Kerala Plus One Question Paper 2025 Mathematics Commerce

KERALA BOARD

QUESTION
PAPER
2025
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Kerala Board of Public
Examinations

FY-355
Reg. No. : ......................................

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

FIRST 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  ‘  ’ .
 ‘  ’    
 .
      .
    .
  , , ,   
.
   .
    .
     
    .

FY-355 1 P.T.O.

Answer any 6 questions from 1 to 7. Each carries 3 scores. (6  3 = 18)

1. (a) {x : x  R, 2  x < 5} = __________. (1)

(A) [2, 5] (B) [2, 5)

(C) (2, 5) (D) (2, 5]

(b) Write all subsets of {1, 2}. (2)

2. (a) If A = {3, 4} and B = {5, 6}, find A  B. (1)

(b) Draw the graph of the function f(x) = | x |. (2)

5
3. (a) radian = __________ degree. (1)
3

(A) 300 (B) 150

(C) 120 (D) 240

3
(b) sin x = , x lies in the second quadrant. Find the values of cos x and tan x. (2)
5

4. Consider the inequality 3(x –1)  2(x – 3).

(a) Solve the above inequality. (2)

(b) Represent the solution on a number line. (1)

5. (a) i12 = _________. (1)

(A) 1 (B) –1

(C) i (D) –i

(b) Solve the equation :

x2 + 3x + 5 = 0 (2)

FY-355 2

1  7    6  .

3  . (6  3 = 18)

1. (a) {x : x  R, 2  x < 5} = __________. (1)

(A) [2, 5] (B) [2, 5)
(C) (2, 5) (D) (2, 5]
(b) {1, 2}    . (2)

2. (a) A = {3, 4}, B = {5, 6}  A  B . (1)

(b) f(x) = | x |   . (2)

5
3. (a)  = __________ . (1)
3

(A) 300 (B) 150
(C) 120 (D) 240

3
(b) sin x = , x   cos x, tan x 
5
 . (2)

4. 3(x –1)  2(x – 3)   .

(a)   . (2)

(b)    . (1)

5. (a) i12 = _________. (1)
(A) 1 (B) –1
(C) i (D) –i

(b) x2 + 3x + 5 = 0     : (2)

FY-355 3 P.T.O.

6. Find the mean deviation about the mean for the data :

3, 6, 8, 9, 13, 14, 17

7. If A and B are two events such that P(A) = 0.42, P(B) = 0.48 and P(A  B) = 0.16
then find

(a) P(not A) (1)

(b) P(A  B) (2)

Answer any 8 questions from 8 to 17. Each carries 4 scores. (8  4 = 32)

8. (a) If A  B, then A  B = ___________. (1)

(b) If A = {1, 2, 3, 4} and B = {3, 4, 5, 6}, find A – B. (1)

(c) If X and Y are two sets such that n(X) = 17, n(Y) = 23 and n(X  Y) = 38, then
find n(X  Y). (2)

9. (a) If (x + 3, y – 1) = (4, 2), find the value of x and y. (2)

(b) Let A = {1, 2, 3, 4, 5, 6} be a set. Define a relation R from A to A by
R = {(x, y) : y = x + 1}. Find the domain and range of R. (2)

10. (a) sin ( – x) = _________. (1)

(b) Prove that :

sin 5 x  sin 3 x
= tan 4x (3)
cos 5 x  cos 3 x

11. Using the principle of mathematical induction, prove that
n
3 –1
1 + 3 + 32 + .... + 3n – 1 = .
2

FY-355 4

6.       
 :
3, 6, 8, 9, 13, 14, 17

7. A, B    P(A) = 0.42, P(B) = 0.48, P(A  B) = 0.16
    
(a) P(not A) (1)
(b) P(A  B) (2)

8  17    8  .
4  . (8  4 = 32)

8. (a) A  B , A  B = ___________. (1)

(b) A = {1, 2, 3, 4}, B = {3, 4, 5, 6}  A – B . (1)

(c) X, Y    n(X) = 17, n(Y) = 23, n(X  Y) = 38 
n(X  Y) . (2)

9. (a) (x + 3, y – 1) = (4, 2)  x, y   . (2)

(b) A = {1, 2, 3, 4, 5, 6}   . R = {(x, y) : y = x + 1}
 A   A    .
R  ,   . (2)

10. (a) sin ( – x) = _________. (1)
sin 5 x  sin 3 x
(b) = tan 4x  . (3)
cos 5 x  cos 3 x

3n – 1
11. 1 + 3 + 32 + .... + 3n – 1 =    
2
  .

