R3 :2u + 3v+4w= 0}
(D) 20
a (B) {(u, v w)€ r.R3 :2u + Jv+dvv= 'ti
Let S, be the permutation group of
{1 , 2, 3}. Then there exists a non-trivial
(C) {(u,v*)
:n :u>0,v>0,
group homomorphism f : S, --> S, . w>o).
such that: (D) {(u, v, *) . R3 : u, v, w are
(A) Kernelf = 11t2), e) rationals)
RS - 24t12 (4) Contd.
Text of page 5
21 The dimension of the vector space (C) lf xe HandyeH,then
A t.Eloverthefield Q is: xye H
(A) 1
(D) Both (B) and (C) are true
(B) €
24. The chqracteristic polynpmialof the
(c) 4
(D) 2 (a 'O 0\
3 x 3 matrixA= lo b ol rs grven
22 Let X and Y be subspaces of finite [oo ")
dimensional vector space V.
by:
LetX * Y= {x + y : x e X, ye Y}. The
dimension of the subspace X + Y is
(A) - 1.3 + (a + b + q),2- (ab +
always equalto : bc+ca)I+abc
(A) dim (X + Y) = dim(x) + dim (Y) (B) 1.3 + (a + b + q)"2 - (ab +
-dim(XnY) bc+ca)tr+abc
(B) dim (X + Y) = dim(X) + dim (Y) (C) -tr3- (a + b + c)12- (ab +
(C) dim (X + Y) = max{dim(X), bc+ca)1.+abc
dim ffi) (D) ).3 + (a + b + c)i.2 + (ab +
(D) dim (X + Y) = dim(X) + dim (Y) bc+ca)l.+abc
+dim(XnY)
The greatest common divisor (gcd)
23. Suppose G is a finitegroup and H is
of 5n + 3 and 7n + 4, for all
a subgroup of G. f [G : H] = 2, then
neNis:
which of the following statement is
true ?
(A) 1
(A) lfxeHandygH,thenry€H (B) 5
(B) ItxeHandysH,then (c) n
xy-1 e H (D) 2
RS - 24t12 (5) (Tum over)
Text of page 6
26. How many three digit numbers are 30. lf 265 = b(mod 19), then b is :
divisible by 6 ? (A) 4
(A) 142 .(B) 1
(B) 150 (c) e
(c) 148 (D) 6
(D) 166
31. Last two decimal digits of 314s2
27. lf gcd (1492,1066) = 2, thenwe have are
tcm (1492, 1 066) = ? (A) 14
(A) 752e36 (B) 41
(B) 7e5326 (c) 10
(c) 7e5236 (D) 40
(D) Can not say
32. Which'of the following equation
28. Let a, b e Z such thatgcd (a, b) = 5. have only finitely many solutions
Then the equation ax + by = c2 has : (integral) ?
(A) Unique solution forany c e Z (A) Forany a e Z,ax + (a + 1)y= $
(B) lnfinitely many solutions (B) Foranya e Z,ax+.a)=a11
if (25, 1
"2)= (C) Forany a e Z,9x+ 3y= a3-a
(C) No solution for any c e Z
(D) For any a e Z, (a + 1)x +
(D) lnfinitely many solutions (a-1)Y=3
(x,y) eZ"* Z tt (25, c27 * 1
33. While writing nrrb"rr'from 1 to
29. Let a > 2. The integer am + 1 is a 10000 how many times the digit 9
prime. Then:
will be written ?
(A) a is even and m is a powerof 2 (A) 4000
(B) a is a power of 2 and m is odd (B) 3600
(C) a is odd and m is even (c) 4100
(D) Cannotsayanything
(D) 3Oo0
RS - 24t12 (6) Contd.
Text of page 7
34. The set of all trancendental numbers (C) Divergent
ts an (D) None of these
(A) Countable set 37. A metric space X is said to be
separable if :
(B) Uncountable set
(A) lt has a countable subsetwhich
(C) Finite set
is dense in X
(D) Nullset
(B) lt has a uncountable subset
35. Let X be the set of all real valued which is dense in X
continuous functions defined on the (C) lt has a countable subsetwhich
is not dense in X
closed interval I a, b]. Define the
mapping d_ and d.,on X x X into R.
