MBOSE Class 12 Question Paper 2020 for Mathematics New Course

Total No. of Printed Pages—11
HS/XII/A. Sc. Com/M/NC/20

2020

MATHEMATICS

( New Course )

Full Marks : 80
Time : 3 hours

The figures in the margin indicate full marks for the questions

General Instructions :

(i) All questions are compulsory.

(ii) This question paper contains 36 questions divided into
four Sections A, B, C and D. Section—A comprises of
20 questions of 1 mark each, Section—B comprises of
6 questions of 2 marks each, Section—C comprises of
6 questions of 4 marks each and Section—D comprises
of 4 questions of 6 marks each.

(iii) There is no overall choice. However, internal choice
has been provided in 7 questions of Section—A,
2 questions of Section—B, 2 questions of Section—C
and 1 question of Section—D. You have to attempt
only one of the alternatives in all such questions.

(iv) Use of calculator is not permitted.

/31 [ P.T.O.

( 2 )

SECTION—A

1. Show that the relation R in the set { 1, 2, 3 } given by
R  { (1, 2),(2, 1) } is symmetric. 1

2. Find the Cartesian equation of the plane

r ·(iˆ  ˆj  kˆ )  2 1

1 1
3. Find the principal value of sin  . 1
 2

4. Construct a 2 × 2 matrix A  [aij ], whose elements are
given by
(i  j )2
aij  1
2

5. Find the values of x, y, z if
 x  y  z  9 
 x  z   5 
    1
 y  z  7 

Or

If
1 2 3  3 1 3 
A  and B   
2 3 1   1 0 2 
then find 2A  B . 1

HS/XII/A.Sc.Com/M/NC/20/31 [ Contd.

( 3 )

6. If a line has direction ratios –18, 12, – 4, then determine
its direction cosines. 1

7. If P ( A )  0·8, P (B )  0·5 and P (B|A )  0·4, find P ( A  B ). 1

8. If A and B are two independent events with P ( A )  0·3
and P (B )  0·4 , find P ( A  B ). 1

9. Show that the function

x if x  1
f (x )  
5 if x  1
is not continuous at x  1. 1

Or
Find the minimum value of f (x )  (2x  1)2  3. 1

10. Differentiate y  sin(x 2  5) with respect to x. 1

Or

Show that the function f (x )  e 2x is increasing on r. 1

11. Find the rate of change of the area of a circle with
respect to its radius r, when r =4 cm. 1

12. Find the slope of the tangent to the curve y  3x 4  4x
at x  4. 1

HS/XII/A.Sc.Com/M/NC/20/31 [ P.T.O.

( 4 )

13. Find g  f and f  g , if f : r  r and g : r  r are
2
given by f (x )  cos x and g (x )  3x . 1

14. Find the order and degree of the differential equation
4
 ds  d 2s
   3s 0 1
 dt  dt 2
Or
/2
Find  cos2 x dx . 1
0

15. Find the unit vector in the direction of the vector

a  ˆj  iˆ  2kˆ 1

Or
Find the vector joining the points P (2, 3, 0) and
Q(–1, –2, –4) directed from P to Q. 1

Choose the correct answer :
16. Which of the given values of x and y make the following
pair of matrices equal? 1
3x  7 5  0 y  2
 y 1 2  3x  and 8 4 
   
1
(a) x , y 7
3
2
(b) x  , y 7
3
(c) Not possible to find
1 2
(d) x , y
3 3

HS/XII/A.Sc.Com/M/NC/20/31 [ Contd.

( 5 )

17. If

x 2 6 2

18 x 18 6

then x is equal to
(a) 6
(b) 6
(c) –6
(d) 0 1
Or
Which of the following is correct? 1
(a) Determinant is a square matrix
(b) Determinant is a number associated to a matrix
(c) Determinant is a number associated to a square
matrix
(d) None of the above

 1 
18. The antiderivative of  x   equals
 x
1 1/3
(a) x  2x1/2  c
3
2 2/3 1 2
(b) x  x c
3 2
2 3/2
(c) x  2x1/2  c
3
3 3/2 1 1/2
(d) x  x c 1
2 2

HS/XII/A.Sc.Com/M/NC/20/31 [ P.T.O.

