AUAT 2024 Question Paper M.Sc in Mathematics

Entrance Exam
2024

QUESTION
PAPER

Question Booklet No. Question Booklet Series : A
AUAT — 2024
2-Year M.Sc. in Mathematics (P13)
( TEST BASED ON MCQ )

Full Marks : 100 Duration : 2 Hours

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Date of Examination : _______________________________________

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contains 100 Multiple Choice Questions in two parts (Part—I and Part—II).

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AUAT-2024 /122- A

PART—I

( Core Subject )

1. The idempotent element in a group 4. The differential equation
is 3xdy − 4ydx = 0 , with initial
condition y(0) = 0 has
[A] inverse element of a group
[A] no solution

[B] identity element of a group [B] infinitely many solutions

[C] more than one but finitely many
[C] any element of a group solutions

[D] unique solution
[D] None of the above
5. If a complete integral of the partial

π m
(
differential equation x p 2 + q 2 = zp ;)
2 sin x dx
2. The integral 0 xn
converges
p=
∂z
,q=
∂z
passes through the
∂x ∂y
curve x = 0, z2 = 4y, then the envelope
[A] m > n + 1 of this family passing through x = 1,
y = 1 has
[B] m < n + 1
[A] z = ±2

[C] n > m + 1 [B] z = ±3

[C] z = ± 2+2 2
[D] n < m + 1

[D] z = ± 3 + 3 3

3. Let A and B are two odd order skew-
symmetric matrices such that 6. The radius of convergence of the
AB = BA. Then the matrix AB is a/an series
∞
x 3n
 n
[A] identity n =0 3
is

[B] skew-symmetric [A] 3

[B] 3 –1
[C] symmetric 1
[C] 33
−1
[D] orthogonal [D] 3 3

AUAT-2024 /122- A 2

7. Let f : R − {0} → R , defined by 10. If u(x,y) is a homogeneous function of
1 x and y of degree n, then the value of
f (x ) = x +. On which of the
x3
following intervals, f is one-one? ∂ 2u ∂ 2u
x 2
+y
∂x ∂x ∂y
[A] (0, 1)
is
[B] (0, 2)
∂u
[A] n
[C] ( 0,+∞) ∂x

∂u
[D] ( −∞, −1) [B] (n − 1)
∂x

8. If z 1 and z 2 be any two complex ∂u
numbers, such that [C] n ∂y

z1 + z 2 = z1 + z 2
∂u
[D] (n − 1) ∂y
then arg z1 – arg z2 is equal to

[A] 0
11. The value of the double integral
[B] π
 x + y dxdy
2 2

D
π
[C]
2 where

[D] −π
{
D = (x , y ) ∈ R 2 | x < x 2 + y 2 < 2x }
9. Let p(x) be a quadratic polynomial is
with p(0) = 1. If p(x) leaves the
remainder 4, when divided by x – 1 [A] 0
and it leaves remainder 6 when it is
divided by x + 1, then what is the
value of p (–3)? 7
[B]
9
[A] 20
17
[B] 30 [C]
9

[C] –20
28
[D]
[D] 40 9

AUAT-2024 /122- A 3 [ P.T.O.

12. What is the equation of the plane, − 2i
15. The principal values of e π is
which is tangent to the surface
xyz = 4 at the point (1, 2, 2)?
[A] e –2
[A] x + 2y + 4z = 12

[B] e 2i
[B] 4x + 2y + z = 12

[C] e –2i
[C] x + 4y + z = 0

[D] e 2
[D] 2x + y + z = 6

13. Let H and K be two finite normal 16. The set of real numbers λ for which
subgroups of co-prime orders of a the boundary value problem
group (G, .). Then for all h ∈ H and d 2y
+ λy = 0, y(0) = 0, y(π) = 0 has non-
k ∈ K, hk is dx 2
trivial solution is

