AUAT 2024 Question Paper B.Sc (Hons) in Mathematics Statistics

Entrance Exam
2024

QUESTION
PAPER

Question Booklet No. Question Booklet Series : A
AUAT — 2024
B. Sc. Honours in Mathematics/Statistics (U13)
(TEST BASED ON MCQ)

Full Marks : 100 Duration : 2 Hours

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Date of Examination : _______________________________________

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AUAT-2024 /91- A

PART—I
( Core Subject )

1. The number of subsets in a set 5. Range of the function
consisting of five elements is f (x )  x  1  1  x is
šòàW¡[i¡ l¡üšàƒàì>¹ &A¡[i¡ ëÎìi¡¹ l¡üšìÎi¡-&¹ Î}J¸à Òº f (x )  x  1  1  x "ìšÛ¡ìA¡¹ šøÎ๠Һ
[A] 5
[A] [1, 1]
[B] 10
[B] (1, 1)
[C] 25
[D] 32 [C] [ 2, 2]

2. Let A  x : x  , B  2x : x   and [D] ( 2, 2)
C  x : x is a prime number . The
number of elements of A  B  C is 6. If 4 f (x )  3 f ( x )  7  3x , then f (x )  ?
A  x : x  , B  2x : x   &¤} ™[ƒ 4 f (x )  3 f ( x )  7  3x ÒÚ, t¡àÒìº
C = { x : x Òº &A¡[i¡ ë³ï[ºA¡ Î}J¸à } Òìº, f (x )  ?
A  B  C -&¹ l¡üšàƒà>-&¹ Î}J¸à Òì¤ [A] 1  3x
[A] finite (ÎÎã³) [B] 1  3x
[C] 3x
[B] 0
[D] 3x
[C] 1
[D]  7. The angle subtended at the centre of a
circle, by a chord whose length is
3. Total number of relation on a set equal to the radius of the circle will be
A  {a, b } is &A¡[i¡ ¤õìv¡¹ ¤¸àÎàì‹¢¹ γà> íƒìQ¢¸¹ &A¡[i¡ \¸à ‡à¹à
A  {a, b } ëÎìi¡¹ l¡üš¹ ë³ài¡ δšA¢¡ A¡Ú[i¡? ¤õìv¡¹ ëA¡ì@ƒø "¤[Ñ‚t¡ ëA¡ào[i¡ Òº
[A] 4 
[A]
2
[B] 8

[C] 16 [B]
3
[D] 32 
[C]
4

4. Domain of the function f (x )  1 
[D]
x x 6

is 8. The minimum value of sec2   cos2 
is
f (x )  1 "ìšÛ¡A¡[i¡¹ Î}`¡à¹ "e¡º Òº sec2   cos2  -&¹ Τ¢[>³¥ ³à> Òº
x x
[A]  [A] 1
[B] 2
[B] 
[C] 0
[C] 
[D] Undefined ("Î}`¡àt¡)
[D]  

AUAT-2024 /91- A 2

12. Which of the following relation is true?
1  a 2  b2 2ab 2b
[>³¥[º[Jt¡ ëA¡à>ô δšA¢¡[i¡ Ît¡¸?
9. 0 1  a 2  b2 2a  ?
b(1  a 2  b 2 ) 2a 1  a 2  b2 [A] sin   7
5
[A] (1  a 2  b 2 )3
[B] cos   7
5
[B] (1  a 2  b 2 )3
[C] tan   7
[C] (1  a 2  b 2 )3 5

[D] (1  a 2  b 2 )3 [D] sec   5
7

10. f (x )  tan1(sin x  cos x )3 is an
0 1 0 0
increasing function in 0 0 1 0
13. If A    , then At (where
f (x )  tan1(sin x  cos x )3 _____-& &A¡[i¡  0 0 0 1
 
¤[‹¢Ìå¡ "ìšÛ¡A¡¡ú 0 0 0 0
At denotes the transpose of A) is
[A] 0, 2  0
0
1 0 0
0 1 0
[B] 0,  ™[ƒ, A  
  ÒÚ, t¡àÒìº At -&¹
4 0 0 0 1
 
[C] 0, 
 0 0 0 0
6 ³à> Òº (ë™Jàì> A t Òº A-&¹ i¡öàXìšà\)

