KEAM 2027 Sample Paper (Paper 2 - Mathematics)

SAMPLE PAPER

S A M P L E Q U E S T I O N P A P E R

KEAM 2027 (Paper 2 - Mathematics)
Modelled on the actual exam pattern
.

QUESTIONS MAX MARKS TIME MARKING

120 480 150 Min +4 / −1

GENERAL INSTRUCTIONS

1. This paper contains 120 multiple-choice questions. All questions are compulsory.

2. Each question has four options — (A), (B), (C) and (D) — of which only one is correct.

3. Each correct answer carries 4 marks; 1 mark is deducted for each wrong answer.

4. Total time allowed is 150 minutes. Manage your time across all sections.

5. Use of calculators, mobile phones or any electronic device is not permitted.

6. Attempt the paper first, then check your answers against the Answer Key at the end.

Candidate Name: Roll No.: Date:

SAMPLE PAPER

KEAM 2027 (Paper 2 - Mathematics)
SAMPLE QUESTION PAPER

Questions 120 Max Marks 480 Time 150 Min Marking +4 / −1

Attempt all questions. Choose the one correct option for each.

MATHEMATICS

1. 2.

The principal argument of the complex number
If then a

unit vector in the direction of is is equal to

(A) (A)

(B)
(B)

(C)

(C)

(D)

(D)

3. 4.

The number of subsets containing exactly 4 elements of
the set { 2, 4, 6, 8, 10, 12, 14, 16, 18} is equal to

(A)

126 (A)

(B)

63

(B)
(C)

189

(D)
(C)
58
1

(D)

0

5. 6.

The three vertices of a triangle are (0, 0), (3, 1) and (1,
The set of all x satisfying the inequality is
3). If this triangle is inscribed in a circle, then the
equation of the circle is
(A)

(A)

(B) (B)

(C)
(C)

(D)

(D)

7. 8.

Consider the linear programming problem:

The value of is equal to

(A)

(B) Then the coordinates of the corner points of the feasible
region are

(A)

(0, 0), (30, 0), (0, 40) and (15, 30)
(C)
(B)

(0, 0), (60, 0), (0, 40) and (15, 30)

(C)
(D)
(0, 0), (30, 0), (0, 60) and (15, 30)

(D)

(0, 0), (30, 0), (0, 40) and (30, 40)

9. 10.

Let L be the line joining (0, 0, 0) and (1, 2, 3) and L be The centre of the ellipse x2 +7y2 —14x-+26y +49 = 0 in
1 2
the line joining (2, 3, 4) and (3, 4, 5). The point of
intersection of L and L is (A)
1 2

(7,0)
(A)

(0, 0, 0) (B)

(7, 4)
(B)

(1, 2, 3) (C)

(7, -2)
(C)

(2, 3, 4) (D)

(-7, 4)
(D)

(3, 4, 5)

11. 12.

Let be a binary operation on Q- {0} defined by a The intercepts of a line with coordinate axes are equal. If
the line passes through (2, 3), then its equation is
b= . Then 1 (2 (3 4)) is equal to

(A)
(A)
2x+3y=5
3/2

(B)
(B)
x+y=5
4/3

(C)
(C)
5x+5y=1
3/4

(D)
(D)
x+y=6
3/8

13. 14.

If then the value of |z| is equal to
is equal to

(A)

(A)
48

5
(B)

(B)
49

25
(C)

(C)
50

1/5
(D)

(D)
51

0

15. 16.

If the radius of the circle x2 +y2 +ax + by + 3 = 0 is 2,
then the point (a, 5) lies on the circle
Let and let Then |B
(A)
T| =

x2 +y2 = 7
(A)
(B)
1/14
x2 +y2 =14
(B)
(C)
14
x2 +y2 = 28
(C)
(D)
10
x2 +y2 = 1
(D)

9

17. 18.

The equation of the plane through the point (1, -5, 3) and

The value of is equal to having a normal vector is

(A)
(A)
2x+2y+z=9
(1/2)25

(B)
(B)
2x-2y-z=11
(1/2)50

(C)
(C)
2x+2y-z=9
(1/2)65

(D)
(D)
2x-2y-z=9
(1/2)60

19. 20.

