KEAM 2026: KEAM Engineering entrance (Mathematics, Physics and Chemistry), session 4
Text of page 1
FINAL ANSWER KEY
Question Paper Code: 11/2026/OL
Exam:KEAM 2026 - 4
Date of Test: 20-04-2026
Let � = {�, �, �, �, �} and � = {�, �, �, �, } . Let P={�: � ⊆ X �� d A\B = {�}}
1. . Then the number of elements in the set P , is
A) 1
B) 2
C) 4
D) 3
E) 8
Correct Answer : Option B
Let � = {1, 2, 3, 4, 5, 6, 7} . Let � = {(1, 1), (1, 2), (1, 3), (5, 6), (6, 7), (7, 5)}
2. be a relation on X. Then the relation R will become reflexive if we include the pairs
A) (2,7),(3,3),(5,5),(6,6) and (7,7)
B) (2,2),(4,3),(5,5),(6,6) and (7,7)
C) (2,2),(3,3),(5,5),(6,6) and (7,6)
D) (2,2),(3,3),(5,5) and (7,7)
E) (2,2),(3,3),(4,4),(5,5),(6,6) and (7,7)
Correct Answer : Option E
����� � ���� ���� ����
Let �(�) = , � ϵ ℝ, � ≠ be a function. Then � (100) , where
3. ����� � ���� ����
� �(�) = �(�(�)) is equal to
A) 10
B) 20028
C) 100
D) 1000
E) 202846
Correct Answer : Option C
4. Let �(�) = � � + 2� + 3, � ≤ − 1 . Then the domain of the inverse of �(�) is
A) [ − 2, ∞)
B) [12, ∞)
C) [ − 22, ∞)
D) [ − 12, ∞)
E) [2, ∞)
Correct Answer : Option E
Text of page 2
�
5. Given that � = − 1 . Then (� �)(� �)(� �) . . . (� ����) is equal to
A) -1
B) 1
C) 2�
D) �
E) -�
Correct Answer : Option E
Given that �
�
= − 1 . If �� = (7 + �√5) � + (7 − �√5) � and �� = (3 + 2�) � − (
6.
3 − 2�) � , then
A) �� is a purely imaginary number and �� is a purely real number
B) �� is a purely real number and �� is a purely imaginary number
C) both �� and �� are purely imaginary numbers
D) both �� and �� are purely real numbers
E) �� +�� is a purely real number
Correct Answer : Option B
7.
A) -16
B) 1+�
C) 1
D) 1-�
E) -16�
Correct Answer : Option A
���� � � ���� ���� ��� � � � ���� ����
8. �
���� � � ����
� +�
���� � � ��� �
� is equal to
A) 1
B) 5
C) 4
D) 16
E) 2
Correct Answer : Option E
�� �
9. Let �� , �� , �� , . . . , �� be in G.P. Then �
��
� is equal to
� �
A) � ��
��
�� �
B) � �
��
Text of page 3
� �
C) � ��
��
�� �
D) � �
��
� �
E) � ��
��
Correct Answer : Option B
���
The sum of the first � terms in a G.P. is �� = 100 − 100 . The common ratio
10.
r is
�
A)
���
�
B)
���
�
C)
��
�
D)
��
�
E)
���
Correct Answer : Option E
�
Let �� , �� , 6, �� be in G.P. Then the value of (6�� + 6�� + �� �� + �� �� ) is
11.
equal to
A) (6 + �� ) �
B) (6 + �� ) �
C) (6 + �� ) �
D) (6 + �� ) �
E) (6 + 2�� ) �
Correct Answer : Option A
12. The two geometric means between 2 and 8 are �� and �� . Then ���+��� is equal to
A) 160
B) 140
C) 180
D) 164
E) 174
Correct Answer : Option A
The number of positive integers up to 100, which are divisible by 3 or 7 but not by both 3
13. and 7, is
Text of page 4
A) 47
B) 33
C) 42
D) 43
E) 45
Correct Answer:-Question Cancelled
14. If ��� = 1365 then the value of � is equal to
A) 13
B) 14
C) 15
D) 12
E) 16
Correct Answer : Option C
15. A polygon has 170 diagonals. The number of sides of the polygon, is
A) 20
B) 24
C) 25
D) 30
E) 28
Correct Answer : Option A
� �
16. The coefficient of � � in the expansion of �� � + √1 − � �� + �� � − √1 − � �� , is
A) -20
B) -10
C) -30
D) -40
E) 20
Correct Answer : Option A
17. If ��� = 6720 and (� − 1)� = 840 , then ��� is equal to
�
A) 346
B) 348
C) 396
D) 376
E) 336
Correct Answer : Option E
�
18. Let � be a square matrix and � be its transpose. Which one of the following is true?
A) � + � � is skew symmetric and � − � � is symmetric.
Text of page 5
� �
B) both � + � and � − � are skew symmetric
� �
C) both � + � and � − � are symmetric
D) � + � � is symmetric and � − � � is skew symmetric
E) −� − � � is skew symmetric
Correct Answer : Option D
19.
A) 100
B) 200
C) 0
D) 6000
E) 60600
Correct Answer : Option C
1 ����
20. Let � = � �. Then �� � is equal to
−���� 1
��� �� − 1
A) � �
−1 ��� ��
��� �� 1
B) � �
1 ��� ��
��� �� 1
C) � �
0 ��� ��
��� �� 0
D) � �
1 ��� ��
��� �� 0
E) � �
0 ��� ��
Correct Answer : Option E
21.