[D] 0, 
 [A] 0
8
[B] 1
[C] 2
 5 8 0 
11. If A   3 5 0  , then A2 is [D] 3
 1 2 1
  14. If A and B are symmetric matrices of
same order then AB  BA is
 5 8 0 
™[ƒ, A   3 5 0  ÒÚ, t¡àÒìº A2 Òº ™[ƒ, A &¤} B &A¡Òü yû¡ì³¹ ƒå[i¡ šø[t¡Î³ ³¸à[i¡öG ÒÚ,
 1 2 1 t¡àÒìº AB  BA Òº
 
[A] idempotent [A] symmetric
γQà[t¡ šø[t¡Î³
[B] nilpotent [B] skew symmetric
Åè>¸Qàt¡ã [t¡™¢A¡ šø[t¡Î³
[C] involutory [C] idempotent

l¡üƒQà[t¡ γQà[t¡
[D] periodic [D] invertible

š™¢àÚ¤õv¡ [¤š¹ãt¡ì™àK¸

AUAT-2024 /91- A 3 [ P.T.O.

15. The system of simultaneous linear  0 1
18. Consider the 2 × 2 matrix T   .
equations 0 0
x  y  z  0 Then det(T  xI )  0 is a polynomial of
x y  z  0 degree
 0 1
has T   Òº &A¡[i¡ 2 × 2 ³¸à[i¡öG, t¡àÒìº
0 0
™åKš; γãA¡¹o‡ìÚ¹ det(T  xI )  0 ¤×šƒ[i¡¹ [l¡Køã Òì¤
x  y  z  0 [A] 1
x y  z  0 [B] 2
[ÎìС쳹 [C] zero (Åè>¸)

[A] no solution in  3 [D] Undefined ("Î}`¡àt¡)
 3 -& γà‹à> ë>Òü 19. Let A be a 3 × 3 nonsingular matrix
[B] a unique solution in  3 such that A3 = A. Then, the
 3 -& &A¡[i¡ &A¡A¡ γà‹à> "àìá determinant of A is
[C] infinitely many solution in  3 ³ì> A¡¹, A &A¡[i¡ 3 × 3 >>-[ÎUåºà¹ ³¸à[i¡öG &³>
 3 -& "Îã³®¡àì¤ &A¡à[‹A¡ γà‹à> "àìá ë™, A3 = A¡ú t¡àÒìº A-&¹ [>o¢àÚA¡ Òº
[D] finite but more than 1 solution in [A] 0
3 [B] 1
 3 -& ÎÎã³ [A¡”ñ 1-&¹ ë¤[Šγà‹à> "àìá
[C] 2
[D] 4
16. If tan   sec   e x , then cos  

™[ƒ, tan   sec   e x ÒÚ, t¡ì¤ cos   ? 20. If a  0, b  0, c  0 are p-th, q-th and
r-th terms of a GP, then the value of
e x e  x
[A] 2
log a 2 ( p  1) 3
x x
e e
[B] 2 log b 4 2(q  1) 6
2 log c 8 4(r  1) 12
[C] e e  x
x

2 is
[D]
ex ™[ƒ &A¡[i¡ P¡ìoàv¡¹ šøK[t¡¹ p-t¡³, q-t¡³ &¤}
r-t¡³ šƒP¡[º a  0, b  0, c  0 Òìº,
17. Every skew-symmetric matrix of odd
order is log a 2 ( p  1) 3
[¤ì\àØl¡ yû¡ì³¹ šøìt¡¸A¡ [t¡™¢A¡ šø[t¡Î³ ³¸à[i¡öG[i¡ Òº log b 4 2(q  1) 6
[A] singular 8
log c 4(r  1) 12
[ÎUåºà¹
[B] non-singular
&¹ ³à> Òº
[A] 1
>>-[ÎUåºà¹
[B] 0
[C] invertible
[C] –1
[¤š¹ãt¡ì™àK¸
[D] None of the above
[D] skew-hermitian
l¡üšì¹¹ ëA¡àì>à[i¡Òü >Ú
[t¡™¢A¡ Òà¹[³[i¡Úà>
AUAT-2024 /91- A 4

21. arg(x ) (where x  0) ; is 25. The coefficient of x20 in the expansion
of (1  x 2 )40  (x 2  2  x 2 )5 is
arg(x ) -&¹ ³à> Òº (ë™Jàì> x  0 )
(1  x 2 )40  (x 2  2  x 2 )5 -&¹ [¤Ñzõ[t¡ìt¡
[A] 0
x20 -&¹ ÎÒK Òì¤

