KCET EXAMINATION – 2022 SUBJECT : MATHEMATICS (VERSION – D3)

[Pages:8]KCET EXAMINATION ? 2022 SUBJECT : MATHEMATICS (VERSION ? D3)

DATE :- 16-06-2022

TIME : 02.30 PM TO 03.50 PM

1. The octant in which the point (2, -4, -7)

1) Eight

2) Third

3) Fourth

4) Fifth

Ans. 1

Sol. Conceptual

x2 1, 0 x 2

2. If f x

the quadratic

2x 3, 2 x 3

equation whose roots are lim f(x) and x 2

lim f(x) is

x 2

1) x2 14x 49 0

2) x2 10x 21 0

3) x2 6x 9 0

4) x2 7x 8 0

Ans. 2

Sol. lim f x lim x2 1 3

x 2

x 2

lim f x lim 2x 3 7

x 2

x 2

x2 x 0

x2 10x 21 0

3. If 3x i 4x y 6 i where x and y are real

numbers, then the values of x and y respectively, 1) 3, 9 2) 2, 4 3) 2, 9 4) 3, 4 Ans. 3 Sol. 3x 6

x2

4x y 1

8 y 1

9 y

4. If all permutations of the letters of the word MASK are arranged in the order as in dictionary without meaning, which one of the following is 19th word 1) KAMS 2) SAMK 3) AKMS 4) AMSK

Ans. NO OPTION

Sol. Original answer SAKM A K M S A 3! K 3!

M 3!

1 SAKM

19

5. If a1,a2,a3,.....a10 is a geometric progression

and a3 25 , then a9 equals

a1

a 5

1) 3 52 2) 54

3) 53

4) 2 52

Ans. 2

Sol. a3 25 a1

ar2 25

a r2 52

a4 ar8 r4 54 a5 ar4

6. If the straight line 2x-3y+17=0 is perpendicular to the line passing through the

point 7,17 and 15, , then equals

1) -5

2) 5

Ans. 2

Sol. m1 m2 1

2 17

1

3 15 7

17 12

3) 29

4) -29

5

7. Let the relation R is defined in N by aRb, if 3a+2b=27 then R is

1) 1,12 3,9 5,6 7,3

2)

0,

27 2

1,12

3,

9

5,

6

7,

3

3) 1,12 3,9 5,6 7,3 9,0

4) 2,1 9,3 6,5 3,7

Ans. 1 Sol. 2b 27 3a

27 3a b

2

R 1,2,3,9,5,6,7,3

3 y3 3

8. lim

y0

y3

1 1)

23

1 2)

32

3) 2 3

Ans. 1

3 y3 3

1

Sol. lim

y0 y 3 y3 3 2 3

4) 3 2

9. If the standard deviation of the numbers 1,0,1,k is 5 where k>0, then k is equal to

5 1) 4

3

2) 6

10 3) 2

3

Ans. 4

Sol.

2 5 , x k 4

1 1 0 1 k2

k2 5

4

16

k2 2 k2 5

4 16

4k2 8 k2 5 3k2 8 80 16

3k2 72

k2 24

4) 2 6

k 24 2 6

10. If the set X contains 7 elements set y contains 8 elements, then the number of bijections from X to Y is

1) 0 Ans. 1

2) 8P7

3) 7!

4) 8!

Sol. n A n B

Number of bijections is zero

2x : x 3

11.

If

f:RR be

defined

by

f

x

x

2

:1

x

3

3 x : x 1

then f 1 f 2 f 4 is

1) 5

2) 10

3) 9

Ans. 3

Sol. f 1 3 1 3

4) 14

f 2 22 4

f 4 24 8

f 1 f 2 f 4 3 4 8 =9

12.

