Kerala Plus One Question Paper 2025 Mathematics Science

KERALA BOARD

QUESTION
PAPER
2025
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Kerala Board of Public
Examinations

FY-327
Reg. No. : ......................................

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

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

FY-327 1 P.T.O.

Answer any 6 questions from 1 to 8. Each carries 3 scores. (6  3 = 18)
1. (a) Write the set A = {1, 4, 9, 16, .... 100} in set builder form. (1)
(b) Write all subsets of the set {1, 2, 3}. (2)

4
2. (a) The degree measure of = __________. (1)
3
3
(b) If cos x = – , x lies in third quadrant. Find the values of the trigonometric
5
functions sin x and tan x. (2)

3. (a) Solve the inequality 3(x – 1)  2(x – 3); x  . (2)
(b) Mark the solution obtained in part(a) to the real line . (1)

4. (a) How many 4 digit numbers can be formed by using the digits 1 to 9, the repetition
not allowed ? (1)
n!
(b) Evaluate for the case n = 9, r = 5. (2)
(n – r)!

5. (a) Which of the following is a point in the 7th octant ? (1)
(A) (2, –2, 2) (B) (2, 2, 2)
(C) (2, 2, –2) (D) (–2, –2, –2)
(b) Find the distance between the points (–2, 3, 5) and (7, 0, –1). (2)

6. Find the centre and radius of the circle x2 + y2 – 4x – 8y – 45 = 0.

tan 2 x
7. (a) Evaluate xlim
0 x (1)

1 – cos x
(b) Prove that xlim
0 =0 (2)
x

8. (a) Write the equation of y-axis. (1)
(b) Find the slope of a line which passes through the origin and the midpoint of the
line segment joining the points P(0, –4) and Q(8, 0). (2)

FY-327 2

1  8    6  . 3 
. (6  3 = 18)
1. (a) A = {1, 4, 9, 16, .... 100}     . (1)
(b) {1, 2, 3}     . (2)

4
2. (a)    = __________. (1)
3
3
(b) x-    (quadrant)  cos x = –
5
, sin x, tan x    
. (2)

3. (a) 3(x – 1)  2(x – 3); x 
   . (2)
(b) (a)     . (1)

4. (a) 1  9     
 4     ? (1)
n!
(b) n = 9, r = 5   . (2)
(n – r)!

5. (a)    7-    ? (1)
(A) (2, –2, 2) (B) (2, 2, 2)
(C) (2, 2, –2) (D) (–2, –2, –2)
(b) (–2, 3, 5), (7, 0, –1)     . (2)

6.       x2 + y2 – 4x – 8y – 45 = 0.

tan 2 x
7. (a)   xlim
0 (1)
x
1 – cos x
(b)  xlim
0 =0 (2)
x

8. (a) y-  . (1)
(b) P(0, –4), Q(8, 0)     
      
. (2)

FY-327 3 P.T.O.

Answer any 6 questions from 9 to 16. Each carries 4 scores. (6  4 = 24)
9. (a) Write the set-builder form of the subset (–1, 2) of . (1)
(b) Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} be the universal set having subsets
A = {1, 3, 5, 7, 9} and B = {5, 6, 7, 8, 9, 10}, then find (3)
(i) AB
(ii) A' and B'
(iii) Verify that (A  B) ' = A'  B'

10. (a) Draw the graph of the function f(x) = 2 – | x |. (2)
(b) Write the domain and range of f(x) in part (a). (2)

3
 i
11. (a) Express z =  3   into a + ib form. (2)
 3
(b) Find the multiplicative inverse of 4 – 3i. (2)

12. (a) Determine the number of 5 card combinations out of a pack of 52 playing cards if
there is exactly one ace in each combination. (2)
(b) If nC9 = nC8, then find nC17. (2)

1
13. (a) Find the equation of a line passing through the point (–2, 3) with slope . (2)
4
(b) Find the distance of the point (3, –5) from the line 3x – 4y – 26 = 0. (2)

14. Find the equation of the ellipse whose length of the major axis is 26 and foci are (5, 0).

15. (a) Simplify (a + b)4 – (a – b)4 using binomial theorem. (2)
(b) Using part (a), evaluate ( 3 + 2 )4 – ( 3 – 2 )4. (2)

16. If A and B are two events such that P(A) = 0.42, P(B) = 0.48 and P(A and B) = 0.16,
then find the following :
(i) P(not A) (1)
(ii) P(not B) (1)
(iii) P(A or B) (2)

