Kerala Plus Two Question Paper 2025 Mathematics Commerce

KERALA BOARD

QUESTION
PAPER
2025
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Kerala Board of Public
Examinations

SY-451
Reg. No. : ......................................
Name : ...........................................

SECOND YEAR HIGHER SECONDARY EXAMINATION, MARCH 2025

Time : 2 Hours
Part – III Cool-off time : 15 Minutes
MATHEMATICS (COMMERCE)
Maximum : 60 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-451 1 P.T.O.

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

x  2 2  4 2
1. (i) If   , find the values of x and y. (1)
 6 4  y – 2 4

(ii) Construct a 2  2 matrix A = [aij] where aij = 2i – j. (2)

1 2 3
2. (i) Let A = 2 3 4 , then A is a (1)
3 4 5 

(a) Scalar matrix (b) Symmetric matrix

(c) Skew symmetric matrix (d) Identity matrix

3 – 2 1 0
(ii) If A =   , show that A2 – A + 2I = 0 where I =  . (2)
4 – 2 0 1 

2 2
3. (i) If = 0, then x is equal to (1)
18 x

(a) –6 (b) 6

(c) 18 (d) –18

(ii) Find the value of k if area of triangle is 4 sq. units and vertices are (k, 0), (4, 0)
and (0, 2). (2)

 x 2 if x  2
4. Find the value of k if the function f(x) =  is continuous at x = 2. (3)
2k if x  2

5. (i) Find the rate of change of the area of a circle with respect to its radius r when
r = 3 cm. (1)

(ii) Show that the function f(x) = x3 + x is increasing on R. (2)

SY-451 2

1  8    6  .
 3  . (6  3 = 18)

x  2 2  4 2
1. (i)  6  ,  x  y   . (1)
 4  y – 2 4

(ii) aij = 2i – j   2  2  A = [aij] . (2)

1 2 3
2. (i) A = 2 3 4  A (1)
3 4 5 

(a)   (b)  
(c)   (d)  

3 – 2 1 0
(ii) A =    I =     A2 – A + 2I = 0 
4 – 2 0 1
. (2)

2 2
3. (i) = 0  x   (1)
18 x

(a) –6 (b) 6
(c) 18 (d) –18
(ii)  (k, 0), (4, 0) and (0, 2)     4 .
  k   . (2)

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

5. (i)     , 
r = 3 .  . (1)

(ii) f(x) = x3 + x    R   . (2)

SY-451 3 P.T.O.

6. Match the following : (3)
2x
(i)  1  x 2 dx (a) log |1 + x2| + C

(ii)  ex (sin x + cos x) dx (b) tan–1 x + C

1
(iii)  1  x2 (c) ex (cos x – sin x) + C

(d) ex sin x + C

a
7. (i)  f(x) dx = __________. (1)
0

a a
(a)  f(a) – f(x) dx (b)  f(a – x) dx
0 0

a a
(c)  f(x – a) dx (d)  f(a) dx
0 0

a
x
(ii) Evaluate :  x  a – x dx (2)
0

8. Let A and B be two independent events with P(A) = 0.3 and P(B) = 0.4. Find (3)
(a) P(A  B) (b) P(A  B) (c) P(A/B)

Answer any 6 questions from 9 to 16. Each carries 4 scores. (6  4 = 24)
9. (i) Which among the ordered pair is to be added to the relation R = {(1, 1), (2, 2),
(2, 1), (1, 3), (2, 3)} defined on the set {1, 2, 3} in order to make it a reflexive
relation ? (1)
(a) (1, 2) (b) (3, 1)
(c) (3, 3) (d) (3, 2)
(ii) Which of the following function f : R  R represents a one-one function ? (1)
(a) f(x) = x2 (b) f(x) = x
(c) f(x) = | x | (d) f(x) = sin x

SY-451 4

6.    : (3)
2x
(i)  1  x 2 dx (a) log |1 + x2| + C

(ii)  ex (sin x + cos x) dx (b) tan–1 x + C
1
(iii)  1  x2 (c) ex (cos x – sin x) + C