A) 2 and 19
B) 1 and 21
C) 1 and 19
D) 1 and -19
Text of page 6
E) -1 and 19
Correct Answer : Option C
�� � ��
22. If
���
≥ � − 5 then � lies in
A) ( − ∞, − 1) ∪ [0, 6]
B) ( − ∞, − 2) ∪ [ − 1, 5]
C) ( − ∞, − 2) ∪ [0, 10]
D) ( − ∞, 0) ∪ [0, 5]
E) ( − ∞, − 2) ∪ [0, 5]
Correct Answer : Option E
23. If 10 < | � + 10 | ≤ 25 , then � lies in
A) [ − 45, − 20) ∪ (0, 15]
B) [ − 35, − 25) ∪ (0, 15]
C) [ − 35, − 20) ∪ (0, 15]
D) [ − 35, − 20) ∪ (0, 25]
E) [ − 35, − 10) ∪ (0, 15]
Correct Answer : Option C
� �
24. The value of 3���( − 405 )���315 − 5���495 ����( − 585 �) is equal to
A) 8
B) -8√2
C) 2
D) -8
E) -2
Correct Answer : Option E
�� �� �� ��
25. ���� � + �� + ���� � + �� − ���� � − �� − ���� � − �� =
A) −2√2����
B) √2����
C) 2√2����
D) 3√2����
E) −√2����
Correct Answer : Option A
�� � � � � �
26. 2 ���( � �� )���( � )���( � � )���( � � ) . . . ���( � �� ) is equal to
A) ����
Text of page 7
B) 2����
C) -2����
D) 4����
E) 8����
Correct Answer : Option B
� � � �
27. The value of ���5 ���10 ���15 ���20 . . . ���240 � is equal to
�
A) �� (���15 � + ���15 �)
�
�
B) (���15 � + ���15 �)
� ��
�
C) �� (���15 � + ���15 �)
�
�
D) (���15 � + ���15 �)
� ��
E) 0
Correct Answer : Option E
28. The domain of the function �(�) = ��� ��(5 − 4� �) + ��� ��(5 − 4� �) is
� �
A) �1, � � ∪ � − � , − 1�
� �
� �
B) �1, � � ∪ � − � , − 1�
� �
� �
C) �1, � � ∪ � − � , − 1�
� �
� �
D) �2, � � ∪ � − � , − 1�
� �
� �
E) �1, � � ∪ � − � , − 2�
� �
Correct Answer : Option A
�� � � ��
29. The value of ��� �1(��� ��� + ��� ��� ) is equal to
� � � �
��
A)
�
�
B)
�
��
C)
�
�
D)
�
��
E)
�
Text of page 8
Correct Answer : Option A
30. If � ≤ 2��� ��� + ��� ��� ≤ � , then � and � respectively, are
�
A) and π
�
B) 0 and 2π
� �
C) and
� �
�
D) and π
�
E) 0 and π
Correct Answer : Option E
Let � be the origin and � be any point on � � = 2� . The locus of the midpoint of the
31. line segment �� , is
A) (� − 1) � = 2�
B) � � = 3� − 1
C) � � + 1 = 2�
D) �� = �
E) � � = 2�
Correct Answer : Option D
A straight line has � -intercept -5 . If it makes 120o with the � -axis, then the equation
32. of the line is
A) √3� + � + 20 = 0
B) √3� + � + 10 = 0
C) √3� − � + 10 = 0
D) √3� + � − 10 = 0
E) √3� + � + 5 = 0
Correct Answer : Option E
33. The nearest point on the line � + 2� = 5 from the point �(7, 9) is equal to
A) (6,1)
B) (7,6)
C) (2,3)
D) (8,3)
E) (3,1)
Correct Answer : Option E
Text of page 9
The equation of a chord to the circle � � + � � = 25 is � + � − 5 = 0 . The equation
34. of a circle whose diameter is � + � − 5 = 0 , is
A) � � + � � − 6� − 5� = 0
B) � � + � � − 5� − 6� = 0
C) � � + � � − 5� − 5� = 0
D) � � + � � − 6� − 6� = 0
E) � � + � � − 3� − 3� = 0
Correct Answer : Option C
35. The length of the latus rectum of the ellipse 225� � + 125� � = 28125 is
��
A)
�
��
B)
�
C) 5
��
D)
�
���
E)
�
Correct Answer : Option A
�
Let the eccentricity of an ellipse be . If �(3, 2) is a focus and � − 9 = 0 is the
�
36. corresponding directrix of the ellipse, then the equation of the ellipse is
A) 3� � + 4� � − 6� − 16� − 29 = 0
B) 3� � + 4� � − 8� − 16� − 29 = 0
C) 3� � + 4� � − 6� − 12� − 29 = 0
D) 3� � + 4� � − 6� − 16� + 29 = 0
E) 13� � + 5� � − 6� − 16� − 29 = 0
Correct Answer : Option A
37. The eccentricity of the hyperbola 25� � − 36� � − 50� − 72� − 911 = 0 , is
√��
A)
�
√��
B)
�
√��
C)
�
√��
D)
�
√��
E)
�
Text of page 10
Correct Answer : Option A
→ → → →
The vectors 2� + 6� and 3� − 7� are position vectors of the points � and �
38. respectively. A point � divides the line segment �� internally in the ratio 3:5 . Then
→
�� =
→ →
�� � ���
A)
�
→ →
�� � ���
B)
�
→ →
�� � ���
C)
�
→ →
�� � ���
D)
�
→ →
�� � ���
E)
�
Correct Answer : Option A
The position vectors of the vertices of a triangle are ^
^ ^
� + 2�^ − 4� , −�^ − 2�^ − 4�
39. ^
^ + 3�^ + 5� . The perimeter of the triangle is
and 2�
A) √20 + √115 + √83
B) √20 + √117 + √83
C) √22 + √115 + √83
D) √20 + √115 + √84
E) √20 + √125 + √83
Correct Answer : Option A
→ → → → →
Let |� | = 2 , |� | = �13 + 6√3 and | � | = 3 . If � − � +→
� = 0 , then
40. → →
the angle between � and � is
�
A)
�
�
B)
�
�
C)
��
�
D)
�
�
E)
�
Correct Answer : Option E
Text of page 11
→ → → | = 2, | →
�, �, →
Let � be three vectors such that |� � | = 3, | →
� | = 4 and →
�.
