Kerala Plus Two Question Paper 2023 Mathematics (Commerce) 551

SECOND YEAR

Kerala Board
Question Paper
2023

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SY-551
SECOND YEAR HIGHER SECONDARY EXAMINATION, MARCH 2023

Part – III
MATHEMATICS (COMMERCE) Time : 2 Hours
Maximum : 60 scores Cool-off time : 15 Minutes

General Instructions to Candidates :
 15 minutes is given as ‘Cool-off time’.
 Use the ‘Cool-off time’ to read the 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.
 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)

1. Construct a 2  2 matrix whose elements are given by aij = 2i + j (3)

3 5 
2. Let A =  
1 – 1

(i) Find A + A' and A – A' (1)

(ii) Express A as the sum of a symmetric and skew symmetric matrices. (2)

2 4 2x 4
3. (i) Find x if  (1)
5 1 6 x

(ii) Find the area of the triangle with vertices (2, 7), (1, 1) and (10, 8). (2)

 kx + 1 if x  5
4. Consider the function f(x) = 
 3x – 5 if x > 5

(i) Find lim – f(x) and lim  f(x). (2)
x5 x5

(ii) Find the value of k if ‘f’ is a continuous function. (1)

5. The radius of a circle is increasing uniformly at the rate of 3 cm/s. Find the rate at
which the area of the circle is increasing when the radius is 10 cm. (3)

1
6. (i)  x dx = _______. (1)

x
(ii) Find  1  x 2 dx (2)

SY-551 2

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

3  . (6  3 = 18)

1.  aij = 2i + j    2  2  . (3)

3 5 
2. A=   
1 – 1
(i) A + A', A – A'  . (1)

(ii) A       
  . (2)

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

(ii) (2, 7), (1, 1), (10, 8)    
. (2)

 kx + 1 x  5
4. f(x) =    (function) .
 3x – 5 x > 5

(i) lim f(x) , lim  f(x)  . (2)
x  5– x5
(ii) ‘f’     k   . (1)

5.    3 cm/s   .
  10 cm    
. (3)

1
6. (i)  x dx = _______. (1)

x
(ii)  1  x 2 dx . (2)

SY-551 3 P.T.O.

^ ^ ^ ^ ^ ^
7. Let a = i + 3 j + 7 k , b = 7 i – j + 8 k

(i) Find a · b (1)

(ii) Find | b | (1)

(iii) Find the projection of a on b . (1)

6 5 7
8. If P(A) = , P(B) = and P(A  B) =
11 11 11
(i) Find P(A  B) (2)
(ii) Find P(A/B) (1)

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

9. Let f : R  R defined by f(x) = 1 + x2
(i) Find f(2) and f(–2) (1)
(ii) Is f one-one. Why ? (1)
(iii) Show that f is not onto (2)

10. Match the following :
A B

1 – (1)
(i) sin–1   (a)
2 4

(ii) tan–1 (–1) 2 (1)
(b)
3

 1   (1)
(iii) cos–1   (c)
 2 6

(iv) sec–1 (–2)  (1)
(d)
3

(e)
4

SY-551 4

^ ^ ^ ^ ^ ^
7. a = i + 3 j + 7 k , b = 7 i – j + 8 k 

(i) a · b  (1)

(ii) | b |  (1)

(iii) a   b   . (1)

6 5 7
8. P(A) = , P(B) = and P(A  B) = 
11 11 11
(i) P(A  B) . (2)
(ii) P(A/B) . (1)

9  16    6  .
4  . (6  4 = 24)
9. f : R  R  f(x) = 1 + x2  .
(i) f(2), f(–2)  . (1)
(ii) f -  ?  ? (1)
(iii) f    . (2)

10.   :
A B
1 – (1)
(i) sin–1   (a)
2 4

(ii) tan–1 (–1) 2 (1)
(b)
3
 1   (1)
(iii) cos–1   (c)
 2 6

(iv) sec–1 (–2)  (1)
(d)
3

(e)
4

SY-551 5 P.T.O.