(d) ex sin x + C

a
7. (i)  f(x) dx = __________. (1)
0
a a
(a)  f(a) – f(x) dx (b)  f(a – x) dx
0 0
a a
(c)  f(x – a) dx (d)  f(a) dx
0 0
a
x
(ii)  x  a – x dx  (2)
0

8. A  B     . P(A) = 0.3, P(B) = 0.4
 (3)
(a) P(A  B)  (b) P(A  B)  (c) P(A/B) 

9  16    6  .
4  . (6  4 = 24)
9. (i)       {1, 2, 3} 
 R = {(1, 1), (2, 2), (2, 1), (1, 3), (2, 3)}   
 . (1)
(a) (1, 2) (b) (3, 1)
(c) (3, 3) (d) (3, 2)
(ii)   f : R  R      
. (1)
(a) f(x) = x2 (b) f(x) = x
(c) f(x) = | x | (d) f(x) = sin x

SY-451 5 P.T.O.

(iii) Let f : A  B defined as in arrow diagram. Is the function f (2)
(a) one-one (b) onto
Justify your answer.

10. (i) If cos–1 x = y, then (1)

 
(a) 0y (b) – y
2 2

 
(c) 0<y< (d) – <y<
2 2

 1  1
(ii) Find the value of tan–1   + cos–1   . (1)
 3 2
  –1 1  
(iii) Evaluate : tan–1 2 cos  2 sin 2  (2)
  

1 3
11. If A = 
4 5

(i) Verify that A + A' is a symmetric matrix and A – A' is a skew symmetric matrix. (3)
(ii) Express the matrix A as a symmetric and a skew symmetric matrix. (1)

12. If f(x) = x3 – 3x + 3
(i) Find f '(x) (1)
(ii) Find the points of local maxima and points of local minima of the function if any.
Also find the local maximum value and local minimum value of the functions. (3)

SY-451 6

(iii)   f : A  B   f (2)
(a) - 
(b)  
 :

10. (i) cos–1 x = y  (1)
 
(a) 0y (b) – y
2 2
 
(c) 0<y< (d) – <y<
2 2
 1  1
(ii) tan–1   + cos–1     . (1)
 3 2
  –1 1  
 2 cos  2 sin 
2    .
(iii) tan–1 (2)
 

 1 3
11. A=   
 4 5
(i) A + A'     A – A'  
   . (3)
(ii) A       
   . (1)

12. f(x) = x3 – 3x + 3 
(i) f '(x)  (1)
(ii)        
.    ,    
. (3)

SY-451 7 P.T.O.

13. (i) Find the direction cosines of a line passing through the two points (3, 5, – 4) and
(–1, 1, 2). (2)

(ii) Find the equation of the line which passes through the point (1, 2, 3) and is

parallel to the vector 3^i + 2^j – 2k.
^ (2)

14. Coloured balls are distributed in two boxes as in the table :

Colour
Box
Black White

I 3 4

II 2 2

A box is selected at random and then a ball is randomly drawn from the selected box. The
colour of the ball is black, what is the probability that ball drawn is from the box II ? (4)

15. Using integration, find the area of the shaded region in the circle x2 + y2 = 4. (4)

dy
16. (i) Find the order and degree of the differential equation – cos x = 0. (1)
dx

(ii) Find the general solution of the above differential equation. (3)

SY-451 8

13. (i) (3, 5, –4), (–1, 1, 2)      
  . (2)

(ii) (1, 2, 3)    3^i + 2^j – 2k^  
   . (2)

14.       
.

Box
 

I 3 4
II 2 2
        
.        II  
 . (4)

15.   x2 + y2 = 4     
 . (4)

dy
16. (i) – cos x = 0    ,   . (1)
dx
(ii)      
. (3)

SY-451 9 P.T.O.

