SSB Odisha Lecturer 2021: Lecturer recruitment for non-government aided colleges, Mathematics
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DO NOT OPEN THIS TEST BOOKLET UNTIL YOU ARE ASKED TO DO
TEST BOOKLET Sl. No.
- - 0 114
ect Code : {8 Subject : Mathematics
LECTURERS FOR NON.GOVT. AIDED COLLEGES OF ODISHA
TimeAllowed:2 Hours Maximum Marks: 150
: INSTRUCTIONS TO CANDIDATES :
1 IiITMEDIATELY AFTER THE.COMMENCEMENT OF THE EXAMINATION, YOU SHOULD
CIIEO(THATT}f,STEST BOO}(LET CONTAINS 23 PAGES AND DOES T.IOTHAVEAI.IYI,'NPRhTTEDOR
TORN OR MISSING PAGES OR TTEIUS ETC. IF SO, GET TT REPI.ACED BY A COMPLETE TEST
BOOI(ET.
2. You have to enter your Roll No. on the Test
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NOTwrite anything else on the Test Booklet.
IIII
3. The Test Booklet contains 100 questions. Each question comprises four answers: You
have to select the correct answer which you want to mark (darken) on the Answer Sheet
(OMR Sheet). ln any case choose ONLY ONE answer for each question. lf more than
one answer is darkened, it will be considered as wrong.
a
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work only in the space provided at the end of the Test Booklet. See instructions in the
Answer Sheet.
r
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tw- 15/26 (Turn over)
202'1-
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1 A non-empty subset H of a group G (c) 0(ab)=0(a)0(b)
isa subgroup ofG ifa e H, b e H +
(D) 0(ab)=0(a)/0(b)
abe Hand:
(A) ae H,be H+ab=ba 4. A homomorphism gfrom G into G is
said to be an isomorphism if :
(B) (ab)c = a(bc) for all a, b, c e H
(C) there exists an identity (A) Q is onto
elementee Gsuchthat (B) Q is into
ea=ae=aforallaeH (C) Q is one to one
(D) ae H=a-1 e Hwherea-1 is
(D) Qismanytoone
the inverse of a in G
5. A field is a :
2. The subgroup N of a group G is a
normalsubgroup of G if and only if :
(A) Division ring
(A) lt has only lefi coset of N in G (B) Commutative ring
(B) lt has only right coset of N in G (C) Gommutativedivisionring
(C) Every left coset of N in G is a (D) !ntergraldomain
right coset of N in G
6. lf is a homomorphism from a ring R
Q
(D) gN *Ng foreveryg € G
into the ring R', then :
3. A mapping Q from a group G into a
: (A) 0(-a)=0(a)
group G is said to be a
(B) 0(-a)=-0(a)
homomorphism ifforall a, b e G :
(A) 0(ab)=0(a)+0(b) (c) O (- a) = [0 (a)l-1
(B) O(ab)=0(a)-00) (D) 0 (- a) = o (a-1)
tw- 15126 Ql Contd.
Text of page 3
7. lf U is an ideal of the ring R then (c) 62
(A) R+Uisaring (D) 63
(B) R/U is a ring 11 A homomorphism of a group into
(C) RU is a ring itself is called :
(D) R-Uisaring (A) An isomorphism
8. The g. c. d. of 12and 30 is : (B) An endomorphism
(A) 4 (C) An automorphism
(B) 5 (D) An abelian
(c) 6 12. The product of anyfourconsecutive
(D) 7 integers is divisible by:
I For what value of p, 3p + 1 is a (A) 22
perfect square ? (B) 24
(A) 17 (c) 25
(B) 13 (D) 26
(c) 7 13. lf S and T are subsets of a vector
(D) 5 spaceVandifScTthen:
10. lf g. c. d. of (a, 30) = 1 then a4 + 59 is (A) L(s) c LO)
divisible by: (B) L(r) c L(S)
(A) 60 (c) L(r) = 11s1
(B) 61 (D) L(S u T) = L(S) - L(T)
tw - 15/26 (3) (Turn over)
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14. lf W is a proper subspace of a finite- (B) dim U < dim R(T)
dimensional vector space V then W (C) dim R(T) < dim U
is finite dimensional and :
(D) dim R(T) sdim U
(A) dim V < dim W
1 -1 2
