Kerala Plus Two Question Paper 2023 Mathematics (Commerce) 555

SECOND YEAR

Kerala Board
Question Paper
2023

Download PDF

Reg. No. : ......................................
Name : ...........................................
SY-555
SECOND YEAR HIGHER SECONDARY EXAMINATION, MARCH 2023

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. Let f :  be a function defined by f(x) = 4x – 1. Prove that f(x) is a one-one
function. Also find inverse of the function f(x). (3)

2 5  0 3
2. Find X and Y if X + Y =   and X – Y =  . (3)
7 0   3 0

2 4 2x 4
3. Find the value of x if  . (3)
5 1 6 x

kx 2 , x  2
4. Find the value of k so that the function f(x) =  is continuous at x = 2. (3)
 5, x  2

1
5. (i)  x dx = ________. (1)

1 – sin x
(ii) Find  cos 2 x dx (2)

6. (i) If l, m, n are direction cosines of a vector then l2 + m2 + n2 = ________. (1)
(a) 0 (b) –1
(c) 1 (d) 2

^ ^ ^
(ii) Find a unit vector in the direction of sum of the vector a = 2 i + 3 j – 5 k and
^ ^ ^
b =3i –2 j + k. (2)

7. Show that the lines :

x 1 y – 2 z – 5 x  3 y –1 z – 5
  ,   are coplanar. (3)
–1 2 5 –3 1 5

SY-555 2

1  7    6  .

3  . (6  3 = 18)

1. f: 
  f(x) = 4x – 1  . f(x)  
   .  f(x)  
  . (3)

2 5  0 3
2. X+Y=   ,X–Y=     X , Y  . (3)
7 0   3 0

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

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

1
5. (i)  x dx = ________. (1)

1 – sin x
(ii)  cos 2 x dx . (2)

6. (i) l, m, n     . 
l2 + m2 + n2 = ________. (1)
(a) 0 (b) –1
(c) 1 (d) 2
^ ^ ^ ^ ^ ^
(ii) a = 2 i + 3 j – 5k , b = 3 i – 2 j + k
      
 . (2)

x 1 y – 2 z – 5 x  3 y –1 z – 5
7.   ,  
–1 2 5 –3 1 5
    . (3)

SY-555 3 P.T.O.

Answer any 8 questions from 8 to 17. Each carries 4 scores. (8  4 = 32)

8. (i) Which among the following is correct if a relation is an equivalence relation ? (1)

(a) reflexive and symmetric but not transitive

(b) reflexive and transitive but not symmetric

(c) symmetric and transitive but not reflexive

(d) reflexive, symmetric and transitive

(ii) R = {(x, y) : y is divisible by x} is a relation defined on A = {2, 4, 6, 8}. Show
that R is reflexive and transitive but not symmetric. (3)

 –1
9. (i) Write the value of cos–1   . (1)
 2 

(ii) tan–1 x + tan–1 y = _________ (1)

2 7 1
(iii) Show that tan–1 + tan–1 = tan–1 . (2)
11 24 2

10. (i) If A is square matrix of order n  n, k is a scalar | kA | = _________ (1)

(ii) Using determinants show that the points (2, 4), (3, 3), (1, 5) are collinear. (3)

dy
11. Find in the following :
dx

(i) y = sec (tan (x2)) (2)

(ii) x = a( – sin )

y = a(1 + cos ) (2)

12. Find the intervals in which the function f(x) = 2x2 – 8x + 7 is strictly increasing and
strictly decreasing. (4)

SY-555 4

8  17    8  .
4  . (8  4 = 32)
8. (i)      
    ? (1)
(a)      
(b)      
(c)      
(d)    .
(ii) R = {(x, y) : y  x   }  A = {2, 4, 6, 8}
    . R ,
    . (3)

 –1
9. (i) cos–1     . (1)
 2 
(ii) tan–1 x + tan–1 y = _________ (1)
2 7 1
(iii) tan–1 + tan–1 = tan–1  . (2)
11 24 2

10. (i) A  n  n     | kA | = _________ (1)
(ii)   (2, 4), (3, 3), (1, 5)   
 . (3)

dy
11.    :
dx
(i) y = sec (tan (x2)) (2)
(ii) x = a( – sin )
y = a(1 + cos ) (2)

12. f(x) = 2x2 – 8x + 7      .
    . (4)

SY-555 5 P.T.O.