0 If A 0

1 0

then

aI bAn

is (where 1 is the

identify matrix of order 2)

1) a2I an1b. A

2) anI n.an1b.A

3) anI n anbA

4) anI bnA

Ans. 2

0 1 Sol. A 0 0

aI

bA 1

a 0

0 0

a

0

b a 0 0

b

a

aI

IA2

a 0

b a

a

0

b a2

a

0

2ab

a2

aI bA3

a2

0

2ab a

a2

0

b a3

a

0

3a 2 b

a3

aI

bAn

a n

0

na n 1b

an

anI

n.a n 1bA

13. If A is a 3 3 matrix such that 5.adjA 5

then A is equal to

1) 1

2) 1/ 25 3) 1/5 4) 5

Ans. 3

Sol. A33 matrix 5. AdjA 5

53

2

A

5

A2

1

52

1 A

5

14. If there are two value of `a' which makes 1 2 5

determinant 2 a 1 86 . 0 4 2a

Then the sum of these numbers is

1) -4

2) 9

3) 4

4) 5

Ans. 1

Sol. 1 2a2 4 2 4a 0 5 8 86

2a2 8a 44 86 0

2a2 8a 42 0

a2 4a 21 0

b

Sum

of

numbers=-4

a

15. If the vertices of a triangle are 2, 6 3, 6

and 1,5 , then the area of the triangle is

1) 40 sq.units

2) 15.5 sq.units

3) 30 sq.units

4) 35 sq.units

Ans. 2

1 2 3 2 1 1 5 3

Sol.

2 6 6 6 5 5 12 1

1

31

5 36 15.5

2

2

16. Domain of cos1 x is, where [.] denotes a

greatest integer function

1) (1,2] 2) 1,2 3) 1,2 4) [1,2)

Ans. 4

Sol. cos1 x 1 x 1 [x]= {-1, 0, 1}

x [1, 2)

17. If A is a matrix of order 3 3 , then A2 1 is

equal to

1) A2 2 2) A1 2 3) A2

4) A 2

Ans. 2

Sol.

A2

1

A1 2

2 1 18. If A 3 2 , then the inverse of the matrix

A3 is

1) A

2) -I

3) I

4) -A

Ans. 1

2 1 Sol. A 3 2

A 1

1 2 1 3

1 2 2 3

1 2 A

A2

2 3

1 2 2 3

1 2

4 3 2 2 1 0 6 6 3 4 0 1 I

A3 A

19. If A is a skew symmetric matrix, then A2021 is

1) Row matrix

2) Column matrix

3) Symmetric matrix

4) Skew symmetric matrix

Ans. 4

Sol. AT A or An is slow symmetric if n is odd

P A2021

PT A2021 T AT 2021

2021

A P

20. If f 1 1,f ' 1 3 then the derivative of

f

f

f

x

f

x

2

at

x

1 is

1) 10

2) 33

Ans. 2

Sol. f 1 1, f '1 3

3) 35

4) 12

d dx

f

f

f

x

f

x

2

f 'f f x.f 'f x.f 'x 2f x.f 'x

f 'f f 1 f 'f 1.f '1 2f 1.f '1

f 'f 1 f '1.3 2.1 3

f '1.3.3 6

= 27+6 = 33

21.

If

y xsinx

sin xx then

dy at x

is

dx

2

4 1)

2) log 3) 1

2

2 4)

2

Ans. 3

Sol. y xsinx sin x x

dy dx

xsin x

sin x x

cos x.log

x

sin x x x cos x log sin x

x

2

2

2

10

0

=1

1 n n

22.

If

An

n

1 n then

A1 A2 ..... A2021

1) -2021

2) 20212

3) 20212

4) 4042

Ans. 2

1 n n

Sol.

An

n

1 n

An 1 n2 n2

1 n2 2n n2

 23. If y 1 x2 tan1 x x then dy is dx

1) 2x tan1 x

tan1 x 2)

x

3) x2 tan1 x

4) x tan1 x

Ans. 1

Sol. y 1 x2 tan1 x x

dy 1 x2

tan1 x.2x 1

dx 1 x2

2x tan1 x

24. If x e sin , y e cos where is a

parameter, then

dy at

1,1 is equal to

dx

1

1

1

1) 0

2)

3)

4)

2

2

4

Ans. 1 Sol. x e sin 1

y e cos 1 ,

x

tan 1

y

4

dy dy / d e sin cos .e cos sin

dx dx / d e cos sin e cos sin

tan

4

0

25.