[B] 2 [A] 30C
5
[C]  [B] 30C
10
1 [C] 30C
[D] 20
2 [D] 30C
25

22. If (1  x )n  a0  a1x  a2x 2  ....  an x n , 26. How many 3-digit even numbers can
be made using the digits 1, 2, 3, 4, 6, 7,
then (a0  a2  ....)2  (a1  a3  ....)2 is if no digit is repeated?
equal to &A¡Òü "S¡ ¤à¹¤à¹ ¤¸¤Ò๠>à A¡ì¹,
1, 2, 3, 4, 6, 7 "S¡P¡[º ‡à¹à 3-"ìS¡¹ A¡Ú[i¡
™[ƒ (1  x )n  a0  a1x  a2x 2  ....  an xn
ë\àØl¡ Î}J¸à ít¡[¹ A¡¹à ë™ìt¡ šàì¹?
ÒÚ, t¡ì¤ (a0  a2  ....)2  (a1  a3  ....)2 &¹
[A] 36
³à> Òº
[B] 30
[A] 2n
[C] 60
[B] 3n
[D] 6!
[C] 2n 1
[D] 3n 1 27. How many words, with or without
meaning, each of 2 vowels and 3
23. If x satisfies the equation consonants can be formed from the
2
(x  2x cos   1)  0 , then the value of letters of the word DAUGHTER?
‘DAUGHTER’ Ŧ[i¡¹ "Û¡¹P¡[º ¤¸¤Ò๠A¡ì¹
x n  1 is 2[i¡ vowel * 3[i¡ consonant ÎÒì™àìK "=¢šèo¢
xn
¤à "=¢Òã> A¡t¡P¡[º Ŧ ít¡[¹ A¡¹à ë™ìt¡ šàì¹?
™[ƒ x, (x 2  2x cos   1)  0 γãA¡¹o[i¡ìA¡ [A] 30
n 1
[·ý¡ A¡ì¹, t¡àÒìº x  n -&¹ ³à> Òº [B] 120
x
[A] 2n cos n [C] 360
[B] 2n cosn [D] 3600
[C] 2 cos n
[D] 2 cosn

1  1  1  1 
28. The value of lim n  n   2  n 1  
n   2 2
24. If  is a cube root of unity, then

1  1 
3  2n  2    
 .....  1 1  is
 1  3  9  27 ....
   2 8 32 128
 is
n 2

™[ƒ  &A¡-&¹ Q>³èº ÒÚ, t¡àÒìº
n   2
n
 
 n  2

 2n 1

 3

 
lim 1  1   1  1   1  1  
2n  2


1  3  9  27 .... 
   2 8 32 128
Òì¤
  
.....  1 1  -&¹ ³à> Òº
n 2 
[A] –1 [A] 0
[B] 0 [B] 1
[C] i [C] e
[D] i3 [D] Does not exist ("[Ñzâ«Òã>)
AUAT-2024 /91- A 5 [ P.T.O.

29. If 4 is a root of the equation 33. The values of m for which exactly one
x 2  ax  12  0 , then the other root of root of the equation
the equation is x 2  2mx  m 2  1  0 lies in the
2
™[ƒ x  ax  12  0 γãA¡¹ìo¹ &A¡[i¡ ¤ã\ 4 interval (2, 4) is
ÒÚ, t¡ì¤ γãA¡¹o[i¡¹ "àì¹A¡[i¡ ¤ã\ Òì¤ m-&¹ ëA¡à>ô ³àì>¹ \>¸
[A] a – 4 2 2
x  2mx  m  1  0 γãA¡¹ìo¹ &A¡[i¡³ày
[B] – 2 ¤ã\ (2, 4) -&¹ "”z¹àìº =àA¡ì¤?
[C] 3
[A] (3,  1)
[D] – 3
[B] (3, 5)
30. If the roots of the equation
2
[C] (,  )
3x  5x  p  0 are equal then p =
[D] (3, 5)  (3, 5)
™[ƒ 3x 2  5x  p  0 γãA¡¹ìo¹ ¤ã\P¡[º γà>
ÒÚ, t¡àÒìº p = ?
34. The equation x  3  4 x 1 
25
[A] x  15  8 x  1  2 has
6

[B]
25 x  3  4 x 1  x  15  8 x  1  2
12
γãA¡¹ìo¹
25
[C]
6 [A] no real root
25 ëA¡àì>à ¤àÑz¤ ¤ã\ ë>Òü
[D]
12
[B] at least one real root
31. If x  y  3 and x  y  9, then the
A¡³šìÛ¡ &A¡[i¡ ¤àÑz¤ ¤ã\ "àìá
minimum value of x is
[C] exactly one real root
™[ƒ x  y  3 &¤} x  y  9 ÒÚ, t¡àÒìº
&A¡[i¡³ày ¤àÑz¤ ¤ã\ "àìá
x-&¹ Τ¢[>³¥ ³à> Òì¤
[D] infinitely many real root
[A] 2
[B] 4 "Îã³ Î}J¸A¡ ¤àÑz¤ ¤ã\ "àìá
[C] 5
35. The product of real roots of the
[D] 6
equation 2x  3 2  3 2x  3  2  0, is