FY-327 4

9  16    6  .
4  . (6  4 = 24)
9. (a)   (–1, 2)     . (1)
(b) U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}    
A = {1, 3, 5, 7, 9}, B = {5, 6, 7, 8, 9, 10}   (3)

(i) AB
(ii) A', B'
(iii) (A  B) ' = A'  B'  

10. (a) f(x) = 2 – | x |    . (2)
(b)  (a)  f(x)    . (2)

3
 i
11. (a) z =  3   - (a + ib) . (2)
 3
(b) 4 – 3i      . (2)

12. (a) 52    (ace)  
 5     . (2)
(b) C9 = C8,  C17- .
n n n (2)

1
13. (a) (–2, 3)       
4
 . (2)
(b) (3, –5)   3x – 4y – 26 = 0   
. (2)

14.    26   (5, 0)  
 .

15. (a)     (a + b)4 – (a – b)4. (2)
(b)  (a)   ( 3 + 2 )4 – ( 3 – 2 )4. (2)

16. A, B   .   P(A) = 0.42,
P(B) = 0.48, P(A and B) = 0.16 .   .
(i) P(not A) (1)
(ii) P(not B) (1)
(iii) P(A or B) (2)

FY-327 5 P.T.O.

Answer any 3 questions from 17 to 20. Each carries 6 scores. (3  6 = 18)
17. (a) sin (765º) = __________. (1)
(A) 0 (B) 1
1 1
(C) (D) –
2 2
(b) Find the value of cos (75º). (2)
(c) Show that tan 3x tan 2x tan x = tan 3x – tan 2x – tan x. (3)

18. (a) Which term of the Geometric Progression 2, 8, 32, .... is 8192 ? (2)
(b) Find the sum to n terms of the sequence 8, 88, 888, ... (2)
(c) Insert three numbers between 1 and 256 so that the resulting sequence is a
Geometric Progression. (2)

19. (a) Find the derivative of tan x using first principle. (4)
dy
(b) Find , if y = sin x cos x. (2)
dx

20. Find Mean, Variance and Standard Deviation of 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

______________

FY-327 6

17  20    3  .
6  . (3  6 = 18)
17. (a) sin (765º) = __________. (1)
(A) 0 (B) 1
1 1
(C) (D) –
2 2
(b)   cos (75º). (2)
(c)  tan 3x tan 2x tan x = tan 3x – tan 2x – tan x. (3)

18. (a) 2, 8, 32, ....      8192 ? (2)
(b) 8, 88, 888, ...   ‘n’   . (2)
(c)      1  256  
3  . (2)

19. (a)    tan x   . (4)
dy
(b) y = sin x cos x  . (2)
dx

20.    , ,  ,
.
Class 30-40 40-50 50-60 60-70 70-80 80-90 90-100

Frequency 3 7 12 15 8 3 2

______________

FY-327 7 P.T.O.

FY-327 8

FY-354
Reg. No. : ......................................

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

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

FY-354 1 P.T.O.

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

1. (i) A  A' = __________. (1)
(A) U (B) A'

(C)  (D) A

(ii) If B = {x : x  N, x is a prime number less than 5}, then write the set B in roster
form. (1)
(iii) Write all subsets of the above set B. (1)

2. (i) 120 degree = _________ radian. (1)

2π π
(A) (B)
3 3
π 3
(C) (D)
2 2
(ii) Find the value of cos 15º. (2)

3. (i) i4 + i10 = _________. (1)
(A) 1 (B) 0
(C) –1 (D) i

(ii) Express the complex number Z = (2 + i)2 in the form a + ib. (2)

4. (i) Solve the inequality 5(x – 1)  4(2x + 1) (2)
(ii) Show the graph of this solution on number line. (1)

5. (i) 3P
3 = ________. (1)

(A) 3 (B) 4
(C) 5 (D) 6
(ii) How many words can be made from the letters of the word MONKEY, if
4 letters are used at a time without repetition of letters ? (2)

FY-354 2

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

3  . (6  3 = 18)

1. (i) A  A' = __________. (1)
(A) U (B) A'
(C)  (D) A
(ii) B = {x : x  N, x  5   } B 
   . (1)
(iii)   B     . (1)

2. (i) 120  = _________ . (1)
2π π
(A) (B)
3 3
π 3
(C) (D)
2 2
(ii) cos 15º   . (2)

3. (i) i4 + i10 = _________. (1)
(A) 1 (B) 0
(C) –1 (D) i
(ii) Z = (2 + i)2    a + ib  . (2)

4. (i) 5(x – 1)  4(2x + 1)    . (2)
(ii)       . (1)

5. (i) 3P
3 = ________. (1)

(A) 3 (B) 4
(C) 5 (D) 6
(ii) MONKEY   , 4   
     ? (2)

FY-354 3 P.T.O.

