Kerala Plus Two Question Paper 2023 Mathematics (Science) 554

SECOND YEAR

Kerala Board
Question Paper
2023

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SY-554
SECOND YEAR HIGHER SECONDARY EXAMINATION, MARCH 2023

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-554 1 P.T.O.

Answer any 6 questions from 1 to 8. Each carries 3 scores. (6  3 = 18)

1. (i) A function f : X  Y is onto if range of f = ________. (1)

1
(ii) Find gof and fog if f(x) = 8x3 and g(x) = x 3 . (2)

 – 2
2. (i) If A =  4  and B = [1 3 –6], what is the order of AB ? (1)
 5 

(ii) Construct a 3  4 matrix whose elements are given by aij = 2i – j. (2)

3 –1 2
3. Evaluate : 0 0 –1 (3)
3 –5 0

4. Check the continuity of the function f given by f(x) = 2x + 3 at x = 1. (3)

5. (i) The derivative of e–x is _______. (1)

dy
(ii) Find if 2x + 3y = sin x (2)
dx

6. Show that the function f given by f(x) = x3 – 3x2 + 4x, x  R is increasing. (3)

SY-554 2

1  8    6  .

3  . (6  3 = 18)

1. (i) f : X  Y     f   = ________. (1)

1
(ii) f(x) = 8x3, g(x) = x 3  gof, fog . (2)

 – 2
2. (i) A =  4  , B = [1 3 –6]  AB   . (1)
 5 

(ii)   aij = 2i – j    3  4 

. (2)

3 –1 2
3.   : 0 0 –1 (3)
3 –5 0

4. f(x) = 2x + 3,   x = 1   . (3)

5. (i) e–x   _______. (1)

dy
(ii) 2x + 3y = sin x  . (2)
dx

6. f(x) = x3 – 3x2 + 4x, x  R     . (3)

SY-554 3 P.T.O.

7. (i) The order of the differential equation :

d2y dy
2x2 2
–3 + y = 0 is (1)
dx dx

(a) 2 (b) 1

(c) 0 (d) not defined

(ii) Verify that the function y = a cos x + b sin x where a, b  R is a solution of
differential equation.

d2y
+y=0 (2)
dx 2

8. (i) Distance between two planes 2x + 3y + 4z = 4 and 4x + 6y + 8z = 12 is

(a) 2 units (b) 4 units

2
(c) 8 units (d) units (1)
29

(ii) Find the vector equation for the line passing through the point (–1, 0, 2) and
(3, 4, 6). (2)

Answer any 8 questions from 9 to 18. Each carries 4 scores. (8  4 = 32)

9. Consider f : R  R given by f(x) = 4x + 3. Show that f is invertible. Find the inverse
of f. (4)

10. (i) tan–1 x + tan–1 y = __________. (1)

(ii) Prove that :

1 1 31
2 tan–1 + tan–1 = tan–1 (3)
2 7 17

SY-554 4

d2y dy
7. (i) 2x2 2
–3 +y=0
dx dx

    : (1)

(a) 2 (b) 1

(c) 0 (d) 

d2y
(ii) y = a cos x + b sin x , a, b  R  + y = 0  
dx 2
   . (2)

8. (i) 2x + 3y + 4z = 4, 4x + 6y + 8z = 12    

(a) 2 units (b) 4 units

2
(c) 8 units (d) units (1)
29

(ii) (–1, 0, 2) , (3, 4, 6)     

 . (2)

9  18    8  .

4  . (8  4 = 32)

9. f : R  R  f(x) = 4x + 3.  . ‘f’  

. f   . (4)

10. (i) tan–1 x + tan–1 y = __________. (1)

(ii)  :

1 1 31
2 tan–1 + tan–1 = tan–1 (3)
2 7 17

SY-554 5 P.T.O.

11. Differentiate xsin x, x > 0. (4)

12. (i) Find the antiderivative of cos 2x. (1)

x3 – 1
(ii) Find  x2 dx (3)

π
13. Find the area bounded by the curve y = cos 2x, x = 0, x = and x-axis. (4)
2

14. (i) Find the general solution of the differential equation : (2)

dy 1  y 2
 .
dx 1  x 2

(ii) Find an Integrating Factor (IF) of the differential equation.

dy
x + 2y = x2 (2)
dx


15. (i) Write the direction ratios of the vector a = i + j – 2k and hence calculate its
direction cosines. (2)

(ii) Find the values of x and y so that the vectors 2i + 3j and xi + yj are equal. (2)

