Kerala Plus Two Question Paper 2024 Maths Commerce 555

Kerala Board

Question Paper
2024

Reg. No. : ......................................
SY-555 Name : ...........................................

SECOND YEAR HIGHER SECONDARY EXAMINATION, MARCH – 2024

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

SY-555 1 P.T.O.

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

1. (a) A function f : X  Y is onto if range of f = _______. (1)

(b) State whether the function f : R  R defined by f(x) = 3 – 4x is bijective. (2)

2. (a) If A is a matrix of order 3  4 and B is a matrix of order 4  2 then AB is a
matrix of order _________. (1)

(b) Construct a 2  2 matrix A = [aij] whose elements are given by aij = 2i + j. (2)

cos  – sin 
3. (a) Evaluate . (1)
sin  cos 

(b) Find the area of the triangle whose vertices are (1, 0), (6, 0) and (4, 3). (2)

3
4. (a) lim x – 8 = ________. (1)
x2 x –2

dy
(b) Find if x2 + 6y = ex. (2)
dx

x
5. (a)  e sec x (1 + tan x) dx = __________. (1)

–1
e tan ( x)
(b) Find  1  x 2 dx (2)

 ^ ^ ^  ^ ^
6. Let a = i + j + k and b = – j + k

Find
 
(a) a + b (1)
 
(b) Find the unit vector along a + b (2)

SY-555 2

1  7    6  .
3  . (6  3 = 18)

1. (a)   f : X  Y   f   = _______. (1)

(b) f : R  R    f(x) = 3 – 4x  
 . (2)

2. (a) A    3  4  B    4  2
  AB    _________ . (1)

(b)   aij = 2i + j    2  2 
A = [aij] . (2)

cos  – sin 
3. (a)   . (1)
sin  cos 

(b) (1, 0), (6, 0), (4, 3)    
. (2)

3
4. (a) lim x – 8 = ________. (1)
x2 x –2

dy
(b) x2 + 6y = ex  . (2)
dx

x
5. (a)  e sec x (1 + tan x) dx = __________. (1)
–1
e tan ( x)
(b)  1  x 2 dx . (2)

 ^ ^ ^  ^ ^
6. a = i + j + k  b = – j + k  
 
(a) a + b . (1)
 
(b) a + b    . (2)

SY-555 3 P.T.O.

7. (a) If l, m, n are the direction cosines of a line, then l2 + m2 + n2 = _______. (1)
(b) If a line has direction ratios 2, –1, –2, determine its direction cosines. (2)

Answer any 8 questions from 8 to 17. Each carries 4 scores. (8  4 = 32)
1
8. (a) If f(x) = 8x3 and g(x) = x 3 . Find gof. (1)
(b) Show that the relation R in the set Z of integers given by R = {(a, b) : 2 divides
a – b} is an equivalence relation. (3)

 3
9. (a) Find the principal value of cos–1  
 2 . (1)
 

1 1 1
(b) Show that tan–1   – tan–1   = tan–1   . (3)
 3 5 8

10. (a) A square matrix A is said to be Skew-symmetric if A' = ________. (1)

1 4 – 1
(b) 
Express the matrix A =  2 5 4  as the sum of a symmetric and a
 – 1 – 6 3 
Skew-symmetric matrix. (3)

a
11. (a)  f (x) dx = _______. (1)
0

a
(A) 0 (B)  f(a – x) dx
0

2a
(C) f(a) (D)  f(x) dx
0



cos 6 x dx
2
(b) Using the above property of definite integrals evaluate  sin 6 x  cos6 x (3)
0

SY-555 4

7. (a)     l, m, n  l2 + m2 + n2 =
_______. (1)
(b)     2, –1, –2  
  . (2)

8  17    8  .
4  . (8  4 = 32)
1
8. (a) f(x) = 8x3, g(x) = x 3   gof . (1)
(b) R = {(a, b) : 2 divides a – b}   Z 
   . (3)

 3
cos–1  
9. (a)
 2     . (1)
 
1 1 1
(b) tan–1   – tan–1   = tan–1    . (3)
 3 5 8

10. (a)    -  A' = ________. (1)
1 4 – 1
(b) 
A= 2 5 4       
 – 1 – 6 3 
-     . (3)

a
11. (a)  f (x) dx = _______. (1)
0
a
(A) 0 (B)  f(a – x) dx
0
2a
(C) f(a) (D)  f(x) dx
0
(b)     


cos 6 x dx
2

 sin 6 x  cos6 x   . (3)
0

SY-555 5 P.T.O.