3 – 2
11. If A =  
4 – 2

(i) Find A2. (2)

(ii) Show that A2 – A + 2I = 0 (2)

12. Consider the function f(x) = x2 + 2x – 5

(i) Find f ʹ(x). (1)

(ii) Find the intervals in which f(x) is increasing or decreasing. (3)

1
1
13. (i) Evaluate :  1 – x 2 dx (2)
0

(ii) Integrate x sin x w.r.t. x. (2)

14. Find the area enclosed by the circle x2 + y2 = r2 using integration. (4)

2
 d 2 y   dy  3
15. (i) The degree of the differential equation  2     = 0 is
 dx   dx 
 

(a) 0 (b) 1

(c) 2 (d) 3 (1)

(ii) Find the general solution of the differential equation

dy
= (1 + x2) (1 + y2). (3)
dx

16. Find the shortest distance between the lines : (4)

^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^
r = ( i + 2 j + 3 k ) + ( i – 3 j + 2 k ) and r = (4 i + 5 j + 6 k ) + (2 i + 3 j + k )

SY-551 6

3 – 2
11. A=   
4 – 2

(i) A2 . (2)

(ii) A2 – A + 2I = 0  . (2)

12. f(x) = x2 + 2x – 5   (function) .

(i) f ʹ(x) . (1)

(ii) f(x)  . (3)

1
1
13. (i)  1 – x 2 dx . (2)
0

(ii) x sin x  x   . (2)

14.   x2 + y2 = r2    . (4)

2
 d 2 y   dy  3
  
 dx 2   dx 
15. (i) = 0    
 

(a) 0 (b) 1

(c) 2 (d) 3 (1)

dy
(ii) = (1 + x2) (1 + y2)    
dx
. (3)

^ ^ ^ ^ ^ ^
16. r = ( i + 2 j + 3 k ) + ( i – 3 j + 2 k ),

^ ^ ^ ^ ^ ^
r = (4 i + 5 j + 6 k ) + (2 i + 3 j + k ) (4)

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

SY-551 7 P.T.O.

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

17. Consider the system of linear equations :

3x – 2y + 3z = 8

2x + y – z = 1

4x – 3y + 2z = 4

(i) Express the system in the form AX = B. (1)

(ii) Find Adj A. (2)

(iii) Solve the system using matrix method. (3)

dy
18. (i) Find if 2x + 3y = sin x (2)
dx

dy
(ii) If x = at2; y = 2at, find . (2)
dx

d2y
(iii) If y = 2 sin x + 3 cos x, prove that +y=0 (2)
dx 2

19. Solve the Linear Programming Problem (LPP) graphically : (6)

Maximise Z = 3x + 2y

subject to x + 2y  10

3x + y  15

x  0, y  0

SY-551 8

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

6  . (3  6 = 18)

17. 3x – 2y + 3z = 8

2x + y – z = 1

4x – 3y + 2z = 4

   .

(i)  AX = B   . (1)

(ii) Adj. A . (2)

(iii)      . (3)

dy
18. (i) 2x + 3y = sin x  . (2)
dx

dy
(ii) x = at2, y = 2at  . (2)
dx

d2y
(iii) y = 2 sin x + 3 cos x  + y = 0  . (2)
dx 2

19.       : (6)

Maximise Z = 3x + 2y

Subject to x + 2y  10

3x + y  15

x  0, y  0

SY-551 9 P.T.O.

20. (i) Given two independent events A and B such that P(A) = 0.3 and P(B) = 0.6

(a) Find P(A and B) (1)

(b) Find P(A and not B) (1)

(ii) Bag – I contains 3 red and 4 black balls and Bag – II contains 4 red and 5 black
balls. One bag is selected and a ball is drawn from it and it is found to be red.
Find the probability that it is drawn from Bag – I. (4)

_____________

SY-551 10

20. (i) A, B     . P(A) = 0.3, P(B) = 0.6



(a) P(A and B)  (1)

(b) P(A and not B)  (1)

(ii) -I  3  4   . -II  4  5
  .       

.   .    -I  

  . (4)

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SY-551 11 P.T.O.

SY-551 12