41. → → → →
� = � . � = 0 If the angle between � and → � is , then →
�
� is equal to
�
� →
A) ± √ (� × →
�)
�
� →
B) ± √ (� × →�)
� �
� →
C) ± √ (� × →�)
� �
� →
D)
± √ (� × →
�
� �
)
� →
E) ± √ (� × →
�)
� �
Correct Answer : Option D
→
The straight line � = (2 − 3�)(�
^ ^
^ + ^�) + (6� + 1)(�^ + �) + (12� − 11)(� + ^� ), �
42.
ϵ ℝ , is parallel to the vector
A) ^
3�^ − ^� + 6�
B) ^
3�^ + ^� + 6�
C) ^
3�^ + ^� − 6�
D) ^
−3�^ + ^� + 6�
E) ^
3�^ + 2�^ + 5�
Correct Answer : Option B
43. The equation of a line passing through (-1,6,5) and (-2,4,3) , is
�� � � ��� �� � �
A) = =
�� �� ��
�� � � ��� �� � �
B) = =
�� �� ��
�� � � ��� �� � �
C) = =
�� �� ��
�� � � ��� �� � �
D) = =
�� � ��
�� � � ��� �� � �
E) = =
�� �� ��
Correct Answer:-Question Cancelled
��� ��� ���
If α , β , γ are the angles made by the straight line = = √ with the x-
44. � � �
axis, y-axis and z-axis respectively, then ��� α , ��� β , cos γ are respectively,
� � √�
A) , ,
√� �√� �√�
Text of page 12
� � �√�
B) , ,
√� �√� �√�
�� � √�
C) , ,
√� �√� �√�
� �� √�
D) , ,
√� �√� �√�
� � √�
E) , ,
√� �√� �√�
Correct Answer : Option E
The shortest distance between the lines
→ ^) + �(2�^ + 3�^ + 4�
� = (2�^ + 6�^ + 3� ^), �
45. ϵ ℝ and
→ ^) + �(�^ + 2�^ + 3�
� = (2�^ − 3� ^), � ϵ ℝ , is
A) 8
B) 5
C) 0
D) 4
E) 1
Correct Answer:-Question Cancelled
46.
A) 55
B) 50
C) 36
D) 40
E) 25
Correct Answer : Option D
47.
A) 6.452
B) 9.126
C) 8.136
D) 9.145
E) 7.111
Correct Answer : Option C
� � �
Let � and � be two events such that �(�) = , �(� / �) = and �(� / �) =
48. � � �
. Then �(�) is equal to
Text of page 13
�
A)
�
�
B)
�
�
C)
��
�
D)
��
�
E)
�
Correct Answer : Option A
A number x is randomly chosen from the set of natural numbers less than or equal to
100. Then the probability of the event that the chosen number satisfies the inequality
49. (� � ��)(� � ��)
≥ 0 , is
� � ��
A) 0.36
B) 0.47
C) 0.48
D) 0.49
E) 0.46
Correct Answer : Option E
�(�) � �(�)
50. Let �(�) = − √49 − � � . Then lim is equal to
�→� ���
�
A)
�√�
�
B)
�√�
�
C)
√�
�
D)
�√�
�
E)
�√�
Correct Answer : Option D
���[�]
Let [�] denote the greatest integer less than or equal to � . Then lim � is
51. �→� [�]
equal to
A) 1
B) -sin1
C) -2
D) 0
E) sin1
Text of page 14
Correct Answer : Option E
52.
A) 1
B) -2
C) -1
D) 3
E) -5
Correct Answer : Option D
The value of lim� � (���� − ����) is equal to
53. �→
�
�
A)
�
�
B)
�
C) 0
D) 2
E) 1
Correct Answer : Option C
�
54. �
Consider the �(�) = �� � function . Which one of the following is not true?
A) �(�) is continuous at � =1
B) �(�) is continuous at � =-1
C) �(�) is continuous at � =2
D) �(�) is continuous at � =-2
E) �(�) is continuous at � =0
Correct Answer : Option E
55.
A) �(�) is differentiable in (-1,1)
B) �(�) is differentiable in (−∞ ,1)
C) �(�) is differentiable at � =-2026
D) �(�) is not differentiable at � =1
E) �(�) is continuous at � =1.5 and � =1.5 differentiable at
Correct Answer : Option D
Text of page 15
56.
A) -310
B) 320
C) -360
D) -320
E) -380
Correct Answer : Option D
�� �
57. If � = � ecθ − ��� θ , � = ��� �� θ − ��� �� θ , then ( ) is equal to
��
�� � �
A) 100 � � �
� ��
�� � �
B) 100 � � �
� ��
�� � �
C) 100 � � �
� ��
�� � �
D) 100 � � �
� ��
�� � �
E) 100 � � �
� ��
Correct Answer : Option A
�
��� � ��
58. Let � = � � . Then at � = 1,
��
is equal to
A) -1
B) 0
C) -2
D) 2
E) 1
Correct Answer : Option E
���� � ���� � ��
59. Let � = ��� ��� �, 0 < � < . Then is equal to
���� � ���� � ��
A) �
B) -1
C) -�
D) 2�
E) -2�
Correct Answer : Option B
Text of page 16
���(� � �)
60. Let �(�) =
���(� � �)
, −2<�<∞ . Then � is
A) decreasing on (-2,∞ )
B) decreasing only on (0,∞ )
C) increasing only on (0,� )
D) increasing on (-2,∞)
E) increasing only on (0,π )
Correct Answer : Option D
Let �(�) = �� + �� � � + �� � � + �� � �, � ϵ ℝ where 0 < �� < �� < �� < ��
61.
