KUET 2023 Question Paper M.Sc. Mathematics

FOR KUET EXAM PREPARATION

KUET 2023
Question Paper · M.Sc.
Mathematics
EXAM YEAR TYPE SUBJECT

KUET 2023 Question Paper M.Sc. Mathematics

Notes · Sample Papers · Previous Year Papers · Mock Tests

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

ENTRANCE TEST-2023
SCHOOL OF PHYSICAL & MATHEMATICAL SCIENCE

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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. :

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Instructions for Candidates :
a Booklet
lEntrance Test Roll Number in the space provided at the top of this page of Question
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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.

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11. Rough work, if any, should be done on the blank sheets provided with the question booklet.
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12. OMR Answer Sheet must be handled carefully and it should not be folded or mutilated in which case it

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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. Let A = {x  R : x < 1} and 5. What is the limit point of the sequence {a n},
B = {x  R : –1 < x  3}. Then : ( 1) n  1
where a n  , n  N.
(A) Both A and B are bounded n!
(B) A is bounded above by 1 and its supremum 1
1  A (A) 
2
(C) The supremum of A belongs to A and the
infimum of B belongs to B (B) –1

(D) The supremum of A does not belong to A (C) 0
and the supremum of B belongs to B (D) None of the above
2. Which of the following is true ?
6. If a n = n2 and bn = –n2 , n  N, then
(A) The set of t he real numbers is t he
neighbourhood of each of its points. (A) the sequence (a n + bn ) diverges
(B) The set Q of rational numbers is the (B) the sequence (a n – bn ) converges
neighbourhood of each of its points.
a 
(C) The closed int erval [ a, b] is a (C) the sequence  n  diverges
neighbourhood of its end points a and b  bn 
 an 
3. What is the set of limit points for the set (D) the sequence  b  converges
 n
 1 3 2 4 
S   1, 1,  , ,  , , .....
 2 2 3 3   2n  3 
7. lim   equals
n 
 n 1 
 1 3
(A)  ,  (A) 0
 2 2
(B) 1
(B) {1, 1}
(C) 2
(C) {0,  1}
(D) –3
8. Which of the following is not true ?
4. Which of the following is not true ?
(A) If a sequence is convergent, it converges
(A) The set of all integers if countable.
to a unique limit.
(B) The set of all ordered pairs of integers is
countable. (B) Every convergent sequence is bounded.

(C) The set of all real numbers is uncountable. (C) Every bounded sequence is convergent.
(D) The set of all rat ional numbers is (D) Every bounded monotonic sequence is
uncountable. convergent.

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 13. The radius of convergence and the exact
9. If the series  n is convergent, then lim a n
a interval of convergence of the power series
n 1 n0

equals : ( n  1)x n
 ( n  2)( n  3) is :
(A) 0
(A) R = 0, [0, 1]
(B) 1

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(B) R = 1, [–1, 1]

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(C) 

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(C) R = 

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m la

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
sin nx 14. Let {fn} be a sequence of functions such

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10. For any fixed value of n, the series 
n 1 n2 that lim f n  f ( x ), x  [a, b] , and let
n 
is :
Mn = sup | f n ( x )  f ( x ) | . Then f  f
(A) absolutely convergent x [ a , b ] n

uniformly on [a, b] if and only if :
(B) divergent
(A) Mn  0 as n  
(C) oscillates finitely
(B) Mn   as n  

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(D) oscillates infinitely

c. o(D) None of the above
(C) Mn  1 as n  
11. If  a n is a positive term series such that

s e: m15. Let {f } be a sequence, where f (x) = nx ,
la
1

g
lim (a )  l , then the series is convergent if
n

a
n 2 2
n n
1 n x n

(A) l < 1 then
(A) Mn  0 as n  
(B) 1 < l < 3
(C) l  3 1
(B) Mn  as n  
2
(C) Mn  1 as n  

m
n 1

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
( 1) (D) None of the above
12. The series A   a n , where a n =
m
.

