KUET 2021 Question Paper M.Sc. Mathematics

FOR KUET EXAM PREPARATION

KUET 2021
Question Paper · M.Sc.
Mathematics
EXAM YEAR TYPE SUBJECT

KUET 2021 Question Paper M.Sc. Mathematics

Notes · Sample Papers · Previous Year Papers · Mock Tests

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Sr. No. .............................

ENTRANCE TEST-2021
SCHOOL OF PHYSICAL & MATHEMATICAL SCIENCES

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MATHEMATICS
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Question Booklet Series

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Total Questions : 60
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Time Allowed : 70 Minutes Roll No. :

s e g l
g l a Instructions for Candidates :
a
a Entrance Test Roll Number in the space provided at the top of this page of Question Booklet
1. Write your
and fill up the necessary information in the spaces provided on the OMR Answer Sheet.
2. OMR Answer Sheet has an Original Copy and a Candidate’s Copy glued beneath it at the top. While
making entries in the Original Copy, candidate should ensure that the two copies are aligned properly
so that the entries made in the Original Copy against each item are exactly copied in the Candidate’s
Copy.
3. All entries in the OMR Answer Sheet, including answers to questions, are to be recorded in the Original

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Copy only.

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4. Choose the correct / most appropriate response for each question among the options A, B, C and D and

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darken the circle of the appropriate response completely. The incomplete darkened circle is not correctly

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read by the OMR Scanner and no complaint to this effect shall be entertained.
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5. Use only blue/black ball point pen to darken the circle of correct/most appropriate response. In no case
gel/ink pen or pencil should be used.
6. Do not darken more than one circle of options for any question. A question with more than one darkened
response shall be considered wrong.
7. There will be ‘Negative Marking’ for wrong answers. Each wrong answer will lead to the deduction
of 0.25 marks from the total score of the candidate.
8. Only those candidates who would obtain positive score in Entrance Test Examination shall be eligible
for admission.
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9. Do not make any stray mark on the OMR sheet.

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10. Calculators and mobiles shall not be permitted inside the examination hall.
m booklet.
em12. OMR Answer Sheet must be handled carefully and it should not be folded l s
aor mutilated in which case it
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11. Rough work, if any, should be done on the blank sheets provided with the question

gla a
a will not be evaluated.
13. Ensure that your OMR Answer Sheet has been signed by the Invigilator and the candidate himself/
herself.
14. At the end of the examination, hand over the OMR Answer Sheet to the invigilator who will first tear off
the original OMR sheet in presence of the Candidate and hand over the Candidate’s Copy to the candidate.
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1. The infimum and supremum of the set 5. The sequence (ρn) converges for :
 ( −1) n 
1 + : n ∈ N  is : (A) ρ > 1
 n 
(B) –1 ≤ ρ < 1
(A) –1, 1
(B) 0, 1 (C) ρ < –1

2 (D) –1 < ρ ≤ 1
(C) –1,
3
6. The sequence (n + (–1)n) is :
2
(D) 0,
3 (A) Convergent

2. For any positive real number a there exists a (B) Divergent
positive integer n such that :
(C) Oscillates finitely
(A) n > a
(D) Oscillates infinitely
(B) n < a
7. The sequence (an) defined by the recursion
(C) n = a
formula a n +1 = 3a n , a1 = 1 is :
(D) None of the above
(A) Monotonically decreasing
3. If A = {1, 2, 3 ... n} then the set of limit points
(B) Monotonically increasing
of A is :
(C) Monotonically increasing and unbounded
(A) Countable
(D) Monotonically increasing and bounded.
(B) Uncountable

(C) Empty 1 1 1
8. The sequence (an) where a n = 1 + + + ... +
2 3 n
(D) Singleton converges to :
4. A subset S of real numbers has a limit point if : (A) 1
(A) S is finite (B) 0

(B) S is countably infinite 1
(C)
(C) S is infinite and bounded 2

(D) S is uncountable (D) None of the above

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13. The maximum value of (x – l)2 ex is attained at :
If ∑n = 1 a n is convergent, then for some finite
∞
9.
number α ≥ 1 the series : (A) e