Answer any 3 questions from 17 to 20. Each carries 6 scores. (3  6 = 18)

1 2 
17. Let A =  
 2 3
(i) Find adj A (1)

(ii) Find A–1 (2)
(iii) Solve the system of equations using matrix method (3)
x + 2y = 5
2x + 3y = 9

dy
18. (i) If y = (3x + 1)3, the find . (2)
dx
dy
(ii) Find , if x = at2, y = 2at (2)
dx

d2y
(iii) If y = 5 cos x + 3 sin x, prove that + y = 0. (2)
dx 2

 
19. If a = ^i + ^j + k^ and b = ^i – ^j + k^
 
(i) Find a unit vector in the direction of the vector a + b (2)
   
(ii) Evaluate ( a + b ) · ( a – b ) (2)
   
(iii) Find ( a + b )  ( a – b ) (2)

20. Solve the following LPP graphically : (6)
Maximise Z = 250x + 75y
Subject to
x + y  60

5x + y  100

x  0, y  0
____________

SY-451 10

17  20    3  .
 6  . (3  6 = 18)

1 2 
17. A=  
 2 3
(i) adj A  (1)
(ii) A–1  (2)
(iii)     . (3)
x + 2y = 5
2x + 3y = 9

dy
18. (i) y = (3x + 1)3  . (2)
dx
dy
(ii) x = at2, y = 2at   (2)
dx
d2y
(iii) y = 5 cos x + 3 sin x  + y = 0  . (2)
dx 2

 
19. a = ^i + ^j + k^  b = ^i – ^j + k^   
 
(i) a + b      (2)
   
(ii) ( a + b ) · ( a – b )    (2)
   
(iii) ( a + b )  ( a – b )  (2)

20.       . (6)
 Z = 250x + 75y
 
x + y  60
5x + y  100
x  0, y  0
____________

SY-451 11 P.T.O.

SY-451 12

SY-455
Reg. No. : ......................................

Name : ...........................................

SECOND YEAR HIGHER SECONDARY EXAMINATION, MARCH 2025

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

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-455 1 P.T.O.

Answer any 6 questions from 1 to 7. Each question carries 3 scores. (6  3 = 18)
1. A function f : R  R given by f(x) = 4x + 3. Show that f is invertible.

2. In the matrix
1 1 – 1
2 0 3 
 
3 – 1 2 
(a) The order of the matrix is _____ . (1)
(b) Write the transpose of the matrix. (2)

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

4. Find the rate of change of area of a circle per second with respect to its radius r when
r = 5 cm.


5. Find the unit vector in the direction of vector a = 2i + 3j + k.

6. Find the vector equation of the line through the point (5, 2, – 4) which is parallel to
the vector 3i + 2j – 8k.

3, 3 1
7. Let E & F be events with P(E) = P(F) = and P(E  F) = . Are E & F
5 10 5
independent ?

Answer any 8 questions from 8 to 17. Each question carries 4 scores. (8  4 = 32)
8. Find gof and fog if
f(x) = cos x
g(x) = 2x2

9. Prove that
2 7 1
tan–1 + tan–1 = tan–1
11 24 2

10. Find x and y if
1 3  y 0  5 6
2   +  =  
0 x  1 2 1 8
SY-455 2

1  7    6  .
3  . (6  3 = 18)
1. f : R  R- f(x) = 4x + 3    
.

1 1 – 1
2. 2 0 3   
 
3 – 1 2 
(a)    _____ . (1)
(b)    . (2)

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

4.   r = 5 cm   
  .


5. a = 2i + 3j + k       
.

6. 3i + 2j – 8k   (5, 2, – 4)     
  .

3 3 1
7. P(E) = , P(F) = , P(E  F) = .  E  F   
5 10 5
 ?

8  17    8  .
4  . (8  4 = 32)
8. f(x) = cos x ,
g(x) = 2x2 
gof, fog .

2 7 1
9. tan–1 + tan–1 = tan–1 .
11 24 2

1 3  y 0  5 6
10. 2   +  =   .  x, y .
0 x  1 2 1 8
SY-455 3 P.T.O.

