Kerala Plus Two Question Paper 2024 Maths Science

Kerala Board

Question Paper
2024

Reg. No. : ......................................
SY-527 Name : ...........................................

SECOND YEAR HIGHER SECONDARY EXAMINATION, MARCH – 2024

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-527 1 P.T.O.

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

1. Let R be a relation on a set A = {1, 2, 3, 4, 5, 6} defined as R = {(x, y) : y = 2x – 1}.

(i) Write R in roster form and find its domain and range. (2)

(ii) Is R is an equivalence relation ? Justify. (1)

 3 1
2. A=  show that
 –1 2 

A2 – 5A + 7I = O

(Where I is the identity matrix)

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

(ii) Determine the value of k so that the function (2)

 kx  1 if x5
f ( x)  
 3x  5 if x 5

is continuous at x = 5.

1
4. (i) Find the principal value of sin–1   (1)
2

  1 
(ii) Find the value of tan–1  2 cos  2 sin 1  (2)
  2 

5. (i) Which of the following function is increasing in its domain :

(A) sin x

(B) cos x

(C) – 2x

(D) log x (1)

SY-527 2

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

1. R   A = {1, 2, 3, 4, 5, 6}  R = {(x, y) : y = 2x – 1} 
.
(i) R    .    . (2)

(ii) R     ?  . (1)

 3 1
2. A=  
 –1 2 

A2 – 5A + 7I = O  .

(I   )

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

 kx  1 , x5
(ii) f ( x)     x = 5    k 
 3x  5 , x5
 . (2)

1
4. (i) sin–1      ________ . (1)
2

  1 
(ii) tan–1  2 cos  2 sin 1    . (2)
  2 

5. (i)       
  :

(A) sin x
(B) cos x
(C) – 2x
(D) log x (1)

SY-527 3 P.T.O.

(ii) Find the intervals in which the function f given by f(x) = x2 – 4x + 6 is

(a) increasing (b) decreasing. (2)


6. (i)

|  | |  |
If  is the angle between two non zero vectors a and b and a  b = a  b then

 = ______. (1)

 
(ii) Find the projection of the vector a = ^i – ^j on the vector b = ^i + ^j. (2)

7. Let A and B are independent events with P(A) = 0.3, P(B) = 0.4, find

(i) P(A  B)

(ii) P(A  B)

(iii) P(A/B)

1
. 4
8. Evaluate 
 5x x5 + 1 dx.
.
–1

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

9. (i) What is the minimum number of ordered pairs to form a reflexive relation on a set
of 4 elements ? (1)

(ii) Let A = R – {3}, B = R – {1}

x–2
Consider the function f : A  B defined by f(x) = .
x–3

Check whether f is one-one and onto. (3)

SY-527 4

(ii) f(x) = x2 – 4x + 6   (a)  (b)  
 . (2)

 
6. (i)  a , b      .

| a  b | = | a  b |   = ______ . (1)

   
(ii) a = ^i – ^j, b = ^i + ^j.  b  a   . (2)

7. A, B    .

P(A) = 0.3, P(B) = 0.4 

(i) P(A  B)

(ii) P(A  B)

(iii) P(A/B)
 .

1
. 4 5
8.  5x x + 1 dx   .
.
–1

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

9. (i) 4       
      ? (1)

(ii) A = R – {3}, B = R – {1}

x–2
f : A  B  f(x) =  .
x–3

f -    . (3)

SY-527 5 P.T.O.

10. (i) If A is a skew symmetric matrix then A' = _____. (1)

 2 –2 –4 
(ii) Express the matrix A =  –1 3 4 
 1 –2 –3 
as the sum of a symmetric and skew symmetric matrix. (3)

11. A wire of length 28 m is cut into two pieces, one of the pieces is to be made into a
square and other into a circle. What should be the length of the two pieces so that the
combined area of square and circle is minimum ?

12. (i) Write the order and degree of the differential equation

d2 y dy2 dy
xy 2 + x dx  – sindx  = 0. (1)
dx    

(ii) Find the integrating factor of the differential equation
dy
x + 2y = x2. (2)
dx

(iii) Solve the differential equation
dy
x + 2y = x2. (1)
dx

 
13. If a = ^i + ^j + k^ , b = ^i + 2^j + 3k^
   
(i) Find a + b and a – b. (1)
   
(ii) Find a unit vector perpendicular to both a + b and a – b. (3)


14. Find the shortest distance between the lines r = ^i + ^j +  (2^i – ^j + k)
^

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

SY-527 6

10. (i) A     A' = _____. (1)

 2 –2 –4 
(ii) A =  –1 3 4       
 1 –2 –3 
   . (3)

11. 28 m     .     
     . 
       
.

d2 y dy2 dy
12. (i) xy 2 + x dx  – sindx  = 0    
dx    
 . (1)

dy
(ii) x + 2y = x2     
dx
. (2)

dy
(iii) x + 2y = x2     . (1)
dx

 
13. a = ^i + ^j + k^ , b = ^i + 2^j + 3k^ 
   
(i) a + b  a – b  . (1)
   
(ii) a + b  a – b      . (3)

 ^ , r = 2^i + ^j – k^ +  (3^i – 5^j + 2k)
14. r = ^i + ^j +  (2^i – ^j + k) ^   

  .

SY-527 7 P.T.O.

15. In a factory which manufactures bolts, machines A, B and C manufacture respectively
25%, 35% and 45% of the bolt. Of their output 5%, 4% and 2% are respectively
defective bolts. A bolt is drawn at random from the product and is found to be
defective. What is the probability that it is manufactured by machine B ?

x2 y2
16. Find the area of the region bounded by the ellipse + = 1 using integration.
16 9

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

17. Solve the following system of equations by matrix method :

2x – 3y + 5z = 11

3x + 2y – 4z = –5

x + y – 2z = –3

dy .
18. (i) sin x + cos y = xy find (2)
dx

dy .
(ii) x = a cos3t; y = a sin3t find (2)
dx

d2y dy
(iii) If y = (sin–1 x)2 then show that (1 – x2) 2 – x = 2. (2)
dx dx

. x–1
19. (i) Find 
 dx. (3)
.(x – 2) (x – 3)


–
4
. 
(ii) Prove that 
 log (1 + tan x) dx = log 2. (3)
. 8
0

SY-527 8

15.      A, B, C  
25%, 35%, 45%  .  5%, 4%, 2% 
   .    
       
  B       ?

x2 y2
16. + = 1    .
16 9

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

17.        
  :

2x – 3y + 5z = 11

3x + 2y – 4z = –5

x + y – 2z = –3

dy
18. (i) sin x + cos y = xy  . (2)
dx

dy
(ii) x = a cos3t; y = a sin3t  . (2)
dx

d2 y dy
(iii) y = (sin–1 x)2  (1 – x2) 2 – x = 2  . (2)
dx dx

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


–
4
. 
(ii)  log (1 + tan x) dx = log 2  . (3)
. 8
0
SY-527 9 P.T.O.

20. Solve the following linear programming problem graphically :

Maximise Z = 60x + 15y

Subject to the constraints

x + y  50

3x + y  90

x  0, y  0

____________

SY-527 10

20.        

 :

x + y  50, 3x + y  90, x  0, y  0

  

Z = 60x + 15y    .

____________

SY-527 11 P.T.O.

SY-527 12

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