Consider the following statements : The sum of the first 24 terms of the series 9+13+17+--- is
equal to
(i) For every positive real number x , x—10 is positive.

(ii) Let n be a natural number. If n2 is even, then n is (A)

even.
1212

(iii) If a natural number is odd, then its square is also
(B)
odd.

1200
Then

(C)
(A)
1440
(i) False, (ii) True and (iii) True

(D)
(B)
1320
(i) False, (ii) False and (iii) True

(C)

(i) True, (ii) True and (iii) True

(D)

(i) False, (ii) True and (iii) False

21. 22.

Let f: R---->R be defined by f(x) = cos x. Then

If , then (A)

f is one - one and odd
(A)

(B)

f is odd but not one - one

(C)
(B)
f is one - one and even

(D)

f is even but not onto

(C)

(D)

23. 24.

If the line y=mx+c is perpendicular to y=1+x and passes
through the point (1, 2), then the value of c is equal to
is equal to

(A)
(A)
1
10
(B)
(B)
-1
-10
(C)
(C)
-3
7
(D)
(D)
3
-7

25. 26.

If A and Bare two events such that P(A)=0.5, P(B)=0.4

then P(A|(AUB)) is equal to
is equal to

(A)

(A)
6/7

(B)

5/7

(B) (C)

8/7

(D)

1/7
(C)

(D)

27. 28.

The equation of the circle touching the x-axis at (5, 0)
Let and
and the line y = 10 is

(A) then the

value of is equal to

(A)

(B)
3

(B)

-3
(C)

(C)

6

(D) (D)

-6

29. 30.

The distance from the point (2, 2, 2) to the plane 2x - y +
3z = 5 is equal to
The value of is equal to

(A)
(A)

18/25

(B)
(B)
24/25

(C)

7/24

(C)
(D)

3/4

(D)

31. 32.

The coefficient of variation (C.V.) and the mean of a
distribution are respectively 75 and 44. Then the
standard deviation of the distribution is

(A)

30
(A)
(B)

31

(C)

(B) 33

(D)

35

(C)

(D)

33. 34.

The equation of the plane passing through the point (-1,- he domain of the function
2,-3) and perpendicular to the x-axis is
is

(A)
(A)
x=-1

(B)

y =-2
(B)

(C)

x = -3

(C)
(D)

x =-4

(D)

35. 36.

The equation of tangent to the circle (x—5)2 +y2 =25 at
(2, 4) is The angle between the planes is

equal to
(A)

(A)
3x-4y4+10=0

(B)

x+y=6

(B)
(C)

2x-y=0

(D)
(C)
3x-2y+2=0

(D)

0

37. 38.

The eccentricity of the hyperbola
If is angle between the lines

then is equal to

(A)
(A)
1/2
5/9
(B)
(B)
3/2
5/8
(C)
(C)
2/3
5/7
(D)
(D)
3/5
5/6

39. 40.

The number of numbers greater than 6000 that can be
formed from the digits 3, 5, 6, 7 and 9 (no digit is
The modulus of is repeated in a number) is equal to

(A)
(A)

264
1

(B)
(B)

720
2

(C)
(C)

192
4

(D)
(D)

132
16

41. 42.

If is equal to
The value of is equal to

(A)

(A)
5

-1/2
(B)

(B)
7

-3/2
(C)

(C)
-5

-5/2
(D)

(D)
-7

-7/2

43. 44.

Let Then the value of c

which satisfies the conclusion of the Mean Value Theorem

(A) for the function f on is equal to

(A)

(B)

(B)

(C)
(C)

(D) (D)

45. 46.

Air is blown into a spherical balloon. If its diameter d is
increasing at the rate of 3 cm/min, then the rate at which If then dy/dx is
the volume of the balloon is increasing when d= 10cm, is
(A)
(A)

(B)
(B)

(C)

(C)

(D)

(D)

47. 48.

If and then dy/dx is The 11th term of the geometric series in

equal to
equal to

(A)
(A)

xy
-4090

(B)
(B)

-x/y
1024

(C)
(C)

-y/x
2048

(D)
(D)

x/y
2050

49. 50.

