Kerala Plus Two Question Paper 2025 Mathematics Science

KERALA BOARD

QUESTION
PAPER
2025
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Kerala Board of Public
Examinations

SY-454
Reg. No. : ......................................

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

SECOND YEAR HIGHER SECONDARY EXAMINATION, MARCH 2025

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-454 1 P.T.O.

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

1. (i) Let f : R  R, f(x) = sin x

g : R  R, g(x) = x3

then gof(x) = _____.

(A) sin x3 (B) sin3 x

(C) 3 sin x (D) sin 3x (1)

(ii) Show that f(x) = | x | is neither one-one nor onto. (2)

2. Find the values of x, y and z if

 x  y  z  9 
 xz  =  5  . (3)
   
 y  z   7 

3. (i) Let A be a square matrix of order 3  3, then | 2 A | is equal to

(A) 2 | A | (B) 4|A|

(C) 8|A| (D) 6 | A | (1)

2x 4 2 4
(ii) Find the value of x if = . (2)
6 x 5 1

kx  1, if x  5
4. Consider the function f(x) = 
 3 x  5, if x  5

(i) Find lim – f(x) (1)
x 5

(ii) Find lim + f(x) (1)
x 5

(iii) If f is continuous, find the value of k. (1)

SY-454 2

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

1. (i) f : R  R, f(x) = sin x

g : R  R, g(x) = x3

 gof(x) = _____.

(A) sin x3 (B) sin3 x
(C) 3 sin x (D) sin 3x (1)

(ii) f(x) = | x | -      . (2)

 x  y  z  9 
2.  x  z  =  5   x, y, z   . (3)
   
 y  z   7 

3. (i) A   3  3      | 2 A | =

(A) 2 | A | (B) 4|A|

(C) 8|A| (D) 6 | A | (1)

2x 4 2 4
(ii) =  x   . (2)
6 x 5 1

kx  1, if x  5
4. f(x) = 
 3 x  5, if x  5
  .

(i) lim f(x)  (1)
x 5–

(ii) lim f(x)  (1)
x 5+

(iii) f   k   . (1)

SY-454 3 P.T.O.

d
5. (i) log (sin x) = _______.
dx
1
(A) log cos x (B)
sin x

(C) tan x (D) cot x (1)
dy
(ii) Find if y + sin y = cos x. (2)
dx

6. Let f(x) = x2 – 4x + 5,

(i) Find f '(x). (1)
(ii) Find the intervals in which f(x) is increasing or decreasing. (2)

7. (i) The degree of the differential equation
3
 d 2 y   dy  4 dy
  +
 dx 2   dx 
+x = 1 is _______.
  dx

(A) 3 (B) 4
(C) 1 (D) 2 (1)
(ii) Find the general solution of the differential equation

dy x
= (2)
dx y

8. Find the equation of the plane passing through the points (1, 1, 0), (1, 2, 1) and
(–2, 2, –1). (3)

Answer any 8 questions from 9 to 18. Each carries 4 scores. (8  4 = 32)
ab
9. Let * be a binary operation on ‘Q’ defined by a * b =
4

(i) Show that * is commutative. (2)

(ii) Find the Identity of * in ‘Q’. (2)

SY-454 4

d
5. (i) log (sin x) = _______.
dx
1
(A) log cos x (B)
sin x
(C) tan x (D) cot x (1)
dy
(ii) y + sin y = cos x  . (2)
dx

6. f(x) = x2 – 4x + 5,

(i) f '(x) . (1)

(ii) f(x) ,    . (2)

3 4
 d2y 
7. (i)   +  dy  + x dy = 1    
 dx 2   dx  dx
 
_______.
(A) 3 (B) 4
(C) 1 (D) 2 (1)
dy x
(ii) =     . (2)
dx y

8. (1, 1, 0), (1, 2, 1), (–2, 2, –1)    
  . (3)

9  18    8  .
4  . (8  4 = 32)

ab
9. ‘Q’  *    a * b =  .
4
(i) *   . (2)

(ii) ‘Q’  *   . (2)

SY-454 5 P.T.O.