32. Total number of terms in the
2x  3 2  3 2x  3  2  0 γãA¡¹o[i¡¹ ¤àÑz¤
expansion of (1  x  x 2 )5 is ¤ã\P¡[º¹ P¡oó¡º Òº
(1  x  x 2 )5 -&¹ [¤Ñzõ[t¡ìt¡ ë³ài¡ šìƒ¹ Î}J¸à Òº [A] 5

5
[A] 9 [B]
2
[B] 10 [C] 2
[C] 11 2
[D]
[D] 12 5

AUAT-2024 /91- A 6

36. If three complex numbers are in 5 ( x 3 sgn x 7)
Arithmetic progression, then they lie 39. f (x )  (tan x ) e is
on 3 7
f (x )  (tan x 5 ) e ( x sgn x ) Òº
™[ƒ [t¡>[i¡ \[i¡º Î}J¸à γà”z¹ šøK[t¡ìt¡ =àìA¡,
t¡àÒìº t¡à¹à "¤Ñ‚à> A¡¹ì¤ [A] an odd function
[A] a circle &A¡[i¡ [¤ì\àØl¡ "ìšÛ¡A¡
&A¡[i¡ ¤õìv¡ [B] an even function
[B] a straight line
&A¡[i¡ ë\àØl¡ "ìšÛ¡A¡
&A¡[i¡ ιºì¹JàÚ
[C] an ellipse [C] neither even nor odd

&A¡[i¡ l¡üš¤õìv¡ ë\àØl¡* >à, [¤ì\àØl¡* >à
[D] None of the above [D] None of the above
l¡üšì¹¹ ëA¡àì>à[i¡Òü >Ú l¡üšì¹¹ ëA¡àì>à[i¡Òü >Ú
2 2
37. The circle (x  4)  (y  3)  9 touches
the 40. The period of f (x )  cos  2  is
5 x


[A] X-axis
f (x )  cos5 x -&¹ š™¢àÚA¡àº Òº
[B] Y-axis 2
[A] 
[C] Both the coordinate axes
[B] 2
[D] None of the coordinate axes [C] 3

[D]
(x  4)2  (y  3)2  9 ¤õv¡[i¡ њŢ A¡¹ì¤ 2

[A] X-"Û¡ìA¡
41. If f :    be such that f ( x )  x  x ,
[B] Y-"Û¡ìA¡
where [a] denotes the greatest integer
[C] l¡ü®¡Ú "Û¡ìA¡
less than or equal to a, then f 1(x ) is
[D] ëA¡àì>à "Û¡ìA¡Òü >Ú
1
[A]
x x
1/2024
 x  [B] | x |  x
38. If f (x )    , then Domain of
1  x 
[C] Not defined
the function D f is   [D] None of the above

 
1/2024 ™[ƒ f :    &³> ë™ f (x )  x  x ,
™[ƒ f (x )   x  ÒÚ, t¡àÒìº
 1  x  ë™Jàì> [a] ë¤àc¡àÚ ëÎÒü Τ¢à[‹A¡ šèo¢Î}J¸à ™à a-&¹
"ìšÛ¡A¡[i¡¹ Î}`¡à¹ "e¡º  D f  Òì¤ ë=ìA¡ ëáài¡ ¤à γà>, t¡àÒìº f 1(x ) Òº

[A]   { 1, 1} 1
[A]
x x
[B] (, 1)
[B] | x |  x
[C] (,  1)  0, 1
[C] "Î}`¡àt¡
[D] None of the above
l¡üšì¹¹ ëA¡àì>à[i¡Òü >Ú [D] l¡üšì¹¹ ëA¡àì>à[i¡Òü >Ú

AUAT-2024 /91- A 7 [ P.T.O.