If

ye

x

x

, x x .......

1

d2y then

dx 2

at x

log

3 e

is

1) 3

2) 5

3) 0

Ans. 1

Sol.

11 1

111

y e e x x x.....

x 2 x 4 .x 8 ......ex2 4 8

4) 1

e e e e 1 1 1

x

2

1

2

4

....

1

x2

x1

x

dy ex dx

d2y dx 2

ex x

log

3 e

elog3e

26. If x is the greatest integer function not

8

greater than x then x dx is equal to

0

1) 28

2) 30

3) 29

4) 20

Ans. 1

8

Sol. [x]dx 1 2 3 .... 7

0

7(7 1)

28

2

2

27. sin cos3 d is equal to

0

8 1)

23

7 2)

23

8 3)

21

Ans. 3

Sol. Put sin t

1 t1/2(1 t2 )dt

8

21

0

7 4)

21

28. If

ey xy e the

ordered

dy d2y

dx

,

dx2

at

x

0

is

equal

to

1 1

1)

e

,

e2

1 1

2)

e

, e2

1 1

3)

e

,

e2

Ans. 4

Sol. x 0 y 1

1 1

4)

e

, e2

dy y

dx ey x

dy

1

dx

(0,1)

e

d2y

1

dx

2

(0,1)

e2

pair

29. The

function

f x log 1 x 2x is

2x

increasing on

1) , 2) , 1 3) 1, 4) ,0

Ans. 3

x2

Sol. f '(x)

>0

(x 1)(2 x)2

x 1 0 x 1

30. The co-ordinates of the point on the

x y 6 at which the tangent is equally

inclined to the axes is

1) 4,4 2) 1,1 3) 9,9 4) 6,6

Ans. 3

dy y

Sol.

1

dx x

yx

x x 6 x 9, y 9

31. The function

f x 4sin3 x 6sin2 x 12sin x 100 is

strictly

1) decreasing in 2 , 2

2) decreasing in 0, 2

3)

increasing

in

3

,

2

4)

decreasing

in

2

,

Ans. 4

Sol. f '(x) (12 sin2 x 12 sin x 12)cos x

f '(x) 12(sin2 x sin x 1)cos x

sin2 x sin x 1 0

x

2

,

cos x 0

32. Area of the region bounded by the curve

y tan x, the x axis and the line x is

3

1

1) log

2) log 2 3) 0

2

4) log 2

Ans. 2

/3

/3

Sol.

A

tan x dx log sec x 0

log 2

0

3

33. Evaluate x2dx as the limit of a sum

2

72 1)

6

53 2)

9

25 3)

7

19 4)

3

Ans. 4

x3

3

1

19

Sol. I (27 8)

3 2 3

3

2 cos x sin x

34. 1 sin x dx is equal to 0

1) log 2 1

2) log 2

3) log 2

4) 1 log 2

Ans. 4

Sol. Sinx=t

2 cos x sin x

1

t

0

1 sin x

dx

0

1

dt t

1

log

2

cos 2x cos 2

35. cos x cos dx is equal to

1) 2sin x x cos c 2) 2sin x x cos c 3) 2sin x 2x cos c 4) 2sin x 2x cos c

Ans. 2 cos2 x cos2

Sol. 2 cos x cos dx 2 (cos x cos )dx

2sin x x cos

1 xex

36.