The minimum value of �(�) is
A) �� + �� + �� + �� .
B) �� + � �
C) ��
D) ��
E) �� + � �
Correct Answer : Option C
�
If |�(�� ) − �(�� ) | ≤ (�� − �� ) , �� , �� ϵ ℝ and if �(0) = 2026 , then �(
62.
2025) is equal to
A) 2025
B) 4051
C) 4050
D) 2026
E) 1
Correct Answer : Option D
Consider the circle � � + � � = 1 . The point on the circle which is nearest from the
63.
point (1,1) , is
� �
A) �−√ ,√ �
� �
� �
B) �√ , − √ �
� �
� �
C) �√ , √ �
� �
� �
D) �−√ , −√ �
� �
E) (0,1)
Correct Answer : Option C
� � � �� � �
64. ∫ �� � �
�� is equal to
Text of page 17
A) � + 2��� | � | − 4��� | 1 + � | + �
B) � + 2��� | � | − 6��� | 1 + � | + �
C) � + 3��� | � | − 6��� | 1 + � | + �
D) � + 2��� | � | + 6��� | 1 + � | + �
E) � + 7��� | � | − 6��� | 1 + � | + �
Correct Answer : Option B
�
65. ∫ ��� � ��� θ ��� θ � θ is equal to
� �����
A) ��� θ − θ − +�
� �
� �����
B) ��� θ − θ − +�
� �
� �����
C) 2��� θ − θ − +�
� �
� �����
D) ��� θ − θ − +�
� �
� �����
E) ��� θ − θ − +�
� �
Correct Answer : Option D
�
66. ∫ (� � � �)��� ��� � � � � �� is equal to
���
���
A) log | ��� ��� �| +�
���
���
B) log | ��� ��� �| +�
���
� ���
C) log | ��� ��� �| +�
� ���
� ���
D) log | ��� ��� �| +�
� ���
� � � ��
E) log | ��� ��� �| +�
� ���
Correct Answer : Option B
67. ∫ � �(2� � + ���� + ���� + 2)�� is equal to
A) � �(� � + ���� + 2) + �
B) � �(� � + ���� + 2�) + �
C) � �(� � + ���� + 2) + �
D) � �(� � + ���� + 1) + �
E) � �(� � − ���� + 2) + �
Correct Answer : Option C
Text of page 18
�
68. ∫ � � � � � ��� is equal to
A) � + ���(1 + � �) + �
B) � − ���(1 − � �) + �
C) 2� − ���(1 + � �) + �
D) � − � ����(1 + � �) + �
E) � − ���(1 + � �) + �
Correct Answer : Option E
69.
A) 0
�
B)
�
�
C)
�
�
D)
��
�
E)
�
Correct Answer : Option D
70.
�
A)
�√�
�
B)
�√�
�
C)
��√�
�
D)
�√�
�
E)
��√�
Correct Answer : Option E
71.
A) 8
B) 6
C) 4
D) 16
E) 12
Text of page 19
Correct Answer : Option C
72.
�
A)
�
�
B)
�
C) 5
�
D)
�
�
E)
�
Correct Answer : Option E
The order and degree of the differential Equation
� �
73. �� � �� � �� � �� �
�1 + + � � =�� + � + + � � respectively, are
�� �� �� ��
A) 4 and 2
B) 3 and 3
C) 3 and 4
D) 2 and 3
E) 3 and 2
Correct Answer : Option C
74. The solution of the differential equation (2� − 1)�� − (� − 2)�� = 0 is
A) 2� + 3��� | � − 2 | = 2� + �
B) 2� + 4��� | � − 2 | = � + �
C) � + 3��� | � − 2 | = � + �
D) 2� + 3��� | � − 2 | = � + �
E) 3� + 2��� | � − 2 | = � + �
Correct Answer : Option D
Consider the Linear Programming Problem (LPP):
Maximize � = 20� + 40� subject to the constraints .
75. 3� + � ≤ 14; � + 3� ≤ 10; � ≥ 0; � ≥ 0 The number of corner points of the
feasible region is
A) 5
B) 4
C) 3
D) 2
E) 6
Text of page 20
Correct Answer : Option B
76. The dimensional formula for the rate of change of momentum of a moving body is
A) �� �� ��
B) ��� ��
C) �� ��� ��
D) ��� ��
E) � ��� ��
Correct Answer : Option D
�� �
If the time period of oscillation of a liquid drop is given by � = �� , where ρ refers
77. �
to its density, � , its radius and � is a dimensionless constant, then σ has the units of
A) surface
B) restoring force
C)
coefficient of viscosity
D) vapour pressure
E) surface tension
Correct Answer : Option B
78. The area under the velocity-time graph of a particle is equal to its
A) acceleration
B) displacement
C) speed
D) velocity
E) angular velocity
Correct Answer : Option B
79. If both x and y components of a vector are equal, its angle with x-axis is
A) 30°
B) 45°
C) 60°
D) 90
E) 0°
Correct Answer : Option B
→
The position vector of a particle is given by � = (� � − 3� � + 2) ^
� , the time at which
80.
the velocity of the moving particle becomes zero is
Text of page 21
A) 1s
B) 2s
C) 3s
D) 4s
E) 5s
Correct Answer : Option B
A particle of mass m is initially at rest. A time varying force � = �� acts on it. Its velocity
81. after time � is
�� �
A)
��
�� �
B)
�
��� �
C)
�
� �� �
D)
�
�� �� �
E)
�
Correct Answer : Option A
A passenger of mass � stands on a weighing scale in an elevator accelerating upward
82. with an acceleration of
�
. The scale would read (� = acceleration due to gravity)
�
���
A)
�
��
B)
�
���
C)
�
D) ��
��
E)
�
Correct Answer : Option A
A block of mass 2 kg slides on a rough horizontal surface (�� = 0.2) with initial speed
83.