11. Find the area of the triangle with vertices (2, 7), (1, 1), (10, 8).

12. Find the intervals in which the function f given by f(x) = x2 – 4x + 6 is
(a) increasing
(b) decreasing.

13. Find the area enclosed by the circle using integration x2 + y2 = a2.

14. (a) The order of the differential equation
3 2
 d2y 
    dy   1  0 is _____ . (1)
 dx 2   dx 
 
(b) Verify that the function y = e–3x is a solution of the differential equation (3)
2
d y dy
2
  6y  0 .
dx dx

15. Show that the points A(2i – j + k), B(i – 3j – 5k), C(3i – 4j – 4k) are the vertices of
right angled triangle.

16. Find the shortest distance between the lines l1 & l2 whose vector equation

r = i + j + (2i – j + k)

r = 2i + j – k + (3i – 5j + 2k)

17. Let A and B be independent events with P(A) = 0.3, P(B) = 0.4. Find
(a) P(A  B)
(b) P(A  B)
(c) P(A / B)
(d) P(B / A)

Answer any 5 questions from 18 to 24. Each question carries 6 scores. (5  6 = 30)
18. Find A2 – 5A + 6I, if
2 0 1 
A = 2 1 3
1  1 0

19. Solve the system of equation
3x – 2y + 3z = 8
2x + y – z = 1
4x – 3y + 2z = 4

SY-455 4

11. (2, 7), (1, 1), (10, 8)    .

12. f(x) = x2 – 4x + 6   (a)  (b)  
 .

13. x2 + y2 = a2       
 .

3 2
 d2 y   dy 
14. (a)       1  0     _____
 dx 2   dx 
 
. (1)
d2y dy
(b) y = e–3x   2
  6y  0  
dx dx
  . (3)

15. A(2i – j + k), B(i – 3j – 5k), C(3i – 4j – 4k)   
   .


16. r = i + j + (2i – j + k)

r = 2i + j – k + (3i – 5j + 2k)
l1 & l2       .

17. A, B     .
P(A) = 0.3, P(B) = 0.4     .
(a) P(A  B)
(b) P(A  B)
(c) P(A / B)
(d) P(B / A)

18  24    5  .
6  . (5  6 = 30)
2 0 1 
18. A = 2 1 3  A2 – 5A + 6I - .
1  1 0

19.   
3x – 2y + 3z = 8
2x + y – z = 1
4x – 3y + 2z = 4
SY-455 5 P.T.O.

dy ,
20. Find if
dx
(a) 2x + 3y = sin x (2)
(b) xx (2)
(c) tan (2x + 3) (2)

21. Match the following :

(a)  dx (i) ex + C

x3
(b)  cos x dx (ii)
3
+C

 e dx
(c) x (iii) x + C

1 (iv) sin x + C
(d)
 x dx
 sin x dx
(e) (v) log x + C

 x dx
(f) 2 (vi) – cos x + C

 
22. (a) Find the angle  between the vectors a = i + j – k and b = i – j + k. (3)
   
(b) Find | a  b | if a = 2i + j + 3k, b = 3i + 5j – 2k. (3)

23. Solve the LPP
Maximize Z = 3x + 4y
subject to x + 2y  8
3x + 2y  12
x  0, y  0

24. Bag I contain 3 red and 4 black balls while another Bag II contain 5 red and 6 black
balls. One ball is drawn at random from one of the bag and it is found to be red. Find
the probability that it is drawn from Bag II.
_____________

SY-455 6

dy
20. .
dx
(a) 2x + 3y = sin x (2)
(b) xx (2)
(c) tan (2x + 3) (2)

21.   :

(a)  dx (i) ex + C

x3
(b)  cos x dx (ii)
3
+C

 e dx
(c) x (iii) x + C

1 (iv) sin x + C
(d)
 x
dx

 sin x dx
(e) (v) log x + C

 x dx
(f) 2 (vi) – cos x + C

 
22. (a) a = i + j – k, b = i – j + k    
. (3)
   
(b) a = 2i + j + 3k, b = 3i + 5j – 2k  | a  b | . (3)

23. LPP  
 Z = 3x + 4y
subject to x + 2y  8
3x + 2y  12
x  0, y  0

24. Bag I- 3   4   , Bag II- 5  
6   .      
.    .  Bag II- 
   .
_____________

SY-455 7 P.T.O.

SY-455 8