(D) lt has a uncountable subset
which is not dense in X
as follows :
38 ln R with the usual metric, which of
d_ (x, y) = ITI?X lx(t)- y(t)l
te[a, b the following statement is true ?
b
u -- 6rd n ,.dr(x, -y) J lx(t) - y(t)ldt (A) The set of integers dense in R
=, a
(B) The set of rationals dense in R
Then
(C) Cantor set is nowhere dense
(A) d_ and d., are metrics on X
in IR
(B) d* and d,, are not metrics on X
(D) None of these
(c) d is a metric and but not d
6 1
(D) d., is a metric and but not d_ 39 Every complete metric space is of :
36. f- ,the space of all bounded (A) First category
sequences in R. is,po!. (B) Second category
(A) Complete (C) Both (A) and (B)
(B) Separable (D) None of these
RS - 24112 (7) ( Tum over )
Text of page 8
40. LetXcNbeanon-emptyfiniteset 43. lf 0<a< 1,then nlg{"1it
and Y c N be an infinite set. (A) 0
DefineA= {x-y : x e X and y € y}.
(B) 1
Then :
(c) oo
(A) lnf(A) 3 -oo and sup (A) = ""
@
(B) lnf(A) E - oo and sup (A) . "" (D) a
(C) lnf(A) ) - oo and sup (A) = "" 44. lf X = (xn) be a convergent
(D) lnf(A) ) -eand sup (A).." sequence of real numbers, then X
":
t!
41. Let X be a non-empty set and A, isa:
B c X. lf (A u B) - (A n B) is a finite (A) Not a Cauchy sequence
set, then : (B) ACauchysequence
(A) BothAand B arefinite set (C) Unbonded
(B) Atleast one of theA, B is a finite (D) None of these
set
45. Which of the following statement is
(C) X is a finite set
nottrue ?
(D) None of these ' ''t
(A) A Cauch sequence of real
42. Let f, g : R + IR be are polynomials numbers is bounded
with deg(fl = 5 and deg (g) = Z.
(B) A sequence of real number is
LetA= {x e R :f(x) = g(x)}.Then :
convergent if and only if it is a
(A) lAl= 5 Cauchysequence
(B) lAlsT (C) Every contractive sequence is
(C) A is finite, but I A I cannot be a Cauchy sequence
arbitrarily large
(D) The sequence 1'+' ( 1)n is a
(D) A can be finite Cauchysequence
RS - 24t12 (8) Contd
Text of page 9
(B) f(x) = 0 PnlY for all x e Z
46. The p-series |,l= ,, ,, ,
$ (C) f(x) = 0 onlY for all x e N
(A) Converges when p > 1 and
(D) None of these
divergeswhen0.p<1
(B) Diverges when p > 1 and 50 Let I - ia, bl and f : [ -r R be
. converges'ivhen0<p<1 continuous on I. lf f(a) < 0 < f(b), or if
(C) Converges for all p f(a) , 0 > f(b), then there exists a
numberc e (a, b) such that:
(D) Diverges for all p
(A) f'1c1= g
47. The series i. In"=, (B) f(c) = 1, ;r, O
fr,, ,
(A) Convergent (C) f(c) = s
(B) Divergent to oo (D) /1c1= k, k > O
(C) Oscillatory 51 Which of the foltowing staterhent is
(D) Divergent to - oo not correct ?
48. LetA c R. Let f, g :A -+ R and let (A) lf f : A -+ R'is a Lipschitz
c e R. a cluster point of A. Suppose function, then f is uniformly
that f(x) s g(x) for all x e A, x * a, continuous onA
implies that:
(B) lf f :A+Risuniformly
(A) lf limr-"f =@,then continuous on a subsetA of R.
limr-agisfinite
and if (xn). is a Cauchy
(B) lf limr-"f =@,then
sequence inA., then f(xn) is not
limx-+avo=oo
aCauchy sequence in IR
(C) lf lirhx-"f =-oo,then
limx-+a-o=*oo (C) lf f and g are each uniformly
continuous on R, the composite
, (D) lf limr-"g--oorthen
limr-agisfinite function fog is uniformly
continuous on R
49. Suppose f : R -+ R is continuous on
R and f(x) = 0 for every rational (D) lf f is uniformlycontinuous on a
number x, then : bounded subset A of R ,then
f is bounded on A
(A) f(x)=0forallxe R'
RS - 24112 (e) (Tum over)
Text of page 10
52. Let f : IR -+ R defined by the series (B) -1
f(x) = >;= o j "o. (3nx) rhen f :
(c) 1
(D) oo
(A) ls continuous atevery point but
derivative exists at finite 55. The value of
number of points 1(.' 1 1)
+-lis
li'n-r- ;|''*r* n)
(B) ls not continuous at every
points (A) 0
(C) ls differentiable at every points (B) 1
t
(D) lscontinuousateverypointbut (C) oo
whose derivative does not (D) e
exists anywhere
56. The Lebesgue measure of R is :
53. lf f : [a , bl -+ ]R is a differentiable
(A) Finite
function and if k is a number between
(@)ano /1u1, then :
(B) 0
(C) Infinite
(A) There is atmost one point
c e (a, b) such that f'(c1 = 1
(D) . None of thesg
,-,1 :i
(B) There is at least one point 57 . Which of the following statement is
c e (a, b) such that f'(c1 = 1 true ?