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n

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n 1
16. What is the identity element of the group of all

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em(A) A is convergent
Then integers Z with operation aob = a + b + 1,

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a, b  Z.

gla (B) A is divergent
a
(A) 0
(B) 1
(C) {a } is monotonically increasing sequence
n
(C) –1
(D) the series | a n | is convergent

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17. The set of all n × n non-singular matrices 21. Let G = {a} be a cyclic group of order six
having their elements as rational numbers is : and H be the subgroup generated by a 2. Then
(A) an infinite abelian group wrt matrix the order of the quotient group (G/H) is :
multiplication
(A) 1
(B) an abelian group wrt matrix addition
(B) 2
(C) an infinite non-abelian group wrt matrix
multiplication (C) 3
(D) 4
18. If  = (1 2 3 4 5) and  = (2 3) (4 5) be two
permutations of five symbols 1, 2, 3, 4, 5 then 22. Which of the following is true for the group
 equals : of order 45 ?
(A) (1 3 5) (A) G has a normal subgroup of order 9
(B) (2 4 5)
(B) G has an element of order 9
(C) (1 2 3)
(C) G is a non-abelian group
(D) G has no proper subgroup of order 5
19. Which of the following is the set of generators
of the cyclic group G = ({0, 1, 2, 3, 4, 5}, +6) 2
are : 23. What is the order of the element  z in the
3
(A) {1, 2} quotient group (G/Z) of the additive group of
(B) {1, 5} rationals ?
(C) {2, 5} (A) 2
(B) 3
20. Which of the following is true ?
(C) 5
(A) The symmetric group P 3 of permutations
of degree three is non-abelian, while its (D) 6
subgroup A3 is abelian
24. If the quotient group (G/Z), where Z is the
(B) The symmetric group P 4 of permutations
centre of the group G, then
of degree four is abelian, while it s
subgroup A4 is also abelian (A) G is abelian
(C) The symmetric group P 3 of permutations (B) G is cyclic
of degree three is non-abelian, while its
subgroup A3 is also non-abelian (C) G is non-abelian

(D) None of the above (D) None of the above

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25. The set F = {0, 1, 2, 3, 4, 5, 6} under addition 29. What is the locus of the extremity of the polar
and multiplication modulo 7 forms : subnormal of the curve r = f() ?
(A) Commutative ring with zero divisors
dr
(A)
(B) Non-commutative ring with unity d
(C) Field

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(D) None of the above (B)  

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26. Let R[x] be the ring of polynomials of a ring

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 

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R, then which of the following statements (C) r  f '    

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 2

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is/are true ?

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(i) R is an integral domain if and only if (D) None of the above
R[x] is an integral domain.
30. What is the angle of intersection of the curves
(ii) If R is an int egral do main, t hen r = a(1 + cos ), r = b(1 – cos ) ?
deg [f(x) g(x)] = deg f(x) + deg g(x),
where f(x), g(x)  R[x]. 
(A)
(A) (i) only 2

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(B) 0

c. o(C) –1
(B) (ii) only

em (D) 
(C) Neither (i) nor (ii)
(D) Both (i) and (ii)
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27. What is the number of proper ideals in a 31. What is the radius of curvature at any point of
field F ? the curve r = ae  cot ?
(A) At least one
(A) r cosec 
(B) Zero
(B) r cot 
(C) Exactly one
(C) r sec 
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 1n
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

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28. lim  a  1 n equals :

s
n  32. The asymptotes of the curve r = a are :

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 

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(A) r sin = a

g
(A) log a

a (B) 1 (B) r cos = a

(C)  (C) cos = r 2

(D) None of the above (D) None of the above

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y U U 37. Let the functions f(x) and g(x) be defined by :
33. If U = 2 2
x  y sin 1
, t hen x  y
x x y
1, if x  0
equals : f (x)  
0, if x  0
(A) 2U
0, if x  0
g( x )  
(B) 0 1, if x  0
(C) U Then
(A) f(x) is continuous at x = 0
(B) g(x) is continuous at x = 0
Z (C) f(x) + g(x) is continuous at x = 0
34. If Z = xy, then equals :
x (D) f(x)g(x) is discontinuous at x = 0
(A) xy log x 38. If x = a cos  and y = b sin , then the value
d2y 
(B) y xy– 1 of 2 at  = is :
dx 2
(C) y e x
b
(A) –
(D) None of the above a2
35. What is the nth-derivative of 3 2x ? (B) ab

(A) yn = 3 2x(log 3) n b
(C) 
a
(B) yn = 2(3 2x) n(log 3)
(C) yn = 2 n3 2x(log 3) n 39. What is the value of the function f(x) = x2 ex ?
(D) None of the above (A) 4e–2
(B) 0
d (C) –4
36. (sin x ) equals :
dx (D) None of the above

(A) cos x 40. Rolle’s theorem cannot be applied to the
function f(x) = |x| in [–1, 1] because :
1 (A) the function is discontinuous at x = 0
(B) cos x
2 x (B) the function is not derivable at x = 0
1
x (C) the function is discontinuous at x =
(C) cos x 2
2
1
(D) None of the above (D) the function is not derivable at x = 
2

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41. The coefficient of x in the Maclaurin’s series 
sin nx
of f(x) = a x is : 45. For odd n,  dx equals :
0
sin x
(A) 1
(A) 0
(B) log a
(C) a (B) –

