Kerala Plus Two Question Paper 2022 Maths Commerce

Reg. No. : .....................................
Name : .........................................
SY-51
SECOND YEAR HIGHER SECONDARY EXAMINATION, MARCH 2022

Part – III
MATHEMATICS (COMMERCE) Time : 2 Hours
Maximum : 60 Scores Cool-off time : 15 Minutes

General Instructions to Candidates :
 15 minutes is given as ‘Cool-off time’.
 Use the ‘Cool-off time’ to read the questions and to plan your answers.
 Read questions carefully before answering.
 Read the instructions carefully.
 Calculations, figures and graphs should be shown in the answer sheet itself.
 Malayalam version of the questions is also provided.
 Electronic devices except non-programmable calculators are not allowed in the
Examination Hall.
 
 15   .
       
.
     .
    .
  , , ,   
.
    .
     
   .

SY-51 1 P.T.O.

PART – I
A. Answer any five questions from 1 to 9. Each carries 1 score. (5  1 = 5)
1. f : x  y is onto if Range of f = _______
(a) x (b) y
(c) R (d) N

2. tan–1x + cot–1x = _______
 
(a) (b)
2 4
(c) 1 (d) 0

4 1
3. = _______
3 2
(a) 5 (b) 4
(c) 3 (d) 1

4. f(x) = sin x is increasing on which of the following intervals ?
(a) (0, ) (b) (0, /2)
(c) (/2, ) (d) (, 2)

5. The area of the region bounded by the curve y = cos x between x = 0 and x = /2 is
1
(a) sq. units (b) 2 sq. units
2
3
(c) 1 sq. units (d) sq. units
2

2
 dy   dy 
6. The order of the differential equation       sin 2 y  0 is
 dx   dx 
(a) 1 (b) 2
(c) 3 (d) 4

7. If a and b are parallel then a  b = _______

x5 y 4 z 6
8. Find the vector equation of the line   .
3 7 2

SY-51 2

PART – I
A. 1  9    5  .
1  . (5  1 = 5)
1. f : x  y    f   = _______
(a) x (b) y
(c) R (d) N

2. tan–1x + cot–1x = _______
 
(a) (b)
2 4
(c) 1 (d) 0

4 1
3. = _______
3 2
(a) 5 (b) 4
(c) 3 (d) 1

4. f(x) = sin x       ?
(a) (0, ) (b) (0, /2)
(c) (/2, ) (d) (, 2)

5. x = 0, x = /2 , y = cos x    
1
(a) sq. units (b) 2 sq. units
2
3
(c) 1 sq. units (d) sq. units
2

2
 dy   dy  2
6.       sin y  0    
d x
    d x
(a) 1 (b) 2
(c) 3 (d) 4

7. a , b   a  b = _______

x5 y 4 z 6
8.       .
3 7 2

SY-51 3 P.T.O.

3 1
9. If A and B are two independent events with P(A) = , P(B) = , then P(A  B) =
5 5
______.
3 3
(a) (b)
25 5
1 1
(c) (d)
5 3

B. Answer all questions from 10 to 13. Each carries 1 score. (4  1 = 4)
1
10. cos–1 = ______
2
 
(a) (b)
2 3
 
(c) (d)
4 6

11. Let A be a square matrix of order 2, then |3A| = ______
(a) 3|A| (b) 4|A|
(c) 2|A| (d) 9|A|

d
12. (log x) = _______
dx
(a) log x (b) ex
1
(c) (d) log ex
x

13. The direction cosines of the plane x + y + z = 1 is
1 1 1
(a) 1, 1, 1 (b) , ,
2 2 2
1 1 1 1 1 1
(c) , , (d) , ,
2 2 2 3 3 3

SY-51 4

3 1
9. A, B     P(A) = , P(B) =  
5 5
P(A  B) = ______.
3 3
(a) (b)
25 5
1 1
(c) (d)
5 3

B. 10  13    . 1  .

(4  1 = 4)
1
10. cos–1 = ______
2
 
(a) (b)
2 3
 
(c) (d)
4 6

11. A   2      |3A| = ______
(a) 3|A| (b) 4|A|
(c) 2|A| (d) 9|A|

d
12. (log x) = _______
dx
(a) log x (b) ex
1
(c) (d) log ex
x

13. x + y + z = 1    

1 1 1
(a) 1, 1, 1 (b) , ,
2 2 2
1 1 1 1 1 1
(c) , , (d) , ,
2 2 2 3 3 3
SY-51 5 P.T.O.

PART – II
A. Answer any two questions from 14 to 17. Each carries 2 scores. (2  2 = 4)
 4 3  y z 
14. If    , then find the values of x, y, z.
 x 5  1 5

15. Find the equation of tangent to the curve y = x2 at (1, 2).

16. Consider the function :
f(x) = e2x
(i) find f '(x) (1)
(ii) show that f(x) is increasing on R (1)

17. Solve the differential equation :
dy 1  y 2

dx 1  x 2

B. Answer any two questions from 18 to 20. Each carries 2 scores. (2  2 = 4)
d2y
18. If y = x2 + 3x + 2 find .
dx 2

dy 2
19. Solve  yx.
dx x

20. Find the distance of the point (2, 5, –3) from the plane 6x – 3y + 2z = 4.

PART – III
A. Answer any three questions from 21 to 24. Each carries 3 scores. (3  3 = 9)
21. Let f : R  R defined by f(x) = 4x + 3. Show that f is one-one and onto.