3
42. lim logsin x sin 2x is
x 0
46. 0 x dx 
[A] 1 [A] 0
[B] e [B] 1
[C] e2 [C] 2
[D] e3 [D] 3

e 1
43. f + g may be a continuous function if 47. 1 dx  ?
x (1  log x )2
f + g "ìšÛ¡A¡[i¡ Δzt¡ Òìt¡ šàì¹, ™[ƒ
1
[A] f is continuous and g is [A]
2
discontinuous
1
f Δzt¡ &¤} g "Δzt¡ ÒÚ [B]
e
[B] f is discontinuous and g is 2
continuous [C]
e
f "Δzt¡ &¤} g Δzt¡ ÒÚ
[C] f and g both are discontinuous [D] 0
f * g l¡ü®¡ÚÒü "Δzt¡ ÒÚ 48. The eccentricity of the hyperbola
[D] None of the above
x 2  y 2  a 2 is
l¡üšì¹¹ ëA¡àì>à[i¡Òü >Ú
x 2  y 2  a 2 š¹à¤õv¡[i¡¹ l¡ü;ìA¡@ƒøt¡à Òº
44. Number of points where
[A] 2
f (x )  min  x  1, x  1 : x  is not [B] 2
differentiable is
[C] a
f (x )  min  x  1, x  1 : x  &[i¡ ë™Òü [D] a
[¤@ƒåP¡[ºìt¡ "¤A¡º>ì™àK¸ >Ú, ëÎÒü [¤@ƒåP¡[º¹ Î}J¸à 49. The equation of xy plane is
Òº xy t¡º[i¡¹ γãA¡¹o Òº
[A] 1 [A] x = 0
[B] 2
[B] y = 0
[C] 3
[C] z = 0
[D] 4
[D] x + y = 0
45. The coordinate of the focus of the
parabola 2x 2  5y is 50. Number of intersecting points of the
conics 4x 2  9y 2  1 and 4x 2  y 2  4
2x 2  5y "[‹¤õv¡[i¡¹ ëó¡àA¡àìι Ñ‚à>àS¡ Òº
is
[A]   5 , 0
8 4x 2  9y 2  1 &¤} 4x 2  y 2  4 A¡[>A¡
[B]  , 0 
5 ƒå[i¡¹ ëáƒ[¤@ƒå¹ Î}J¸à Òº
2
[A] 0
[C] 0,
8 
5
[B] 1

[D] 0,
2
5 [C] 2
[D] 3

AUAT-2024 /91- A 8

x 2
lim (1  3x ) x  54. Let A and B be two mutually exclusive
51.
x 0 events such that P (A )  3 and
8
1
[A] e P (B )  , then P  A  B c  
3
[B] e2 ™[ƒ, A * B ƒå[i¡ š¹Ñš¹ šõ=A¡ Qi¡>à &³> ë™
[C] e4 P (A )  3 &¤} P (B )  1 ÒÚ, t¡àÒìº
8 3
[D] e6 P  A  B c   ?

dx  0, 17
52. If then the tangent to the [A]
dy 24
curve y  f (x ) at the point (x , y ) is
7
[B]
24
dx
™[ƒ dy  0 ÒÚ, t¡àÒìº (x , y ) [¤@ƒåìt¡
13
[C]
y  f (x ) ¤yû¡ì¹J๠њŢA¡ 24

[A] parallel to x-axis 1
[D]
24
x-"ìÛ¡¹ γà”z¹àº Òì¤
55. The area bounded by the two curves
[B] parallel to y-axis
y 2  x  4 and x  2y  4, is
y-"ìÛ¡¹ γà”z¹àº Òì¤
y 2  x  4 &¤} x  2y  4 ¤yû¡ì¹Jà‡Ú ‡à¹à
[C] passing through origin
ë¤[Ê¡t¡ "e¡ìº¹ ëÛ¡yó¡º Òº
ëA¡@ƒøKà³ã Òì¤ [A] 9
[B] 18
[D] None of the above
[C] 36
l¡üšì¹¹ ëA¡àì>à[i¡Òü >Ú
[D] 72
53. If v is the variance and  is the
standard deviation then 56. The order of differential equation
3
™[ƒ v šøA¡¹o &¤} γA¡ šà=¢A¡¸ ÒÚ, t¡àÒìº  d 4y  d 3y dy
 4   3
 1 is
dx
1  dx  dx
[A] v  2
 3
 d 4y  d 3y dy
 4   3  1  "¤A¡º γãA¡¹ìo¹
dx
1  dx  dx
[B] v 
 yû¡³ Òº
[A] 3
[C] v  2 [B] 4
[C] 6
[D] v 2   [D] 7

AUAT-2024 /91- A 9 [ P.T.O.