(B) –e
∑
∞
(i) a
n =1 n
(C) 1
∑
∞
(ii) αa n

m
n =1 (D) –1

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(A) (i) is convergent (ii) is divergent

c. (ii) m
π
e


s
log x − 
m la
(B) (i) is divergent
e
 2

s
14. lim =

a a g
π tan x
l
(C) Both are convergent x→

ag
2

(D) Both are divergent
π
(A)
p 2
 1

10. The series ∑n = 1 ( n 3 + 1) 3 − n 
∞
is :
  π
(B) –
2
(A) Convergent for p = 1
(C) 0
(B) Divergent for p = 1

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c
(D) 1

15.. For which of the following functions Rolle’s
(C) Divergent for all values of p

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e theorem fails ?
s
(D) None of the above

g l a
a
x2 x3
11. The series x + + + ... : (A) sin x ; x ∈ [–π, π]
2 3
(A) Converges for x > 1 sin x
(B) ; x ∈ [0, π]
ex
(B) Converges for x < 1
(C) | x | ; x ∈ [–1, 1]
(C) Converges for x = 1
(D) x3 – 4x ; x ∈ [–2, 2]
(D) Converges for all values of x
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16. Which of the following functions doesn’t have
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1 1 1
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12. The series 1 − p + p − p ... converges for : the Maclaurin’s expansion :
2 3 4
s e
s em(A) p > 1 l a
(A) log x

gla (B) p < 1 (B) log(1 + x) ag
a (C) p > 0
(C) log sec x

(D) log cos x
(D) p < 0

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17. The angle between tangent and radius vector 20. Which of the following statements is true for
π the curve x4 + y4 = 4a2xy ?
for the curve r = a(1 + cos θ) at θ = is :
2 (A) The curve is symmetrical w.r.t. x + y = 0
π (B) The curve is symmetrical w.r.t. x = 0
(A)
2
(C) The curve is symmetrical w.r.t. y = 0
3π (D) The curve is symmetrical w.r.t. x = y
(B)
2
nx
π 21. The functional sequence f n ( x ) = is :
(C) 1 + n2x2
4
(A) Uniformly convergent in [0, 1]
3π
(D) (B) Uniformly convergent in [–1, 1]
4
(C) Uniformly convergent in [–1, 0]
x 2 y2
18. The curve 2 − 2 = 1 has the asymptotes :
a b (D) Uniformly convergent in [1, 2]

22. The series ∑n = 1 f n ( x ) is uniformly convergent
∞
(A) y = ±bx

(B) y = 0 on [a, b] if there exists a convergent series of
positive numbers Σ M n such that for all
(C) x = ±by
x ∈ [a, b] :
(D) x = 0
(A) | fn(x) | ≤ Mn ∀ n
19. Which of the following is true for the functions
(B) | fn(x) | ≥ Mn ∀ n
(i) y = ex
(C) | fn(x) | ≤ Mn ∀ n
(ii) y = log x
(D) | fn(x) | ≥ M n ∀ n
(A) (i) is convex upwards (ii) is concave
upwards 23. The limit of a sequence of functions is continuous
(B) (i) is concave upwards (ii) is convex if :
upwards (A) Every function of the sequence is continuous
(C) (i) is convex downwards (ii) is concave (B) The convergence is uniform
downwards
(C) Either (A) or (B)
(D) (i) is concave downwards (ii) is convex
downwards (D) Both (A) and (B)

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3x − 2 , x ≤ 0 dx
24. The function f ( x ) =  is : 28. ∫ =
1 − 4 cos2 x
x + 1 , x > 0
1 tan x + 3
(A) Continuous (A) log
2 3 tan x − 3
(B) Discontinuous at x = –1
1 tan x − 3
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log

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(B)
2 3 tan x + 3
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(C) Discontinuous at x = 2

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(D) Discontinuous at the origin tan x + 2 3
s e m
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1

