KUET 2025 Question Paper Mathematics

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KUET 2025 Question
Paper Mathematics
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KUET 2025 Question Paper

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Question Booklet No. : .............................

ENTRANCE TEST-2025
SCHOOL OF PHYSICAL AND MATHEMATICAL SCIENCES

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MATHEMATICS
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Total Questions : 60 Question Booklet Series

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Time Allowed : 70 Minutes Entrance Test Roll No. :

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l compulsorily use only blue/ black ball point pen. In no case gel/ink pen or pencila should be
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Important Instructions for Candidates :

used.
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1. Candidates shall

2. Compulsorily write your entrance test roll number in the space provided at the top of this page of the question
booklet.
3. Fill up the necessary information in the spaces provided on OMR Answer Sheet including Question Booklet
Number and Question Booklet Series.
4. 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

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entries made in the original copy against each item are exactly copied in the candidate's copy.

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5. All entries in the OMR Answer Sheet, including answers to questions, are to be recorded in the Original Copy

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only.

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6. Choose only one correct/most appropriate response for each question among the options A, B, C and D
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and darken the circle of the appropriate response completely. Incompletely darkened circle is not correctly read
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darkened response shall be considered wrong.
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be properly evaluated by the scanning machine.

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13. Rough work, if any, should be done on the blank sheets provided with the question booklet.

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14. Ensure that the OMR Sheet is signed by the Examinee as well as by the invigilator.

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separate the Original OMR Sheet from the Duplicate Copy.
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x 5. In the Maclaurin’s series expansion of the function
1. lim is equal to :
x 
4x 2  1 – 1 f  x   ex cos x, the coefficient of x3 is :
(A) 1 1
(B) –1 (A)
2
1
(C) 1
2 (B) –
2
(D) 
1
 1 (C)
 x; 0x 3
2 1
2. The function f  x    at x = is:
1– x; 1  x  1 2 1
 (D) –
2 3
(A) Continuous but not differentiable log x
(B) Differentiable 6. The function  x  0  has :
x
(C) Discontinuous
(D) None of these (A) Minimum value = e
(B) Maximum value = e
 x– y u u
If u = sin –1  , then  ,
 x  y 
3. 1
  x y (C) Minimum value 
e
where  is equal to : 1
(D) Maximum value 
y e
(A) 
x
7. The value of 02 sin 4  cos6  dθ is :
x
(B) 
y (A)
256
y
(C) – 3
x (B)
256
x 3
(D) – (C)
y 512
4. Which one of the following is true for x > 0 ? 5
(D)
x 1024
(A)  x  log 1  x  x
1 x 8. The value of the integral  dx is :
x 1
x (A) 2 x – tan –1 x  c
(B) log 1  x   x
1 x (B) 2x – tan –1 x  c
x
(C)  log 1  x   x (C) 2 x  2 tan –1 x  c
1 x
(D) None of these (D) 2 x – 2 tan –1 x  c

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
cos x 14. If A and B are square matrices of the same order,
9. The value of  2
 dx is : then adj (AB) is equal to :
–
2 1  ex
(A) adj(A)adj(A)
1
(A) (B) adj(B) adj(A)
2
(C) adj(A) + adj(B)
(B) 2
(D) None of these
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15. Which one of the following is not ture ?
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(A) If A is a symmetric matrix, then adj(A) is also

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(D) 1

e m symmetric
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a s g
dy
a
  3x  1 y  xe –2x is :
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10. The integrating factor of x (B) If A is a non-singular matrix of order n,

ag
dx
then |adj(A) | = |A|n–1
(A) xe 3x
(C) Product of two idempotent matrices is an
(B) 3xex
idempotent matrix
(C) xex
(D) Product of two non-singular matrices is a non-
(D) x3ex
singular matrix
11. A particular solution of the differential equation 16. The system of equations
(D3–3D2 + 3D – 1) y = ex cos 2x is :
2x  y  5
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1 x x – 3y  –1
.
(A) – e sin 2x

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3x  4y  k
(B)
1 x
e sin 2x
l a s is consistent when k is :

a g
8
(A) 1
1 x (B) 3
(C) – e cos 2x
8 (C) 6
1 x (D) 10
(D) e cos 2x
8 17. The elements on the principal diagonal of a Skew-
12. The differential equation 2ydx – (3y – 2x)dy = 0 is : Hermitian matrix are :
(A) Exact and homogeneous but not linear (A) Purely real or purely imaginary

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(B) Homogeneous and linear but not exact (B) Purely imaginary or zero

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(C) Exact and linear but not homogeneous (C) Zero only

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(D) Exact, homogeneous and linear (D) Complex numbers

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If the eigen values of a 3×3 matrix are 1, 2 and 3, 18. Which one of the following is not true ?