(C) There is no pointc e (a, b) such (A) Lebesgue measure is not
that/1c1= I translation invariant
(D) None of these (B) Open intervals are Lebesgue
measurable
54. Let f : LR -+ R be defined by f(x) = 1 +
(C) Lebesgue measure of Cantor
^1 for x * 0 and f(0) = 0. Then
2x'sin; x set non-zelo,
,. . - ,:riri ,,, i
/10; is : (D) The Cantor set has countably
(A) o manyelements
RS - 24112 (10) Contd
Text of page 11
58. Let p be a rneasure on a set Q and (C) Dominated conver$ence
letf, g : f) -+ [0, -] measurable. Then theoem
which of the following statement is (D) None of there
false ? 61 The function f(x) - - log ( I x l) is
(A) lf A c Q is measurable and integrable over any compact interval
f(x) = 0 foralmost every x € A, in the sense of :
then lofdpr = O (A) Both Riemann and Lebesgue
(B) lf A c Q is a null set, then (B) Riemann only
Iorop = g (C) Lebesgue only
(c) lf f < g, then lofdp s JoOdu (D) None of these
(D) lf Ac B cQ are measurable, 62 Every countable set has measure :
then Iofdp > IoOdr (A) 1
59. A measure p on a metric spac€
(B) 0
(Cl, d) is called a Borelmeasure, if :
(C) Non-zero.
(D) Uncountable
(A) F(a) < ""
(B) Alltsorelsetsare.measurable 63 Which of the following statement is
wrong ?
(C) lf for every x e Q , there is a
(A) lffisameasurablefunction and
r > 0 such that p(8,, (x)) <
"" ' let fog = 0 a.e., then g is
(D) Noneof these
measurable
60. Let f., , f 2 ...... be a measurable (B) Monotone functions are not
functions such thatfn -+f p-a.e.there
measurable
exists a p-summable function g such
(C) lf f is a measurablefunction and
that I fn | < g, then : es.s Sup P I f I ( oo, then f is
Jg1llfn -f ldp=oanontim Jf,op=Jtop
essentially bounded
(D)
' lf f is a measurable function,
This represent as :
(A) Fatou's Lemma then ess sup fs sup f.
(B) Monotone converge theorem
RS - 24t12 ( 11 ) (Tum over)
Text of page 12
(C) Lipschitz at x = 0
64. The value of dx
{* (D) Differentiable at x = 0
(A) 0
(B) 1
68. The solution of y'+ y = ry4 is given
(c) oo by:
(D) None of these (A) ,*:r*{*ce3*
65. Let f e BV [a, b] (f is a function of (B) y3=x*1*."3t
bounded variation on I a, b]) and (C) y{=x***oe-3t
x e (a, b), then :
(D)'y3=x+*+6-3x
(A) f(x-) exists but f(x+) does not
exists 69. The general solution of d) + 1)dx
(B) f(x+) and f(xl exist * ($V * tr1 dy = 0 passing through
(C) f(x ) does not exists but f(x+) (a O1 is given by:
exists (A) ,-lf * {fy = o
(D) We can not say anything about
(B) x*lf*{fy=6,
f(x+) and f(x )
(c) x*lf-lrty=o
66. BV [a, b] is a vector space over :
(D) -x* +f * {x3y = 6,
(A) Realnumbers
(B) Complex numbers 70. The initial value problem
(C) Rationals only
(D) lrrationals only # =x3r2(t),x(o)=ohas
(A) Unique solution
67. Thefunctionf(x) = I x lP, 0 < p < 1 is :
(B) Two solution
(A) Not Lipschitz at x = 0
(c) lnfi nitely many solutions
(B) Lipschitz at x = 0 with Lipschitz
constant 1 (D) None ofthese
RS - 24112 (12) Contd.
