KUET 2022 Question Paper M.Sc. Mathematics

FOR KUET EXAM PREPARATION

KUET 2022
Question Paper · M.Sc.
Mathematics
EXAM YEAR TYPE SUBJECT

KUET 2022 Question Paper M.Sc. Mathematics

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co m .
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Sr. No.
453

ENTRANCE TEST-2022
m
SCHOOL OF'PHYSICAL & MATHEMATICAL SCIENCES
c o
60 .c
om Question Bookl"t s""i". [T-lm
.
MATHEMATICS

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Total Qu€stions

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:

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TimeAllowed
l
70 Minutes
: Roll No. :

ag Instructions for Candidates :

L WriteJ_ our Entrance Test Roll Number in the space provided at the top ofthis page ofeuesrion Booklet
and fill up the necessary information in the sp-ces provided on the OVRAnsiv; She;t.
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making entries in lhe Original Copy. candidate should ensure thaiihi rwo copies are alisned oroperlr
so lhat the entrjes made in the Original Copy againsr each item are exacrly copied in rhi Canaldare s
Lopy.

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Copyonly.

e m The
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and

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foradmission.
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em1l Rough work, ifany, should done on the blank provided with gthelaquestion booklet.
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5. What is the radius of curvature of the curve
L hg- equals : s = a sec\y tan\U + a log (secv + tan\') at the
point (s, V) ?
(A) 0 (A) 2a secrtY
(B) -l (B) a secrv
(c) e (C) 2 secsv
(D) I (D) None of the above
6. What is the angle between the radius vector and

2. Let I t, x>o
f(x)' = I the tangent to the curve r = a(l + sino) at
|.-1. x<0.
l-1, x>o 0= 2
s(x) = i
=
I l. x<u 1l
(A) 6
Then which of the following is tlue at x = 0 ?

(A) f and g are contrnuous (B) t
(B) fg is discontinuous
1l
(C) f+ g is continuous (c) ;
(D) None of the above (D) None of the above
./ v\ Az*I;:Az 7. What is the pedal equation of the curve
1 19 z = x'y'sin-'l
' \xr'" -'-x;
'- l. then
dx equals
oY
:
al =12cos20 ?
(A) a' = rP
(A) 6z
(B) a=rP
(B) 32
(C) a=r+P
(c) %
(D) 6
8. What is the number of asymptotes of the cuNe
4. Letz= x2 - 3yz + 20, where x = 2 cost and a^
tdz equals
y = 2 sint. Al t=-.
I -COSV
:

- (A) 2
(A) 16 (B) I
(B) -16 (c) 0
(c) -8 (D) None of the above
(D) 8 'l

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9. On which of the following functions Rolle's dx
theorem cannot be applied ? 13. J eouals ---
J2x-x'
(A) f(x) = sinx in [0, n] (A)
(, cos '(x - 1)

sinj(x - 1)
m
(B) f(x) = Ja4 in t-2,21 (B)
0

(c) f(x) = lxl in [-1, 1]m . co
(c) sin(x - 1)

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t\

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m s
(D) cos(x - 1)

a
0

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lasthe following intervals, all the t4. J (e- -1)(e- +2) dx equals :
(D) (x) = x'? in [-1, l]

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10. ln which of
conditions of the Lagrange's mean value iheorem
I
is satisfied ?
(A) -
le'+1lj
(A) t2, 4l 'n l"r 2l
(B) tl,31
(c) I-2, 2l ,",,1#l
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em (c)
las
l.* -1li
.. ,. /tn1t +xr') In
F ----11

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ll. uml- | eouals '
"_o\ sinx J
(A) -1
A
(B) I -- | (t +oroI eouals:
dx
(c) 0
(A) -x(cosax + 6) sin x
m
(B) -x(cosax + 6) cos x

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12. What is the maximum val,e of the function (C) {cosax + 6) cos x

s em
f(x) = 2x3 .- 9x2 - 24x - 20 2

em (A) I

s g l a equation (ay, + x + xr)dx
la - y a+ bxy)dy = 0 is exact if
16. The differential

ag (B) 7
+ (yB

(A) b=a
:

(c) -7
(B)a=landb=3
(D) None of the above - (C)a=3andb=1
(D) b=2a

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is the general solution of the diff-erent
t'7 . What is the integrating factor for the diff€rential 20. What
equation equauon

(x+1;4I-y = 6:"1"*1'1u 2 i9Il-rl9Il+rz=o r
dx \dx] I
\ dx

(A) (y-3x-c)(Y 4x c)=0
(A) (B) (y+ 3x-c) (Y-4x-c) = 0
t+x-
(C) (y- 3x - c) (Y + 4x - c) : 0
I
(B) . (D) None of the above
l+x
21' what is the general solution of the difteren
l
(c) l-x
equation p = log(px - v), *trere p = l| I
(D) I +x
(A)Y=cx-e-
18. What is the particular integral for the differential
(B) Y=cx-e'
equation
(C) y=c-e-
(D+2XD-l)rY=(e-+1)? (D) None of the above
I .. I 22. If d is the degree and p is the order of
(A) l sx-e- --
differential equation
(B) t8x'e- -2
I x-e {,d'vl'=f,*grl:
(C) ,* ", --I [d','J I dxJ

