Kerala Plus Two Question Paper 2023 Mathematics (Science) 527

SECOND YEAR

Kerala Board
Question Paper
2023

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Reg. No. : ......................................
Name : ...........................................
SY-527
SECOND YEAR HIGHER SECONDARY EXAMINATION, MARCH 2023

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 A = {1, 2, 3}
B = {2, 3, 4, 5}
and f : A  B defined by f(x) = {(x, y) : y = x + 1}.
(i) Write f in roster form. (1)
(ii) Check whether ‘f’ is one-one and onto. (2)

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

3. Find the equation of line through the points A(1, 3) and B(0, 0) using determinants. (3)

4. Find the value of ‘a’ and ‘b’ if
 10 if x  3
f(x) =  ax + b if 3<x<4
 20 if x  4

is a continuous function. (3)

π
5. Find the local maxima or local minima of the function f(x) = sin x + cos x, 0 < x <
2
if it exists. (3)

6. Consider the vectors :
 ^ ^ ^
a = i + 2 j + 3k
 ^ ^ ^
b =3i +2 j + k
 
(i) Find a · b . (1)
 
(ii) Find the angle between a and b . (2)

SY-527 2

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

3  . (6  3 = 18)

1. A = {1, 2, 3},
B = {2, 3, 4, 5}
 f : A  B  f(x) = {(x, y) : y = x + 1}  .
(i) f    . (1)
(ii) f  -  -   . (2)

7 0 
2. X+Y=   
 2 5
3 0 
X–Y=    X, Y   . (3)
 0 3

3.   A(1, 3), B(0, 0)    
   . (3)

 10 if x  3
4. f(x) =  ax + b if 3 < x < 4
 20 if x  4
    ‘a’  ‘b’   . (3)

π
5. f(x) = sin x + cos x, 0 < x <
2
        
. (3)

 ^ ^ ^
6. a = i + 2 j + 3k
 ^ ^ ^
b = 3 i + 2 j + k   .
 
(i) a · b . (1)
 
(ii) a  b    . (2)

SY-527 3 P.T.O.

7. Find the Vector and Cartesian equation of the line passing through (1, 2, 3) and parallel
^ ^ ^
to the vector 3 i + 2 j – 2 k . (3)

1 7 1
8. If A and B are two events such that P(A) = , P(B) = and P(A'  B') = .
2 12 4
(i) Find P(A  B). (1)
(ii) Check whether A and B are independent events. (2)

Answer any 6 questions from 9 to 16. Each carries 4 scores. (6  4 = 24)
9. (i) Let R be a relation on the set Z, set of integers defined by
R = {(a, b) : 2 divides (a – b)}
Choose the right answer :
(A) (2, 4)  R (B) (3, 8)  R

(C) (7, 6)  R (D) (8, 7)  R (1)
(ii) Check the above relation R is an equivalence relation. (3)

1
10. (i) The principal value of sin–1   is _________. (1)
2

  1 
(ii) Find tan–1 2 cos  2 sin –1  (3)
  2 

11. (i) Which among the following is not true ?
(A) (A')' = A (B) (A + B)' = A' + B'
(C) (AB)' = A' · B' (D) (kA) ' = k · A' (1)

1 5
(ii) If A =   , then verify that (A + A') is symmetric and (A – A') is
6 7 
skew-symmetric.
[A' denotes the transpose of the matrix A] (3)

SY-527 4

^ ^ ^
7. (1, 2, 3)      3 i + 2 j – 2 k  
     
. (3)

1 7 1
8. P(A) = , P(B) = , P(A'  B') =     A  B
2 12 4
, 
(i) P(A  B) . (1)
(ii) A  B      . (2)

9  16    6  .
4  . (6  4 = 24)
9. (i) R = {(a, b) : 2 divides (a – b)}
     Z  
,   .
 :
(A) (2, 4)  R (B) (3, 8)  R
(C) (7, 6)  R (D) (8, 7)  R (1)
(ii)   R     
  . (3)