[A] h 2
[A] ( −∞,0)
2
[B] k
[B] {n2 : n is a positive integer}
[C] kh

[C] { n : n is a positive integer}
[D] 1

[D] ℝ
14. The singular solution of the nonlinear
first order differential equation
17. The work done in moving a particle
under the action of force
dy  3 2
8x   dy  − 27x = 0
 − 12y   
 dx   dx  F = 5x 2iˆ + ( zx − y ) ˆj + 3zkˆ along the
straight line joining ( 0,0,0) to (1,1,1)
is is

[A] 27x 3 + 4y 3 [A] 0

[B] 27y 3 + 4x 3 [B] 1

[C] 27x 2 + 4y 2 [C] 3

[D] 27y2 + 4x2
[D] 4
AUAT-2024 /122- A 4

18. Let G be a cyclic group of order 24. 21. Order of Newton-Raphson method is
The total number of group
isomorphism of G onto itself is
[A] 2

[A] 7
[B] 0

[B] 17
[C] 1

[C] 8
[D] 5

[D] 24

 1− i 
 1 2  . Then det e P is
19. The function f(z) = log(z + 2) is analytic 22. If P =  
1 + i 0 
 2 
[A] everywhere in the complex plane

[A] 2i sin 2
[B] everywhere except x ≤ −2 , and
y = 0
[B] e 2

[C] everywhere except x ≤ 2, and
y = 0
[C] e −2 2

[D] everywhere except x ≤ −2 , and
[D] e
y ≤ −2

23. What is the degree of the first order
20. The function f (z ) = z is analytic forward difference of a polynomial of
degree n?

[A] everywhere
[A] n

[B] no where
[B] n – 2

[C] only at z = 0
[C] n – 1

[D] everywhere except at z = 0
[D] n – 3
AUAT-2024 /122- A 5 [ P.T.O.

26. The nature of the equilibrium point
2 1 
24. Given a matrix M =  , which of of the following linear dynamical
1 2 system
 πM 
the following represents cos  ? dx dy
 6  = x + 3y, = 3x + y is
dt dt

[A] stable
1 1 2
[A]
2 2 1 
[B] unstable

[C] center
3  1 −1
[B]
4  −1 1
[D] saddle

3 1 1 27. The degree of precision of Simpson’s
[C]
4 1 1 1/3rd rule is

[A] 3

1 1 3
[D]   [B] 2
2 3 1

[C] 1

[D] 4
25. Let f : X → X such that f ( f (x )) = x
for all x ∈ X . Then f is

28. If the Integral Domain D is of finite
characteristic, then its characteristic
[A] one-one and onto is

[A] odd number
[B] one-one but not onto

[B] even number

[C] onto but not one-one
[C] prime number

[D] neither one-one nor onto [D] natural number

AUAT-2024 /122- A 6

29. The pay-off matrix of a two person 32. Which of the following is not a convex
zero-sum game is set?

1 2 1
[A] {(x, y ) : x ≤ 2, y ≤ 2}
0 4 1
1 3 2
[B] {(x, y ) : x 2 + y 2 ≤ 1}
The number of saddle point is
[C] {(x, y ) : y 2 ≥ 4x }
[A] 1 [D] {(x, y ) : y 2 ≤ 4x }
[B] 2 33. The probability density function of a
random variable X is f (x ) = 1− |1 − x |,
[C] 3
0 < x < 2 and f (x) = 0 elsewhere. The
variance of the distribution of X is
[D] 4
1
[A]
6
30. Let A be a 3 × 3 matrix with
1
eigenvalues 1, –1, –3. Then [B]
12
[A] A 2 + A is non-singular 1
[C]
3
1
[B] A 2 − A is non-singular [D]
4

[C] A 2 + 3 A is non-singular 34. A stable asymptotic solution of the
3
[D] A 2 − 3A is non-singular equation x n +1 = 1 + is 2. If we
1 + xn

take x n = 2 + εn and x n +1 = 2 + εn +1,
   
31. If a +b = a −b . Then the angle where ε n and εn +1 are both small, the
  εn +1
between the vectors a and b is ratio is approximately
εn
[A] 0
1
[A] −
2
π
[B]
4 1
[B] −
4
π
[C] 1
3 [C] −
3

π 2
[D] −
[D] 3
2

AUAT-2024 /122- A 7 [ P.T.O.