Then 3€( + 50
19. what is the genenl solution of the dftrrd
(B) 18
equatl0n

(D'z+ l)y = cos2x + e" ?
(c) 28
23. The differential equation (y - px)'? : a:(l+
1
(A) crcosx + crsinx -lcos3x +
2
,dy
wnere p =
(B) (cr + c2xte* * Iaor* * I ;;, rs :

(A) Linear equation
.l
{L) crcosx + c:slnx
I (B) Lagrange's equation
-JCOSlx +
'a
2 (C) Clairaut's equation
(D) None of the above (D) None of the above

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What is the expansion of x2 in tems of the 27. What is the partiril di{fefential equation obtained
Legendre polynomial ? ftom z = f(x'? - y'?) by eliminating the arbitrary
function ?
I
(A);[2p,(x)+po(x)] Az Az
(A) Y;=x- v!
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1

Az 6z
. e
(B) + 3po(x)]

s
;[2p,(x) /D\
\v/ X-=V-
m
Ax 'ay
.,,se g l a
la a
l_^.- Az Az
g
(C)
a
Jlzprlxl+Po(x)l /t1\y-=z-
Av
(D) None of the above (D) None of the above
25. IfJ,(x) is the Bessel fimction, then J-1(x) equals : 28. What is the general solution of the partial
6z dz
2
'dxoy:- +:- = srnx '
differential equation
(A)
' -ttx (A) O(x-y, z+cosx)=0
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(B) o(xy,z+cosx)=0

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(B)
t (C) 0(x-v, z+sinx)=0

l a s
g 29. What is the complete integral of the partial
(D) 0(x - y, zcosx) = 0
a
t2
{C) ,l:sinx
ll rx
differcntial equation z = px + qy - 2164 .

(A) z=(ax+,&y)'- ZJut
26. What is the geneml solution of the Bcssel's
differcntial equation
(B) z= ary- zJ[
(C)z=ax+by-ZJ"a
m
,r'd'v) /dv\/,9)
x'l :+ l+xl :1 lrlx'-; ly = 0.
co
m .
\ax- / \ox,/ \ Lo)
co
.y=AJr(x)+BJ r(x) e m
30. At all points, the partial differential equation

em s
),la,fa'u)_"
s
/a\
la 1p' = AJ3(x)+BJ 3(x) ag
16 6'u . o'1u
16
"( -"
g
a<t-1a-ayJ-'lartj
a 11
Y
(A) is hlTerbolic
(B) is elliptio
(c) y=AJ4(x)+BJ_1(x) (C.) is parabolic
lr
(D) None of the above (D) None of the above

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31. Which ofthe following statements is/are true ? 35. what is the number oflimit points ofthe sequer
{1.2, l'4'1.6. 1,...}?
t I I I I
(1) rhe set
\-t,-2,- t'-Z'....i is bounded. (A) one
(B) more than one but finite
lt 4 9 t6 I
bounded. (c) infinite
(2) The set
|t' l' ,'i' -j is
(D) 0
(A) Both (l) and (2) are true
(B) Only (1) is true 36 The setuence f_ 1l'
{l+-1
(c) only (2) is true (A) converges to zero
(D) Neither (1) is true nor (2) is true (B) converges to I
32. Which of the following is order complete ? (C) converges to 2
(A) set of rational numbers (D) diverges to infinity
(B) set of inational numbers ,. /(a, -a, -.... .a.
37' If lima"=9' then I's-
(c) set of integers [-
equals :
(D) set of real numbers
(A) 0
33. What is number of limit points in a finite set ?
(B) 3
(A) I
(c) e
(B) 2 (D) None of the above
(C) infinite 38. Which of the following is NOT true ?
(D) None of the above (A) A necessary and sufficient condilion
of a monotonic seque
34. Which of the following is uncountable ? lh"-con:e:g:nce
is that it is bounded.
(A) set of all integers
(B) Evqy noootonic decre€sing sequence wr
(B) set P" ofall polynomial functions with integer is bouded below diverges.
coefficients (c) Every monotonic infieasing sequence wl
(C) set {1, 4,9, 16, .....) is not bounded above diverges.

(D) set of all real numbers (D) A monotonic increasing bounded ab
,a tequ€nce converges.

