Kerala Plus Two Question Paper 2024 Maths Science 554

Kerala Board

Question Paper
2024

Reg. No. : ......................................
SY-554 Name : ...........................................

SECOND YEAR HIGHER SECONDARY EXAMINATION, MARCH – 2024

Part – III Time : 2½ Hours
MATHEMATICS (SCIENCE) Cool-off time : 15 Minutes
Maximum : 80 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-554 1 P.T.O.

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

1. If f : R  R given by f(x) = cos x and g : R  R given by g(x) = 3x2 then find gof and

fog.

2. Construct a 2 × 3 matrix whose elements are given by aij = 2i + j.

3. (i) Let A be a square matrix of order 3  3, then | k A | = _______.

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

(C) k3 | A | (D) 3k | A | (1)

3 x 3 2
(ii) Find the value of x for which = (2)
x 1 4 1

kx  1, if x  5
4. Find the value of k so that the function f(x) = 
 3 x  5, if x  5

is continuous.

d
5. (i) sin x = _______.
dx

(A) cos x (B) 2 sin x

cos x sin x
(C) (D) (1)
2 sin x 2 cos x

dy
(ii) Find if y + sin y = cos x (2)
dx

SY-554 2

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

3  . (6  3 = 18)

1. f : R  R, f(x) = cos x g : R  R, g(x) = 3x2  gof, fog 

.

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

3. (i) A  3  3      | k A | = _______.

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

(C) k3 | A | (D) 3k | A | (1)

3 x 3 2
(ii) =  x   . (2)
x 1 4 1

kx  1, if x  5
4. f(x) =      k  
 3 x  5, if x  5

.

d
5. (i) sin x = _______.
dx

(A) cos x (B) 2 sin x

cos x sin x
(C) (D) (1)
2 sin x 2 cos x

dy
(ii) y + sin y = cos x  . (2)
dx

SY-554 3 P.T.O.

6. Let f(x) = x2 – 4x + 6

(i) Find f '(x) (1)

(ii) Find the interval in which f is increasing (2)

7. (i) Order of the differential equation

3
d2y  dy 
2
+ x    y  0 is
dx  dx 

(A) 1 (B) 2

(C) 3 (D) 4 (1)

(ii) Find the general solution of the differential equation

dy
= (1 + x2) (1 + y2) (2)
dx

8. Find the vector equation for the line passing through the points (–1, 0, 2) and (3, 4, 6)

Answer any 8 questions from 9 to 18. Each carries 4 scores. (8  4 = 32)

9. Let f : R  R defined by f(x) = 2x + 3.

(i) Show that f is one-one. (2)

(ii) Is f invertible ? Then find the inverse of f. (2)

SY-554 4

6. f(x) = x2 – 4x + 6 

(i) f '(x)  (1)

(ii) f    . (2)

3
d2y  dy 
7. (i) + x    y  0    
dx 2  dx 

(A) 1 (B) 2

(C) 3 (D) 4 (1)

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

. (2)

8. (–1, 0, 2), (3, 4, 6)     

 .

9  18    8  .

4  . (8  4 = 32)

9. f : R  R, f(x) = 2x + 3 

(i) f -  . (2)

(ii) f   ?  f   . (2)

SY-554 5 P.T.O.

1
10. (i) The principal value of cos–1 = __________.
2

 
(A) (B)
3 6

 2
(C) (D) (1)
4 3

1 2 3
(ii) Show that tan–1 + tan–1 = tan–1 (3)
2 11 4

dy
11. (i) Find if x = log t ; y = sin t (2)
dx

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

a
12. (i)  f( x) dx  _______.
0

0 a
(A)  f( x) dx (B)  f( x  a) dx
a 0

2a a
(C)  f( x) dx (D)  f(a  x) dx (1)
0 0

π/2
sin x 
(ii) Show that  sin x  cos x dx = 4 (3)
0

13. Find the area of the region bounded by the two parabolas y = x2 and y2 = x.

SY-554 6

1
10. (i) cos–1    = __________.
2

 
(A) (B)
3 6

 2
(C) (D) (1)
4 3

1 2 3
(ii) tan–1 + tan–1 = tan–1  . (3)
2 11 4

dy
11. (i) x = log t ; y = sin t  . (2)
dx

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

a
12. (i)  f( x) dx  _______.
0

0 a
(A)  f( x) dx (B)  f( x  a) dx
a 0

2a a
(C)  f( x) dx (D)  f(a  x) dx (1)
0 0

π/2
sin x 
(ii)  sin x  cos x dx = 4  . (3)
0

13. y = x2, y2 = x     (Area) .