Text of page 13
71. lf x., and x2 are any solutions t
- ," sin at
of x" + p(t)x'+ q(t)x = 0 on a given
t
interval l, where p(t) and q(t) are (D) - ," cos at
continuous,,then Wronskian of x.,
7i. !f P- is the Legendre polynomial of
n
and x, is given by
1
(A) W(0 = sefn(t)ot . degree n, then J lentx)J2dx is given
(B) W(t)=-sJe(t)ot
r' ',
by:
' '.'l*i 'r' ' : ' ',r
(C) W(t) = ."- lottlot (A)
2
n+1'
(D) 0
1
(B) n = 0, 1,2, ..........
The general solution of 2r, + 1.
{' -zx' * x = et is girren by :
(c)
2n
2n+1 , n = 0, 1,2, ..........
(A) x=(cl*c,t)et*1"'
2n
(B) ,=(cr+crt)e-t**"* (D)
2n+1 ,n=0, 1.,2,
(C) ,=(cr+crt)e-t-1"* 75. The Bessel's function Jr(x) for
p = 112 is given by :
(D) x=(cr+crt)et*1"'
t
(A)
E sln x
^l-
Yn
13. The value of cos at IS
D\
given by:
^r--1-
(B)
E
^l- sln x
Ynx
(A)
t
,, sin at (c)
E
^l- cos x
Ynx
t D
(B)
," cos at "l-
Yn
cos x
RS - 24t12 (13) (Turn over)
Text of page 14
76. The inverse Laplace transform of
(C) f *e=
p +7 $sint
is given by
p2 +2p+5
^3
(D) f *o= l-cost
3
(A) et(cos 2l+ 3sin 2t)
(B) etlsin 2t + 3 cos 2t)
79. The eigen functions of the Sturm-
d2v
(c) e-t(cos 2t+ 3sin 2t) Liouvilleproblem - ly=0,
#
(D) e-t1sin 2t + 3 cos 2t) Y(o) = o, Y'(rr) = o is given bY :
77. The Laplace transform of a (A) v =cn sin (2n - il X, n = 1,
2
I
periodicfunction f(t) with period T >
2,3,........
0 is: (2n - 1\
(B) Yn = cn cos ,rx, n = 1,
T
(A) 1-Fr Je -"tt(t) dt
1
2,3;,........
0
(2n+1
(C) . yn = cn sin n 1
T 2
1
(B) iF J
ettf( t)dt """"
0 .2,3,
(2n + 1)
(D) yn = cn co,s X, n = 1,
T 2.
(c) 1- dr Je -'tf(t)dt
1
0
2,3,........
1
T 80. All theeigen-valuesof regularSturm-
e'tf1t1 d t
(D)
1-t I 0 Liouville Problem are:
78. The convoltrtio.n of f(t) = t2 and g(t) = (A) 0
sin t is given by:
(B) Complex
(A) f *s= (C) Real
lsint
(D) None of these
(B) f*g=tssint
RS - 24t12 (14) Contd
Text of page 15
81 . lf u is a function of x, y and z satisfies 1
(B) z=ax+by+
the partial differential equation "-b
(v-.)ff +@-,)# +(x-v)fl =o (C) z=ax-by+ a*b
1
is given by:
(D) None of these
(A) u=f(xy +yz+zr,:x2*f *.2)
(B) u=f(1y2, **tf **1 84. The orthogonal trajectories of the
, i .(G), u=f(x+ y +z,x2f*1 family of rectangular hyperbolas
(D) u=f(x *y+2,x2+rf +721 c1
v is
x
82. Thd complete integlalof the equation
(A) f-*=c
pq = 1, where
* = e, # = q iS
(B) ,f+*=c
given by:
(c) *f ="
(A) z=ax+I+b
a
x2
(D)
7
(c) z=ax- Y -o 85. A particular integral of the
a
equation (D2 - D')z = 2y - x2 is
'a I +b
(D) i.=-^*+
given by:
83. The complete integralof the equation (A) ,f
(p+qxz-xp-yq)=1
(B) fv
dz p, dz
where
a, : ay =
q'i" givert,by : (C) *,f
(A) z=ax+by+ a_b
1 (D) 0
RS -24t12 (15) (Tum over)
Text of page 16
86. The one dimensionalwave equation gg The solution of the lnitial Value
't2- ':'2-
d-z .> d-z Problem (lVP);
0x' ,, (where c is a
0y'
-=c-
u*=c2ur*,-oo(X,t<-
constant)is:
with u(x, 0) = sin x and u,(x, 0) = 0
(A) Parabolic
is given by:
(B) Hyperbolic
(C) Elliptic (A) u(x, t) = *tcos(x - ct) +
cos (x + c$I
(D) None of these
(B) u(x, t) = |1.in1, - ctj +
87 . The solutio r'f = 4xe2Yis given
cos (x + ct)l
^ f-*N
. by: (C) u(x, t) = |1rin1, - ct) +
(A) z= xe2Y + Er(x) + E (V) sin (x + c1)l
,
(B) z=x2eY +Er(x) +E $) (D) u(x, t) = tlcos(x - ct) +
(C) z= x2e2Y * Er (x) + Ez(y) sin (x + c0)
(D) z=xeY * 6r(x) + Er$)
90 Let Q be a bounded domain in R2
88. The Laplace transform of {t2 with boundary 04. The solution
is: of the Dirchlet's Problem Au(x, y) =
(A)
G -f(x, y) in O, with u(x, y) = g(x, y) on
2s"t'
aO . lf it exists then'it is :
(B)
G
sD
5
(A) Notunique
G (B) Unique
(c)
ffi (C) ffinitelymanysdilion
(D)
G
JE (D) Trivialsolution
RS - 24112 (16) Contd.
