Kerala Plus Two Model Question Paper Maths (Commerce)

SAMPLE
PAPER

Sample Paper-Group 4 Reg No———-
Attingal Cluster Name———-

SECOND YEAR SAMPLE QUESTION PAPER- 2023
MATHEMATICS (COMMERCE)
Maximum: 60 (Score)
Time: 2 Hours

Cool-off time: 15 minutes

GENERAL INSTRUCTIONS

• There is a ’Cool-off time’ of 15 minutes in addition to the writing time of 2 hours

• You are not allowed to write your answers or to discuss anything with others during the ’Cool-off time’

• Use ’Cool-off time’ to get familiar with questions and to plan your answers

• Read questions carefully before answering.

• All questions are compulsory and only internal choice is allowed.

• When you select a question, all the sub-questions must be answered from the same question itself.

• Calculations, figures and graphs should be shown in the answer sheet itself.

• Give equations wherever necessary.

• Electronic devices except non-programmable calculators are not allowed in the Examination Hall.

Answer any 6 questions from 1 to 8. Each carries 3 marks. (6 × 3 = 18)
 
6 x 2
 
1. (a) If A = −2 3 −1 is a symmetric matrix, then the value of x is —— (1)
 
 
2 −1 3
   
−2 3 −1 0
(b) If A′ =   and B =   then find (A + 2B)′ (2)
1 2 1 2

2. (a) Construct a 2 × 2 matrix A = [aij ] whose entries are given by aij = 2i + j (1)

(b) Find A2 (2)

x 1
3. (a) If = 15, then find x (1)
1 x
(b) Find the area of triangle with vertices (2,7), (1,1) and (10,8) (2)
dy
4. Find if x = sin t and y = cos(2t) (3)
dx
5. If y = (tan−1 x)2 , then prove that (x2 + 1)2 y2 + 2x(x2 + 1)y1 = 2 (3)

6. Given two independent events A and B such that P (A) = 0.3, P (B) = 0.6. Find
(i) P (A and B) (ii) P (A or B) (iii) P (A and not B) (3)

7. Find a unit vector perpendicular to each of the vectors ~a + ~b and ~a − ~b, where ~a = 3î + 2ĵ + 2k̂ and
~b = î + 2ĵ − 2k̂ (3)

8. An edge of a variable cube is increasing at the rate of 3cm/s. How fast is the volume of the cube increasing
when the edge is 10 cm long. (3)
Answer any 6 questions from 9 to 17. Each carries 4 marks.

9. (a) Let R be a relation defined on A = {1, 2, 3} by R = {(1, 3), (3, 1), (2, 2)}. Then R is
(i) reflexive (ii) symmetric (iii) transitive (iv) reflexive but not transitive (1)

(b) Prove that the function f : R → R given by f (x) = 2x + 3 is a bijection (3)
x+1 y+1 z+1 x−3 y−5 z−7
10. Find the shortest distance between the lines = = and = = (4)
7 −6 1 1 −2 1
 
−1 2π
11. (a) Find the value of sin sin (2)
3
   
−1 −1
(b) Find the value of tan−1 (1) + cos−1 + sin−1 (2)
2 2
12. Find the area bounded by the circle x2 + y 2 = 9 using integration (4)
 3
d2 y dy
13. (a) The degree of the differential equation 2x2 2 + x4 + 7 = 0 is
dx dx
(i) 4 (ii) 3 (iii) 2 (iv) 1 (1)
dy y

(b) Solve the differential equation + = x2 (3)
dx x
14. Find the intervals in which the function given by f (x) = 2x2 − 3x is (i) increasing (ii) decreasing (4)


kx + 1 , x ≤ 5

15. (a) Find the value of k so that the function f (x) = is continuous at x = 5 (2)
3x − 5

,x > 5
dy
(b) Find if 2x + 3y = sin y (2)
dx
 
−2 −2 −4
 
16. Express the matrix A = −1 3 4  as the sum of a symmetric and a skew-symmetric matrix. (4)
 
 
1 −2 −3
Answer any 3 questions from 17 to 20. Each carries 6 marks
 
3 −2 3
 
17. If A = 2 1 −1
 
 
4 −3 2

(a) Find A−1 (3)

(b) Use A−1 from part (a) to solve the system of equations
3x − 2y + 3z = 8, 2x + y − z = 1, 4x − 3y + 2z = 4 (3)

18. Solve the following Linear Programming Problem Graphically. (6)
Maximise Z = 3x + 2y
Subject to

x + 2y ≤ 10

3x + y ≤ 15

x ≥ 0

y ≥ 0

19. Integrate
x
Z
(a) dx (2)
(x − 1)(x − 2)
Z π2
sin4 x
(b) 4 dx (2)
sin x + cos4 x
Z0
(c) x log xdx (2)

20. (a) If A and B are two events such that P (A) = 0.8, P (B) = 0.5, P (B/A) = 0.4 then find P (A/B). (2)

(b) A bag contains 4 red balls and 4 black balls. Another bag contains 2 red and 6 black balls. One of
the two bags is selected at random and a ball is drawn from the bag which is found to be red. Find
the probability that the ball is drawn from the first bag. (4)