FY-355 5 P.T.O.

12. (a) Conjugate of 3 + i is ________. (1)

(b) Represent the complex number 3 + i in polar form. (3)

13. (a) nC
n = ________. (1)

(b) If nC8 = nC2, then find n. (1)

(c) How many 3 digit numbers can be formed with the digits 1, 2, 3, 4 where

repetition of the digits is not allowed. (2)

14. Find (a + b)4 – (a – b)4. Hence evaluate ( 3 + 5 )4 – ( 3 – 5 )4.

15. Find the sum to n terms of the series 4 + 44 + 444 + ......

16. (a) A point in the XY plane is (1)

(A) (0, 1, 2) (B) (1, 0, 2)

(C) (1, 2, 0) (D) (1, 0, 9)

(b) Show that the points A(0, 7, 10), B(–1, 6, 6) and C(–4, 9, 6) are the vertices of a

right triangle. (3)

17. (a) Write the negation of the statement :

‘ 5 is not a complex number’ (1)

(b) Verify by the method of contradiction 3 is irrational. (3)

FY-355 6

12. (a) 3 + i   ________ . (1)

(b) 3 + i      . (3)

13. (a) nC
n = ________. (1)

(b) nC nC
8= 2  n   . (1)

(c) 1, 2, 3, 4    

  3    (2)

14. (a + b)4 – (a – b)4 ,  ( 3 + 5 )4 – ( 3 – 5 )4 

 .

15. 4 + 44 + 444 + ......    ‘n’   .

16. (a)   XY   (1)

(A) (0, 1, 2) (B) (1, 0, 2)

(C) (1, 2, 0) (D) (1, 0, 9)

(b) A(0, 7, 10), B(–1, 6, 6), C(–4, 9, 6)   

 . (3)

17. (a) ‘ 5 is not a complex number’

   . (1)

(b) 3 is irrational     . (3)

FY-355 7 P.T.O.

Answer any 5 questions from 18 to 24. Each carries 6 scores. (5  6 = 30)
18. Let U = {1, 2, 3, 4, 5, 6, 7}, A = {2, 4, 6} and B = {3, 4, 5}
(a) Find A' and B' (2)
(b) Find A  B and (A  B)' (2)
(c) Verify that (A  B)' = A'  B' (2)

19. Solve the following system of inequalities graphically :
5x + 4y  20
x 1
y1

20. Consider the straight line 3x + 4y – 12 = 0
(a) Slope of the line is (1)
3 3
(A) – (B)
4 4
4 4
(C) (D) –
3 3
(b) Write the above equation in intercept form and find its x and y intercepts. (3)
(c) Find the equation of a line parallel to the above line and passes through the
point (1, 2). (2)

21. (a) Length of latus rectum of the parabola y2 = –8x is (1)
(A) 8 (B) –8
(C) 4 (D) –4

x2 y2
(b) Consider the ellipse  = 1. Find
16 9
(i) Foci (2)
(ii) Vertices (1)
(iii) Length of major and minor axis (2)

FY-355 8

18  24    5  .
6  . (5  6 = 30)
18. U = {1, 2, 3, 4, 5, 6, 7}, A = {2, 4, 6}, B = {3, 4, 5} 
(a) A', B'  . (2)
(b) A  B, (A  B)'  . (2)
(c) (A  B)' = A'  B'   . (2)

19.        :
5x + 4y  20
x 1
y1

20. 3x + 4y – 12 = 0   .
(a)    (1)
3 3
(A) – (B)
4 4
4 4
(C) (D) –
3 3
(b)     x, y 
. (3)
(c)    (1, 2)    
  . (2)

21. (a) y2 = –8x      (1)
(A) 8 (B) –8
(C) 4 (D) –4
x2 y2
(b)  = 1   
16 9
(i)   (2)
(ii)   (1)
(iii)      . (2)

FY-355 9 P.T.O.

d
22. (a) (tan x) = __________. (1)
dx

(b) Find the derivative of f(x) = sin x with respect to x from first principle. (3)

(c) Find the derivative of x sin x. (2)

23. Find the mean and variance of the following data :

Class 0 – 10 10 – 20 20 – 30 30 – 40 40 – 50

Frequency 4 7 16 16 7

24. (a) If S is a sample space, then P(S) = ________. (1)

(b) Consider the random experiment “Two coins are tossed one”.