The feasible region for a L.P.P. is shown in the figure
below, Let z = 50x +15y be the objective function, then
the maximum value of z is

Let If f(x) is

differentiable at then the values of a and b are

respectively

(A)

(A)

900
(B)

(B)

1000

(C)
(C)

1250

(D)

(D)
1300

51. 52.

If the lines 2x-3y+5=0, 9x-Sy+14=0 and 3x-7y+ =0 are
concurrent, then the value of is equal to
Let for n=1, 2, 3, «-:. Then t is
10

equal to (A)

(A) 7

7/600 (B)

(B) 8

231/100 (C)

(C) 10

209/600 (D)

(D) 9

11/200

53. 54.

The equation of the straight line parallel to y = -3x and
The normal to the curve at the point (25, 5) passing through the point (3, -2) is

intersects the y-axis at
(A)

(A) y= -3x + 7

(0, 245)
(B)

(B) y=-3x+9

(0, 255)
(C)

(C) y =-3x-11

(255, 0)
(D)

(D) y=-3x-7

(245, 0)

55. 56.

If -1+7i, -1+xi and 3 + 3i are the three vertices of an
if and , then
isosceles triangle which is right angled at —1+.xi, then
the value of x is equal to

(A)
(A)
-1
530
(B)
(B)
3
580
(C)
(C)
-3
630
(D)
(D)
7
680

57. 58.

In an AP. there are 18 terms and the last three terms of
If then the angle the A.P. are 67, 72, 77. Then the first term of the A.P. is

between is equal to (A)

(A) -7

(B)

-8

(B) (C)

-9

(D)

(C) 9

(D)

59. 60.

If x and y are both non-negative and if x+y= then the
maximum value of 5sinxsin y is equal to
If then the value of

(A)

1

is equal to (B)

5
(A)
(C)
2
-5
(B)
(D)
-2
0
(C)

-3

(D)

3

61. 62.

The range of the function f(x) = 2sin(3x) +1 is equal to
If are position vectors of the points ( , 3, 0)
and (1, 0, 0) respectively - 2 and if the angle between the
(A)

[-1 1]
vectors then the value of is equal

(B) to

[-2 1]
(A)

(C) 1

[-1 -2]
(B)

(D) 2

[-1 3]
(C)

3

(D)

4

63. 64.

If x sin y+ y sinx = then dy/dx at is equal

to
The value of is equal to

(A)

(A) 0

1 (B)

(B) 1

2 (C)

(C) -1

0 (D)

(D) -2

3

65. 66.

(A) If then the value

-72x-7

(B) of is equal to

72x-7
(A)
(C)
3
12x-7
(B)
(D)
-3
82x-7
(C)

9

(D)

-9

67. 68.

If x+13y=40 is normal to the curve at The solution set of the inequality is

the point (1, 3), then the value of is equal to
(A)

(A)

15
(B)
(B)

-15

(C)
(C)

6

(D) (D)

-6

69. 70.

If is a finite number, then the

value of a is equal to

(A) (A)

2 1

(B) (B)

3 2

(C) (C)

4 4

(D) (D)

5 0

71. 72.

If are two acute angles of a right triangle,

then

(A)
(A)

(B)
(B)

(C)
(C)

(D)

(D)

73. 74.

The equation of the line through the point (1, -1, 1) and
If of 7 is equal to parallel to the line joining the points (-2, 2, 0) and (-1, 1,
1) is
(A)
(A)
49
1-x=1+y=1-z
(B)
(B)
50
1-x=2+y=1-z
(C)
(C)
45
1-x=1+y=2-z
(D)
(D)
46
1-x=1+y=3-z

75. 76.

The values of x satisfying the equation

The value of is equal to
are

(A)
(A)

2, -4

(B)
(B)

1, 2

(C)

-1, 2
(C)

(D)

-2, 4

(D)

77. 78.

The points of intersection of the line y=x+2 and the circle If the solution set of the inequality
(x-2)2 + y2 = ——

(A) then the value of a is equal to

(2, 0) (2, 4)
(A)

(B)
-1

(-2, 4), (2, 0)
(B)

(C)
-2

(4, 0), (4, 2)
(C)

(D)
4

(4, 6), (4, -2)
(D)

-4

79. 80.