( 6 )

Or

sin2 x  cos2 x
 sin2 x cos2 x dx is equal to
(a) tan x  cot x  c

(b) tan x  cosec x  c

(c)  tan x  cot x  c

(d) tan x  sec x  c 1

19. The value of iˆ·( ˆj  kˆ )  ˆj ·(iˆ  kˆ )  kˆ·(iˆ  ˆj ) is

(a) 0

(b) –1

(c) 1

(d) 3 1

20. The maximum value of Z  3x  4y subject to the
constraints x  y  4, x  0, y  0 is

(a) 8

(b) 16

(c) 10

(d) 4 1

HS/XII/A.Sc.Com/M/NC/20/31 [ Contd.

( 7 )

SECTION—B

21. Find the value of

  1 
tan1 2 cos  2sin1   2
  2 

Or

Show that
8 3 77
sin1  sin1  tan1 2
17 5 36

1
22. Find  dx . 2
1  tan x

23. Verify that y  x sin x is a solution of the differential
equation xy   y  x x 2  y 2 (x  0 and x  y or x  y ). 2

24. If y  (tan1 x )2 , show that

(x 2  1)2 y2  2x (x 2  1) y1  2 2

25. If x and y are connected parametrically by the equations
x  a (  sin ) and y  a (1  cos )
dy
find without eliminating the parameter. 2
dx

HS/XII/A.Sc.Com/M/NC/20/31 [ P.T.O.

( 8 )

26. Find the general solution of
y log y dx  x dy  0 2
Or
Find the general solution of

e x tan y dx  (1  e x )sec2 y dy  0 2

SECTION—C

27. Show that the relation R in the set A = { 1, 2, 3, 4, 5 }
given by R = {(a, b) : |a – b | is even } is an equivalence
relation. 4

28. Find the value of k so that the function
1  cos kx
 x sin x if x 0
f (x )  
 1
if x 0
 2
is continuous at x =0. 4

29. Find the equation of all lines having slope 2 and being
2
tangent to the curve y  0. 4
x 3
Or

2x
Show that y  log (1  x )  , x  1 is an increasing
2x
function of x throughout its domain. 4

HS/XII/A.Sc.Com/M/NC/20/31 [ Contd.

( 9 )

(x 2  1) e x
30. Find  dx . 4
(x  1)2

Or

Find the area of the region bounded by the two
2 2
parabolas y  x and y  x . 4

31. Let

a  iˆ  4 ˆj  2kˆ

b  3iˆ  2 ˆj  7kˆ

c  2iˆ  ˆj  4kˆ
 
Find a vector d which is perpendicular to both a and
 
b, and c ·d  15 . 4

32. Solve the following graphically : 4

Minimize Z  3x  5y

subject to the constraints

x  3y  3
x y 2
x, y  0

HS/XII/A.Sc.Com/M/NC/20/31 [ P.T.O.

( 10 )

SECTION—D

33. The sum of three numbers is 6. If we multiply third
number by 3 and add second number to it, we get 11.
By adding first and third numbers, we get double of the
second number. Represent it algebraically and find the
numbers using matrix method. 6

Or

Obtain the inverse of the following matrix using
elementary operations : 6

0 1 2 
A  1 2 3
3 1 1 

34. Evaluate : 6

 x sin x
0 1  cos2 x dx

35. Find the vector equation of the plane passing through
the intersection of the planes

r ·(2iˆ  2 ˆj  3kˆ )  7

r ·(2iˆ  5 ˆj  3kˆ )  9

through the point (2, 1, 3). 6

HS/XII/A.Sc.Com/M/NC/20/31 [ Contd.

( 11 )

36. State Bayes’ theorem on probability. Use it to solve the
following :

A laboratory blood test is 99% effective in detecting a
certain disease when it is in fact present. However, the
test also yields a false positive result for 0·5% of the
healthy persons tested (i.e., if a healthy person is tested,
then, with probability 0·005, the test would imply he has
the disease). If 0·1% of the population actually has the
disease, then what is the probability that a person has
the disease given that his test result is positive? 6



HS/XII/A.Sc.Com/M/NC/20/31 20K