Text of page 17
91. The solution of the Cauchy Problem 94. !f g(x) is a continuous function on
1t + f)u, + xyuy = 0, u(0, y) = y2 some interval [a, b] and differentiable
is given by: on (a, b)and if g(a) = 0 and g(b) = 0,
x2 then there is at least one point c
(A) T;7
inside (a, b) forwhich :
x (A) g'1c1* o
(B) T;7
(B) 9(c) = 0
Y2
(c) (C) g'(c) = 0
1+ x2
(D) g(c) + 0
v.
(D)
1+ x2 95. The unique polynomial of degree
92. The rate of convergence of Secant 2 or less, such that (0) = 1, f(1) = 3,
method for finding the roots of f(3) = 55, using Newton's divided
f(x)=g1t' difference interpolation is :
(A) .618
1 (A) 8x2 -6x + 1
(B) 1.168 (B) 8x2+6x+1
(c) 1.816 (c) 8f -6x- 1
(D) 2 (D) 8x3-6x- 1
93. Given l(2) = 4,,1(2.5\= 5.5, the linear 96. The relation between Central
interpolating polynomial by using difference operator 6 and Shift
Lagrange's interpolation is : operator is :
(A) 3x+2 (A) $= E1t2 -E-1t2
(B) 3x-2 (B) a= *.(e'2 + E-1t2,
(c) 3x- 1 (C) 6= 1-E-1
(D) 3x+ 1 (D) 6=1+E-1
RS - 24t12 (17 ) (Tum over)
Text of page 18
97 . The Chebyshev polynomials of the (B) (- -, 0)
first kind Jn(x) defined on [1, -1] is (c) (--, -)
orthogonal with respect weight
(D) None of these
function :
'
100. Which of the following iterative
W
1
(A) method is more efficient for solving
simu ltaneous eq uations ?
W
1
(B) (A) Jacobi's method
(B) Gauss-Seidalmethod
1
(C) (C) Relaxation method
F
(D) None of these
1
(D)
F 101. A lattice is a partially ordered set in
which:
1
dx (A) inf(a, b)and a v b =
' 98. The approximate value of I
0
1 + x "^P=
sup(a, b) exist for qny pair of
by using Gauss-Legendre three elements a and b
point formula is given by:
(B) an b= inf(a, b)exLtonly
(A) 0.693122
(C) avb=sup(a, b)existonly
(B) 0.6e3100
(D) None of these
(c) 0.63s122
(D) 0.6e3 102. Everyfinite Lattice L is :
(A) Unbounded
99. The Hermite Polynomtals are
orthogonal with respect to the weight
(B) Bounded
funtion on (c) Bounded above only
"-*2 '
(A) (0, -) (D) Bounded belowonly
RS - 24112 (18) Contd.
Text of page 19
103. LetN={1 ,2,_3,.........}beorderedby '106. Let T be a finite non-empty tree,
divisibility. Which of the following then:
statement is ture ?