10 ms−1. The distance travelled by the block before it stops is (g = 10 ms-2)
A) 5m
B) 12 m
C) 15 m
D) 21 m
E) 25 m
Correct Answer : Option E
Text of page 22
84. A non-conservative force is one for which
A) work depends only on initial position
B) work depends only on final position
C) work is always zero
D) work depends on path
E) energy is conserved
Correct Answer : Option D
A flywheel in a steam engine ensures a smooth ride for the passengers on the vehicle
85. because it has
A) low mass
B) high speed
C) low acceleration
D) large moment of inertia
E) high friction
Correct Answer : Option D
Two bodies A and B of masses 80 g and 120 g move with same speed, 6 cm s-1 in a
86. plane. The speed of the centre of mass of the system is
A) 6.2 cm s-1
B) 6.0 cm s-1
C) 2.4 cm s-1
D) 2.0 cm s-1
E) 3.2 cm s-1
Correct Answer:-Question Cancelled
A satellite revolves around a planet in a circular orbit of radius R with the time period � . If
87. the orbital radius is 3R, then its time period of revolution is
A) �
B) 2�
C) √27�
D) √3�
E) 2√2�
Correct Answer : Option C
The maximum and minimum distances of a planet from the sun are respectively �� and
�� . If it moves with a maximum speed of �� and the minimum speed of �� around
88.
the sun then the ratio �� :�� is
A) ��� : ���
B) �� : ��
C) � � :� �
Text of page 23
D) ���: ���
E) ��� :���
Correct Answer : Option C
If a capillary tube is dipped vertically in water of surface tension T, the capillary rise is 4
�
89. cm. If the same tube is dipped in a liquid of density 800 kgm-3 of surface tension , then
�
the capillary rise will be
A) 2.5 cm
B) 3.2 cm
C) 1.2 cm
D) 2.4 cm
E) 5.0 cm
Correct Answer : Option A
A brass rod has a length 1m linear coefficient of thermal expansion, 2 × 10
��
/� �
90. ��
and Young’s modulus 1 × 10 �� �� . When it is heated by 50 °C, the thermal stress
developed in it (in �� �� ) due to heating is
A) 1 × 10 �
B) 1 × 10 �
C) 2 × 10 �
D) 2 × 10 �
E) 3 × 10 �
Correct Answer : Option A
Two different liquids of having same mass at temperatures �� and �� and of specific
91. heat capacities, S and 1.2 S, respectively, are mixed in an insulated system. If �� >
�� , then the temperature of the mixture is
��� � ���
A)
��
��� � ���
B)
��
C) �� + ��
D) �� − ��
�� � ��
E)
�
Correct Answer : Option A
If the kinetic energy of 1 mole of a monoatomic gas is 24.942 kJ, then its temperature is
92. (R = 8.314 J kg-1K-1)
A) 200 K
Text of page 24
B) 300 K
C) 400 K
D) 2000 K
E) 3000 K
Correct Answer : Option D
One mole of a monoatomic gas is heated at constant volume such that its temperature
93. increases from � to 3� . The heat supplied is (�- Universal Gas constant)
A) 3��
B) ��
�
C) ��
�
�
D) ��
�
E) 5��
Correct Answer : Option A
If the rms speed of ideal gas molecules increases by 20% at constant volume, the
94. percentage increase in their pressure is
A) 20 %
B) 40 %
C) 44 %
D) 10 %
E) 32 %
Correct Answer : Option C
A gas has pressure 2.76 ×105 Pa at 400 K. If Boltzmann constant
95.
k = 1.38×10-23 JK-1, its number density is
A) 5.5 × 10 ��
B) 5.0 × 10 ��
C) 6.5 × 10 ��
D) 6.5 × 10 ��
E) 5.0 × 10 ��
Correct Answer : Option E
If the instantaneous displacements of any two particles executing S.H.M are given by x1
96. = � sin ωt and x2 = � cosωt, then the phase difference between them is
�
A)
�
�
B)
�
Text of page 25
�
C)
�
�
D)
�
�
E)
��
Correct Answer : Option C
Two springs of spring constants k and 2k, are connected in series and suspended
97. vertically. If the mass m attached at its free end is allowed to oscillate vertically, then its
time period of oscillation is
��
A) 2π �
�
��
B) 2π �
��
��
C) 2π �
��
�
D) 2π �
��
�
E) 2π �
��
Correct Answer : Option C
The force between two equal and opposite charges kept at a certain distance of
98. separation in air medium is �. If the charges are kept at the same separation distance in
a medium of dielectric constant � , then the force between the charges becomes
�
A)
�
B) ��
�
C)
�
D) �
�
E)
��
Correct Answer : Option C
An oil drop of mass � carrying a charge � is descending under its own weight. If it is
99. made to remain stationary by applying an electric field of intensity � , then the value of
� is (� = acceleration due to gravity)
A) ���
��
B)
�
��
C)
�
Text of page 26
�
D)
��
��
E)
��
Correct Answer : Option B
The electric field at a point just outside a positively charged perfect spherical conductor
100. is
A) zero
B) parallel to the surface
C) perpendicular to the surface and directed away from the surface
D) at any random direction to the surface
E) perpendicular to the surface and directed towards the surface
Correct Answer : Option C
64 small identical globules of mercury, each of radius r carrying a charge q each combine
101. to form a big globule. The ratio of the surface charge density of each small globule to that
of the big globule is:
A) 1:2
B) 2:1
C) 16 : 1
D) 1:4
E) 4:1
Correct Answer : Option D