cos 4 x
2
13. Evaluate :  sin 4 x  cos 4 x dx (4)
0

14. Find the area enclosed by the circle x2 + y2 = 4, using definite integral. (4)

^ ^ ^
15. The position vectors of the vertices A, B, C of triangle ABC are 2 i – j + k ,
^ ^ ^ ^ ^ ^
i – 3 j – 5 k , 3 i – 4 j – 4 k respectively. Prove the triangle is right angled triangle. (4)

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

^ ^ ^ ^ ^ ^
r = (2 i – j – k ) + (2 i + j + 2 k )

^ ^ ^ ^ ^ ^
r = ( i + 2 j + k ) + ( i – j + k )

17. 12 cards are numbered 1 to 12 and placed in a box. One card is drawn randomly and it
is known to be more than 5. What is the probability that it is an even number ? (4)

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

18. (i) Construct the matrix A = [aij] of order 3  3 such that aij = 2i – j. (2)

(ii) Express A as the sum of a symmetric and a skew symmetric matrices. (4)

19. Solve the following system of linear equations using matrix method : (6)

x + 2y – 4z = 3

2x – y – z = 3

x + y – 2z = 3

SY-555 6


cos 4 x
2
13.  sin 4 x  cos 4 x dx   . (4)
0

14.    x2 + y2 = 4   
. (4)

15.  ABC  A, B, C    
^ ^ ^ ^ ^ ^ ^ ^ ^
 2 i – j + k , i – 3 j – 5 k , 3 i – 4 j – 4 k .
   . (4)

^ ^ ^ ^ ^ ^
16. r = (2 i – j – k ) + (2 i + j + 2 k )
^ ^ ^ ^ ^ ^
r = ( i + 2 j + k ) + ( i – j + k )
     . (4)

17.   1  12   12 .  
     5   
       ? (4)

18  24    5  .
6  . (5  6 = 30)
18. (i) aij = 2i – j  A = [aij]   3  3   . (2)
(ii) A      
 . (4)

19.       
  . : (6)
x + 2y – 4z = 3
2x – y – z = 3
x + y – 2z = 3

SY-555 7 P.T.O.

d2y dy 2y
20. (i) If y = ex(x2 – 1), show that 2
  (4)
dx dx x – 1

(ii) Find the point at which the slope of the tangent to the curve y = x2 – x is 5. (2)

21. (i) Find the order and degree of the differential equation :

2
 d 2 y   dy  3
     – y dy  = 0 (2)
 dx   dx 
2
 dx 
 

(ii) Find the Integrating Factor of the linear differential equation :

x dy – (2x2 + y) dx = 0

Hence find general solution of this equation. (4)

^ ^ ^ ^ ^ ^
22. (i) Find the projection of the vector a = i + 2 j + k on the vector b = 2 i + 3 j + 2 k . (2)

^ ^ ^
(ii) Find area of the triangle whose adjacent sides are a = i + 2 j + k and

^ ^ ^
b = 2 i + 3 j + 2k . (4)

23. Consider the following LPP : (6)

Maximize

Z = 4x + 5y

subject to the constraints

2x + 3y  6

2x + y  4

x  0, y  0

Draw the feasible region of LPP and find the solution.

SY-555 8

d2y dy 2y
20. (i) y = ex(x2 – 1)  2
   . (4)
dx dx x – 1

(ii) y = x2 – x      5   
     ? (2)

21. (i)      
 :
2
 d 2 y   dy  3  dy 
  
 dx 2   dx 
– y  = 0 (2)
   dx 

(ii) xdy – (2x2 + y) dx = 0    
  .
     . (4)

^ ^ ^ ^ ^ ^
22. (i) a = i + 2 j + k   b = 2 i + 3 j + 2 k  
   . (2)

^ ^ ^ ^ ^ ^
(ii) a = i + 2 j + k , b = 2 i + 3 j + 2 k    
    
. (4)

23.   LPP  : (6)

Maximize
Z = 4x + 5y
Subject to the constraints
2x + 3y  6
2x + y  4
x  0, y  0
   LPP   .

SY-555 9 P.T.O.

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

X 1 2 3 4 5

P(X) k 1 1 1 1
4 8 16 16

(i) Find the value of k. (1)

(ii) Find mean of random variable X. (2)

(iii) Find variance of X. (3)

____________

SY-555 10

24. X      
 :

X 1 2 3 4 5

P(X) k 1 1 1 1
4 8 16 16

(i) k   . (1)

(ii)   X  . (2)

(iii)   X  . (3)

____________

SY-555 11 P.T.O.

SY-555 12