(A) T is non-planar
(A) {2A,2,6} is linearly ordered
(B) Tisaplanar
(B) [3, 15, 5] is linearly ordered
(C) T has one less vertexthan edge
(C) (2,'8, 32, 4I is not linearly
(D) None of these
ordered
(D) None of these 107. Let a finite non-empty connected
plannar graph have vertex, edge and
104. A Boolean algebra B is a :
face sets V E and F respectively,
(A) Bounded, distributive and
then:
complemented lattice
(B) Unbounded, distributive and. (A) lv!*lFl=lEl-1
complemented lattice (B) lvl-lFI=lEl-1
(C) Bounded, commutative and (c) lVl+lFl=lEl+1
complemented lattice
(D) None of these
(D) Bounded, non-commulative
108. The numberof faces in anytwo plane
and complemented lattice
representations of the'same finite
105. For some positive integer n, finite
planargraphs are :
Boolean algebra has:
(A) Notequal
(A) 2n elements
(B) Equal
(B) n!elements
(C) 2n!elements (c) 0
(D) 2n+ l elements (D) None of these
RS - 24t12 ( 1e ) (Turn over)
Text of page 20
109. Any finite planar graph G can be 112. lf a partially ordered set (P, S) has
coloured using: atleast n2 + l elements, then it has a
totally ordered subset of size :
(A) Six colours
(A)
(B) Five colours orfewer
(B)
(C) Seven colours (c) n+1
(D) None of these (D)
110. ln anygraph: 11 3. Which of thefollowing graphsare not
(A) There is an odd number of connected ?
vertices of even degree
(B) There is an even number of even
vertices of 'even deg ree
(C) There is an odd number of prime
vertices of odd degree (C) Y =2, m, n joined if lm-nl= 5
(D) There is an even number of or7
vertices of odd'degree (D) None of these
111 . There is an Eulerian circuit in an finite 114. lna finite loopfree graph, the sum of
connected graph if and only if all its the degrees of the vertices is
vertices have :
(A) Odd degree (A) 0
(B) Both odd and even degree (B) lnfinite
(C) Even degree (C) Thrice the number of edges
(D) None of these (D) Twicethe numberof edges
RS - 24112 (20) Contd.
Text of page 21
I
115. A graph is bipartite if and only if it has 118. The North West corner rule :
no circuit of : (A) !s used to tinO initial feasible
solution
(A) Even length
(B) ls used to find an optimal
(B) Odd length
solution
(C) Both even and odd length
(C) ls based on the concept of
(D) None of these minim izing opportu n ity cost'
116. A feasible solution to an L.P. (D) 'None of these
problem: 119. ln simplex method, slack, surplus and
(A) Mustsatisff all ofthe problem's artificial variables are restricted to
be:
. constr:aints simultaneously
(A) Multiplies
(B) Need not satisfy all of the
(B) Negative
constraints, only some of them
(C) Non:negative
(C) Must be corner point of the
(D) Divided
feasible region
120. According to algebra oi simplex
(D) Mustoptimizethevalue ofthe
method, slack variabtes assigned
objective function
zero coefficlents because :
117. Which of the following is a valid (A) No contribution in objective
objective function for linear fundion
programming problem ? (B) High contribution in objective
fundion
(A) Max Sry
(C) Divisorcontribution inobjective
(B) Min4x +3V+Z
function
(C) Max 5x2 * 6f
(D) Less contribution in objective
(D) Min (x., +xr)lx, function
RS - 24112 (21',) (Turn over)
Text of page 22
121. Which of the following is a method (B) F. N. Trefethen
for improving an initial solution in a
(C) F. M. Adams
transportation problem ?
(D) Both (A) and (B)
(A) Stepping-Stone
125. Which of the following models;
(B) North West Corner
consider as one of the important
(C) South East Corner
variable ?
(D) lntuitivb Lowest Cost
(A) Static models
122. Variable in dual problem which can
(B) Dynamic models
assume negative values, posiitive
(c) Both (A)and (B)
values or zero valueb is classified
(D) None of these
as:
(A) Unrestricted constant 126. MODI method is used to obtain
(B) Restrictedconstant (A) Optimalsolution
(C) Restrictedvariable (B) Optimaltest
(D) Unrestrictedvariable (c) Both (A) and (B)
123. The assignment model is a special (D) Optimization
case ofthe :
127. Abasic solution which also satisfies
(A) Maximum flow model the condition in which all the basic
(B) Transportationmodel vaiiables are non-negative is called
(C) Shortest-routemodel AS:
(D) None of these (A) Basic feasible solution
124. Who coined the term Operation (B) Feasible solution
Research ? (C) Optimalsolution
(A) J. F. Mc Closkey (D) None of these
RS - 24t12 (22 ) Contd.