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(C) log

25. For u = log(tan xa+stan y), sin 2 x g
2 3 tan x − 2 3

l a
∂u ∂u

ag
+ sin 2 y =
∂x ∂y 1 tan x − 2 3
(D) log
2 3 tan x + 2 3
(A) 0
dx
(B) 1 29. ∫ (x + 1) x + 2 =
(C) 2
x+2
(A) log −1
(D) 4 x+2
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26. Leibnitz theorem is used to find :
c. o(B) log xx ++ 22 + 1
(A) n derivative of sum of two functions em
a s
th

g l
a
x + 2 +1
(B) n derivative of subtraction of two functions
th
(C) log
x + 2 −1
(C) nth derivative of product of two functions
x + 2 −1
(D) log
(D) None of the above x + 2 +1
π/2
1
∫ cos x cos nx dx then :
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27. lim (1 + x ) =x 30. If f(m, n) =
x →0 0

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(A) 0 n

c. o
(A) f ( m, n ) = f ( m + 1, n + 1)

e m
m+n

em 1 s
(B) 1

s
m
l a
la (C) e ag n
(B) f ( m, n ) = f ( m + 1, n + 1)
m+n

ag (C) f ( m, n ) = f ( m − 1, n − 1)
(D) e m+n
m
(D) f ( m, n ) = f ( m − 1, n − 1)
m+n
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31. For integer n, the Bessel functions Jn(x) and 35. What is the order of the dihedral group of
J–n (x) are connected with the relation : square ?
(A) J– n (x) = –Jn(x) (A) 2
(B) J– n (x) = (–1)nJn(x) (B) 4
(C) J– n (x) = –J–n (x) (C) 6
(D) J– n (x) = (–1)nJ–n (x) (D) 8

36. Which of the following is not a group under
32. ∫ J ( x ) dx =
n
multiplication ?
2 (A) R − {0}
(A) − J 2 ( x ) − J 2 (x)
x
(B) R
2
(B) J 2 ( x ) − J 2 (x) (C) {1, –1, i, –i}
x
(D) {1, ω, ω2}
2
(C) − J 2 ( x ) + J 2 (x) 37. The identity of the set of positive rationals
x
ab
(D) None of the above w.r.t. the binary operation a * b = is :
3
33. The Legendre polynomial for m ≠ n,
(A) 0
1

∫ P ( x ) P ( x ) dx =
−1
m n (B) 1

(C) 3
(A) –1
1
(B) 1 (D)
3
(C) 0
38. If G is a finite group of even order
(D) None of the above
(i) order of subgroup of G divides order of G
34. Which of the following sets is a group under
(ii) G has an element of order two.
multiplication modulo 8 ?
(A) Only (i) is true
(A) {1, 3, 5, 7}
(B) Only (ii) is true
(B) {1, 2, 3, 4}
(C) Both are true
(C) {2, 4, 6, 8}
(D) Both are false
(D) {1, 2, 3, 4, 5, 6, 7}

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39. If H and K are subgroups of a group G of finite 43. If R is a ring (i) C = {x ∈ R : xy = yx,
order and O(H) > O(G) , O(K) > O(G ) , ∀ y ∈ R}, (ii) N(a) = {r ∈ R : ar = ra}
then :
then :
(A) C is a subring, N(a) is not
(A) O(H ∩ K) = O( G )
(B) N(a) is a subring, C is not

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(B) O(H ∩ K) = 1

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(C) Both are subrings

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(C) O(H ∩ K) < 1

(D) O(H ∩ K) > 1 . s e
(D) None is a subring

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40. If H ⊆ G be aa subgroup of a group G then a g
44. In a ring M of matrices of order 2 over integers

which of a g following is not true ?
the


the set S = 
 a 0  
 : a , b ∈ Z  is a :
 b 0    
(A) Ha = H if a ∈ H

(B) Ha = H if a ∈ G (A) Right ideal

(C) Ha = Hb if ab–1 ∈ H (B) Left ideal

(D) ab–1 ∈ H if Ha = Hb (C) Two sided ideal

o m None of the above
. c
41. Which of the following statements is true ? (D)

(A) Any finite group of order n is cyclic if item
s which of the following statements is true ?
45. If R[x] denotes a polynomial ring over R then

g l a
a
has an element of order n.