16. Find the angle between the two planes 3x – 6y + 2z = 7 and 2x + 2y – 2z = 5. (4)

17. Find the shortest distance between the lines : (4)

x 1 y 1 z 1 x –1 y – 5 z – 7
  and  
7 –6 1 1 –2 1

SY-554 6

11.  xsin x, x > 0. (4)

12. (i) cos 2x   . (1)

x3 – 1
(ii)  x2 dx . (3)

π
13. y = cos 2x, x = 0, x = x-     
2
. (4)

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

dy
(ii) x + 2y = x2     
dx
. (2)

15. (i) a = i + j – 2k     .
    . (2)

(ii) 2i + 3j  xi + yj      x , y 
. (2)

16. 3x – 6y + 2z = 7, 2x + 2y – 2z = 5
     . (4)

17.          : (4)

x 1 y 1 z 1 x –1 y – 5 z – 7
  ,  
7 –6 1 1 –2 1

SY-554 7 P.T.O.

3 1 1
18. (i) E and F are two events with P(E) = , P(F) = , P(E  F) = . Are E and F
5 3 5
independent ? (1)

(ii) Two balls are drawn at random with replacement from a box containing
10 black and 8 red balls. Find the probability

(a) Both balls are red

(b) First ball is black and second is red

(c) One of them is black and other is red (3)

Answer any 5 questions from 19 to 25. Each carries 6 scores. (5  6 = 30)

19. (i) (A')' = ________. (1)

1 – 1
(ii) Find the transpose of the matrix   (2)
2 3 

 3 1 2
(iii) If A =   show that A – 5A + 7I = 0. (3)
 – 1 2 

2 1 3
20. Find the inverse of the matrix  4 – 1 0 .
 (6)
 – 7 2 1

21. (i) Find the equation of the tangent of the following curve y = 3x4 – 4x at x = 4. (3)

(ii) The volume of a cube is increasing at the rate of 9 cm3/s. How fast is the surface
area increasing when the length of an edge is 10 cm ? (3)

SY-554 8

3 1 1
18. (i) P(E) = , P(F) = , P(E  F) =      E  F
5 3 5
.  E  F    ? (1)

(ii) 10   8       
     . 
  .

(a)   .

(b)     

(c)     (3)

19  25    5  .
6  . (5  6 = 30)

19. (i) (A')' = ________. (1)

1 – 1
(ii) 2 3     . (2)
 

 3 1 2
(iii) A =    A – 5A + 7I = 0  . (3)
 – 1 2 

2 1 3
20.  4 – 1 0    . (6)
 
 – 7 2 1

21. (i)      .

y = 3x4 – 4x at x = 4. (3)

(ii)    9 cm3/s    -
.    10 cm 
    . (3)

SY-554 9 P.T.O.

22. Integrate :

(i)  (4e3x + 1) dx (2)

dx
(ii)  x 2 – 16 (2)

3x 2
(iii)  x6  1 dx (2)

23. (i) Find the area of a triangle having the points A(1, 1, 1), B(1, 2, 3) and C(2, 3, 1)
as its vertices. (2)
   
(ii) Evaluate : (3 a – 5 b ) · (2 a + 7 b ) (2)
   
(iii) Find a unit vector perpendicular to each of the vector ( a + b ) and ( a – b )
 
where a = i + j + k, b = i + 2j + 3k. (2)

24. Solve the LPP graphically : (6)
Maximize Z = 3x + 2y
subject to

x + 2y  10

3x + y  15

x, y  0

6 5 7
25. If P(A) = , P(B) = and P(A  B) = find (6)
11 11 11

(i) P(A  B)
(ii) P(A|B)
(iii) P(B|A)

____________

SY-554 10

22.   :

(i)  (4e3x + 1) dx (2)

dx
(ii)  x 2 – 16 (2)

3x 2
(iii)  x6  1 dx (2)

23. (i) A(1, 1, 1), B(1, 2, 3), C(2, 3, 1)   
  . (2)
   
(ii) (3 a – 5 b ) · (2 a + 7 b )   . (2)
     
(iii) a = i + j + k, b = i + 2j + 3k , a + b  a – b 
   . (2)

24.       
 : (6)

Maximize Z = 3x + 2y
subject to
x + 2y  10
3x + y  15
x, y  0
6 5 7
25. P(A) = , P(B) = , P(A B) = (6)
11 11 11
   .
(i) P(A B)
(ii) P(A|B)
(iii) P(B|A)

____________

SY-554 11 P.T.O.

SY-554 12