Text of page 23
128. The method used to solve an (C) Optimalitytest
assignment problem is : (D) None of these
(A) Hungarian
132. Once the initial basic feasible
(B) American
sol.ution has been computed, what is
(C) German the next step in the problem ?
(D) Allare incorrect (A) vAM
129. Optimal solution is a feasible solution (B) Modified distribution method
(not necessarily basic) which (C) Qptimality test
minimizesthe: (D) None of these
(A) Timetaken 133. lf the totalsupply is less than the total
(B) Partialcost demand, a dummy source (row) is
(C) Totalcost included in the cost matrixwith :
(D) None of these (A) Dummy Demand
(B) DummySupply
130. VAM stands for
(C) Zero Cost
(A) Vogeal's . Approximation
Method
(D) Both (A) and (B)
(B) Voage's Approximation 134. Let LP1a, b1, the class of all p-
Method integrable functions over [a, b]. Let p
and q be non-negative extended real
(C) Vangel's Approximation
Method numbers such that !*!= 1. lf f e
pq
(D) Vogel'sApproximationMethod
LP1a, bland g e L{a, bl, then
131. One can find initial basic feasible (A) f ge L11a,bl
solution by using :
(B) f g e LP[a, b]
(A) vAM (:) f s e Lq[a, b]
(B) MoDr (D) None of these
RS -24112 (23) ( Turn over )
Text of page 24
135. Let0 < p < 1 andf, g e LP[a, b], such (C) Not a normed space
thatf> 0 and g > 0. Then : (D) None of these
(A) Ilf -g lle< llrllp * lls llo 138. Let.Y be a subspace of a normed
(B) llf+g lle>lltllo*lls ll, space X. Then Y is complete implies
(C) llf+g llr+11rllo*lls llo that:
(D) None of these (A) Y is open
(B) Y is closed .
136. Let L-[a, b] space of measurable
' (C) Y is bounded
essentially bounded functions on
(Dl Y is semi-open
[a, b].Then which of the following is
true ? 139. LP[a; b], 1.p<oo,is:
(A) LP1a, b1c L-[a, b], 1 < p < od (A) Not separable
(B) L-[a, b]= LP[a, b], 1 < P < oo (B) Separable sometime
(C) L-[a, b]c LP[a, b], 1 <p.-
(C) Separable
.
(D) Both separable and not
- (D) None of these
separable
137. The real linear space C1[0, 1] of all
140. LetX and Y be normed spaces over
continuously d ifferentiable functions
the field IR and T : X -+Y be a linear
defined on [0, 1] equipped with the
operator. Then T is continuous if and
norm given by:
only if
llxll-= sup lr(t)l (A) T is not bounded
xeIo,t]
isan: (B) suPT=oo
(A) Cornplete normed space (C) infT=oo
(B) incomplete normed space (D) T is bounded
RS - 24t1,2 (24) Contd
Text of page 25
141. LetXandYbeBanachspacesover 144. Dualspaceof fp(n), 1.p."", is
the field K and T : X --+ Y be a linear (A) rp(n)
operator. Then T is closed if and
(B) feln;, 1<q<oo,"no **=,
only if f
(A) T is open (C) f-(n)
(B) T is unbounded (D) 11(n)
(C) T is bounded
145. Let H is a Hilbert space and Y is a
(D) T is linear
subspace of H. Then Y is complete
a
142. LetX and Y be normed spaces and
if and only if :
T:X-+X be a linearoperator. Then
(A) Y is closed in H
T has a closed graph, then :
(B) Y is open ih H
(A) T has a closed nullspace
(B) T has a open nullspace (c) Y is neither open and nor
(C) T is one-one closed in H
(D) T*l (D) None of these
143. ThespacefoisaHilbertspaceifand 146. lff (r)=
*, 1<x<_,then
only if
(A) f e L211, o.)and ll f ll = 1
(A) p is even
(B) f e L21t, oo)and ll f ll= O
(B) p>1
(9) t e t211, o-)and ll f ll = 1
(c) P=-
(D) P=2 (D) f e t21t, "o)and ll f ll =.0
RS -24112 (25) (Tum over)
Text of page 26
147 . .lf f e L2(0, -). and lim,
--
f(x) exists (B) f is analytic in C
and is equalto : (C) Derivative of f exists only at
(A) 1 z=0
(B) 0 (D) None.of these
:
(c) € 151. Which of the following statement is
(D) None of these correct ?
148. A Banach space
(A) lf f(z) is continuous at zo, then
f(z) is analytic atzo.