1
10. (i) Find the principal value of cos–1 . (1)
2

1 2 3
(ii) Show that tan–1 + tan–1 = tan–1 . (3)
2 11 4

11. (i) Differentiate xsinx with respect to x. (2)
dy
(ii) Find if x = a cos  ; y = b sin  (2)
dx

12. (i) .ex (sin x + cos x) dx = _________.
.
(A) ex + C (B) ex sin x + C
(C) ex cos x + C (D) sin x + C (1)

. x
(ii) Find . dx. (3)
 (x – 1) (x – 2)

x2 y2
13. Find the area bounded by the ellipse + = 1. (4)
a2 b2

dy y
14. Consider the differential equation :   x2
dx x
(i) Find the integrating factor. (2)
(ii) Solve the differential equation. (2)

15. Consider the vectors
 
a = ^i – 2^j + 3k^ and b = 3^i – 2^j + k^
–
(i) Find a– . b. (1)
 
(ii) Find the projection of a on b. (2)

(iii) Find the unit vector in the direction of b. (1)

SY-454 6

1
10. (i) cos–1    . (1)
2

1 2 3
(ii) tan–1 + tan–1 = tan–1  . (3)
2 11 4

11. (i) xsinx  x   . (2)

dy
(ii) x = a cos  ; y = b sin  . (2)
dx

12. (i) .ex (sin x + cos x) dx = _________.
.
(A) ex + C (B) ex sin x + C
(C) ex cos x + C (D) sin x + C (1)

(ii) . x
dx . (3)
. (x – 1) (x – 2)

x2 y2
13. + = 1    . (4)
a2 b2

dy y
14.   x 2    .
dx x
(i)   . (2)

(ii)   . (2)

 ^ 
15. a = ^i – 2^j + 3k, b = 3^i – 2^j + k^
  .
–
(i) a– . b . (1)
 
(ii) a   b   . (2)

(iii) b    . (1)

SY-454 7 P.T.O.

16. Consider the line
x+3 y–4 z+8
= =
3 5 6

(i) Write the equation in vector form. (1)

(ii) Write any point on the line. (1)
(iii) Find the Cartesian equation of the line passing through the point (1, 2, 3) and
parallel to the given line. (2)

17. Find the shortest distance between the lines :

r = ^i + ^j +  (2^i – ^j + k)
– ^

r = 2^i + ^j – k^ + (3^i – 5^j + 2k)
– ^ (4)

18. A bag contains 4 red and 4 black balls, another bag contains 2 red and 6 black balls.
One of the two 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 is drawn from the first bag. (4)

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

 3 3 1 
19. (i) Express the matrix   2  2 1  as the sum of a symmetric and skew-

  4  5 2 

symmetric matrices. (3)

 3 1
(ii) If A =   , show that A2 – 5A + 7I = 0. (3)
 – 1 2

20. Solve the following system of equations by matrix method :
3x – 2y + 3z = 8

2x + y – z = 1
4x – 3y + 2z = 4 (6)

SY-454 8

x+3 y–4 z+8
16. = =   .
3 5 6
(i)    . (1)
(ii)     . (1)
(iii) (1, 2, 3)      
    . (2)

17. r = ^i + ^j +  (2^i – ^j + k)
– ^

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

     . (4)

18.   4 , 4   .   2
, 6   .    
     .   .
       
. (4)

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

 3 3 1 
19. (i)   2  2 1       
 
  4  5 2 
    . (3)

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

20. 3x – 2y + 3z = 8
2x + y – z = 1
4x – 3y + 2z = 4
     . (6)

SY-454 9 P.T.O.

21. (i) Use differentials to approximate 36.6 . (3)
(ii) Consider the curve y = 3x
(a) Find the slope of the tangent at x = 4. (1)
(b) Find the equation of tangent at x = 4. (2)

 
22. Let a = 3^i + ^j + 4k^ , b = ^i – ^j + k^
–
(i) Find a– × b. (2)
 
(ii) Find the area of the parallelogram with adjacent sides a and b. (2)
 
(iii) Find a unit vector perpendicular to both a and b. (2)

.
23. (i) Find .x2 log x dx (2)

π/2
sin x 
(ii) Prove that  sin x  cos x dx = 4 (4)
0

24. Solve the Linear Programming Problem (LPP) graphically :
Maximize Z = 4x + y
subject to
x + y  50

3x + y  90

x  0, y > 0 (6)

25. Three coins are tossed simultaneously, let X denotes the number of heads
(i) Write the probability distribution of X. (3)
(ii) Find the mean and variance of X. (3)

____________

SY-454 10

21. (i) 36.6     . (3)
(ii) y = 3x   (curve) .

(a) x = 4    . (1)

(b) x = 4    . (2)

22.
 ^ 
a = 3^i + ^j + 4k, b = ^i – ^j + k^
–
(i) a– × b  (2)
 
(ii) a , b       (2)
 
(iii) a , b      . (2)

23. (i) .x2 log x dx . (2)
.
π/2
sin x 
(ii)  sin x  cos x dx = 4  . (4)
0

24.   (LPP)   :
Maximize Z = 4x + y
subject to
x + y  50
3x + y  90
x  0, y > 0 (6)

25. 3    . X   
.
(i) X    . (3)

(ii) X  ,   . (3)

____________

SY-454 11 P.T.O.