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(C) 
m
(D) (log a) 2

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1

e l a
42.  x  x 3 dx equals :
(A) log 1  lxa
s ag
46. What is the general solution of the differential

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2 equation

dy
(B) log x  log 1  x 2  e x  y  x 2e  y ?
dx
(C) log x  log(1  x 2 )
x3
(A) y   ex  c
(D) log x  log 1  x 2 3

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2
x3
y
43.  log x dx equals : (B) e   ex  c

em (C) y  e  e  c
3

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1

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x y
(A) log 4 – 1
(B) 2 log 2 (D) None of the above
(C) 2 log 2 + 1
47. What is the particular integral of the differential
(D) None of the above equation
n
44. The reduction formula of I n   tan x dx is (D 2 – 3D + 2)y = e x + e 2x ?

m
e x xe 2 x

c. o
tan n 1 x (A) 

m
(A) I n   In  2

c. o tan x
5 6

m
n 1

s e
em(B) I  n  2  I l a
n 2
xe 2 x

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(B) –xe  x c

a
n n 1
4

ag 2 tan x n (C) xe – xe 2x x

(C) I  n  I n 3
n

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48. What is the complementary function of the 52. For a first order non-linear differential equation
differential equation  dy 
f  x , y,   0, which o f t he fo llowing
4 2
(D – D )y = 2 ?  dx 
(A) c1 ex + c 2e –x statements is/are true ?
(i) Its general solution must contain only one
(B) c1 + c 2x + c 3e x
arbitrary constant.
(C) c1 + c 2x + c 3e x + c 4 e– x (ii) Its singular solution can be obtained by
(D) None of the above subst it uting part icular values of the
49. What is the general solution of the differential arbitrary constant in its general solution.
equation (iii) Its singular solution is an envelope of its
general solution which also satisfies the
2
dy dy equation.
x 2 2  3x  4y  0 ?
dx dx (A) (i), (ii) and (iii)
2
(A) y = (c 1 + c 2 log x)x (B) (i) and (ii)
(B) y = (c 1e x + c 2e –x)x (C) (i) and (iii)
(D) (ii) and (iii)
(C) y = (c 1 + c 2log x)
53. Which o f t he following is a non-linear
(D) None of the above differential equation ?
50. What is the singular solution of the differential
d2 y dy
(A)  3  ex
a dy dx 2
dx
equation y = px + , where p = ?
p dx
dy
2 (B)  3y  e x
(A) y = 4ax dx
(B) y = 4ax2 2
 dy 
(C)    5y  0
(C) y = 4ax  dx 
(D) None of the above (D) None of the above
1
51. What is the general solution of the differential
54. Let I   Pm ( x ) Pn ( x ) dx , where Pm(x) and Pn(x)
dy 1
equation (px – y) 2 = p 2 + m2, where p = .
dx are Legendre polynomials, then
(A) I = 0, if m  n
(A) y  cx  m 2  c 2
(B) I = 0, if m = n
(B) y  cx  m 2
(C) I = , if m  n
2n  1
(C) y  m 2  4ac
2
(D) None of the above (D) I = , if m  n
2n  1
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 d ( x  1)  58. The partial differential equation obtained from
n 2 n

55. For n Legendre polynomial Cn 
th
, z = f(x) + e yg(x) by eliminating the arbitrary
 dx n 
functions is :
the value of C n is :
(A) p = q
1
(A)
n! 2 n (B) t = q

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n! (C) r = s
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(B)

m
2n
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(C) n! 2 n

s g
59. What is the general solution of the partial

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differential equation p – 2q = 3x2 sin(y + 2x) ?
56. Which of t he following is t he Bessel’s
equation ? (A) x3 sin(y + 2x) – z = (y + 2x)
2 (B) sin(y + 2x) = z
2 d w   dw  2 2
(A) z 
 dz 2   z  dz   ( z  n ) w  0
    (C) (y – 2x) = cos(y + 2x)
 d 2 w   dw 
2 2 2
(B) z  2   z    (z  n ) w  0 60. The partial differential equation
 dz   dz 
m
c. oxy xu  (x  y ) xuy   xy yu   y xu
2 2 2

em
2 2 2
2 d w   dw  2 2 2 2

s
(C) z 
 dz 2   z  dz   ( z  n ) w  0

a
  
l

a g x
u
y
 2( x 2  y 2 )
57. What is the Wronskian of x and xe 2 ?
(A) is parabolic at all points
(A) 0
(B) is hyperbolic at all points
(B) x – x2e x
(C) is elliptic at all points
(C) x2e x

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a

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ROUGH WORK

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a

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