0

2

x

3

dx

is

equal

to

11 1) .e

27 8

11 2) .e

27 8

11 3) .e

94

11 4) .e

94

Ans. 4

1

Sol. ex

1

2 dx

2

3

0 x 2 x 2

1

ex e 1

x 22 0 9 4

dx

37. If x 2 x2 1

a log 1 x2 b tan1 x 1 log x 2 c, then 5

1 2 1) a , b

10 5

1

2

2) a , b

10 5

1 2

3) a , b

10

5

1

2

4) a , b

10

5

Ans. 1

1

A Bx C

Sol.

x 2 x2 1 x 2 x2 1

1 1 2

A ,B ,C

5

5

5

38. If a 2 and b 3 and the angle between a

and b is 1200 , then the length of the vector

2

1a 1b

is

23

1

1) 2

2) 3

3)

4) 1

6

ORIGINAL QUESTION

38. If a 2 and b 3 and the angle between a

and b is 1200 , then the length of the vector

2 ab

is 23

Ans. 2

2

ab

2

a

2

b

ab

Sol. 2 . = 3

2 3 4 9 23

2

39. If a b a.b 36 and a 3 then b is

equal to

1) 9

2) 36

3) 4

4) 2

ORIGINAL QUESTION

2 2

39. If a b a.b 36 and a 3 then b is

equal to

Ans. 4

2

2

Sol. a b a b 36

2

a

2

b

36

b2

4,

b

2

40.

If

^i 3^j ,

^i 2^j k^

then

express

in

the

from 1 2 where 1 is parallel to and 2

is perpendicular to then 1 is given by

1) 5 (^i 3^j) 8

3) ^i 3^j

2) 5 (^i 3^j) 8

4) ^i 3^j

Ans. NO OPTION

Sol.

Correct

answer

is

1

^i 3^j

2

1 2

2 .

2 . 0

41. Then sum of the degree and order of the

differential equation (1 y12 )2/3 y2 is

1) 4

2) 6

3) 5

4) 7

Ans. 3

Sol.

1 y12

3

y2

235

42. If dy y x2 then 2y(2) y(1) dx x

11 1)

4

15 2)

4

9 3)

4

Ans. 2

Sol.

x4 y.x C

4

2y 2 y 1 15

4

13 4)

4

43. The solution of the differential equation

dy (x y)2 is dx

1) tan1(x y) x c

2) tan1(x y) 0

3) cot1(x y) c

4) cot1(x y) x c

Ans. 1

Sol. x y z dz 1 z2 dx

1

1

dz z2

1dx

Tan1 x y x c

44. If y(x) be the solution of differential equation

dy x log x y 2x log x , y(e) is equal to

dx

1) e

2) 0

3) 2

4) 2e

Ans. 4

Sol. I.F= logx,

y log x 2x log x 1 c

If x =e then y=c then y(e)=2e

45. A dietician has to develop a special diet using two foods X and Y. Each packet (containing 30g) of food. X contains 12 units of calcium, 4 units of iron, 6 units of cholesterol and 6 units of vitamin A. Each packet of the same quantity of food Y contains 3 units of calcium, 20 units of iron, 4 units of cholesterol and 3 units of vitamin A. The diet requires atleast 240 units of calcium, atleast 460 units of iron and most 300 units of cholesterol. The corner points of the feasible region are 1) (2, 72), (40, 15), (15, 20) 2) (2, 72), (15, 20), (0, 23) 3) (0, 23), (40, 15), (2, 72) 4) (2, 72), (40, 15), (115, 0)

Ans. 1 Sol.

46. The distance of the point position vector is (2^i ^j k^ ) from the plane r.(^i 2^j 4k^ ) 4 is

8 1)

21

8 2) 8 21 3)

21

Ans. 1

2 2 4 4 8

Sol. Dis tan ce

1 4 16 21

8 4)

21

47. The co-ordinate of foot of the perpendicular drawn from the origin to the plane 2x 3y 4z 29 are

1) (2, 3, 4)

2) (2, -3, -4)

3) (2, -3, 4)

4) (-2, -3, 4)

Ans. 3

Sol. verification (2, -3, 4)

48. The angle between the pair of lines

x 3 y 1 z 3

x 1 y 4 z 5

and

is

3

5

4

1

4

2

1)

cos1

27 5

2)

cos1

8

3

15

3)

cos1

19 21

Ans. NO OPTION Sol. Original answer

4)

cos1

5

3

16

31 5 4 4 2

31

cos

32 52 42 12 42 22 5 42

cos1 31 5 42

49. The corner points of the feasible region of an LPP are (0, 2), (3, 0), (6, 0), (6, 8) and (0, 5) then the minimum value of z 4x 6y occurs at

1) finite number of points 2) infinite number of points 3) only one point 4) only two points Ans. 4 Sol. At (0,2), (3, 0), z=12

Hence minimum at 2 points.