57. The integrating factor of the differential 61. Number of diagonals in a convex n-gon
dy is
equation x log x  y  2 log x is
dx x
&A¡[i¡ l¡üv¡º ¤×®å¡ì\¹ A¡ìo¢¹ Î}J¸à Òº
dy
x log x  y  2 log x "¤A¡º γãA¡¹o[i¡¹ [A] n (n  1)
dx x
Òü[@i¡ìKø[i¡} ó¡¸àC¡¹ Òº
[B] n (n  3)
[A] x2
n (n  3)
[B] log x [C]
2
n (n  1)
1 [D]
[C] 2
x
1 62. If the position vector of the points P
[D]
x2  
and Q are a and b respectively,

58. The rate of change of the function then PQ is
y  f (x ) w.r.t. x at the point x is
x [¤@ƒåìt¡ x-&¹ ÎàìšìÛ¡ y  f (x ) -"ìšÛ¡A¡[i¡¹ 
™[ƒ P [¤@ƒå * Q [¤@ƒå¹ Ñ‚à>àS¡ 뮡C¡¹ ™=àyû¡ì³ a *
š[¹¤t¢¡ì>¹ Ò๠Һ  
b ÒÚ, t¡àÒìº PQ Òº
[A] f (x )
 
f (x ) [A] a b
[B]
2
 
[C] 2 f (x ) [B] a b
f (x )
[D]
f (x )  
[C] b a
     
59. ( a  b )  (5 a  6b )  [D] a  b

[A] 0    
63. If a  4, b  2 3 and a × b  12,
[B] a2 + b2 
then the angle between the vectors a
[C] a2 + 5b2

[D] None of the above and b is
l¡üšì¹¹ ëA¡àì>à[i¡Òü >Ú    
™[ƒ a  4, b  2 3 &¤} a × b  12 ÒÚ,
60. Number of zeros at the end of the  
decimal representation of the number t¡àÒìº a 뮡C¡¹ * b 뮡C¡ì¹¹ ³‹¸¤t¡ã¢ ëA¡ào A¡t¡?
100! is

[A]
100! Î}J¸à[i¡¹ ƒÅ[³A¡ ¹ê¡š-&¹ ëÅìÈ ë³ài¡ Åèì>¸¹ 3
Î}J¸à Òº 
[B]
4
[A] 20

[B] 21 [C]
6
[C] 23

[D]
[D] 24 2

AUAT-2024 /91- A 10

 
64. If a straight line in the yz – plane 67. If a and b are vectors with

makes an angle with the positive z- magnitude a and b respectively, then
3
axis, then it’s direction cosines are  
a b2 
 
yz t¡ìº &A¡[i¡ ιºì¹Jà ™[ƒ ‹>àuA¡ z "ìÛ¡¹ ™[ƒ a 뮡C¡¹ * b 뮡C¡ì¹¹ ³àyà ™=àyû¡ì³ a * b
 
Îàì= 3 ëA¡ào Kk¡> A¡ì¹, t¡àÒìº t¡à¹ [ƒA¡ ëA¡àÎàÒü> ÒÚ, t¡àÒìº a  b 2  ?
Òº  
[A] a 2b 2  ( a  b )2
 
[B] ab  a  b
[A] 0, 1 , 3
2 2  
[C] a 2b 2  ( a  b )2
3, 1  
[B] 0, [D] ab  a  b
2 2
68. Area of a triangle formed by the
[C] 3 , 1, 0   
2 2 vectors a , b and c is
  
1, 3 , 0 a, b * c 뮡C¡ì¹¹ ‡à¹à K[k¡t¡ [y®åè¡ì\¹
[D]
2 2 ëÛ¡yó¡º Òº
65. Angle between the planes 1    
[A] (a  b )  (a  c )
2
x  y  2z  9 and 2x  y  z  7 is
1    
[B] |( a  b )  ( a  c )|
x  y  2z  9 &¤} 2x  y  z  7 t¡ºƒå[i¡¹ 2
  
³‹¸¤t¡ã¢ ëA¡ào Òº [C] ( a  b  c )
[A] 60° 1   
[D] (a  b  c )
[B] 45° 2

[C] 90° 69. The number of solutions of the
equation tan4x  cos x ; for 0  x  ;
[D] 30° is
66. A card is drawn at random from a tan 4x  cos x ; 0  x   , γãA¡¹ìo¹
well-shuffled pack of 52 cards. The A¡t¡P¡[º γà‹à> ¹ìÚìá?
probability of getting a heart or a [A] 1
diamond is [B] 2
[C] 5
®¡àìºà®¡àì¤ &ìºàì³ìºà A¡¹à 52 [i¡ A¡àìl¢¡¹ &A¡[i¡ [D] 8
š¸àìA¡i¡ ë=ìA¡ &A¡[i¡ A¡àl¢¡ ™ì=Zᮡàì¤ ë>*Úà Òº¡ú
&A¡[i¡ ºàº šà> ¤à l¡àÚ³“¡ šà*Ú๠δ±à¤>à Òº 70. If f (x )  log x , x  0, then f (x ) 
™[ƒ f (x )  log x , x  0 ÒÚ, t¡àÒìº
[A] 1
f (x )  ?
1 1
[B] [A]
2 x
1
[B]
3 x
[C] 1
13
[C]
x
1 1
[D] [D]
26 x

AUAT-2024 /91- A 11 [ P.T.O.