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39. rne senes
t.2 3.4 5.6 42. lf >U, is a positive term sefles such that
Jtt+---* 2,,8, *.....
(A) oscillates finitely lgg(U"); = t, then the series converges if
(B) oscillates infinitely (A)t=1
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(C) is divergent (B) t> I

.c (c) t< I
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(D) is convergent

s e (D) t= 2
g l
g l a
40. Which of the following statements is true ? 43. Which of the following statemenrs ls/are truc
a
a .-l
( r, Ine Inllnrle series
for the sequence F"(x) = x" on [0, l] ?
) is divergent (l) Uniformly convergent
-
no (2) Pointwise convergent
.-l (A) Both (l) and (2)
tz, tne rnllnrle series
) _ is convergenr
(B) only (l)
nn
(c) only (2)
(D)m
co
(A) Both (1) ard (2) are true
.
Neither (l) nor (2)

em
(B) Only (1) is true 44. Ifa sequence {f"} converges uniformly in
s
[a. bl
(C) Only (2) is true
g l a and x0 is a point of[a, b] such that
]q f"tx) = a,,
a
,

(D) Neither (l) is true nor (2) is true n = I,2,3,......., then :

4t
(A) {a"} diverges
Which of the following statements is true ,/
(B) {a"} converges
(11 The infinite serie. 1-l+1-l tC) lim f(x) * lim a-
2 3 4'... r.
conditionally convergent (D) Iim f(x) + a"
o m
om convergent
t-l*l-1*.... c
exact interual ofconvergence .ofthc power
45. If R is the radius of convergence and p is thc
c. absolutely m
(2) The infinire ..r;",

e
2" 3' 4' ,"

m I(*-l)".th.n, gla
s
serics

s e(A)
gla Both (1) and (2) are true ")^ a
a (B) Only (l) is tiue (A) R=2andp=(_1,3)
(B) R=4andp=(_1,s)
(c) Only (2) is true
(C) R=coandp=(_1,3)
(D) Neither (l) is true nor (2) is true (D) None of the above

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46. Which of the following forms a group ? 50. Which of the following is NOT true for a group
(A) 11, 2,3) under multiplication modulo 4 G of order 7 ?

(B) {1, -1, i, -i} under addition (A) G is cvclic
(C) {0, 1, 2, 3 } under multiplication modulo 4 (B) G is abelian
(D) {1, 2, 3, 4} multiplication modulo 5 (C) c has no proper subgroups
(2 6\ (D) G has a unique proper subgroup
47. ln the $oup cL(z, Z!), the inverse of | | -- __-.
!5 5J 51. What is thernumber of proper subgroups of a
finite cyclic group of order 12 ?
i k k)
it 1,. . , ,. , l. rhen 5k + 3 equats : (A) One
\K +1 K-ll
(B) Two
(A) 45
(C) Four
(B) 46
(C) 47 (D) Six
(D) None of the above 52' What is the order of element 3 in the group
48. In the multiplicative group {1,-1, i. -i}. ifx is
(o' t' z':' +1, +s) z
the inverse of-l and y is the inverse of i, then (A) 4
xy equals :
(ts) 5
(A) i
(c) 6
(B) -i
(c) I
(D) -l
53 which ofthe following statcments is/are true ?
49. For an abelian group Gwitha, b e Gandnu (1) The intersection ofany collection olnormal
non-negative integer, which of the following subgroups is itself a normal subgroup'

statements is/are true ? (2) .If H and K are subgroups of an abcliau
(1) (ab) = a"b" group G, then HK is a subgroup of G

(2) (ab|' = a t6-t (A) Both (l) and (2)
(A) Both (l) and (2) ' (B) Only (1)
(B) only (l) (C) only (2)
(c) only (2) (D) Neither (1) nor (2)
(D) Neither {l) nor (2) ,.

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54. Which ofthe following statements is/are true ?
57. The ring of integers is :
(l) A subgroup H of index 2 in a group G is (A) A field
a normal subgroup of G
(B) An integral domain
(2) If H and N are subgroups of a group e
m
co
with N normal in q then H n N is normal (C) A divisior ring

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(A) Only (1)
s e l
58. The ring of all 2 x 2tnelrices over rationals is

a g
:

l a
(B) only (2)
g
(A) An inregral dom;in
(c) Both (l) anda(2) (B) A commutative ring
(D) Neither (l) nor (2) (c) A field
55. lvhich one ofthe following is NOT true ? (D) None of the above
(A) A quotient group of an abellan group is 59.
I abelian.
Which ofthe following is NOT order ofa finite
field ?
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m
(B) A euotient group ofa cyclic group is cyclic.
o
| (A) 8l
. c
m
.(C) The euotient group p,/A, is an abelian group,
|
e
(B) 125

3 and ,{3 is (he alternating group of as(c) 32
I where P3 is the symmetric group of degree

| order 3. gl (D) a 36
I (D) The symmetric goup
| P3 of degree 3 is 60. Which ofthe following statem€nts is/arc true .1

f.aqbetiu. (l) A dng has more than one unity.
F The set M ofall nxn matrices with their elernents
as real numbers is a
(2) A unit of a ring divides every element of
:
I (A) Commutative the ring.

m
| ring with udry

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(A) Both (1) and (2)

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ring with unity (B) Only (l)
.
m *ittout zero divisors s
ening a
I tC) Non-commutarive
(D) Neither (l) nor (2)gl
ring without unity (c) Only (2)
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