50. If A and B are two independent events such

that P A 0.75, P A B 0.65, and

P B x , then find the value of x

5 1)

14

8 2)

15

9 3)

14

Ans. 2

1

3

13

Sol. P(A) , P(B) , P(A B)

4

4

20

1

1 13

x .x

4

4 20

7 4)

15

3 13 5 8 x

4 20 20 20

84 8 x

20 3 15

51. Find the mean number of heads in three tosses

of a fair coin

1) 1.5

2) 4.5

3) 2.5

4) 3.5

Ans. 1

Sol.

X

0

1

2

3

P(x)

1

3

3

1

8

8

8

8

3 6 3 12 3 Mean 1.5

888 8 2

52. If A and B are two events such that

P A 1 ,P B 1

2

3

and

A 1

P

B

4

,

then

P A ' B' is

1 1)

4

3 2)

16

1 3)

12

3 4)

4

Ans. 1

1

1

11 1

Sol. P(A) , P(B) , P(A B)

2

3

4 3 12

P(A B) P(A B)

P(A B) 1 P(A B)

1 1 1 P(A B) 1 2 3 12

6 4 1 P(A B) 1 12

9 P(A B) 1

12

1 P(A B)

4

53. A pandemic has been spreading all over the world. The probabilities are 0.7 that there will be a lockdown, 0.8 that the pandemic is controlled in one month if there is a lockdown and 0.3 that it is controlled in one month if there is no lockdown. The probability that the pandemic will be controlled in one month is 1) 0.65 2) 1.65 3) 1.46 4) 0.46

Ans. 1 Sol. P(E1 ) probability of there is lockdown =0.7

P(E2 ) probability of there is lockdown=0.3 A is the event controlled in one month

P A / E1 0.8 , P A / E2 0.3

P(A) 0.7(0.8) (0.3)(0.3)

0.56 0.09 0.65

 54. The degree measure of is equal to

32

1) 50 30 ' 20 ''

2) 50 37 '20 ''

3) 50 37 '30 ''

4) 40 30 ' 30 ''

Ans. 3

Sol.

1800

5037 ' 30 ''

32 32

5 55. The value of sin sin is

12 12

1

1) 0

2) 1

3)

2

Ans. 4

5

Sol. sin . sin

12

12

1 sin

26

11 1

22 4

1 4)

4

56. 2 2 2 2 cos 8

1) sin 2

2) 2 cos

3) 2sin

Ans. 2

Sol.

1

cos

2

cos2

2

4) 2 cos

2

2 2 2 2 cos 8 2 cos

59. The domain of the

1

f x

x 2 is

log10 1 x

function

1) [2,0) 0,1

2) [2,1)

3) [2,1)

4) [2,0) 0,1

Ans. 4

Sol. 1-x>0, 1-x 1

x 1 0 x 0 x 2 0

x 1

x 2

x [2,0) 0,1

60. The trigonometric function y tan x in the II quadrant 1) Decreases form 0 to 2) Decreases form to 0 3) Increases from 0 to 4) Increases from to 0

Ans. 4 Sol. By graph

57. If A 1,2,3,.....10 then the number of

subsets of A containing only odd numbers is

1) 31

2) 27

3) 32

4) 30

Ans. 3

Sol. Odd number 1,3,5,7,9

No. of sub sets 25 32

58. Suppose that the number of elements in set A is p, the number of elements in set B is q and the number of elements in A B is 7 then p2 q2 ______

1) 50

2) 51

Ans. 1

Sol. n(A) p , n(B) q

3) 42

4) 49

n(A B) 7

pq 7

p2 q2 72 12 or 12 72

p2 q2 50

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