PART—II
( Islamic History and Culture, General English & General Knowledge )
71. The third Caliph was 76. The shape of Kaaba inside Masjid al-
Haram is like a
t¡õt¡ãÚ J[ºó¡à¹ >à³
³Î[\ƒåº Òà¹àì³¹ "®¡¸”zì¹ A¡à¤à¹ "àAõ¡[t¡ Òº
[A] Bilal ibn Rabah
[A] Cube
[B] Khalid ibn al-Walid
[B] Sphere
[C] Uthman ibn Affan [C] Cylinder
[D] Zayd ibn Haritha [D] Cone

72. Insha’ allah means 77. Prophet Muhammad (PBUH) was a
Òü>Åà"àÀàÒ A¡=๠"=¢ Òº >¤ã ³åÒà´¶ƒ (Îà–) [áìº>
[A] Glory be to Allah [A] Translator
[B] If Allah wills [B] Merchant
[C] Peasant
[C] Praise be to Allah
[D] Mason
[D] As per Allah’s wills
78. Who among the following prophets
73. According to Islam, after death was born miraculously?
Òüκೠ">åÎàì¹ ³õt塸¹ šì¹ [>ì³¥ l¡üìÀ[Jt¡ >¤ã샹 ³ì‹¸ A¡à¹ \@µ "ìºï[A¡A¡®¡àì¤
[A] human beings will be resurrected ÒìÚ[áº?
[B] reward or punishment will be given [A] Moses
[C] Both of the above [B] Jesus
[C] Yaqub
[D] Nothing happens
[D] Yusuf
74. Sayings of Prophet Muhammad (PBUH)
are called 79. At the beginning of any task, Muslims
say
>¤ã ³åÒà´¶ƒ (Îà–) &¹ ¤àoãγèÒìA¡ ¤ºà ÒÚ
ëA¡àì>à A¡àì\¹ Ç¡¹ç¡ìt¡ ³åÎ[º³¹à ¤ìº
[A] Nasihat
[A] bismillah
[B] Sifat [B] inshaallah
[C] Akhlaq [C] mashaallah
[D] Hadith [D] subhanallah

75. The Holy Qur’an has _____ chapters. 80. The city of Madina was known in the
past as
š[¤y A塹"àì> _____ [i¡ "‹¸àÚ "àìá¡ú
"t¡ãìt¡ ³[ƒ>à ÅÒ¹ ë™ >àì³ š[¹[W¡t¡ [áº, t¡à Òº
[A] 30
[A] Taif
[B] 99 [B] Tabuk
[C] 114 [C] Yathrib
[D] 786 [D] Dammam

AUAT-2024 /91- A 12

81. What did Allah (SWT) command the 86. During Ramadan, Muslims refrain
angels to do to Adam (AS)? from which of the following?
"àÀàÒ (Îå¤Òà>à× *Úà t¡àÚàºà) "àƒ³ ("à–)-&¹ ¹³\à> ³àìÎ ³åκ³à>¹à [>ìW¡¹ ëA¡à>[i¡ ë=ìA¡ [¤¹t¡
šø[t¡ ëó¡ì¹Åt¡à샹 A¡ã "àìƒÅ A¡ì¹[áìº>? =àìA¡?
[A] Test his patience [A] Sleeping
[B] Prostate
[B] Speaking
[C] Degrade
[C] Eating and drinking
[D] Bring him food
[D] Working out
82. Where did Adam (AS) and Hawwa
reside before coming to earth? 87. What is the meaning of “Allahu Akbar”
in English?
šõ[=¤ãìt¡ "àÎ๠"àìK "àƒ³ ("à–) * Òà*Úà ëA¡à=àÚ
=àA¡ìt¡>? Òü}ì¹[\ìt¡ ‘"àÀà× "àA¡¤¹’&¹ "=¢ A¡ã?
[A] Sahara [A] Allah is the most praiseworthy
[B] Jahannam [B] Allah is the most merciful
[C] Madinah [C] Allah is the greatest
[D] Jannah [D] None of the above