(A) Have a denumerable (Hamel)
(B) lf f(z) is analytic atzo,theri(z)
basis
is continuoudat zo.
(B) Cannot have a denumerable
(C) lf f(z) is analytic atzo, then f(z)
(Hamel)basis
is not continuous atzo.
(C) ls a normed linear space which
(D) lf f(z) is analytic at zo, then
is not complete
f(z)= g
(D) None of these
152. lf f(z)= u(x, y) + iv(x,,y),is.asalytic in
149. A subset of a Hilbert space is weak the region S), then u and v are
bounded if and only if it is : ' hrrronic in ft if :
(A) Unbounded (A) Both u and v are continuous
(B) Bounded in S)
(C) Empty .
(B) Both u and v are differentiable
in et
(D) None of these
(C) They have continuous second
'150. Let f : C -+ C such that f(z) = Z fpr all
order partialderivative in Q
- ze C. Then :
(D) Both u and v are piecewise
(A) f is non-analytic in C I
continuous in S)
RS - 24t12 (26) Contd.
Text of page 27
153. Let f : C -+ C such that f(z) =
156. The value ot fc where C is
*,
_T
sin€
forall ze Then:C.
any simple closed curve and z = a is
(A) z= 0 is notaremovablesingular inside C is given by :
point (A) 2n
(B) z = 0 is essential singular point (B) - 2ni
(C) z=0isazerof(z) . (C) Zni
(D) z= 0cannotbea branch point (D) 0
154. Let f : C -+ C such that f(z) = 157. Letf : C -+ C such thatf(z) = sin zfor
,in-1 1 'for a{l ze C,then f has :
allzeC.Then:
z
(A) -1<sinz<1
(A) Essentialsingularity atz = 0
(B) sin z is bounded
(B) Removablesingularityatz= 0
(C) sin z is constant
(C) Pole at z = 1
(D) sin z is not bounded in C
(D) None of these
158. lf f(z) is analytic inside and
155. if t (z) be continuous in a simply-
on a closed curve C and is not
connected region I'and suppose
identically equal to a constant,
that fc fQ)dz = 0 around a simple
then the maximum value of lf(z)l
closed curve C in f, then :
occurs:
(A) f(z) is analytic in I- (A) ln C
(B) f(z)= 0 in I- (B) At center of C
(C) f(z) is constant in f (C) on C
(D) f(z) is not analytic in I. (D) None of these
RS - 24t12 (27 ) (Tum over)
Text of page 28
I
159. lf f(z) and g(z) are analytic 161. Every polynomial equation p(z) of
inside and on a simple closed degree n has exactly:
curve C and if I g(z) I . If(z) | on C, (A) n roots
then (B) n-l roots
(A) f(z) - g(z) and f(z) have same (C) More than n roots.
numberof zeroes inside C
(D) Less than n roots
(B) f(z),g(z) and f(z) have same
162. ltf(z) = u(x, y) + iv(x, y) is analytic in
numberof zeroes inside C
the region Sl, then u and v are
(C) f(z) - g(z) = 0 inside C
satisfies
(D)' f(z) + g(z) and f(z) have same
(A) Waveequation
numberof zeroes inside C
(B) Laplace equation
160. lf f(z) is analytic inside and (C) Heatequation
on the boundary C of a simply-
(D) None of these
connected region l, then
163. The equation ztanz= ?,? > 0 has
r thl dz is:
9c .-*
v-a)- (A) Exactly n roots
(A) 0 (B) Exactly n imaginary roots
(B) 2ni( @) (C) No roots
(c) 1
(D) lnfinitly many roots but no
(D) 2nif(a) imaginary roots
RS - 24t12 (28) Contd.
Text of page 29
sinx 165. lf w = f(z) = u(x, y) + iv(x, y) is
164. The value of i _dx
t
0
x
analytic in the region o, then
##
(A) n
is equalto :
(B) oo
(A) lf'G)12
TE
(c)
2
(B) t2(=)
(c) f@)(@)
7t
(D)
4 (D) 0
!ltlltrrrlttttr
la
RS;24112 (2e) ( Turn over )
Text of page 30
SPACE FOR UGH WORK RS -24t12 (30) Contd.
Text of page 31
SPACE FOR H WORK
a
RS - 24112 (2,250) (31 ) Mathematics
Text of page 32
a U, l rn l-
Official key not published for this paper.
Page 1 of 32
Source: State Selection Board. Question paper published by State Selection Board.