Kerala Plus Two Question Paper 2022 Maths Science

Reg. No. : .....................................
Name : ..........................................
SY-56
SECOND YEAR HIGHER SECONDARY EXAMINATION, MARCH 2022

Part – III Time : 2½ Hours
MATHEMATICS (SCIENCE) Cool-off time : 15 Minutes
Maximum : 80 Scores

General Instructions to Candidates :
 There is a ‘Cool-off time’ of 15 minutes in addition to the writing time.
 Use the ‘Cool-off time’ to get familiar with 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.
 Give equations wherever necessary.
 Electronic devices except non-programmable calculators are not allowed in the
Examination Hall.
 
    15  ‘  ’ .
 ‘  ’    
 .
      .
    .
  , , ,   
.
   .
    .
     
    .

SY-56 1 P.T.O.

PART-I
A. Answer any 4 questions from 1 to 6. Each carries 1 score. (4  1 = 4)
1. Let R be the relation in the set {1, 2, 3, 4} given by R = {(1, 1), (2, 2), (3, 3), (4, 4),
(1, 2), (1, 3), (3, 2)}
Choose the correct answer.
(a) R is reflexive and symmetric, but not transitive.
(b) R is reflexive and transitive, but not symmetric.
(c) R is symmetric and transitive, but not reflexive.
(d) R is an equivalence relation.

2. sin–1 x + cos–1 x = ________.
–π
(a) 0 (b)
2
π
(c) (d) 
2

3. The slope of the tangent to the curve y = x2 + 2 at x = 2 is _______.

4
 dy  d2y
4. The order of the differential equation   + 3y 2 = 0 is ________.
 dx  dx
(a) 1 (b) 2
(c) 3 (d) 4

  ^  
5. If a = ^i + ^j and b = 3 ^j + k , then a · b = ________.

^ ^
If vector equation of a line is r = (–3 ^i + 5 ^j – 6 k ) +  (2 ^i + 4 ^j + 2 k ), then its

6.
Cartesian equation is ________.

B. Answer all questions from 7 to 10. Each carries 1 score. (4  1 = 4)
7. Principal value of tan–1 (1) is ________.
π π
(a) (b)
6 4
π π
(c) (d)
3 2
SY-56 2

PART-I
A. 1  6    4  .
1  . (4  1 = 4)
1. R = {(1, 1), (2, 2), (3, 3), (4, 4), (1, 2), (1, 3), (3, 2)}   {1, 2, 3, 4}
  .   
.
(a) R    ,  .
(b) R    ,  .
(c) R   ,  .
(d) R    .

2. sin–1 x + cos–1 x = ________.
–π
(a) 0 (b)
2
π
(c) (d) 
2

3. y = x2 + 2   x = 2     _______ .

4
 dy  d2y
4.   + 3y = 0     ________ .
 dx  dx 2
(a) 1 (b) 2
(c) 3 (d) 4

  ^  
5. a = ^i + ^j , b = 3 ^j + k  a · b = ________.

  r = (–3 ^i + 5 ^j – 6 k^ ) +  (2 ^i + 4 ^j + 2 k^ )   

6.
  ______ .

B. 7  10    . 1  .
(4  1 = 4)
7. tan–1 (1)    ________ .
π π
(a) (b)
6 4
π π
(c) (d)
3 2
SY-56 3 P.T.O.

8. Derivative of e2x w.r.t. x is ______.

    
9. For any two vectors a and b , [ a , a , b ] = ________.

10. The direction ratios of the line passing through two points (2, 1, –2) and (1, 2, –3) are
________.

PART-II
A. Answer any 3 questions from 11 to 15. Each carries 2 scores. (3  2 = 6)
1
11. Find fog if f(x) = 8x3 and g(x) = x 3 , where f and g are real functions.