83. As a young boy Prophet Muhammad 88. What do we say when we hear the
(SAW) used to work as a name of Prophet Muhammad?
ëáài¡ ¤àºA¡ ¤ÚìÎ >¤ã ³åÒà´¶ƒ (Îà–) ëA¡à> A¡à\ >¤ã ³åÒശ샹 >à³ Ç¡>ìº "à³¹à A¡ã ¤[º?
A¡¹ìt¡>?
[A] Salallahu Alayhi Wasallam
[A] Merchant
[B] Sadaqallahul Azim
[B] Scientist
[C] Radi Allah Anhu
[C] Trader
[D] None of the above
[D] Shepherd
89. What is the meaning of the term ‘Dua’
84. What is the relation of Aisha (RA) with
in Arabic?
Prophet Muhammad (SAW)?
Ò™¹t¡ ³åÒà´¶ƒ (Îà–) &¹ Îàì= "àìÚÅà (¹à–) &¹ "à¹[¤ìt¡ ƒå"à Å즹 "=¢ A¡ã?
δšA¢¡ A¡ã? [A] Supplication
[A] Siblings [B] Charity
[B] Husband and wife [C] Prayer
[C] Father and daughter [D] Fasting
[D] None of the above
90. How many times does the term
85. What is the meal to break the fast ‘Allahu Akbar’ appear in the call to
called? prayer (Adhan)?

ë¹à\à ®¡àR¡à¹ Jà¤à¹ìA¡ A¡ã ¤ìº? "à™àì> A¡t¡¤à¹ ‘"àÀà× "àA¡¤¹’ Ŧ[i¡ &ìÎìá?
[A] Suhoor [A] 2
[B] Sadaqah [B] 4
[C] Tarawih [C] 5
[D] Iftar [D] 6
AUAT-2024 /91- A 13 [ P.T.O.

91. Identify the correct alternative : 96. LED stands for what?

[A] Misanthropist — one who hates LED ³àì> A¡ã?
or distrusts mankind [A] Light Emitting Device
[B] Misogynist — one who hates men [B] Low Emitting Diode
[C] Atheist — one who believes in the [C] Light Electronic Diode
existence of God
[D] Light Emitting Diode
[D] Philanthropist — one who writes
life story of another person 97. Computer is connected to Internet by
which device?
92. She dealt with the problem of her
client so _____, that even the judge A¡[´šl¡üi¡à¹ ëA¡à>ô ™ì”|¹ ³à‹¸ì³ Òü@i¡à¹ì>ìi¡¹ Îàì=
was _____. Î}™åv¡û¡ ÒÚ?
Fill in the blanks with the correct [A] Modem
alternatives.
[B] Mouse
[A] agreeably, satisfied [C] CPU
[B] inspiringly, appreciating [D] RAM
[C] diligently, happy
98. Which is the smallest planet in our
[D] dexterously, impressed
solar system?

93. A _____ of arrows. Fill in the blank "à³à샹 ëÎï¹\Kìt¡¹ ΤìW¡ìÚ ëáài¡ KøÒ ëA¡à>[i¡?
with the appropriate collective noun.
[A] The Earth
[A] quiver
[B] Mars
[B] fleet
[C] Mercury
[C] swarm
[D] Saturn
[D] cluster
99. The first British Viceroy of India was
94. Select the correct sentence from the
given alternatives :
®¡à¹ìt¡¹ šø=³ [¤ø[i¡Å ®¡àÒüÎ¹Ú [áìº>
[A] Lord Curzon
[A] The convict was hung this morning.
[B] Lord Irwin
[B] These houses are for the poor to live.
[C] Lord Canning
[C] Neither of the books were
interesting. [D] Lord Tom
[D] What is the time by your watch?
100. Which of the following is not a feature
of the Indian Constitution?
95. A collection of poems or other pieces
of writing is called [>ìW¡¹ ëA¡à>[i¡ ®¡à¹t¡ãÚ Î}[¤‹àì>¹ í¤[ÅÊ¡¸ >Ú?
[A] analogy [A] Parliamentary form of Government
[B] anthology [B] Independence of Judiciary

[C] philology [C] Presidential form of Government

[D] cryptology [D] Federal Government
AUAT-2024 /91- A 14

SPACE FOR ROUGH WORK

AUAT-2024 /91- A 15 [ P.T.O.

SPACE FOR ROUGH WORK

AUAT-2024 /91- A 16 NNN24(119)—75×4

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