If Z and Z are two different complex numbers with |Z |
1 2 1

=1,then = is equal to
If A=[2 0 6] and then AB =

(A)
(A)
0
[ 42 46]
(B)
(B)
1/2
[ 17 19]
(C)
(C)
1/4
[13 12]
(D)
(D)
1
0

81. 82.

If n (AU B)=97, and n(A-B) =39, then
n(B) is equal to The solutions of the equation in

the interval are
(A)

52 (A)

(B)

55

(B)
(C)

58

(D)

65 (C)

(D)

83. 84.

There are 37 men and 33 women at a party. If a prize is Let S be the sum of the first n terms of the series a +a
n 1 2
given to one person chosen at random, then the +....+a +.... If S =n +4n, then the nth term a , is
n n 2 n
probability that the prize goes to a woman is
(A)
(A)
2n + 3
33/70
(B)
(B)
2n - 1
32/70
(C)
(C)
n -1
31/70
(D)
(D)
0
30/70

85. 86.

Which of the following statements is not correct?

The period of the function is (A)

equal to
log 10 = 1
10

(A)
(B)

2
log (2 + 3) = log (2 x 3)

(B)
(C)

3
log 1=0
10

(C)
(D)

4
log (1 + 2 + 3) = log 1 + log 2 + log 3

(D)

5

87. 88.

A particular solution of the differential equation dy/dx =
xy2 with y(0)=1 is
If A is non-singular matrix and if

(A)
then adj(A) =

(A)

(B)

(B)

(C)

(C)

(D)

(D)

89. 90.

There are 4 red, 3 blue and 3 yellow marbles in an urn. If If (-1, 0) and (3, 0) are foci of an ellipse and the length of
three marbles are drawn simultaneously, then the the major axis is 6, then the length of the minor axis is
probability that the number of yellow marbles will be less
than 2 is equal to (A)

(A)

97/120

(B)
(B)
5
49/60

(C)
(C)
10
50/80

(D)
(D)

45/57

91. 92.

The function is increasing in the interval
In the binomial expansion of the coefficient

of x6y6 is equal to
(A)

(A)

-672

(B)
(B)

672

(C) (C)

336

(D)
(D)

-336

93. 94.

f n term of a series is n+ (-1)n-1 ,n=1,2,3,..., then the sum
of first 40 terms of the series is
Let where and

(A) c is a constant. The values of c for which f is continuous
on R are
810

(B) (A)

820 -7,1

(C) (B)

830 -1, 7

(D) (C)

840 1, 7

(D)

1,1

95. 96.

If the points (1, 0, 0), (0, 3, 0) and (0, 0, 2) lie on a plane,

then the unit normal vector to the plane is

Let ,Then (A)

(A)

f (x) is not continuous at x = -1
(B)
(B)

f(x) is continuous at x =1

(C)

f(x) isnot continuous at x = 5 (C)

(D)

f(x) is not continuous at x = 2

(D)

97. 98.

The end points of the major axis of an ellipse are (2,4)
and (2, -8). If the distance between foci of this ellipse is
4, then the equation of the ellipse is
Let and let B=|A| adj (A). Then |B| =
(A)

(A)

256

(B)
(B)

2345

(C)

1024

(C)
(D)

1023

(D)

99. 100.

he minimum value of |z+1| +|z-2| is equal to
If then the value of sin @
(A)
is equal to
1
(A)
(B)

2

(C)

(B)
3

(D)

4

(C)

(D)

101. 102.

Let A(-1, 2), B(1, 3) and C(a, b) be collinear. If B divides The general solution of the differential equation
AC such that BC =8 AB, then the coordinates of C are
is

(A)
(A)
(11 , 15)

(B)

(17, 11)
(B)
(C)

(10, 11)

(D)
(C)
(5, 6)

(D)

103. 104.

If the first term of a GP. is 3 and the sum of second and Three fair dice are rolled simultaneously. Let a, b, ¢ be
third terms is 60, then the common ratio of the GP. is the numbers on the to dice. Then the probability that
min(a, b, c) =6 is
(A)
(A)
4 only
1/216
(B)
(B)
4 or -5
1/215
(C)
(C)
-5
1/214
(D)
(D)
5 only
1/123

105. 106.