�� ��
102. A copper wire of temperature coefficient of resistance 4 × 10 � has resistance 10
Ω at 20° C. Its resistance at 80° C is
A)
1.22 Ω
B) 12.2 Ω
C) 24.4 Ω
D) 38.8 Ω
E) 2.44 Ω
Correct Answer : Option B
When the electric field applied across a conductor is doubled without changing its
103. temperature, the drift velocity
A) doubles
B) triples
C) quadruples
D) becomes half
E) remains unchanged
Text of page 27
Correct Answer : Option A
Two cells of emfs 2 V and 6V each with internal resistance of 1 Ω are connected in series
104. to an external resistance 8 Ω . Then the current in the circuit is
A) 2A
B) 0.5 A
C) 1A
D) 0.8 A
E) 1.5 A
Correct Answer : Option D
Two long straight conductors carrying currents 1 A and 2 A are kept parallel to each other
105. in air medium with a separation of 10 cm. The force acting on each conductor per unit
length is
A) 4 × 10 ���� ��
B) 4 × 10 ���� ��
C) 4 × 10 ���� ��
D) 2 × 10 ���� ��
E) 2 × 10 ���� ��
Correct Answer : Option B
A bar magnet of length l has magnetic moment M. If its length is halved, its magnetic
106. moment becomes
�
A)
�
B) �
�
C)
�
D) 2�
E) 4�
Correct Answer : Option A
�
A solenoid has 1000 turns per metre and carries a current of A. The magnetic field
107. �
inside the solenoid is
A) 7.0 × 10 ���
B) 7.0 × 10 ���
C) 4.0 × 10 ���
D) 1.4 × 10 ���
E) 2.8 × 10 ���
Text of page 28
Correct Answer : Option E
A proton and an alpha particle are accelerated in the same cyclotron using the same
108. magnetic field. The ratio of their respective cyclotron frequencies is
A) 1:2
B) 2:1
C) 1:4
D) 4:1
E) 8:1
Correct Answer : Option B
In an ac circuit containing an ac source of frequency � , a capacitor offers a reactance
109. which is proportional to
�
A)
�
�
B)
��
C) �
D) ��
E) ��
Correct Answer : Option A
A circular coil of radius 10 cm with 200 turns is placed perpendicular to a uniform
110. magnetic field of 0.4 T. If the magnetic field is reduced to zero uniformly in 0.2 s, then the
average induced emf (in volt) is
A) 4π
B) 120
C) 20
D) 15
E) 40π
Correct Answer : Option A
If the total energy transferred to a completely absorbing surface by an EM wave in unit
111. time is 3.6 J, then the radiation pressure exerted by the wave on the surface is
A) 1 × 10 ��� ��
B) 1.8 × 10 ��� ��
C) 1 × 10 ��� ��
D) 1.2 × 10 ��� ��
E) 1.2 × 10 ���� ��
Correct Answer : Option E
Text of page 29
In a single slit diffraction experiment, if the slit width is reduced to half while wavelength
112. and screen distance remain constant, the angular width of the central maximum will
A) remain unchanged
B) become half
C) become double
D) become four times
E) become one-fourth
Correct Answer : Option C
113. Sustained interference is observed with
A) two independent sources
B) two coherent sources
C) a single coherent source
D) a single independent source
E) a single monochromatic source
Correct Answer : Option B
An aperture of size 2 mm is illuminated by a parallel light beam of wavelength 400 nm.
114. The distance up to which ray optics in a good approximation is
A) 20 m
B) 10 m
C) 15 m
D) 30 m
E) 50 m
Correct Answer : Option B
If an electron and a proton have same kinetic energy and their de Broglie wavelengths
115. are λ
e and λp, respectively, then the ratio λp ∶ λe is
A) 1:1
B) 1 : 1836
C) 1836 : 1
D) √1836 : 1
E) 1: √1836
Correct Answer : Option E
Stopping potentials for the metals A and B are 0.4 V and 1.6 V, respectively. When
116. illuminated by same light, the difference in their work functions is:
A) 2.0 eV
B) 1.2 eV
C) 6.4 eV
D) 0.4 eV
Text of page 30
E) 2.0 V
Correct Answer : Option B
Identify the isotones among the given nuclei
117.
a. ��� � b. ��� � c. ��� � d. ��� �
A) a and c only
B) b and d only
C) a, c and d only
D) b and c only
E) c and d only
Correct Answer : Option C
In Bohr’s theory of hydrogen atom, if the speed of an electron in its first orbit is � , then
118.
its speed in its 3rd orbit is
A) �
�
B)
�
�
C)
�
D) 3�
�
E)
�
Correct Answer : Option B
119. Identify the incorrectly matched pair about the characteristics of a pn junction diode
A) Cut-in voltage – Voltage up to which voltage-independent reverse current exists
B) Reverse bias - Leakage current region below breakdown voltage
C) No bias - No net current due to diffusion and drift
D) Breakdown Voltage - Sudden large current due to avalanche effect occurs
E) Forward bias - moderate ohmic region above barrier voltage
Correct Answer : Option A
120. Ripple frequency in full-wave rectifier of input frequency � is:
A) �
�
B)
�
C) 2�
D) 4�
E) 3�
Correct Answer : Option C
Text of page 31
The mass of sodium acetate required to prepare 250 mL of 0.4 M aqueous solution is
121. (Molar mass of sodium acetate = 82 g mol-1)
A) 4.1 g
B) 8.2 g
C) 16.4 g
D) 2.05 g
E) 0.82 g
Correct Answer : Option B
A 250 watt bulb emits monochromatic light of wavelength 198.78 nm. How many photons
122. are emitted by the bulb per second?