las then the characteristic polynomial of the matrix is :
a g
(A) The rank of a unit matrix of order n is n

ag (A)  3 – 6 2 –11 – 6
(B)  3  6 2  11 – 6
(B) The rank of a non-singular matrix of order n is n
(C) If A is a matrix of order m × n, then
(C)  3 – 6 2  11 – 6 Rank (A)  min{m, n}
(D)  3 – 6 2  11  6 (D) The rank of a skew-symmetric matrix cannot
be one

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1 1 0  24. The sequence Sn ,
 
19. The distinct eigen values of the matrix 1 1 0  are : 1 1 1
0 0 0  where Sn    ...  is :
n 1 n  2 nn
(A) 0 and 1
(A) Convergent
(B) 0 and –1
(B) Divergent
(C) 1 and 2 (C) Unbounded
(D) 0 and 2 (D) Bounded but not convergent
 0 0 1 22 x 2 33 x 3 44 x 4
  25. The series x     ...,  x  0  ,
20. For the matrix A   2 1 1 , A–1 is given by : 2! 3! 4!
1 1 1 converges for :
(A) A2 + 2A – 3I
1
(B) A2 – 2A + 3I (A) x 
e
(C) A2 – 2A – I
(D) A2 – 3A + I 1
(B) x
21. Which one of the following is not true ? e
(A) A finite set has no limit point 2 3
(B) The set Z of integers have no limit point (C) x
e e
(C) The set Q of rational numbers have no limit point
1 2
(D) x
 1 1  e e
(D) The set S  1, , ,... has only one limit point
 2 3 
0 which does not belong to S x x 2 x3
26. The series 1+    ... is :
1! 2! 3!
22. The value of the limit
(A) Conditionally convergent
1 (B) Absolutely convergent
lim 1  2  3  ...  n  is :
1/2 1/3 1/n
n  n (C) Divergent
(A) 1 (D) None of these
1 n 2 –1 n
(B) 27. The series  x ,  x  0 :
2 n2 1
(C) 0 (A) Converges for x  1
(D)  (B) Diverges for x  1
23. The limit superior and limit inferior of the sequence (C) Converges for x  1
(D) Diverges for x  1
 n  1  28. Which one of the following is not true ?
 –1 1    are respectively :
  n  (A) The sequence{n(–1)n} oscillates infinitely
1 (B) The sequence {1+ (–1)n}oscillates finitely
(A) 1 and
2   –1 n 
(B) 2 and –2 (C) The sequence  n  oscillates infinitely
1 1  
(C) and –
3 3   –1 n 
(D) 1 and –1 (D) The sequence  n  has only one limit point
 
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29. For what value of  does the line y = x +  touches 34. The condition for the diameters y = m1x and y = m2x
the ellipse 9x2 + 16y2 = 144 ? x 2 y2
of the hyperbola 2 – 2  1 to be conjugate is :
(A)   3 a b
2
a
(B)   5 (A) m1m 2  2
b
(C)   7
b2

m
(B) m1m 2  2

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(D)   9 a

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2
30. The equation of the chord of contact of tangents of m1 a
. e

s
(C)
m
m2 b2

la
tangents drawn from (2, 3) to the parabola y2 = 4x is :

s e g
a a
m1 b 2
(B) 2x – y +l4 = 0
(A) x – 3y – 4 = 0

g
(D)

(C) 2x –a3y + 4 = 0
m2 a 2
35. Condition that the plane  x + my + nz = 0 may touch
the cone 4x2 – y2 + 3z2 = 0 is :
(D) 2x – 3y – 4 = 0
(A) 3 2  4m 2 – n 2  0
31. The pole of the line y = 2x with respect to the parabola
(B) 3 2 – m 2  4n 2  0
y2 = 2x is :
(C) 3 2  12m 2  4n 2  0
(A) (0, 2)
(D) 3 2 –12m 2  4n 2  0
(B) (0, 1)
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36. If a right circular cone has three mutually perpendicular
(C) (1, 2) generators, then the semi-vertical angle is :
 1
s em (A) 3
a
(D)  0, 
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ag of
 2
32. Sum of the squares of semi-conjugate diameters 
(B)
x 2 y2 6
the ellipse   1 is : (C) tan –1 2
a 2 b2
(A) a2 + b2  1 
(D) tan –1  
(B) a + b  2
(C) a2 – b2 37. If a and b are integers such that g.c.d.(a,b)=1 then

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g.c.d. (a + b, a – b) is :

m .
(D) a 2  b2 (A) 1

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33. The distance between the foci of a hyperbola is 16 (B) 2