(B) dim W < dim V 18. The nullity of the matrix 3
-2 5
(C) dim V = dim W IS
(D) dim W S dim V (A) 0
15. lf V be the vector space of all2 x 2 (B) 1
matrices over the field F, then (c) 2
dimension of V is :
(D) 3
(A) 3
(B) 4 19. The eigen values of the matrix
(c) 5 0 0l
ol are
I
1
(D) 6 2 oj
16. lf T : Vn - V, be a linear trans- (A) 2,3,4
formation defined byT(x, L,t,xl) = (B) 2, 1,0
(xr -xq'\+ x3' x3 - xo) then its rank
(c) 1,2,3
is:
(D) 1, 1, 0
(A) 2
(B) 3 20. lf A is an m x n matrix and B is an
n x p matrixthen (AB)T is:
(c) 4
(D) 5 (A) ArBr
(B) BrAl
17. ltT: U -rV be a linear map and if U
is finite dimensionalthen :
(c) (BA)r
(A) dim U = dim R(T) (D) BrA
tw- 15/26 (4) Contd
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21. The union of an arbitrary family of 25. A necessary and sufficient condition
open set is : for the convergence of a sequence
is that it is bounded and :
(A) closed
(A) has no limitpoint
(B) open
(B) has finite number of limit
(C) closed and open
points
(D) neitherclosednoropen
(C) has a unique limit point
22. lf S and T are subsets of real (D) has infinite number of limit
numbersthen (S uT)'is: points
(A) S'n T' 26. The sequence {bn} where bn =
(B) SUT'
11
'
:-a
1l fI
(C) S'u T' JF +1' JFi .6';J
(D) SnT converges to :
23. The set R of realnumbers is: (A) 4
(A) open (B) 3
(B) closed (c) 2
(C) open as well as closed (D) 1
(D) neitheropen norclosed 2x-3 when 0<x<2
27. lf f(x) =
24. The set of all order pairs of positive x2 -3 when 2<x<4
integers is: then at x=2,the function is :
(A) uncountable (A) Not derivable
(B) countable (B) Derivable
(C) finite (C) Notcontinuous
(D) uncountable and finite (D) Derivable and continuous
tw-15t26 (5) (Tum over)
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28. The function f(x) = (x - g)5 (x + D4
has:
31 rhe rimit or .rg *[(*)'.(;)'. .
(A) maxima at x = 3
(B) minima atx = 3 (;)'l IS
(C) neither maxima nor minima at
1
(A)
x=3 2
(D) has point of inflection at (B)
1
3
x=3
1
(c)
29 ln the interval 10, 1[, the function 4
1
1
(D)
f(x) = - is: 5
(A) notuniformlycontinuous 32. !f the set E c [a, b] is measurable
(B) uniformlycontinuous and if c and d are in (a, b) such that
c < d then [c, d]is :
(C) continuous
,
(A) not measurable
(D) notcontinuous
(B) measurable
30. ln any interval containing zero, the (C) onlyoutermeasureexists
nx
sequence ffnlwhere fn(x) =
iW (D) only inner measure exists
is 33. For any two partitions P, P, :
(A) uniformlyconvergence (A) L(P1' 0 > U(P2' 0
(B) not uniformly convergence (B) L(P1'0<U(P2'0
(c) convergen@ (c) L(P1' 0 < u(P2' 0
(D) notconvergence (D) L(P1' 0 > U(P2' 0
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34. Every finite subset of any metric (B) differentiable on [a, b]
space is : (C) not integrable on [a, b]
(A) not bounded (D) integrable on [a, b]
(B) not closed
38. lf f is a non-negative continuous
(C) notcompact
(D) compact
function on [a, b]and
f fox= 0, then
forallx e[a, b]
35, Every bounded infinite subset of R
has: (A) f(x) = g
(A) no limit point (B) f(x) = 2
(B) only one limit point (c) f(x) = 3
(C) at least one limit point
(D) f(x) = 4
(D) two limit points
39 A series of function Xn converges
36. A subset S of the set of real numbers
R is compact if : uniformly on [a, b] if there exists
(A) S is closed a convergent series )Mn of positive
(B) S is bounded numbers such that for all x e
(C) S is closed and bounded
la, bI
(D) S is not closed and not
(A) Mn < | fn(x) I for all n