35. If two roots of the equation 38. Two cards are drawn successively
2x3 – x2 – 18x + 9 = 0 are equal in from a pack of cards without replacing
magnitude and opposite in sign, then the first. If the first card is a spade,
the roots are then the probability that the second
card will also be a spade is

1
[A] 2, –2, 1
2 [A]
52

1
[B] 2, –2, 1
4 [B]
26

1 1
[C] 3, –3,
2 [C]
13

1 4
[D] 3, –3, [D]
4 17

36. If three positive numbers a, b and c 39. If (G , ) be a group with a2 = e, for all
are in Arithmetic Progression, such
a ∈ G , then G is
that abc = 8. Then the minimum
possible value of b is
[A] Abelian

[A] 4
[B] Non-Abelian

[B] 2 [C] Ring

[D] Field
[C] 8

[D] 2 40. Let S and T be two linear
transformations on ℝ 3 , defined by
S(x, y, z ) = ( x, x + y, x − y − z ) and
37. If p(x + 1) – 2p(x) + p(x – 1) = 4 for all
x, then the expression of p(x) will be T (x,y, z ) = ( x + 2y,y − z, x + y + z ).
Then
[A] 2x
[A] S is invertible but T is not
[B] –2x
[B] T is invertible but S is not

[C] 2x 2
[C] both S and T are invertible

[D] –2x 2 [D] neither S nor T is invertible
AUAT-2024 /122- A 8

41. The dimension of the vector space 44. Suppose that |3x | + |2y | ≤ 1. Then
{ ( )
W = A = aij
n ×n
: a ij ∈ ℂ,a ij = −a ji } maximum value of 9x + 8y is

over the field ℝ is
[A] 3
[A] n 2
[B] 2
[B] n(n – 1)

[C] 4
[C] n2 – 1

[D] 8
n2
[D]
2

45. Let A be a matrix such that A3 = –I.
Then which of the following numbers
2n +1 + 3n +1 can be an eigenvalue of A?
42. lim equals to
n →∞ 2n + 3n
[A] 1
[A] 1

[B] 0
[B] 2

[C] –1
1
[C]
2

1 3
[D] 3 [D] +i
2 2

43. The equation of the right circular
cone, whose vertex is at the origin, 46. What is the value of k for which the
axis is the x axis and semi-vertical equations x 4 + kx 2 + 1 = 0 and
3
π x + kx + 1 = 0 have a common root?
angle is , is
6
[A] 2
[A] 3x 2 = y 2 + z 2

[B] 3
[B] 3y 2 = z 2 + x 2

[C] 3z 2 = x 2 + y 2 [C] –2

[D] x 2 = 3(y 2 + z 2 ) [D] –3
AUAT-2024 /122- A 9 [ P.T.O.

47. In the Fourier series of the periodic 50. If the straight line x +y +2= 0
function
touches the parabola y 2 = µx , then
∞ the value of µ is
f (x ) = sin x =  (an cos nx + bn sin nx )
n =0
Which of the following coefficients are [A] 2
nonzero?
[B] 4
[A] an for odd n

[B] an for even n [C] 8

[C] bn for odd n [D] 16

[D] bn for even n

51. The number of units in ℤ 12 is
iπ
48. The matrix A2×2 has eigenvalues e 5
[A] 1
iπ
and e6 . The smallest n such that
An = I2 is [B] 3

[A] 20
[C] 4
[B] 30
[D] 6
[C] 60

[D] 120 52. A particle describes a path
r cosh (nθ) = a under a force F to pole.
49. If y(x) satisfies the differential The law of force is proportional to
dy
equation
dx
(
= y 1 + ( ln y ) 2 ) and
1
[A]
 π r
y(0) = 1 for x ≥ 0, then y   is
 2

[A] 0 1
[B]
r2
[B] 1

1
π [C]
[C] r3
2

[D] ∞ [D] None of the above

AUAT-2024 /122- A 10

53. Let A ∈M 2 ( ℝ) be such that its 99
1
eigenvalues are 1 and –1. Which of
56. The sum  n + 1 + n is equal to
n =1
the following statements is correct?
[A] 11
[A] A–1 = A