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(C) 1 or 2

s
and its eccentricity is 2 , then the equation of

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(D) None of these
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hyperbola is :
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38. The remainder obtained by dividing 1! + 2! + 3! +
a (A) x2 – y2 = 16
(B) x2 – y2 = 8
...+ 100! by 12 is :
(A) 3
(B) 5
(C) x2 – y2 = 32 (C) 7
(D) y2 – x2 = 16 (D) 9

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39. Which one of the following is a reduced residue 46.
In the group G = {2, 4, 6, 8} under multiplication
system (mod 9) ? (modulo 10), the identity element is :
(A) {1, 2, 3, 4, 5, 6} (A) 6
(B) {1, 2, 3, 4, 5, 7} (B) 8
(C) {1, 2, 4, 5, 6, 7} (C) 4
(D) {1, 2, 4, 5, 7, 8} (D) 2
40. The number of solutions in integers of the equation 47.
The order of the permutation (1 2) (3 4 5) in S5 is :
2x + 4y = 7 is :
(A) 2
(A) Two
(B) 3
(B) Three
(C) Four (C) 5
(D) None of these (D) 6
41. The number of positive integers less than 720 and 48.
Up to isomorphism, the number of abelian groups of
prime to it is : order 108 is :
(A) 42 (A) 12
(B) 120 (B) 8
(C) 192 (C) 6
(D) 208 (D) 4
42. Which one of the following is the order of 3 49. Which one of the following statements is not true ?
(modulo 11) ? (A) Every ideal of a ring is its subring
(A) 2 (B) Every subring of a ring is its ideal
(B) 3 (C) Every division ring is a simple ring
(C) 4 (D) Every field is a simple ring
(D) 5 50.
Which one of the following is a ring without unity ?
43. The number of primitive roots (modulo 25) is
(A) The set Z of integers under usual addition and
equal to :
multiplication
(A) 5
(B) The set E of even integers under usual addition
(B) 6
and multiplication
(C) 7
(D) 8 (C) The set M of 2 × 2 matrices over integers under
matrix addition and multiplication
 3 (D) A division ring
44. Value of the Legendre symbol   is :
 19  51. In the ring E of even integers, the ideal (4) is :
(A) 1 (A) Maximal
(B) –1 (B) Prime
(C) 2 (C) Both maximal and prime
(D) –2 (D) Neither maximal nor prime
45. The number of generators of a cyclic group of 52. The number of homomorphisms from Q (ring of
order 8 is : rational numbers) into itself is :
(A) 1 (A) 1
(B) 2 (B) 2
(C) 3
(C) 3
(D) 4 (D) Greater than 3

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57. The Laplace transform of f(t) = tn, n is a positive
 xy
 2 ;  x, y    0, 0  integer is :
If f  x, y    x  y
2
53. then :
 0 ;  x, y    0, 0  1
 (A) n 1
s
(A) fx(0,0) = 0 = fy(0,0)
(B) fx(0,0) = 1 , fy(0,0) = –1 1
(B) – n 1
s
m
1

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(C) fx(0,0) = 1 , fy(0,0) =
m
n!

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2

m
(C)

e
(D) fx(0,0) = –1 = fy(0,0) s n 1

e m xy  x – y  la s
s
n!

a g
2 2
– n 1
a
(D)
l
 ; x 2  y2  0

ag
54. The function f  x, y    x 2  y2 is : s
 s
 0 ; xy0
58. The inverse Laplace transform of is :
(A) Continuous but not differentiable at (0,0) s  a2
2

(B) Continuous with continuous partial derivatives eiat  e –iat
but is not differentiable at (0,0) (A)
2
(C) Differentiable at (0,0)
(D) None of these eiat – e –iat
(B)
2
u u u

o(C)m e 2ie
2 2 2
1
55. If u= then   

c
iat –iat
x 2 y 2 z 2
.
x y z ’
2 2 2

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(A) 0

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(B) 1 eiat – e –iat

ag
(D)
1 2i
(C)
2 59. The Fourier series for f(x) = sin2 x defined over the
(D) 2 range –   x   is :

 x 2 y2 1 cos 2x
 ;  x, y    0, 0  (A) 
56. For the function f  x, y    x 2  y2 , 2 2
 0 ;  x, y    0, 0 
 1 cos 2x
(B) –

m
which of the following is true ? 2 2

m .c o (C) cos 2x

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(A) fxy (0,0)  fyx (0,0) and the conditions of
(D) –cos 2x
m
ean odd periodic function
s
Schwarz theorem are satisfied

em (B) f (0,0) = f (0,0) and the conditions of Schwarz 60. contains l a
The Fourier series of

as g
l (A) Only sineaterms
xy yx
:

ag
theorem are not satisfied
(C) f (0,0) = f (0,0) and the conditions of Schwarz
xy yx
(B) Only cosine terms
theorem are satisfied
(C) Both sine and cosine terms
(D) None of these
(D) Constant term only

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ROUGH WORK

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