bounded
(B) 1fn(x) l< Mnforalln
37. A bounded function f having a finite
numberof points of discontinuity on (C) 1fn(x) l. Mnforalln
[a, b]is:
(D) Mn . I fn (x) I for all n
(A) not differentiable on [a, b]
aw - 15t26 (7',) ( Tum over )
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7 _1"
40 If-1<x<1,then*.#. (B)
1
2
E 2 +E 2
4x3 1
7_7
(c) E2_E2
2
il*+.....is
(D) el
1 -e-l
(A)
1+x
43. ln Secant method, the (n+1)th
1 approximation xn * ., is wriften as :
(B)
1-x
xn- Xn-1
(A) Xn * f(xn)+f(xn_r) f(xn)
2
(c)
1+x Xn f Xn_1
(B) xn - f(xn) f(xn)
- f(xn_r)
2
(D)
1-x - Xn Xn-1
(C) Xn * f(xn)-f(xn_r) f(xn)
41. The central difference operator 6
- Xn Xn-1
is defined in terms of shift operator (D) xn - f(xn)-f(xn_r) f(xn)
Eas:
44. If E, p and 6 are shift operator,
7 _-1
(A) E2 +E 2 averaging operator and central
_L -1
d ifterence operator respectively then
(B) E 2 -E2
1
E2 can be written as :
7 _l_
(c) E2_E2
(A) *- *u
1
_L L
(D) E 2 +E 2 1^
2 (B) p*
zb
42. The averaging operator p is defined (c) u. **
in terms of shift operator E as :
(A) el *e-L (D) 6- Zr
1
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45. ln the differentiat equation (D2 - 47. Laplace transform of eat sinh bt
IS
2D + 4)y = e'cos x, the particular
integral is: s+a
(A)
(s+a)2+b2
(A) 2ex cos x
(B) ex cos x s-a
(B) _b2
(. -,
1
(c) ex cos x
2 s+a
(c)
(s+a _b2
(D)
J"'"o"
s-a
(D)
(s-a)2 +b2
46. ln the differential equation (x2D2 +
xD + 1)y = logx sin(log x), the
particular integral is :
48. tf L (+) trn-t[1)
\s/
then
(sinat\
(log x)2 cos(log x) +- log tt,]
1 1
(A) L is
4
(log x)
1
(A)
(B) -
3 tan '(:)
f tns x)2 cos(log x)
(B) 1t"n-' (:)
a
(c)
I ,on (log x) sin(los x)
(c) tan-l ?
S
(D) -
]f,on
x)2 cos(tos x) +-4 log
1
(log x) sin(log x) (D) ,t"n-'[3)
\.s/
rw- 15/26 (e) (Tum over)
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49. Laplace transform of t sin at is
s1. tnverie transform o, $|fl
:
i,
1 AS
3 gt2 *1to
(A) 2
(,'*a')' (A)
2 4
24 -!(
(B) 1+t2
AS
(B)
(s2 + a2 9+t2 *1to
(c)
24
2as (D) !-t'-!(
24
(c)
(,'*"')'
52. Solution of (D2 + DD' - 6D'272 =
y cosx is :
4as
(D)
(.'*a')2 (A) fr(y + 3x) + t2$ -2x\ + sinx +
y cosx
50. Laplace transform of (B) f1(y - 3x) + tz(y -2x)'+ sinx -
ycosx
(""t - cosnt)f is :
(C) f.,(v + 3x) + tr(Y + 2x) + sinx -
s-a ycosx
(A) (n
ffi (D) f1(y - 3x) + f ,(Y + 2xl + sinx -
y cosx
s2+ b2
(B) (n s-a 53. lnverse transform of
S
IS
(,'*,')2
S
2
+b2 (A) 2ro(t sin <ot)
(c) -!n s-a (B) rr{t sin ot)
(c) 16 sin cot)
(D) -,,(fi#) o
1
(D) tt sin ot)
^
rw- 15/26 (10) Contd
Text of page 11
U. The Ldplace transform of te-tsin t
e -a+iaor +e-a-laor
is (B)
(t + ior).,En
2s+2
(A) _
(s2+zs+z)2 (c)
@
"a-iaro "a+iaot
2s +2
_
(B) (D)
(s2-zs+z)2 "a-iao "-a+iaor
(1-r',o).8-
2s-2 57. The solufon of rys = 1 is :
(c) 2
(s2-zs+z)
(A) z= logr+ logy+qtfi) +020)
2s +2 (B) z= log x-log y+ Ql(x)* 0e0)
(D)
(s2 -zs+z)2
(C) z= log xlogy+ qr(x) *02(y)
55. The solution of y' + 3y = 10 sin t,
' Ioox
(D) ,= to;t*01(x)+020)
Y(0) = 0 is:
(A) y=e3t+6gst+3sint 58. The solution of fr * 2rys + y2t = 0
by Monge's method is :
(B) !=g-3t-cost+3sint
(c) Y=e3t-cost-3sint (A) =-*(;) =,-(;)
(D) Y=e-3t+cost-3sint
56. The Fourier transform of (B) ,. r(I) =, *(l)
f(x) =
e' if-agxca
0 otherwise is (c) z- g ( v_x =.(l)
)
*
(A) "a+iao "-a+iato
(r+ io)Jzn (D) =. r(I) = -(I)
tw - 15t26 ( 11 ) (Tufiover)
-
Text of page 13
62. The minimum value of Z=4x,,+x, (B) Zero-sum game
Subiect to 3x, + \= 3 (C) Two person zero sum game '.