SY-554 7 P.T.O.

dy y
14. Consider the differential equation :   x2
dx x

(i) Find the integrating factor. (1)

(ii) Find the general solution of the differential equation. (3)

15. Find the shortest distance between the lines :

–r = ^i + ^j +  (2^i – ^j + k)
^

–r = 2i + j – k + (3i – 5j + 2k)

16. Let ^ b– = 2^i – 7^j + k^
–a = ^i – ^j + 3k,

–
(i) Find a– × b (2)

–
(ii) Find the area of the parallelogram with adjacent sides a– and b. (2)

17. (i) Find the Cartesian equation of the plane –r (^i + ^j – k)
^ = 2. (2)

(ii) Find the distance of a point (2, 5, –3) from the plane 6x – 3y + 2z – 4 = 0. (2)

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

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

(ii) Find P (A or B) (2)

(iii) Find P (neither A nor B) (1)

SY-554 8

dy y
14.   x 2    .
dx x

(i)   . (1)

(ii)     . (3)

15. –r = ^i + ^j +  (2^i – ^j + k)
^

–r = 2i + j – k + (3i – 5j + 2k)

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

16. ^ b– = 2^i – 7^j + k^ 
a– = ^i – ^j + 3k,

–
–a × b
(i)  (2)

–
– b     .
(ii) a, (2)

–r (^i + ^j – k)
^ = 2
17. (i)   (Plane)  
. (2)

(ii) 6x – 3y + 2z – 4 = 0    (2, 5, –3)  

 . (2)

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



(i) P (A and B)  (1)

(ii) P (A or B)  (2)

(iii) P (neither A nor B) . (1)

SY-554 9 P.T.O.

Answer any 5 questions from 19 to 25. Each carries 6 scores. (5  6 = 30)

 3 1 2
19. (i) If A =   show that A – 5A + 7I = 0. (3)
 – 1 2 

1 5
(ii) Express the matrix   as the sum of a symmetric and skew-symmetric
6 7

matrices. (3)

20. Consider the system of equations

3x – 2y + 3z = 8

2x + y – z = 1

4x – 3y + 2z = 4

(i) Write the system of equations in the form AX = B (1)

(ii) Find Adj. A. (2)

(iii) Solve the system of equations. (3)

21. (i) Consider the curve y = x3 – x

(a) Find the slope of the tangent at x = 2 (2)

(b) Find the equation of tangent at x = 2 (2)

(ii) Use differential to approximate 36.6 (2)

SY-554 10

19  25    5  .

6  . (5  6 = 30)

 3 1
19. (i) A=    A2 – 5A + 7I = 0  . (3)
 – 1 2

1 5
(ii) 6 7      -
 

   . (3)

20. 3x – 2y + 3z = 8

2x + y – z = 1

4x – 3y + 2z = 4   .

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

(ii) Adj. A . (2)

(iii)  . (3)

21. (i) y = x3 – x   (curve) .

(a)   x = 2  . (2)

(b) x = 2    . (2)

(ii)   36.6    . (2)

SY-554 11 P.T.O.

1
e tan x
22. (i) Integrate with respect to x. (2)
1 x2

1
(ii) Find  2
dx (2)
x – 6 x 13

(iii) Find  x log x dx (2)

23. Consider the vectors

^ b– = ^i – ^j + k^
a– = ^i + ^j – k,

–
(i) Find a– . b (2)

–
(ii) Find the angle between a– and b. (2)

–
(iii) Find the projection of a– on b. (2)

24. Solve the following Linear Programming Problem (LPP) graphically :

Maximize Z = 4x + y

subject to

x + y  50

3x + y  90

x  0, y > 0

SY-554 12

1
e tan x
22. (i)  x   . (2)
1 x2

1
(ii)  x 2 – 6 x 13 dx . (2)

(iii)  x log x dx . (2)

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

–
(i) a– . b . (2)

– –
(ii) a, b    . (2)

–
(iii) a–   b   . (2)

24.      (LPP) .

Maximize Z = 4x + y

subject to

x + y  50

3x + y  90

x  0, y > 0

SY-554 13 P.T.O.

25. Consider the random experiment of tossing 3 coins simultaneously, let X denotes the

number of heads obtained

(i) Find the probability distribution of X (2)

(ii) Find the mean and variance of X. (4)

____________

SY-554 14

25. 3       .

X      .

(i) X    . (2)

(ii) X  ,   . (4)

____________

SY-554 15 P.T.O.

SY-554 16

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