Let A={1,2,3,4,5} and let B={1,2,3,4}. If the relation
if then the
R:4—B is given by if and only if a+b is even,

value of is equal to then n(R) is equal to

(A)
(A)

10
1

(B)
(B)

16
2

(C)
(C)

20
3

(D)
(D)

12
0

107. 108.

A fair coin is tossed twice. Given that the first toss
resulted in head, then the probability that the second
toss also, would result in head is

(A)
(A)

1/8

(B)

1/4
(B)
(C)

1/2

(D)
(C)

0

(D)

109. 110.

If the line 2x-3y +c =0 passes through the focus of the In a box there are four marbles and each of them is
parabola x2 = — 8y, then the value of c is equal to marked with distinct number from the set a {1, 2, 5, 10}.
If one marble is randomly selected four times with
(A) replacement and the number on it noted, then the
probability that the sum of number equal 18 is
4

(A)
(B)

1/64
-6

(B)
(C)

3/16
5

(C)
(D)

3/32
-4

(D)

1/32

111. 112.

The integrating factor of the differential equation

is

The value of is equal to
(A)

e4y
(A)

(B) 0

e3y
(B)

(C) 1/100

e
(C)

(D) 1

e2y
(D)

1/7

113. 114.

The number of arrangements containing all the seven
if is
letter of the word ALRIGHT that begins with LG is

angle between and then
(A)

720 (A)

(B) 1

120 (B)

(C) 2

600 (C)

(D) 3

250 (D)

0

115. 116.

Let and let

Then is equal to (A)

-cosx+C
(A)

{1, 2, 3} (B)

-sinx+C
(B)

{2, 3} (C)

x-cosx+C
(C)

{1, 2, 3, 4} (D)

x-sinx+C
(D)

{0, 1, 2, 3}

117. 118.

If 11P = 7920 and 11C = 330, then the value of p is
r r
equal to
is equal to

(A)

(A)
2

1
(B)

(B)
3

2
(C)

(C)
4

3
(D)

(D)
5

0

119. 120.

The area bounded by the curve y =x ( 2-x) and the line y
Let
= x is
where a , a ,......,a are constants. Then the value of
1 2 10
(A)
is equal to
1/6
(A)
(B)
220
1/3
(B)
(C)
210
1/2
(C)
(D)
310
1/7
(D)

211

ANSWER KEY

1. (C) 2. (B) 3. (A) 4. (D) 5. (D) 6. (C) 7. (D)

8. (A) 9. (B) 10. (C) 11. (D) 12. (B) 13. (C) 14. (C)

15. (B) 16. (C) 17. (B) 18. (D) 19. (A) 20. (D) 21. (A)

22. (D) 23. (C) 24. (B) 25. (C) 26. (B) 27. (A) 28. (D)

29. (C) 30. (D) 31. (C) 32. (C) 33. (A) 34. (D) 35. (A)

36. (C) 37. (C) 38. (B) 39. (B) 40. (C) 41. (A) 42. (D)

43. (B) 44. (A) 45. (B) 46. (C) 47. (C) 48. (C) 49. (A)

50. (C) 51. (B) 52. (C) 53. (B) 54. (A) 55. (B) 56. (C)

57. (B) 58. (B) 59. (C) 60. (B) 61. (D) 62. (C) 63. (A)

64. (C) 65. (A) 66. (A) 67. (B) 68. (C) 69. (D) 70. (D)

71. (B) 72. (D) 73. (B) 74. (A) 75. (D) 76. (A) 77. (A)

78. (D) 79. (D) 80. (A) 81. (C) 82. (C) 83. (A) 84. (A)

85. (B) 86. (B) 87. (B) 88. (B) 89. (B) 90. (D) 91. (A)

92. (D) 93. (B) 94. (B) 95. (B) 96. (D) 97. (C) 98. (A)

99. (D) 100. (C) 101. (B) 102. (C) 103. (B) 104. (A) 105. (D)

106. (A) 107. (B) 108. (C) 109. (B) 110. (C) 111. (A) 112. (A)

113. (B) 114. (D) 115. (A) 116. (A) 117. (D) 118. (C) 119. (A)

120. (C)