(B) Every cyclic group need not be abelian. (A) R is commutative if R[x] is commutative

(C) The group of integers is not cyclic. (B) R[x] is commutative if R is commutative

(D) None of the above (C) R has no proper zero divisors if R[x] has
no proper zero divisors
 a b (D) All the above
42. If M is a set of all matrices of the type  
m


c. o
 − b a 
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46. The union of two subgroups N1 and N2 of a

c. o e m
where bars denote the complex conjugates, then group G is normal if :

em
M is a :
l a s
las (A) N1 ∩ N2 = ϕ
ag
g
(A) Division ring

a
(B) N1 ∩ N2 ≠ ϕ
(B) Ring but not a division ring
(C) N1 ∪ N2 ≠ ϕ
(C) Field
(D) None of the above
(D) Not a ring

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47. Which of the following is true for the groups 51. If M and N are homogenous in Mdx + Ndy = 0
G1 = {1, –1, i, –i} and G2 = ( Z , +) ? then the integrating factor is :
(A) All subgroups of G1 are normal (A) Mx + Ny
(B) All the subgroups of G2 are normal (B) Mx – Ny
(C) Both (A) and (B)
1
(D) None of the above (C)
Mx + Ny
48. If N is a normal subgroup of a group G then :
1
G (D)
(A) O  = O(G ) – O( N ) Mx – Ny
N
1 −x
G 52. e V=
(B) O  = O(G ) + O( N ) f ( D)
N
–x 1
G (A) e V
(C) O  = O(G ) . O( N ) f ( D)
N
–x +1 1
 G  O( G ) (B) e V
(D) O  = f ( D)
 N  O( N )
dx –x 1
49. + Qx = Px n is a Bernoulli’s differential (C) e V
dy f ( D + 1)

equation where P and Q are : –x 1
(A) Functions of x only (D) e V
f ( D – 1)
(B) Functions of x or constants 53. The solution of the ordinary differential
(C) Constants only equation y2 log y = xyp + p2 is :
(D) Functions of y or constants (A) log x = cy + c2
1 (B) log y = cx + c2
50. cos 2x =
D +42
(C) log y2 = cx + c2
x (D) log x2 = cy + c2
(A) sin 2x
4
54. The solution of the ordinary differential equation
x p2 = y2 + 1 is :
(B) sin 2x
−4
(A) y = sin(c ± x)
x
(C) cos 2x (B) y = cos(c ± x)
4
(C) y = sinh(c ± x)
x
(D) cos 2x (D) y = cosh(c ± x)
−4

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55. What is the singular solution of the differential 58. What is the general solution of the partial
equation 2px + y2p3 ? y 2z
differential equation p + xzq = y 2 .
c3 x
(A) y = cx +
8 (A) ϕ(x3 – y3, x3 – z3) = 0

c3 (B) ϕ(x3 + y3, x3 + z3) = 0

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(B) y = cx –

.co
8
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(C) ϕ(x3 – y3, x2 – z2) = 0

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3
c
m
(D) ϕ(x3 + y3, x2 + z2) = 0
e la
(C) y2 = cx –

s
8

a
lc a g
59. The complete solution of the partial differential

ag 8
3
(D) y2 = cx + equation p + q = 1 is :

56. The singular solution of the ordinary differential (
(A) z = −ax + 1 − a y + c )
2

a
equation y = xp +
p
is :
(
(B) z = ax – 1 − a y + c )
2

(A) y2 = 4ax
(
(C) z = – ax – 1 − a y + c )
2

m
c. o(D) z = ax + (1 − a ) y + c
(B) y2 = – 4ax

em
2
(C) x2 = 4ay
(D) x2 = – 4ay
l a s 60. What is the solution of the partial differential
a g
57. The partial differential equation by eliminating equation x + qy = pq ?
the arbitrary constants of the equation
1
x 2 y2 (A) az = (y + ax)2 + b
2z = 2 + 2 is : 2
a b
(A) z = px + qy 1
(B) az = (y – ax)2 + b
2
(B) 2z = px + qy

o m
c
(C) az = (y + ax)2 + b
m (D) az = (y – ax) + b .
(C) 2z = px – qy

c. o em
2

s
(D) z = px – qy

s em g la
gla a
a

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ROUGH WORK

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c. o s e m
s em l a
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a

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