 4 – 2  2 3
12. Find A if 2A + B =   and B =  .
– 1 3  1 2 

13. Show that the function f(x) = 4x + 3 is strictly increasing in .

 ^  ^
14. Find a vector perpendicular to both a = 5 ^i – ^j – 3 k and b = ^i + 3 ^j – 5 k .

 ^  ^
15. Find the angle between the vectors a = ^i – 2 ^j + 3 k and b = 3 ^i – 2 ^j + k .

B. Answer any 2 questions from 16 to 18. Each carries 2 scores. (2  2 = 4)
ab
16. Let ‘*’ be a binary operation on the set Q of rational numbers defined by a * b = .
4
Check whether ‘*’ is commutative or not.

17. Find the distance of the point (2, 3, 1) from the plane x + 2y + 3z = 9.

18. The random variable X has a probability distribution P(X) of the following form :
k, if x0
2k, if x 1

P(X) = 
3k, if x2
0, otherwise
Determine the value of k.
SY-56 4

8. e2x   ______ .

    
9. a , b      [ a , a , b ] = ________.

10. (2, 1, –2), (1, 2, –3)     
  ________ .

PART-II
A. 11  15    3  .
2  . (3  2 = 6)
1
11. f(x) = 8x3, g(x) = x 3     fog .

 4 – 2  2 3
12. 2A + B =   ,B=    A .
– 1 3  1 2 

13. f(x) = 4x + 3       .

 ^  ^
14. a = 5 ^i – ^j – 3 k , b = ^i + 3 ^j – 5 k     
 .

 ^  ^
15. a = ^i – 2 ^j + 3 k , b = 3 ^i – 2 ^j + k   
.

B. 16  18    2  .
2  . (2  2 = 4)
ab
16. ‘*’    Q  a * b =   
4
 . ‘*’    .

17. x + 2y + 3z = 9    (2, 3, 1)   
.

18.   X      P(X) :
k, if x0
2k, if x 1

P(X) = 
3k, if x2
0, otherwise
k   .
SY-56 5 P.T.O.

PART-III
A. Answer any 3 questions from 19 to 23. Each carries 4 scores. (3  4 = 12)
19. Consider f :  given by f(x) = 2x + 3. Show that f is invertible and find the inverse
of f.

20. Find two positive numbers x and y such that their sum is 15 and sum of whose squares
is minimum.

21. Find the area of the region bounded by x2 = 4y, y = 2, y = 4 and the y-axis in the first
quadrant.

dy
22. Find the general solution of the differential equation x + 2y = x2, x  0
dx

23. Find the shortest distance between the lines :
^
r = ^i + ^j + (2 ^i – ^j + k ) and


^ ^
r = 2 ^i + ^j – k + (3 ^i – 5 ^j + 2 k )


B. Answer any 1 question from 24 to 25. Carries 4 scores. (1  4 = 4)
24. Find the equation of the line joining the points (1, 2) and (3, –1) using determinants.

25. Find the area between the curves y2 = x and y = x2.

PART-IV
A. Answer any 3 questions from 26 to 29. Each carries 6 scores. (3  6 = 18)
 2 
26. (i) Find the value of sin–1 sin   (2)
 3 
(ii) Prove that :
2 7 1
tan–1 + tan–1 = tan–1 (4)
11 24 2

SY-56 6

PART-III
A. 19  23    3  .
4  . (3  4 = 12)
19. f:  , f(x) = 2x + 3   . f  
. f   .

20.  15        
 x  y  .

21. x2 = 4y, y = 2, y = 4, y-    
 .

dy
22. x + 2y = x2, x  0     
dx
.

^
r = ^i + ^j + (2 ^i – ^j + k )

23.
^ ^
r = 2 ^i + ^j – k + (3 ^i – 5 ^j + 2 k )     


.

B. 24  25     .
4 . (1  4 = 4)
24. (1, 2), (3, –1)      
  .

25. y2 = x , y = x2    .