12. (a) Draw a rough sketch of the curve x2 + y2 = 9. (1)

(b) Find the area bounded by the above curve using integration. (3)

dy y
13. Consider the differential equation  = x2.
dx x

(a) What is its integrating factor ? (1)

(b) Find its general solution. (3)

 ^ ^ ^  ^ ^ ^
14. Consider the vectors a = i – 7 j + 7 k , b = 3 i – 2 j + 2 k .

 
(a) Find a · b (1)

 
(b) Find the area of parallelogram with adjacent sides a and b . (3)

15. (a) Let A be a square matrix of order 3  3, then | kA | is equal to (1)

(A) k| A | (B) k2 | A|

(C) k3 | A | (D) 3k | A |

yk y y
(b) Using properties of determinants prove that y yk y = k2 (3y + k) (3)
y y yk

16. Find the shortest distance between the lines

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

SY-555 6

12. (a) x2 + y2 = 9     . (1)

(b)      . (3)

dy y
13.  = x2    .
dx x

(a)     ? (1)

(b)    . (3)

 ^ ^ ^  ^ ^ ^
14. a = i – 7 j + 7 k , b = 3 i – 2 j + 2 k   .

 
(a) a · b . (1)

 
(b) a  b    
. (3)

15. (a) A  3  3    | kA |   . (1)

(A) k| A | (B) k2 | A|

(C) k3 | A | (D) 3k | A |

yk y y
(b)    y yk y = k2 (3y + k)
y y yk

. (3)

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

     .

SY-555 7 P.T.O.

1
17. (a) If P(A) = , P(B) = 0, then P(A/B) is _______. (1)
2

1
(A) 0 (B)
2

(C) not defined (D) 1

5 2
(b) Evaluate P(A  B), if 2P(A) = P(B) = and P(A/B) = . (3)
13 5

Answer any 5 questions from 18 to 24. Each carries 6 scores. (5  6 = 30)

18. Solve by matrix method :

x+y+z=2

x – 2y – z = 1

2x – y + z = 5

1 – 1
19. (a) Using elementary operation, find the inverse of A =   if it exists. (3)
2 3 

 3 – 2 1 0 2
(b) If A =   and I = 0 1  . Find k so that A = kA – 2I. (3)
 4 – 2   

dy
20. (a) Find if x = sin t, y = cos t. (3)
dx

(b) Verify Rolle’s theorem for the function y = x2 in the closed interval [–2, 2]. (3)

21. (a) Use differential to approximate 36.6 . (3)

(b) Find two positive numbers such that their sum is 8 and the sum of their squares
is minimum. (3)

SY-555 8

1
17. (a) P(A) = , P(B) = 0   P(A/B) = _______. (1)
2
1
(A) 0 (B)
2
(C) not defined (D) 1
5 2
(b) 2P(A) = P(B) = , P(A/B) =  P(A  B) . (3)
13 5

18  24    5  .
6  . (5  6 = 30)

18. x+y+z=2
x – 2y – z = 1
2x – y + z = 5
   .

1 – 1
19. (a)    A =   
2 3 
   . (3)

 3 – 2 1 0 2
(b) A= 
4 – 2  ,I= 
0 1   A = kA – 2I   k   . (3)
   

dy
20. (a) x = sin t, y = cos t  . (3)
dx
(b) y = x2 in [–2, 2]      
. (3)

21. (a) 36.6     . (3)

(b)  8        
  . (3)

SY-555 9 P.T.O.

22. (a) Show that the points A(1, 2, 7), B(2, 6, 3) and C(3, 10, –1) are collinear. (3)

 ^ ^ ^  ^ ^ ^  ^ ^
(b) If a = 2 i + 2 j + 3 k , b = – i + 2 j + k and c = 3 i + j are such that
  
a +  b is perpendicular to c , then find the value of . (3)

23. Solve the linear programming problem graphically :

Maximise Z = 250x + 75y

Subject to

5x + y  100

x + y  60

x  0, y  0

24. A random variable X has the following probability distribution :

X 0 1 2 3 4

P(X) 0 k 2k 3k 4k

(a) Find the value of k. (2)

(b) Using the value of k, find mean and variance of the random variable X. (4)

______________

SY-555 10

22. (a) A(1, 2, 7), B(2, 6, 3), C(3, 10, –1)    
. (3)
 ^ ^ ^  ^ ^ ^  ^ ^  
(b) a = 2 i + 2 j + 3 k , b = – i + 2 j + k , c = 3 i + j  a +  b 

 c     . (3)

23.       
 .
Maximise Z = 250x + 75y
Subject to
5x + y  100
x + y  60
x  0, y  0

24. X      
.
X 0 1 2 3 4
P(X) 0 k 2k 3k 4k
(a) k   . (2)

(b) k    X    
 . (4)

______________

SY-555 11 P.T.O.

SY-555 12

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