[B] 99 − 1
[B] A–1 = –A

[C] 9
−1 1
[C] A = A
2
1
[D]
[D] No conclusion can be drawn 99 − 1
for A–1

∞
1
54. The coefficient of ( x − 1) 2 in the Taylor 57. The series  n p is convergent if
n =1
x
series expansion of f (x ) = xe ( x ∈ ℝ)
about the point x = 1 is [A] p > 1

1 [B] p = 1
[A]
2
[C] p > 0

3e
[B] [D] |p| < 1
2

e 58. The equation
[C]
2

x n + nx n −1 + n (n − 1) x n −2 + ... + n (n − 1)
[D] 3e
...2. x + n ! = 0

has
55. Center of the symmetric group S3 is

[A] trivial subgroup [A] multiple root with multiplicity 2

[B] two-element subgroup [B] multiple root with multiplicity 3

[C] three-element subgroup [C] multiple root with multiplicity 2

[D] None of the above [D] no multiple root

AUAT-2024 /122- A 11 [ P.T.O.

59. A particle moves according to the law 62. The coordinate transformation
v2 = 4(x sin x + cos x), where v is the x ′ = 0.8x + 0.6y, y ′ = 0.6x + 0.8y
velocity at a distance x. Then it’s
represents
acceleration is

[A] 0 [A] a translation

[B] 2x sin x [B] a proper rotation

[C] 2x cos x [C] a reflection

[D] 2x [D] None of the above

60. The value of the real number α
63. How many elements does the set
for which the curves z 2 + αy 2 = 1 and
y = x2 intersect orthogonally is
{z ∈ ℂ | z 60 = −1, z k ≠ −1, for 0 < k < 60}
[A] 2 have?

[B] –2 [A] 45

[B] 32
1
[C]
2
[C] 30

1
[D] − [D] 24
2

61. The fixed points of the complex valued 64. The possible set of eigenvalues of a 4
orthogonal skew-symmetric matrix is
2iz + 5
function f (z ) = are
z − 2i
[A] ±i
[A] 1 ± i
[B] ± 2i
[B] ±1 + i

[C] ± 1, ± i
[C] 1 ± 2i

[D] ±1 + 2i [D] 0, ± i

AUAT-2024 /122- A 12

65. If a is an element of a group (G , ) of 68. Consider the initial value problem
order n and p is prime to n, then the ∂z ∂z
+2 = 0, with z ( 0, y ) = 4e −2y .
order of the element ap is ∂x ∂y
Then the value of z(1, 1) is
[A] n
[A] 4e –2
[B] np
[B] 4e 2
[C] n p
[C] 2e –4
[D] 1
[D] 4e 4

66. Let ℝ be a commutative ring with 1
(unity) but not a field. Let I be a proper ∞
( 2n ) !
ideal of ℝ such that every element of 69. The series  (n !) 2
n =1
ℝ not in I is invertible in ℝ . Then
the number of maximal ideal of
subfields of ℝ is [A] converges to 1

[A] 1 [B] diverges

[B] 2
1
[C] converges to
7
[C] 3

1
[D] infinite [D] converges to
2

dy 70. If a set S contains n elements, then
67. The differential equation + y = 0,
dx the power set contains _____ number
with initial condition y(0) = 1 has of subsets.

[A] no solution [A] n

[B] infinite solutions [B] n 2

[C] unique solution [C] 2 n

[D] n n
[D] None of the above
AUAT-2024 /122- A 13 [ P.T.O.

PART—II
( Islamic History and Culture, General English & General Knowledge )
71. Alhamdulillah means 76. Mother of Prophet Muhammad (PBUH)
"àºÒೃå[ºÀàÒ &¹ "=¢ died when he was _____ year(s) old.

