Kerala Plus Two Question Paper 2024 Maths Commerce

Kerala Board

Question Paper
2024

Reg. No. : ......................................
SY-551 Name : ...........................................

SECOND YEAR HIGHER SECONDARY EXAMINATION, MARCH – 2024

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-551 1 P.T.O.

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

3 5 6 
1. Let A = 1  1 5 
 2 3  1

(i) Find A + AT and A – AT. (2)

(ii) Express A as sum of symmetric and skew symmetric matrices. (1)

2. (i) Let A be a square matrix of order 3 and |A| = 4. Then the value of |2A| = _____ (1)

x 2 6 2
(ii) If = , find value of x. (2)
18 x 18 6

dy
3. (i) If y = sin (2x + 3) then = _____ . (1)
dx

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

2 x if x  5
f(x) =  is continuous.
 k if x  5

4. (i) Let f be continuous on [a, b], differentiable on (a, b) and if f ' (x) > 0 for each x (a,

b) then (1)

(a) f is increasing in [a, b]. (b) f is decreasing in [a, b].

(c) f is constant in [a, b]. (d) None of these

(ii) Find the intervals in which the function given by f(x) = x2 – 4x + 6 is increasing. (2)

SY-551 2

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

3 5 6 
1. A = 1  1 5  
 2 3  1

(i) A + AT, A – AT  . (2)

(ii) A       
  . (1)

2. (i) A   3     .  |A| = 4
 |2A|   = _____ (1)

x 2 6 2
(ii) =  x   . (2)
18 x 18 6

dy
3. (i) y = sin (2x + 3)  = _____ . (1)
dx

2 x , x  5
(ii) f(x) =   
k , x  5
  k   . (2)

4. (i) f  [a, b]  , (a, b)   .

 x (a, b)  f ' (x) > 0 . (1)

(a) f, [a, b]   .

(b) f, [a, b]   .

(c) f, [a, b]   .

(d) 

(ii) f(x) = x2 – 4x + 6      . (2)

SY-551 3 P.T.O.

5. Find

.
(i)  (sin x + cos x) dx (1)
.

. x
(ii)  xe dx (2)
.

dy y
6. (i) The order of the differential equation = is _____ (1)
dx x

dy y
(ii) Find the general solution of = . (2)
dx x

5 2 3 6 
7. Find X and Y if X + Y =   and X – Y =  
0 9 0 1

8. (i) If A and B are independent events then P (A  B) = _____ . (1)

(a) P(A)·P(B) (b) P(A) + P(B)

(c) 0 (d) None of these

2 5 6
(ii) If P(A) = , P(B) = and P(A  B) = then find P(A  B) and P(A/B). (2)
7 7 7

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

9. Let S :  such that S = {(x, y) : x – y is divisible by 2}. Show that S is an
equivalence relation.

SY-551 4

.
5. (i)  (sin x + cos x) dx . (1)
.

. x
(ii)  xe dx . (2)
.

dy y
6. (i) =     _____ (1)
dx x

dy y
(ii) =   . (2)
dx x

5 2 3 6 
7. X+Y=    X – Y =    X, Y .
0 9 0 1

8. (i) A  B      P (A  B) = _____ . (1)

(a) P(A)·P(B) (b) P(A) + P(B)

(c) 0 (d) 

2 5 6
(ii) P(A) = , P(B) = , P(A  B) =  P(A  B)  P(A/B) 
7 7 7

. (2)

9  16    6  .

4  . (6  4 = 24)

9. S :   S = {(x, y) : x – y  2  } 

. S     .

SY-551 5 P.T.O.

x2 y2
10. Using integration find the area of the region bounded by the ellipse + = 1.
4 9


11. (i) Write the principal value of sin–1 sin   . (1)
 3

(ii) Prove that sin–1 (3x – 4x3) = 3 sin–1x. (3)

 3  2 1 0 2
12. If A =   and I =   , find k so that A = kA – 2I.
 4  2   0 1 

dy
13. (i) If y = sin–1x then = _____ . (1)
dx

(ii) Find second order derivative of the function y = x3 + 3x2 + 5x. (3)

14. Graph of derivative of the function f(x), f'(x) is given below.

SY-551 6

x2 y2
10.   + = 1   .
4 9


11. (i) sin–1 sin      . (1)
 3

(ii) sin–1 (3x – 4x3) = 3 sin–1x  . (3)

 3  2 1 0
12. A =    I =    A2 = kA – 2I  k   .
 4  2 0 1 

dy
13. (i) y = sin–1x  = _____ . (1)
dx

(ii) y = x3 + 3x2 + 5x      . (3)

14. f(x)     f'(x)   .

SY-551 7 P.T.O.

(i) Find the points of local maxima and local minima of the function f(x). (2)

(ii) Find the intervals in which the function f is (2)

(a) increasing

(b) decreasing

15. Find

. 2x + 1
(i)  dx (2)
.x2 + x + 2

3
. 2
(ii)  x dx (2)
.
2

16. Find the shortest distance between the lines whose vector equations are

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

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

1 2
17. (i) Find inverse of the matrix  . (3)
 2 3

(ii) Solve the following system of equations by matrix method : (3)

x + 2y = 2

2x + 3y = 3

SY-551 8

(i) f(x)       

. (2)

(ii) f   (a)  (b)   

. (2)

. 2x + 1
15. (i)  dx . (2)
.x2 + x + 2

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

 ^  r = 2^i – ^j – k^ + (3^i – 5^j + 2k)
16. r = ^i + ^j + k^ + (2^i – ^j + k) ^ 

    .

17  20    3  .

6  . (3  6 = 18)

1 2
17. (i)  2 3    . (3)
 

(ii) x + 2y = 2

2x + 3y = 3     . (3)

SY-551 9 P.T.O.

 
18. (i) If a = 2^i – ^j + 3k^ and b = 4^i + 2^j – 2k^ are perpendicular to each other then
find . (2)

(ii) Vertices of triangle ABC is given as A(1,1,1), B(1,2,3), C(2,3,1). Find vectors
 
AB, AC. (2)

(iii) Find area of ABC. (2)

19. Solve the following linear programming problem graphically :

Maximise Z = 3x + 2y

Subject to x + 2y  10,

3x + y  15,

x, y  0.

20. 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 drawn is from the first bag.

______________

SY-551 10

 
18. (i) a = 2^i – ^j + 3k^  b = 4^i + 2^j – 2k^    

  . (2)

(ii) A(1,1,1), B(1,2,3), C(2,3,1)    ABC 
 
AB, AC  . (2)

(iii) ABC   . (2)

19.      .

Maximise Z = 3x + 2y

Subject to x + 2y  10,

3x + y  15,

x, y  0.

20.   4  4     2 
6   .       
.       
  .

______________

SY-551 11 P.T.O.

SY-551 12

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