(i) Write the sample space. (1)

(ii) Write the events

A : First coin shows head

B : No head appears (2)

(iii) Find P(A) and P(B). (2)

______________

FY-355 10

d
22. (a) (tan x) = __________. (1)
dx

(b) f(x) = sin x  with respect to x    

 . (3)

(c) x sin x   . (2)

23.     ,   :

Class 0 – 10 10 – 20 20 – 30 30 – 40 40 – 50

Frequency 4 7 16 16 7

24. (a) S     P(S) = ________. (1)

(b) “    ”   

.

(i)    . (1)

(ii) A :    

B :    .

  . (2)

(iii) P(A), P(B) . (2)

______________

FY-355 11 P.T.O.

FY-355 12

FY-351
Reg. No. : ......................................

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

FIRST 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  ‘  ’ .
 ‘  ’    
 .
      .
    .
  , , ,   
.
   .
    .
     
    .

FY-351 1 P.T.O.

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

1. (i) The shaded region in the given Venn diagram is (1)

(a) A' (b) B–A

(c) A–B (d) B'

(ii) If A = {–1, 0, 1}, write all subsets of the set A. (2)

2. (i) If (x – 1, y + 2) = (2, 1), then the value of x and y are ________. (1)

(ii) If A = {0, 1}, write A  A  A. (2)

3. Find the number of arrangements of the letters of the word ‘MATHEMATICS’. How

many of these start with H ?

4. (i) Name the octant in which the point (4, –5, 6) belongs. (1)

(ii) Find the distance between the points A(2, 3, 5) and B(4, 3, 1). (2)

FY-351 2

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

3  . (6  3 = 18)

1. (i)        (1)

(a) A' (b) B–A

(c) A–B (d) B'

(ii) A = {–1, 0, 1},  A    . (2)

2. (i) (x – 1, y + 2) = (2, 1),  x  y   ________ . (1)

(ii) A = {0, 1},  A  A  A . (2)

3. ‘MATHEMATICS’      

 ?   H   ?

4. (i) (4, –5, 6)     . (1)

(ii) A(2, 3, 5), B(4, 3, 1)    . (2)

FY-351 3 P.T.O.

5. Consider the following circle :

(i) Write the centre of the circle. (1)

(ii) Find the equation of the circle. (2)

6. (i) lim sin x
x  0 x = __________. (1)

sin ax  bx
(ii) Evaluate xlim
0 . (2)
ax

x –1 dy
7. If y = , then find .
x2 dx

8. Consider the data :

5, 7, 6, 9, 4, 11, 8, 6

(i) Find the mean for the data. (1)

(ii) Also find the Mean Deviation about its Mean. (2)

FY-351 4

5.     :

(i)   . (1)

(ii)   . (2)

6. (i) lim sin x
x  0 x = __________. (1)

(ii) lim sin ax  bx   . (2)
x0 ax

x –1 dy
7. y= ,  .
x2 dx

8.     :

5, 7, 6, 9, 4, 11, 8, 6

(i)    . (1)

(ii)      . (2)

FY-351 5 P.T.O.

Answer any 6 questions from 9 to 16. Each carries 4 scores. (6  4 = 24)
9. (i) A  A' = ________. (1)
(ii) If U = {1, 2, 3, 4, 5, 6}, A = {2, 3, 4} and B = {3, 4, 6}, then verify that
(A  B) ' = A'  B'. (3)

10. (i) Draw the graph of the function, f :  defined by f(x) = | x – 2 |. (2)

x–3
(ii) Find the domain and range of f(x) = . (2)
x–4

   1
11. (i) Show that sin2 + cos2 – tan2  – . (2)
6 3 4 2
cos 7 x  cos 5 x
(ii) Prove that = cot x (2)
sin 7 x – sin 5 x

12. (i) Find the multiplicative inverse of the complex number z = 3 – 4i. (2)
1 i
(ii) Express the complex number in x + iy form. (2)
1– i

2 x – 1 3x – 2
13. (i) Solve the linear inequality  . (3)
3 4
(ii) Represent the solution in real line. (1)

n
14. (i) Cr = __________. (1)

n! n!
(a) (b)
r! r!(n – r)!

n! (n – r)!
(c) (d)
(n – r)! r!