8 0 2 1 
22. Let A =   B  
3 1 3 2
(i) Find A + B (1)
(ii) Find 2A (1)
(iii) Find A' (1)

SY-51 6

PART – II
A. 14  17    2  .
2  . (2  2 = 4)
 4 3  y z 
14.  x 5  1 5  x, y, z   .
   

15. (1, 2)   y = x2    
.

16. f(x) = e2x   .
(i) f '(x) . (1)
(ii) R- f(x)   . (1)

dy 1  y 2
17.      .
dx 1  x 2

B. 18  20    2  .
2  . (2  2 = 4)
2
d y
18. y = x2 + 3x + 2  .
dx 2

dy 2
19.    yx.
dx x

20. 6x – 3y + 2z = 4    (2, 5, –3)   
.

PART – III
A. 21  24    3  .
3  . (3  3 = 9)
21. f : R  R  f(x) = 4x + 3  . f -,  
.

8 0 2 1 
22. A=   B  
3 1 3 2
(i) A + B . (1)
(ii) 2A . (1)
(iii) A' . (1)
SY-51 7 P.T.O.

23. Let a  î  ĵ, b  î  ĵ
 
(i) Find a . b (1)
 
(ii) Find the projection of a on b . (2)

24. A Random Variable X has the following probability distribution :
X 0 1 2
P(x) k 2k 3k
(i) Find the value of k (2)
(ii) Find P(x < 2) (1)

B. Answer any two questions from 25 to 27. Each carries 3 scores. (2  3 = 6)
25. Let * be a binary operation on N given by a * b = ab.
(i) Find 5 * 7 (1)
(ii) Is * commutative ? (1)
(iii) Find the identity element of * in N. (1)

1  1
26. Find the inverse of the matrix A =   using elementary transformations.
2 3 

27. Two balls are drawn at random without replacement (one after the other) from a box
containing 10 black and 8 red balls. Find the probability that first ball is black and
second is red.

PART – IV
A. Answer any three questions from 28 to 31. Each carries 4 scores. (3  4 = 12)
28. (i) tan–1x + tan–1y = _____. (1)
1 2 3
(ii) Show that tan–1 + tan–1 = tan–1 . (3)
2 11 4

29. Find the value of k so that :
kx 2 , if x  2
f(x) =  is continuous at x = 2.
3 , if x  2

30. Find the intervals in which f(x) = 2x2 – 3x is
(i) increasing (2)
(ii) decreasing (2)
SY-51 8

23. a  î  ĵ, b  î  ĵ 
 
(i) a . b . (1)
 
(ii) a   b   . (2)

24. X      
 :
X 0 1 2
P(x) k 2k 3k
(i) k   . (2)
(ii) P(x < 2) . (1)

B. 25  27    2  .
3  . (2  3 = 6)
25. *    N  a * b = ab  .
(i) 5 * 7 . (1)
(ii) *   ? (1)
(iii) N- *    . (1)
1  1
26.    A =    
2 3 
 .

27. 10  8       2  
(  ) .    
   .

PART – IV
A. 28  31    3  .
4  . (3  4 = 12)
28. (i) tan–1x + tan–1y = _____. (1)
1 2 3
(ii) tan–1 + tan–1 = tan–1  . (3)
2 11 4

kx 2 , if x  2
29. f(x) = 
3 , if x  2
  x = 2    k   .

30. f(x) = 2x2 – 3x  
(i)   (2)
(ii)  
   (2)
SY-51 9 P.T.O.

31. Find the shortest distance between the lines :
r  î  ĵ  λ (2 î  ĵ  k̂)
r  2 î  ĵ  k̂  μ (3 î  5ĵ  2k̂)

B. Answer any one question from 32 to 33. Each carries 4 scores. (1  4 = 4)
32. Prove that :
 x  y y  z z  x
 z x y   0

 1 1 1 

33. A fair coin is tossed 5 times. Find the probability of
(i) Exactly 4 heads. (2)
(ii) Atleast 4 heads. (2)

PART – V
Answer any two questions from 34 to 36. Each carries 6 scores. (2  6 = 12)
34. Consider the following system of equations
x–y+z=4
2x + y – 3z = 0
x+y+z=2
(i) Express the system of equations in the form AX = B. (1)
(ii) Find A–1. (3)
(iii) Solve the system of equations using matrix method. (2)

1x
e tan
35. (i) Evaluate  1  x 2 dx . (2)

(ii) Integrate x sin x with respect to x. (2)
1
2x
(iii) Evaluate 0 x 2 1 dx . (2)

36. Solve the L.P.P. graphically
Maximise Z = 3x + 2y
Subject to x + 2y < 10
3x + y < 15
x, y > 0
___________

SY-51 10

31. r  î  ĵ  λ (2 î  ĵ  k̂)
r  2 î  ĵ  k̂  μ (3 î  5ĵ  2k̂)
      .

B. 32  33     .
4 . (1  4 = 4)
 x  y y  z z  x
32.  z x y   0  .

 1 1 1 

33.   5   .   
.
(i)  4  . (2)
(ii) 4   . (2)

PART – V
34  36    2  .
6  . (2  6 = 12)
34.    
x–y+z=4
2x + y – 3z = 0
x+y+z=2
(i)  AX = B   . (1)
(ii) A–1 . (3)
(iii)      . (2)

1x
e tan
35. (i)  1 x 2 dx . (2)

(ii) x sin x  x   . (2)
1
2x
(iii) 0 x 2 1 dx . (2)

36. L.P.P.    .
Maximise Z = 3x + 2y
Subject to x + 2y < 10
3x + y < 15
x, y > 0
___________

SY-51 11 P.T.O.

SY-51 12