4xr+3L > 6
(D) Non rectangular game
x,r+ 2xr< 4 65. The value of the game whose pay off
and x,t,\> 0 is :
B
matrix is A
51 is given by:
20
(A) T 34
30 17
(B) 7 (A)
5
16
(c) 18s (B)
5
29 19
(D) (c)
T 5
63. The variable which represents (D) 215
amount by which solution value
exceed a resource is: 66. The contrapositive of the statement
"l come to class whenever there is
(A) Slack variable
going to be a quiz" is:
(B) Surplus variable
(A) lf t do not come to class then
(C) Dependent variable therewillbe aquiz
(D) lndependentvariabte (B) lf I come to the class then
there will be a quiz
64. A game with two players where the
(C) lf there will not be a quiz then I
gain of one player equals the loss of
do not come to class
the other is known as :
(D) lf I do not come to class then
(A) Two person games there willnot be a quiz
tw - 15t26 ( 13 ) (Tumover)
Text of page 14
67. lf n is an integer and 3n + 4 iS odd, 70. Alattice is a partiallyorderset (L, <)
then:
in which every pair of .elements
(A) n is prime a,'b e L has
(B) n is odd
(A) a greatest lowerbound
(C) n is even
(B) a least upper bound
(D) nisanaturalnumber
(C) a glb and a lub
68. On the set of integers, the relation
.greater
(D) a glb or a lub
than or equal" is :
(A) Only reflexive 71. ln a distributive lattice (L, <), if
(B) Only symmteric b,ce Landb,rc=0then:
(C) Only antisymmetric (A) csb
(D) Only poset (B) bcc
69. By the principle of mathematical (C) c<b
induction, 12 * 32 + 52 +....+ (D) bcc
(2n- 1)z is :
72. The number of different Boolean
n(n + +1)
(A)
6 function of degree n is:
n(2n-1X2n+1) (A) 2n-2
(B)
3
(B) 2n-1
n(2n -1X2n+1)
(c) (c) 2n
6
n(n+1X2n+1) (D) 2n+1
(D)
3
tw- 15/26 (14) Contd.
Text of page 15
73. lf G = M E) be an undirected graph 76. lf n is an integer and 3n + 2 is even,
with e edges, then the sum of the then n is:
degrees of vertices is : (A) odd
(A) 5e (B) even
(C) prime
(B) 4e
(D) perfect
(c) 3e
77. Atreewith 10 vertices has:
(D) 2e
. (A) 7 edges
74. ln a simple graph, the maximum
(B) 8 edges
numberof edgeswith n vertices is:
(C) 9 edges
n(n + 1)
(A)
2
(D) 10 edges
n(n - 1) 78. Afull m-arytreewith i intemalvertices
(B)
2
contains:
(n-1)(n-2)
(c)
2 (A) mi-2 vertices
(n-2Xn-3) (B) mi- 1 vertices
(D)
2
(C) mi vertices
75. lf a graph has 21 edges; 3 vertices
(D) mi+ 1 vertices
of degree 4 and other vertices of
' degree 3, then the total number of 79. The value of the Boolean function
vertices are : represented by f(x, y, z) =yry + f is :
(Al 13 (A) 1
(B) 14 (B) 0
(c) 15 (c) 2
(D) 16 (D) 3
rw - 15/26 ( 15 ) (Tum over)
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80. lf G be a connected planar simple 83. ff f@) is an analytic function then
graph with'e' edges and'v' vertices
then the number of regions 'r' is :
t*.#)tr(=)t'i''
(A) -e+v-2 (A) llz)12
(B) zlflz)
(B) e+v-2 12
(c) 3lflz)12
(C) e-v+2
(D) af @)12
(D) e+v+2
84. lf f(z) be a function which is analytic
81. The derivative of (iz3 + 322\3 at 2i inside and on a simple closed curve
IS '
C and zo be any point in the interior
(A) 3
(B) 2
of c, then [ft[= ,r,
c-
(c) 1
(A) nif(z)
(B) Znit(zo)
(D) 0
(C) 3zrif(zo)
82. lf a function f(z) = u(x, y) + iv(x, t), (D) 4nil(zo)
then the Cauchy-Riemann cond itions
85. The value
" t-I*
are where C is the
(A) U*=vy and u =v x circle lzl= zi counter clockwise is :
v
(B) ur=.vrand uyx
=v (A) ni
(B) Zni
(c) uxy
=v and 'y = -',
(C) 3rci
(D) -ux = v, and 'y = -', (D) 4ni
lw - 15/26 (16) Contd.