6. Consider the line joining the points (3, 4) and (–1, 2).
(i) Slope of this line is ___________. (1)
1
(A) 2 (B) –
2
1
(C) –2 (D)
2
(ii) Find the equation of a line passing through point (–3, 4) and perpendicular to
the given line. (2)

7. (i) Centre of the circle (x + 2)2 + (y – 3)2 = 16 is __________. (1)
(A) (2, –3) (B) (–2, 3)
(C) (2, 3) (D) (–2, –3)
(ii) Find the equation of the parabola with vertex (0, 0), passing through (–2, 4) and
axis is along x-axis. (2)

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

1
(A) 1 (B)
a
(C) a (D) 0

lim  x3 – 3 if x  2
(ii) Find x  2 f(x) where f(x) =  2 (2)
 x  1 if x  2

Answer any 8 questions from 9 to 18. Each carries 4 scores. (8  4 = 32)
9. (i) If (2 – x, 3 – y) = (4, 3), find the values of x and y. (2)
(ii) If A = {3, 4}, B = {–1, –2}, find A  B. (2)

10. Using the principle of mathematical induction, prove that
n(n  1)
1 + 2 + 3 + ..... + n = for all n  N.
2
FY-354 4

6. (3, 4), (–1, 2)      .

(i)    ___________. (1)
1
(A) 2 (B) –
2
1
(C) –2 (D)
2
(ii)    (–3, 4)    
   . (2)

7. (i) (x + 2)2 + (y – 3)2 = 16    __________. (1)
(A) (2, –3) (B) (–2, 3)
(C) (2, 3) (D) (–2, –3)
(ii)  (0, 0)  (–2, 4)      
x-   . (2)

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

1
(A) 1 (B)
a
(C) a (D) 0
 x3 – 3 if x  2
(ii) f(x) =  2  xlim
 2 f(x) . (2)
 x  1 if x  2

9  18    8  .
4  . (8  4 = 32)
9. (i) (2 – x, 3 – y) = (4, 3)  x, y   . (2)
(ii) A = {3, 4}, B = {–1, –2}  A  B . (2)

10.     
n(n  1)
1 + 2 + 3 + ..... + n = for all n  N  .
2

FY-354 5 P.T.O.

11. Consider the complex number Z = –8 + 6i.

(i) Find Z . (1)
(ii) Find | Z |. (1)
(iii) Find the multiplicative inverse of Z. (2)

12. Solve the following system of inequalities graphically :

x + 2y  8

2x + y  8

x  0, y  0

13. (i) If nC3 = nC7, then value of n is _________. (1)

(A) 7 (B) 3
(C) 4 (D) 10
(ii) In how many ways a committee of 4 men and 3 women can be selected from
6 men and 5 women ? (3)

2n
 1
14. (i) The total number of terms in the expansion of  x   is __________. (1)
 x

(A) 2n + 1 (B) 2n
(C) n+1 (D) n

(ii) Find (x + 1)4 + (x – 1)4. (3)

15. (i) y intercept of the line having the equation y = 2x – 3 is __________. (1)
(A) 2 (B) 3
(C) –2 (D) –3
(ii) Find the equation of the line passing through the point (–1, 3) and cutting off
equal intercepts on the coordinate axes. (3)

FY-354 6

11. Z = –8 + 6i    .

(i) Z . (1)
(ii) | Z | . (1)
(iii) Z    . (2)

12.       
 :
x + 2y  8
2x + y  8
x  0, y  0

13. (i) nC nC
3= 7  n   _________ . (1)
(A) 7 (B) 3
(C) 4 (D) 10
(ii) 6   5   4  3 
     . (3)

2n
 1
14. (i) x       __________
 x
. (1)
(A) 2n + 1 (B) 2n
(C) n+1 (D) n
(ii) (x + 1)4 + (x – 1)4 . (3)

15. (i) y = 2x – 3    y  ________ . (1)
(A) 2 (B) 3
(C) –2 (D) –3
(ii) (–1, 3)
    
    
 . (3)

FY-354 7 P.T.O.