(Planck’s constant = 6.626 x 10-34 J s and velocity of light = 3 x 108 ms-1).
A) 2.5 x 1020 s-1
B) 2.5 x 1018 s-1
C) 2 x 1020 s-1
D) 2.2 x 1016 s-1
E) 2.6 x1020 s-1
Correct Answer : Option A
Which of the following statements are incorrect about boundary surface diagram of
orbital?
(i) A boundary surface is drawn in space for an orbital on which the value of probability
density| Ψ |2 is constant.
2
123. (ii) The boundary surface for | Ψ | and | Ψ | are not identical.
(iii) The density of the dots represents the probability density of finding the electron in
the region.
(iv) In 1s and 2s orbitals, the probability of finding electron at a given distance is not equal
in all the directions.
(v) The 2p orbitals in the boundary surface diagram are not spherical.
A) (i), (iv)
B) (iii), (iv)
C) (ii), (iv)
D) (ii), (iii)
E) (i), (ii)
Correct Answer : Option C
124. Which of the following ions are isoelectronic?
A) K+, Al3+, Ca2+, Sc2+.
B) O2-, Al3+, Cl-, Sc2+.
C) K+, S2-, Ca2+, Cl-.
D) Na+, Mg2+, Ca2+, Sc2+.
E) Mg2+, Al3+, Ca2+, S2-.
Text of page 32
Correct Answer : Option C
125. The correct order of metallic radius of the elements is
A) Be < B < Al < Mg < Li
B) Mg < Li < Al < B < Be
C) Li < Mg < B < Be < Al
D) B < Be < Al < Li < Mg
E) Li < Mg < Be < B < Al
Correct Answer : Option D
The correct increasing order of number of electrons around the central atom in the
126. following molecules is
A) SF6 < SCl2 < PF5
B) SCl2 < PF5 < SF6
C) PF5 < SCl2 < SF6
D) SCl2 < SF6 < PF5
E) PF5 < SF6 < SCl2
Correct Answer : Option B
127. Which of the following molecule has double bond with two ᴫ bonds?
A) Ne2
B) O2
C) B2
D) C2
E) Li2
Correct Answer : Option D
The quantity of heat necessary to raise the temperature of 60 g of metal ‘X’ from 30˚C to
128. 54˚C is (Molar heat capacity of metal, ‘X’ is 25 J mol-1 K-1 and molar mass is 30 g mol-1)
A) 1.02 kJ
B) 1.20 kJ
C) 1.25 kJ
D) 2.40 kJ
E) 1.80 kJ
Correct Answer : Option B
The ΔG˚ for the reaction 3X(g) + 2Y(g) → 3Z(g), at 298 K is (ΔH˚ = -13 kJ mol-1 and ΔS˚=
129.
‒ 45 J K-1mol-1)
A) – 0.41 kJ mol-1
B) – 0.21 kJ mol-1
Text of page 33
C) +0.41 kJ mol-1
D) – 0.14 kJ mol-1
E) +0.32 kJ mol-1
Correct Answer : Option C
For the equilibrium, 2A(g)⇋ 2B(g)+C(g), the value of the equilibrium constant, Kp is 0.1662
130. atm at 1000 K. The value of Kc for the equilibrium at the same temperature is (R =
0.0831 lit atm mol-1)
A) 1 x 10-3 mol lit-1
B) 2 x 10-3 mol lit-1
C) 0.4 x 10-3 mol lit-1
D) 1.6 x 10-3 mol lit-1
E) 0.6 x 10-3 mol lit-1
Correct Answer : Option B
CO2 is taken in a closed container at 1500 K and at a pressure of 0.6 bar. When a certain
amount of carbon is added to the system at equilibrium some of CO2 is converted into
131. CO and the equilibrium pressure is 0.9 bar. What is the Kp value at 1500 K for the
equilibrium CO2(g) + C(s) ⇋ 2CO(g)?
A) 0.1 bar
B) 0.2 bar
C) 0.3 bar
D) 0.4 bar
E) 0.5 bar
Correct Answer:-Question Cancelled
In the reaction, O2(g) + 4H+(aq) + 4 e- →2H2O(l) , the quantity of electricity needed to
132.
reduce one mole of gaseous oxygen is (E▫cell = 1.23V)
A) 386000 Coulombs
B) 96500 Coulombs
C) 24125 Coulombs
D) 19300 Coulombs
E) 579000 Coulombs
Correct Answer : Option A
Which of the following metal is incorporated into the electrodes of fuel cell to increase the
133. rate of reaction?
A) Nickel
B) Zinc
C) Palladium
D) Tin
E) Magnesium
Text of page 34
Correct Answer : Option C
3 g of benzoic acid dissolved in 25 g of benzene shows a depression in freezing point
134. equal to 2.5 K. What is the experimental molar mass of benzoic acid in benzene? (Molal
depression constant for benzene is 5 K kg mol-1).