Text of page 17
86. The residue at the pole of the (B) 2ni
az
function--
/ .\J^;"'
(c) - ni
\z- n)
(D) - 2ni
1
(A)
2 89. Using Cauchy's residue, the value of
(B)
1
rdx
l-
2 J x2 +1 ls
1
(c)
3 fi
(A)
4
1
(D)
3
fi
(B)
3
87. The residue at the pole of the
function cot z is : fi
(c)
2
(A) 1
(D) fi
(B) -2
(c) 1
90. The function ZZ + 1 in the form
(D) 2
u(x, Y) + iv(x, y) is:
88 The value of the complex integration
(A) e-f+1)+oi
3z- 4
I z(z-1)(z-2) dz where C is the
c- (B) <**f-1\+zi
circlelrl=]i, (c) t**f+1)+oi
(A) ni (D) *-f-1)+si
tw - 15t26 (17 ) (Tum over)
Text of page 18
91. Two normed linear spaces X and Y (C) ordinary addition and
are said to be isometrically multiplication
isomorphic if :
(D) ordinary multiplication and
(A) there is a one to one
subtraction
correspondence between the
elements of X and Y which 93. A Hilbert space is an inner-product
makes the two spaces space which is :
. isomorphic (A) finite dimensional and
(B) there is a many to one incomplete
cbrrespondence between the
(B) finite dimensional and
elements of X and Y
complete
(C) there is a one to many
(C) infinite dimensional and
correspondence between the
incomplete
elements of X and Y
(D) there is a many to many (D) infinite dimensional and
clrrespondence between the complete
elements of X and Y
94. lf x and y are any two vectors in a
92. The set C of allcomplex numbers is Hilbert space then :
a complex linear space under: (A) I (x, v) I = llx ll llv ll
(A) ordinary addition and (B) I (x, y) I , llx ll llv ll
subtraction
(c) I (x, v) < llx ll llv ll
I
(B) ordinary mu{tiPlication and
(D) I (x, y) I . llx ll llv ll
division
tw - 15/26 (18) Contd.
Text of page 19
95. The space trwilh the inner product (C) x->(x,e,)e,=e,
Of VeCtOfS y = {X1,
\,
.....,Xn, ......} and (D) x - Xx, e,)e,is perpendicular
y = {y1 , y2, ....., yn, ......} defined by to e.
I
98. Every.non-zero Hilbert space
(x, y)= I
n=1
*nyn it
contains
(A) notconvergent (A) onlyorthonormalset
(B) convergent (B) an incomplete set
(C) divergent (C) a complete orthonormalset
(D) oscillating (D) an incomplete orthonormalset
96. lf {e,} is an orthonormalset in a Hilbert 99. !f N is a finite-dimensional normed
space H, then for every vector linearspace of dimension n, then the
xinH: dimension of N* is :
(A) )l(r,",)l' = llrll' (A) n-2 ir
(B) n-1
(B) Il(*,",)l' = llrll'
(c) n
(c) Il(r,",)l' = llrll' (D) n+1
(D) Il(r,",)l' ,llrll' 100. lf B is a Banach space and if B is
97. lf {ei} is an orthonormal set in a reflexive then B* is :
Hilbertspace H and if xis an arbitrary (A) reflexive
vector in H, then for each j : (B) symmetric
(A) x- X(x,e,)e,< e, (C) antisymmetric
(B) x- X(x, e,)e,is parallelto e, (D) transitive '
llrItItIItlttIt
lw - 15t26 (1e) (Tum over)
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SPACE FOR ROUGH WORK tw - 15/26 (20) Contd
Text of page 21
SPACE FOR ROUGH WORK lw- 15/26 (21) (Tum over)
Text of page 22
SPACE FOR ROUGH WORK
I
I
lw- 15/26 (22) Contd
Text of page 23
SPACE FOR ROUGH WORK tw - 15t26 (3,495) (23) Mathematics
Text of page 24
@
rn
r
IOfficial key not published for this paper.
Page 1 of 24
Source: State Selection Board. Question paper published by State Selection Board.