1
10. (i) sin–1      _________ . (1)
2
  1 
(ii) tan–1 2 cos  2 sin –1    . (3)
  2 

11. (i)     ?
(A) (A')' = A (B) (A + B)' = A' + B'
(C) (AB)' = A' · B' (D) (kA) ' = k · A' (1)
1 5
(ii) A=    (A + A')   (A – A') -
6 7 
  .
[A'   A   .] (3)

SY-527 5 P.T.O.

x 2 y2
12. Using integration find the area enclosed by the ellipse  = 1. (4)
9 4

3
 d 2 y   dy 2  dy 
13. (i) The degree of the differential equation  2        + 1 = 0.
 dx   dx   dx 
 

(A) 1 (B) 2

(C) 3 (D) Not defined (1)

dy
(ii) Solve the differential equation = (1 + x2) (1 + y2). (3)
dx

14. Consider the vectors :

 ^ ^ ^
a = i – 7 j + 7k

 ^ ^ ^
b = 3 i – 2 j + 2k

 
(i) Find a  b . (2)

 
(ii) Find the unit vector perpendicular to both a and b . (1)

 
(iii) Find the area of parallelogram whose adjacent sides are a and b . (1)

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

 ^ ^ ^ ^ ^ ^
r = ( i + 2 j + k ) + ( i + j + k ) and

 ^ ^ ^ ^ ^ ^
r = (2 i – j + 4 k ) + (2 i + j + 2 k )

SY-527 6

x 2 y2
12.    = 1     
9 4
 . (4)

3
 d 2 y   dy 2  dy 
   
 dx 2   dx   dx 
13. (i) + 1 = 0    
 

(A) 1 (B) 2

(C) 3 (D) Not defined (1)

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

 ^ ^ ^
14. a = i – 7 j + 7k

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

 
(i) a  b . (2)

 
(ii) a , b      . (1)

 
(iii) a, b     
. (1)

 ^ ^ ^ ^ ^ ^
15. r = ( i + 2 j + k ) + ( i + j + k )

 ^ ^ ^ ^ ^ ^
r = (2 i – j + 4 k ) + (2 i + j + 2 k )

      . (4)

SY-527 7 P.T.O.

16. Bag-I contains 3 red and 4 black balls, while Bag-II contains 5 red and 6 black balls.
One of the bag is selected at random and a ball is drawn out of it. If the ball drawn is
found to be red, find the probability that it was from Bag-II. (4)

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

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

x+y+z=3

2x + y + z = 4

2x – y + z = 2 (6)

dy
18. (i) If y = xx find . (2)
dx

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

(iii) The radius of a circle is increasing uniformly at the rate of 5 cm/sec. Find the rate
at which the area of the circle is increasing when the radius is 8 cm. (2)

1
19. (i)  x2 – a 2 dx = ________. (1)

1
(ii) Find :  x2  4 x – 5 dx (2)

3
x
(iii) Evaluate :  1  x 2 dx (3)
2

SY-527 8

16. -I  3   4  , -II  5 
 6   .   
       .
       -II 
   . (4)

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

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

x+y+z=3
2x + y + z = 4
2x – y + z = 2 (6)

dy
18. (i) y = xx  . (2)
dx
dy
(ii) x = at2 , y = 2at   . (2)
dx
(iii)    5 cm/sec   .  8 cm
    
. (2)

1
19. (i)  x2 – a 2 dx = ________. (1)

1
(ii)  x2  4 x – 5 dx . (2)

3
x
(iii)  1  x 2 dx   . (3)
2

SY-527 9 P.T.O.

20. Solve the LPP graphically : (6)

Maximize

Z = 250x + 75y

subject to

5x + y  100

x + y  60

x0

y0

____________

SY-527 10

20. LPP      : (6)

Maximize

Z = 250x + 75y

subject to

5x + y  100

x + y  60

x0

y0

____________

SY-527 11 P.T.O.

SY-527 12