[A] Praise be to Allah šÚK´¬¹ ³åÒà´¶ƒ(Îà.) ³àtõ¡Òà¹à Ò> _____ ¤á¹
[B] Glory be to Allah
¤ÚìΡú
[A] 1
[C] Allah is almighty
[B] 3
[D] Allah is the most gracious
[C] 6
72. As per Islam, which of the following is [D] 12
true about intoxicants?
77. Which of the following is a festival of
Òüκೠ">åÎàì¹ ³àƒA¡‰ì¤¸¹ ¤¸àšàì¹ ëA¡à>ô ¤v¡û¡¤¸[i¡ sacrifice?
Î[k¡A¡? [>ì´•¹ ëA¡à>[i¡ t¡¸àìK¹ l¡ü;Τ?
[A] Totally allowed [A] Laylat al-Qadr
[B] Marginally allowed [B] Muharram
[C] Marginally forbidden [C] Eid al-Fitr
[D] Strongly forbidden [D] Eid al-Adha

73. The Holy Qur’an is revealed 78. The first woman martyr in Islam is

‘š[¤y A塹"à>’ "¤t¡ão¢ ÒìÚìá Òüκàì³¹ šø=³ ³[Òºà ÅÒムÒìº>
[A] at a time [A] Asma
[B] Nusaybah
[B] over a period of 12 years
[C] Al-Khansa
[C] over a period of 23 years
[D] Sumayyah
[D] over a period of 32 years
79. Hijri calendar is enumerated from the
74. The saying ‘Cleanliness is half of faith’ time of
is made by
ëA¡à>ô Î³Ú ë=ìA¡ [Ò\[¹ A¡¸à캓¡à¹ Ko>à A¡¹à ÒÚ?
‘ š[¤yt¡à #³àì>¹ "ì‹¢A¡’ &Òü l¡ü[v¡û¡[i¡ A¡ì¹ìá>
[A] birth of Prophet Muhammad
[A] Prophet Muhammad [B] migration of Prophet Muhammad
[B] Hazrat Umar to Madinah
[C] Ibn Taymiyya [C] journey of Prophet Muhammad
into heaven
[D] Imam Bukhari
[D] death of Prophet Muhammad
75. The prayer at the noon is called 80. Islam acknowledges
ƒåšåì¹¹ >à³à™ìA¡ ¤ºà ÒÚ ÒüκೠѬãAõ¡[t¡ ëƒÚ
[A] Fajr [A] only the last Prophet
[B] Zuhr [B] only the last four Prophets
[C] Asar [C] only the last ten Prophets
[D] Isha [D] all the Prophets
AUAT-2024 /122- A 14

81. On which day Muslims pray Jummah 86. Which Country is called the ‘Land of
Prayer? Prophets’?
ëA¡à>ô [ƒì> ³åκ³à>¹à \å³à¹ >à³à™ šìØl¡? ëA¡à>ô ëƒÅìA¡ ‘>¤ã샹 ëƒÅ’ ¤ºà ÒÚ?
[A] Sunday [A] Saudi Arabia
[B] Saturday [B] Syria
[C] Tuesday [C] Palestine

[D] Friday [D] Iraq

82. The term Waqf refers to _____. 87. Which word is mentioned most times
in the Holy Qur’an?
*ÚàA¡ó¡ Ŧ[i¡ ë¤àc¡àÚ _____¡ú ‘š[¤y A塹"àì>’ ëA¡à>ô Ŧ[i¡ ΤìW¡ìÚ ë¤[Å ¤à¹
[A] Bill of exchange l¡üìÀJ A¡¹à ÒìÚìá?
[B] What is due to the State treasury? [A] Qul
[C] Promissory note [B] Allah
[D] Charitable foundation or trust [C] Rahman
–
83. The Arabic term Haq means [D] Amana