SY-454 12

SY-427
Reg. No. : ......................................

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

SECOND YEAR HIGHER SECONDARY EXAMINATION, MARCH 2025

Part – III Time : 2 Hours
MATHEMATICS (SCIENCE) Cool-off time : 15 Minutes
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-427 1 P.T.O.

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

1. (a) A function f defined on , the natural numbers as f(n) = 2n; n . Then f is ____. (1)

(A) one-one and onto (B) one-one but not onto

(C) not one-one and not onto (D) onto but not one-one

(b) R = {(1, 4), (2, 5), (3, 6)} is a relation from A = {1, 2, 3} to B = {4, 5, 6, 7}.
Check whether R is bijection or not. Justify your answer. (2)

2. (a) If A is a square matrix such that A2 = A, then (I + A)3 – 7A is ______. (1)

 1 3   x+y 0   8 6 
(b) Find x and y if 2  +  =   (2)
 0 x   1 2   1 8 

3. (a) The graph below depicts : (1)

(A) y = sin–1 x (B) y = cos–1 x

(C) y = cosec–1 x (D) y = cot–1 x

 1 + x2 – 1
(b) Write the simplest form of tan–1   ; x  0. (2)
 x 

SY-427 2

1  8    6  .
3  . (6  3 = 18)
1. (a) f(n) = 2n; n   (1)
(A) -     
(B) -    
(C) -     
(D)   - 

(b) R = {(1, 4), (2, 5), (3, 6)}   A = {1, 2, 3}  
B = {4, 5, 6, 7}  . R  
   . (2)

2. (a) A  A2 = A,      (I + A)3 – 7A
 . (1)
 1 3   x+y 0   8 6 
(b) 2  +  =    x, y . (2)
 0 x   1 2   1 8 

3. (a)      : (1)

(A) y = sin–1 x (B) y = cos–1 x
(C) y = cosec–1 x (D) y = cot–1 x

 1 + x2 – 1
(b) tan–1      ; x  0. (2)
 x 

SY-427 3 P.T.O.

4. (a) Which of the following is an increasing function in ? (1)

(A) cos x (B) x3

(C) x2 (D) | x |

(b) Find the interval in which the function f(x) = x2 – 4x + 6 is increasing or
decreasing in . (2)

5. Let f :  be defined by

 kx + 1 if x  
f(x) = 
 cos x if x > 

(a) For what value of k, f is continuous ? (1)

(b) Find f(2). (2)

.
6. (a) The value of the integral .ex (sin x + cos x)dx is ______ (1)


. tan–1 x
(b) Evaluate . dx (2)
 1 + x2
0

     
7. (a) If  is the angle between two vectors a and b with | a · b| = | a  b|, then  is : (1)


(A) 0 (B)
2


(C) (D) 
4

(b) Write a unit vector in the direction of a = ^i – k.
^ (2)

8. If P(A) = 0.8, P(B) = 0.5 and P(B/A) = 0.4, then find

(a) P(A  B) (1)

(b) P(A/B) (1)

(c) Check whether A and B are independent events (1)

SY-427 4

4. (a)      
 ? (1)

(A) cos x (B) x3
(C) x2 (D) | x |
(b) f(x) = x2 – 4x + 6      
 . (2)

5. f:  
 kx + 1 if x  
f(x) =  
 cos x if x > 
(a) f   k   . (1)
(b) f(2) . (2)

6. (a) .ex (sin x + cos x)dx . (1)
.

. tan–1 x
(b) . dx   . (2)
 1 + x2
0

     
7. (a) a  b       | a · b| = | a  b|   
  : (1)

(A) 0 (B)
2

(C) (D) 
4

(b) a = ^i – k^      . (2)

8. P(A) = 0.8, P(B) = 0.5, P(B/A) = 0.4 ,   

(a) P(A  B) (1)
(b) P(A/B) (1)
(c) A  B     . (1)
SY-427 5 P.T.O.