16. Find the coordinates of the foci, vertices, the eccentricity and the length of latus
x2 y2
rectum of the ellipse  = 1.
169 25

17. (i) Which of the following point lie on the 7th octant ? (1)
(A) (2, 1, 3) (B) (–2, 1, –3)
(C) (2, –1, –3) (D) (–2, –1, –3)
(ii) Show that the points A(–2, 3, 5), B(1, 2, 3) and C(7, 0, –1) are collinear. (3)

18. (i) Write the negation of the statement : (1)
“Today is Monday”

(ii) Using the method of contradiction, prove that 5 is irrational. (3)

Answer any 5 questions from 19 to 25. Each carries 6 scores. (5  6 = 30)
19. (i) If U = {0, 1, 2, 3, 4, 5, 6}, A = {0, 2, 4}, B = {0, 1, 2, 3}, verify that
(A  B)' = A'  B'. (3)
(ii) In a class of 60 students, 35 like football, 15 like both football and volley ball.
How many students like volleyball ? (3)

20. Consider the set A = {–1, 0, 1, 2, 3}.

(i) A relation R from A to A is defined by R = {(x, y) : x, y  A and y = x – 1}.
Write down the domain, co-domain and range of R. (3)

(ii) Draw the graph of the function f : R  R defined by f(x) = | x |. Write domain
and range of f(x). (3)

cos 8 x  cos 6 x
21. (i) Prove that = cot x. (3)
sin 8 x – sin 6 x
(ii) Find the general solution of the trigonometric equation cos 4x = cos 2x. (3)

FY-354 8

x2 y2
16.  = 1   ,  
169 25
 , ,    
.

17. (i)   7-   
. (1)
(A) (2, 1, 3) (B) (–2, 1, –3)
(C) (2, –1, –3) (D) (–2, –1, –3)
(ii) A(–2, 3, 5), B(1, 2, 3), C(7, 0, –1)    
. (3)

18. (i) “Today is Monday”    . (1)

(ii)    5   . (3)

19  25    5  .
6  . (5  6 = 30)
19. (i) U = {0, 1, 2, 3, 4, 5, 6}, A = {0, 2, 4}, B = {0, 1, 2, 3}  (A  B)' = A'  B'
  . (3)
(ii) 60   , 35   , 15 
   .   
 ? (3)

20. A = {–1, 0, 1, 2, 3}   .
(i) R   A   A  R = {(x, y) : x, y  A and y = x – 1}
 . R  ,  , 
 . (3)
(ii) f : R  R  f(x) = | x |     .
f(x)    . (3)

cos 8 x  cos 6 x
21. (i) = cot x  . (3)
sin 8 x – sin 6 x
(ii) cos 4x = cos 2x     . (3)

FY-354 9 P.T.O.

22. (i) The sum of a certain number of terms of the A.P. 10, 14, 18, ..... is 960. Find the
number of terms. (3)

(ii) Find the sum to n terms of the sequence 4, 44, 444, 4444, ..... (3)

23. (i) Using the first principle of derivatives, find the derivative of the function
1
f(x) = . (4)
x

dy x
(ii) Find , if y = . (2)
dx 1  tan x

24. (i) If A and B are mutually exclusive events, then P(A  B) = _________. (1)

(A) 0 (B) 1

(C) –1 (D) 2

(ii) A and B are two events such that P(A) = 0.42, P(B) = 0.48 and P(A and B) = 0.16.
Find P(A') and P(A  B). (3)

(iii) A coin is tossed three times. Write the sample space and also find the
probability of getting two heads. (2)

25. Find the mean, variance and standard deviation of the following frequency distribution :

Classes 0 – 10 10 – 20 20 – 30 30 – 40 40 – 50

Frequencies 5 8 15 16 6

____________

FY-354 10

22. (i) 10, 14, 18, .....  A.P.     960 .
  . (3)

(ii) 4, 44, 444, 4444, .....   n   . (3)

1
23. (i)      f(x) =  
x
 . (4)

x dy
(ii) y=  . (2)
1  tan x dx

24. (i) A  B      P(A  B) = _______. (1)

(A) 0 (B) 1

(C) –1 (D) 2

(ii) A, B    P(A) = 0.42, P(B) = 0.48, P(A and B) = 0.16
. P(A'), P(A  B)  . (3)

(iii)   3   .    .
     . (2)

25.     , , 
   :

Classes 0 – 10 10 – 20 20 – 30 30 – 40 40 – 50

Frequencies 5 8 15 16 6

____________

FY-354 11 P.T.O.

FY-354 12