"à¹[¤ Ŧ ÒA¡ &¹ "=¢ Òìºà 88. What is the name of the fountain
which each person of paradise will
[A] legal rights drink before entering?
[B] credit proposal
\àÄàìt¡ šøìt¡¸A¡ ¤¸[v¡û¡ šøì¤ìŹ šè줢 ë™ c¡o¢à¹ \º
[C] taking full responsibility šà> A¡¹ì¤ t¡à¹ >à³ [A¡?
[D] kind of sale [A] Zamzam
84. How many gates of Jannah are there? [B] Kawsar
[C] Tasneem
\àÄàìt¡¹ ƒ¹\à A¡Ú[i¡?
[D] None of the above
[A] Six
[B] Seven 89. How many times does ‘Bismillah’
repeat in the Holy Qur’an?
[C] Eight
‘š[¤y A塹"àì>’ A¡t¡¤à¹ ‘[¤Î[³ÀàÒ’ &ìÎìá?
[D] Nine
[A] 112
85. Name the Angel who was appointed to [B] 113
deliver messages to Prophet
Muhammad (SAW) from Allah (SWT)? [C] 114
[D] None of the above
"àÀàÒ¹ (Îå¤àÒà>à× *Úà t¡àÚàºà) šÛ¡ ë=ìA¡ >¤ã
³åÒà´¶ƒ(Îà.)-&¹ A¡àìá ¤àt¢¡à ëš]ìá ëƒ*Ú๠\>¸ 90. Which is the longest Surah in the Holy
Qur’an?
[>™åv¡û¡ ëó¡ì¹Åt¡à¹ >à³ [A¡?
[A] Jibreel (AS)
‘š[¤y A塹"àì>¹’ ƒãQ¢t¡³ Îè¹à ëA¡à>[i¡?
[A] Surah Al - Maidha
[B] Mikael (AS)
[B] Surah An - Nisa
[C] Israfil (AS)
[C] Surah Al - Imran
[D] Azrael (AS)
[D] Surah Al - Baqarah
AUAT-2024 /122- A 15 [ P.T.O.

91. The study of various aspects of ageing 96. Who is called the first female teacher
is called of India?
[A] Psephology ®¡à¹ìt¡¹ šø=³ ³[Òºà [ÅÛ¡A¡ A¡àìA¡ ¤ºà ÒÚ?
[B] Chronology [A] Sarojini Naidu
[C] Gerontology [B] Savitribai Phule
[D] Entomology [C] Rani Lakshmibai
[D] None of them
92. The fear of being without your mobile
phone is called 97. Mount Etna is a famous volcano
[A] Philophobia located in
[B] Claustrophobia ³àl¡ü@i¡ &i¡>à &A¡[i¡ [¤J¸àt¡ "àìN¥Ú[K[¹ "¤[Ñ‚t¡
[C] Ophidiophobia [A] Argentina
[D] Nomophobia [B] Italy
[C] Mexico
93. His courage won him honour. Identify [D] Philippines
the type of the underlined noun.
[A] Proper 98. The Battle of Kalinga was fought in
[B] Collective the year

[C] Abstract A¡[ºìU¹ ™å‡ý¡ A¡ì¤ Î}Q[i¡t¡ ÒìÚ[áº?
[D] Common [A] 540 BC
[B] 320 AC
94. Fill in the blank with the correct [C] 440 BC
alternative.
[D] 261 BC
When I met her _____ years after, she
looked old and haggard.
99. In which year the First World War was
[A] a few fought?
[B] the few šø=³ [¤Å¬™å‡ý¡ A¡t¡ Îàìº Î}Q[i¡t¡ ÒìÚ[áº?
[C] few [A] 1917 – 1920
[D] some [B] 1914 – 1918
[C] 1912 – 1916
95. Choose the correct alternative from [D] 1910 – 1914
the given sentences :
[A] Those never fail who dies in great 100. Which country is the largest tea
cause producer in the world?
[B] They never fail who die in a great
[¤ìŬ¹ ¤õÒv¡³ W¡à l¡ü;šàƒ>A¡à¹ã ëƒÅ ëA¡à>[i¡?
cause
[A] India
[C] They never fails who dies in a
great cause [B] China

[D] They never fails who die in a [C] Bhutan
great cause [D] Nepal

AUAT-2024 /122- A 16

SPACE FOR ROUGH WORK

AUAT-2024 /122- A 17 [ P.T.O.

SPACE FOR ROUGH WORK

AUAT-2024 /122- A 18

SPACE FOR ROUGH WORK

AUAT-2024 /122- A 19 [ P.T.O.

SPACE FOR ROUGH WORK

HHH
AUAT-2024 /122- A 20 NNN24(141)—30×4

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