Answer any 6 questions from 9 to 16. Each carries 4 scores. (6  4 = 24)

9. (a) What is the minimum number of ordered pairs to form a reflexive relation on

A = {1, 2, 3} ? (1)

(b) Show that the relation R = {(a, b) : | a – b | is even } defined on

A = {1, 2, 3, 4, 5} is an equivalence relation. (3)

 3 1 
10. (a) Express A =   as the sum of symmetric and skew symmetric matrix. (2)
 –1 2 

(b) Prove that A in part(a) satisfies A2 – 5A + 7I = 0. (2)

11. Show that of among all rectangles inscribed in a given circle, the square has the

maximum area. (4)

 ^ and 
12. Given lines r = 3^i + ^j – 2k^ + (^i – ^j – 2k) r = 2^i – ^j – 56k^ + (3^i – 5^j – 4k)
^

(a) Obtain a vector perpendicular to both lines. (2)

(b) Find the equation of the line perpendicular to given lines and passing through the

point (1, 2, 3). (2)

13. Using integration, find the area of the region bounded by the ellipse

x2 y2
+ = 1 in the first quadrant. (4)
4 9

SY-427 6

9  16    6  .
4  . (6  4 = 24)

9. (a) A = {1, 2, 3}       
  . (1)

(b) R = {(a, b) : | a – b |   }  

A = {1, 2, 3, 4, 5}     
 . (3)

 3 1 
10. (a) A=     , 
 –1 2 
  . (2)

(b)  (a)  A  A2 – 5A + 7I = 0   . (2)

11.     
    . (4)


12. r = 3^i + ^j – 2k^ + (^i – ^j – 2k)
^


r = 2^i – ^j – 56k^ + (3^i – 5^j – 4k)
^

 

(a)      . (2)

(b)    (1, 2, 3)   
 . (2)

x2 y2
13.   + = 1    
4 9
 . (4)

SY-427 7 P.T.O.

14. (a) If ^i, ^j, k^ are unit vectors in the coordinate system, then the value of
^i(^j  k)
^ + ^j(^i  k)
^ + k(
^ ^i  ^j) is _______. (1)

(A) 0 (B) –1

(C) 1 (D) 3

(b) Using vectors, find the area of the triangle with vertices (1, 1, 1), (1, 2, 3) and
(2, 3, 1). (3)

15. Given three identical boxes I, II and III, each containing two coins. In box I, both coins
are gold, in box II, both are silver and in box III, there is one gold and one silver coin.
A person chooses a box at random and takes out a coin. If the coin is of gold, what is
the probability that the other coin in the box is also of gold ? (4)

16. (a) Prove that y = cos x + A, (A is an arbitrary constant) is a solution of the
dy
differential equation + sin x = 0. (1)
dx

dy
(b) Consider the differential equation x + 2y = x2
dx

(i) Find the integrating factor. (1)

(ii) Write the general solution. (2)

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

c d2 y
17. (a) If x = ct, y = , then find 2 (2)
t dx

dy
(b) Find of the following :
dx
1 1 1
(i) 3 3
x +y =a3 (2)

(ii) y = (sin x)cos x (2)

SY-427 8

14. (a) ^i, ^j, k^    ^i(^j  k)
^ + ^j(^i  k)
^ + k(
^ ^i  ^j) 

. (1)

(A) 0 (B) –1
(C) 1 (D) 3
(b)   (1, 1, 1), (1, 2, 3), (2, 3, 1)  
  . (3)

15.     .    
      
     . 
       . 
     
 . (4)

dy
16. (a) y = cos x + A  + sin x = 0   
dx
  . (A is an arbitrary constant) (1)

dy
(b) x + 2y = x2    .
dx
(i)   . (1)

(ii)   . (2)

17  20    3  .
6  . (3  6 = 18)
c d2 y
17. (a) x = ct, y =  2 . (2)
t dx
dy
(b)  :
dx
1 1 1
(i) 3 3
x +y =a3 (2)
(ii) y = (sin x)cos x (2)

SY-427 9 P.T.O.

18. Consider the system of equations,
2x + 3y + 3z = 5
x – 2y + z = – 4
3x – y – 2z = 3
(i) Write its matrix form. (1)
(ii) Prove that the system is consistent. (1)
(iii) Solve the system using matrix method. (4)

19. Evaluate the following :

(i) . x
dx (1)
.(x + 1)(x + 2)

(ii) . 1
dx (1)
.x2 – 6x + 13


.
(iii) .| x – 1 | dx (4)

0

20. Using graphical method, solve the Linear Programming
Maximize, Z = 3x + 4y
subject to the constraints
x + 2y < 10
3x + y < 15; x > 0, y > 0 (6)
______________

SY-427 10

18.      
2x + 3y + 3z = 5
x – 2y + z = – 4
3x – y – 2z = 3
(i)    . (1)

(ii)    . (1)

(iii)     . (4)

19.   :

(i) . x
dx (1)
.(x + 1)(x + 2)

(ii) . 1
dx (1)
.x – 6x + 13
2


.
(iii) .| x – 1 | dx (4)

0

20.       
.
Maximize, Z = 3x + 4y
subject to the constraints
x + 2y < 10
3x + y < 15; x > 0, y > 0 (6)

______________

SY-427 11 P.T.O.

SY-427 12