A) 122 g mol-1
B) 244 g mol-1
C) 240 g mol-1
D) 242 g mol-1
E) 250 g mol-1
Correct Answer : Option C
The rate constant of a reaction at 400 K and 500 K are 0.04 min-1 and 0.08 min-1
135. respectively. The activation energy of the reaction is about
( 2.303 R = 19 J K-1 mol-1, log2 = 0.3010)
A) 8.5 kJ
B) 6.5 kJ
C) 10.5 kJ
D) 11.5 kJ
E) 15 kJ
Correct Answer : Option D
A reaction is third order with respect to a reactant. If the concentration of the reactant is
136. doubled, the rate of reaction increased by
A) 2 times
B) 3 times
C) 4 times
D) 8 times
E) 5 times
Correct Answer : Option D
137. The correct order of standard electrode potentials of the following transition elements is
A) Cr < V < Fe < Co
B) V < Cr < Fe < Sc
C) Fe < Cr < V < Co
D) Co < V < Cr < Fe
E) Cr < Fe < V < Co
Correct Answer:-Question Cancelled
138. IUPAC name of [CoCl2(en)2]Cl is
A) Bis(ethane-1,2-diamine)dichloridocobalt(III)chloride
B) Dichloridobis(ethane-1,2-diamine)cobalt(II)chloride
C) Dichloridobis(ethylenediamine)cobalt(III)chloride
Text of page 35
D) Dichloridobis(ethane-1,2-diamine)cobalt(III)chloride
E) Dichloridobis(ethylenediamine)cobalt(II)chloride
Correct Answer : Option D
139. The order of field strength of the following ligands in the spectrochemical series is
A) NCS-< CN-<SCN-< S2-< OH-
B) NCS-< OH-< S2-< NCS-< CN-
C) CN-< S2-< OH-< NCS-< SCN-
D) SCN-< S2-< OH-< NCS-< CN-
E) SCN-< S2-< CN-< NCS-< OH-
Correct Answer : Option D
The technique used to purify liquids having very high boiling points and decompose at or
140. below their boiling point is
A) steam distillation
B) distillation
C) differential extraction
D) distillation under reduced pressure
E) fractional distillation
Correct Answer : Option D
One mole of an alkene on ozonolysis gives one mole of propanal and one mole of pent-3-
141. one. The alkene is
A) 3-ethylpent-2-ene
B) 4-ethylhex-3-ene
C) 2-methylpropene
D) 3-ethylhex-2-ene
E) 3-ethylhexene
Correct Answer:-Question Cancelled
142. The number of possible structural isomers of C3H6Cl2 are
A) 3
B) 2
C) 5
D) 4
E) 6
Correct Answer : Option D
Match the following:
Compound Use
(a) Methylene chloride (i) Solvent for iodine
143. (b) Chloroform (ii) Manufacture of propellents
(c) Iodoform (iii) Metal cleaning
(d) Carbon tetrachloride (iv) Antiseptic
Text of page 36
A) (a)-(iii), (b)-(i), (c)-(iv), (d)-(ii)
B) (a)-(i), (b)-(ii), (c)-(iv), (d)-(iii)
C) (a)-(iii), (b)-(ii), (c)-(i), (d)-(iv)
D) (a)-(i), (b)-(iv), (c)-(ii), (d)-(iii)
E) (a)-(iv), (b)-(i), (c)-(iii), (d)-(ii)
Correct Answer : Option A
144. The major products formed by heating C6H5CH2–O-C6H5with HI are
A) C6H5CH2Iand C6H5-CH2OH
B) C6H5CH3 and C6H5OH
C) C6H5CH2 I and C6H6
D) C6H5-CH2OH and C6H5I
E) C6H5CH2I and C6H5OH
Correct Answer : Option E
Match the following reactions with the corresponding reagents
Reactions Reagents
(a) Oxidation of secondary alcohols to ketones (i) 85% H3PO4, 440 K
145. (b) Dehydration of secondary alcohols to alkenes (ii) Na2Cr2O7/H2SO4
(c) Reduction of ketones to secondary alcohols (iii) Chromic anhydride
(d) Oxidation of phenol to benzoquinone (iv) NaBH4
A) (a)-(iii), (b)-(iv), (c)-(i), (d)-(ii)
B) (a)-(iii), (b)-(i), (c)-(iv), (d)-(ii)
C) (a)-(iii), (b)-(ii), (c)-(i), (d)-(iv)
D) (a)-(i), (b)-(iv), (c)-(ii), (d)-(iii)
E) (a)-(iv), (b)-(i), (c)-(iii), (d)-(ii)
Correct Answer : Option B
In aldol condensation reaction, 1, 3-diphenylprop-2-en-1--one is obtained as major
146. product in the reaction between
A) benzaldehyde and benzophenone
B) benzaldehyde and acetophenone
C) propan-2-one and benzaldehyde
D) acetophenone and propan-2-one
E) benzaldehye and butan-2-one
Correct Answer : Option B
Which of the following is NOT correctly matched for the compound and use mentioned
147. below?
A) Esters of Benzoic acid - Perfumery
B) Hexanedioic acid - Detergents
C) Ethanoic acid - Food industry
Text of page 37
D) Methanoic acid - Electroplating
E) Sodium benzoate - Food preservative
Correct Answer : Option B
An aromatic compound ‘A’ on treatment with ethanolic NaCN forms compound ‘B’ which
148. on reduction with H2/Ni gives a compound ‘C’ of molecular formula, C8H11N. The
compounds A, B and C are
A) chlorobenzene, phenylnitrile and 2-phenyl methanamine
B) chlorobenzene, phenylnitrile and 2-phenyl ethanamine
C) chlorobenzene, phenylethane nitrile and 2-phenyl methanamine
D) chlorophenyl methane, phenylethane nitrile and 2-phenyl ethanamine
E) dichlorobenzene, phenylnitrile and 2-phenyl methanamine
Correct Answer : Option D
149. Which of the following amine has lowest pKb value in aqueous phase?
A) Ethanamine
B) Methanamine
C) N-Methylmethanamine
D) N-Ethylethanamine
E) N, N-Diehtylmethanamine
Correct Answer : Option D
The linkage formed between 5’ and 3’ carbon atoms of the pentose sugar in nucleotides
150. is
A) glycosidic
B) phosphomonoester
C) amide
D) peptide
E) phosphodiester
Correct Answer : Option E
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Source: Commissioner for Entrance Examinations. Question paper published by Commissioner for Entrance Examinations.