PART-IV
A. 26  29    3  .
6  . (3  6 = 18)
 2 
26. (i) sin–1 sin     . (2)
 3 
2 7 1
(ii) tan–1 + tan–1 = tan–1  . (4)
11 24 2
SY-56 7 P.T.O.

dy
27. Find
dx
(i) 2x + 3y = sin y (3)
(ii) x = sin t, y = cos 2t (3)

28. Integrate the following :
1
(i) (3)
x 2 – 6 x  13
(ii) x log x (3)

29. Solve the following Linear Programming Problem graphically :
Maximise Z = 3x + 2y
Subject to x + 2y  10
3x + y  15
x  0, y  0

B. Answer any 2 questions from 30 to 32. Each carries 6 scores. (2  6 = 12)
dy
30. (i) Find if y = xsin x (3)
dx
2
(ii) If y = (tan–1 x)2, then show that (1 + x2) y2 + 2x(1 + x2)y1 = 2 (3)

2
31. (i) Find  x2 dx as the limit of a sum. (4)
0

π
4
(ii) Evaluate  sin x dx (2)
0

32. Consider the differential equation (x – y) dy – (x + y) dx = 0
(i) Show that it is homogeneous. (2)
(ii) Solve this different equation. (4)

SY-56 8

dy
27. 
dx
(i) 2x + 3y = sin y (3)
(ii) x = sin t, y = cos 2t (3)

28.     :
1
(i) 2
(3)
x – 6 x  13
(ii) x log x (3)

29.       
 :
Maximise Z = 3x + 2y
Subject to x + 2y  10
3x + y  15
x  0, y  0

B. 30  32    2  .
6  . (2  6 = 12)
dy
30. (i) y = xsin x  . (3)
dx
2
(ii) y = (tan–1 x)2  (1 + x2) y2 + 2x(1 + x2)y1 = 2  . (3)

2
31. (i)  x2 dx     . (4)
0
π
4
(ii)  sin x dx   . (2)
0

32. (x – y) dy – (x + y) dx = 0    .
(i)    . (2)

(ii)     . (4)

SY-56 9 P.T.O.

PART-V

Answer any 2 questions from 33 to 35. Each carries 8 scores. (2  8 = 16)

2 0 1
33. Let A = 2 1 3
1 – 1 0

(i) Express A as the sum of a symmetric and a skew symmetric matrix. (4)

(ii) Find A2 – 5A + 6I (4)

34. Consider the matrix

1 1 1
A = 0 1 3
1 – 2 1

(i) Find Adj A (2)

(ii) Prove that A · AdjA = | A | I (3)

(iii) Solve the following system of equations using matrix method :

x+y+z=6

y + 3z = 11

x – 2y + z = 0 (3)

35. (i) Given two independent events A and B such that P(A) = 0.3 and P(B) = 0.6. Find

(a) P(A or B) (2)

(b) P (neither A nor B) (2)

(ii) A bag contains 4 red and 4 black balls, another bag contains 2 red and 6 black balls.
One of the bags is selected at random and a ball is drawn from the bag which is
found to be red. Find the probability that the ball drawn is from the first bag. (4)

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SY-56 10

PART-V
33  35    2  .
8  . (2  8 = 16)

2 0 1
33. A = 2 1 3 .
1 – 1 0

(i) A        
 . (4)
(ii) A2 – 5A + 6I . (4)

1 1 1
34. A = 0 1 3   .
1 – 2 1

(i) Adj A . (2)
(ii) A · AdjA = | A | I  . (3)
(iii)        
   :
x+y+z=6
y + 3z = 11
x – 2y + z = 0 (3)

35. (i) A, B    .
P(A) = 0.3, P(B) = 0.6 
(a) P(A or B) (2)
(b) P (neither A nor B) (2)
 .
(ii)   4   4     2 
 6   .    
    .    
     . (4)

__________
SY-56 11 P.T.O.

SY-56 12