(ii) In how many ways can one select a cricket team of eleven from 17 players in
which only 5 players can bowl if each cricket team of 11 must include exactly
4 bowlers ? (3)

FY-351 6

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

9. (i) A  A' = ________. (1)
(ii) U = {1, 2, 3, 4, 5, 6}, A = {2, 3, 4}, B = {3, 4, 6},  (A  B) ' = A'  B'
 . (3)

10. (i) f:   f(x) = | x – 2 |    . (2)
x–3
(ii) f(x) =     . (2)
x–4

   1
11. (i) sin2 + cos2 – tan2    . (2)
6 3 4 2
cos 7 x  cos 5 x
(ii) = cot x  . (2)
sin 7 x – sin 5 x

12. (i) z = 3 – 4i      . (2)

1 i
(ii)  x + iy  . (2)
1– i

2 x – 1 3x – 2
13. (i)       . (3)
3 4
(ii)    . (1)

n
14. (i) Cr = __________. (1)

n! n!
(a) (b)
r! r!(n – r)!

n! (n – r)!
(c) (d)
(n – r)! r!

(ii) 5   17     11
      ? (3)

FY-351 7 P.T.O.

15. (i) Number of terms in the expansion of (a + b)n is (1)

(a) n (b) 2n

(c) 2n (d) n+1

5
 x 1
(ii) Expand using Binomial theorem  –  . (3)
3 x

16. An ellipse has its foci at (5, 0) and length of major axis 26.

(i) Find the length of the minor axis. (1)

(ii) Find the length of the latusrectum and eccentricity of the ellipse. (2)

(iii) Also write the equation of the ellipse. (1)

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

17. (i) How many terms of the G.P. 3, 32, 33, ..... are needed to give the sum 120 ? (2)

(ii) The 5th, 8th and 11th terms of a G.P. are p, q and s respectively. Show that q 2 = ps. (2)

1 1 1
(iii) Find the sum to infinite terms of the G.P., 1, , , , .... (2)
2 4 8

18. (i) The equation of x-axis is (1)

(a) x=k (b) x=0

(c) y=k (d) y=0

(ii) Find the equation of the line through (2, 3) and perpendicular to the line through
(2, 5) and (–3, 6). (3)

(iii) Find the distance of the point (2, –1) from the line 4x – 3y – 1 = 0. (2)

FY-351 8

15. (i) (a + b)n     (1)

(a) n (b) 2n

(c) 2n (d) n+1
5
 x 1
(ii)     –  . (3)
3 x

16.    (5, 0)     26  .

(i)    . (1)

(ii)      . (2)

(iii)    . (1)

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

17. (i) 3, 32, 33, .....  G.P.     120  ? (2)

(ii) 5th, 8th, 11th   p, q, s . q2 = ps  . (2)

1 1 1
(iii) 1, , , , ....   G.P.    . (2)
2 4 8

18. (i) x-axis   (1)

(a) x=k (b) x=0

(c) y=k (d) y=0

(ii) (2, 5), (–3, 6)       , (2, 3)
       . (3)

(iii) 4x – 3y – 1 = 0    (2, –1)   
. (2)

FY-351 9 P.T.O.

19. Consider the following data :

Class 30-40 40-50 50-60 60-70 70-80 80-90 90-100

Frequency 3 7 12 15 8 3 2

(i) Find the mean. (2)

(ii) Find the variance. (3)

(iii) Find the Standard Deviation. (1)

2 1 1
20. (i) If two events A and B such that P(A) = , P(B) = and P(A  B) = , then
5 2 5

find P(A  B). (2)

(ii) A bag contains 8 red and 5 white balls. Three balls are drawn at random. Find the
probability that (4)

(a) All the three balls are white

(b) All the three balls are red

(c) One ball is red and two balls are white

_____________

FY-351 10

19.     :

Class 30-40 40-50 50-60 60-70 70-80 80-90 90-100

Frequency 3 7 12 15 8 3 2

(i)  . (2)

(ii)  . (3)

(iii)   . (1)

2 1 1
20. (i) P(A) = , P(B) = , P(A  B) =   
5 2 5

 A  B   P(A  B) . (2)

(ii)   8   5   . 

 3   ,  

 . (4)

(a)   

(b)   

(c)   ,    

_____________

FY-351 11 P.T.O.

FY-351 12