MBOSE Class 12 Question Paper 2023 for Mathematics

Total No. of Printed Pages—12
HS/XII/A. Sc. Com/M/23

2023

MATHEMATICS

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 9 questions of Section—A, 5 questions
of Section—B, 5 questions of Section—C and 3 questions
of Section—D. You have to attempt only one of the
alternatives in all such questions.
(iv) Use of calculator is not permitted.

SECTION—A

1. Define an equivalence relation. 1

Or

A relation R in the set N of natural numbers is defined as
R = {(x , y ) : y = x + 5 and x < 4}. Find the range of R. 1

/52 [ P.T.O.

( 2 )

2. Find the principal value of cos -1 æç - ö÷.
1
1
è 2ø

Or

Evaluate : 1
æ1 ö æ1 ö
cos -1 ç ÷ + 2 sin -1 ç ÷
è2ø è2ø

3. Find the value of x, if
2 3 x 3
= 1
4 5 2x 5

Or

é0 -1ù é3 5 ù
If A = ê ú and B = ê ú, find AB and BA. 1
ë0 2 û ë0 0 û

4. What is an objective function of a linear programming
problem? 1

5. Evaluate : 1
p /2
ò0 cos 2x dx

Or

Evaluate : 1
1 dx
ò0 1 + x 2
HS/XII/A. Sc. Com/M/23/52 [ Contd.

( 3 )

6. Differentiate : 1
2 cot x

Or
dy
Find , if y = sec (tan x ). 1
dx

7. Write the order and degree of the differential equation
4
æ ds ö d 2s
ç ÷ + 3s 2 = 0 1
è dt ø dt

8. Prove that y = Ax is a solution of the differential equation
xy ¢ = y , (x ¹ 0)
and A is a constant. 1

9. Show that the function f : R ® R given by f (x ) = x 3 is
injective, where R is the set of real numbers. 1

Or

Show that the modulus function f : R ® R given by
f (x ) = |x| is not one-one, where R is the set of real
numbers. 1

dy
10. Find , if y = log (cos e x ). 1
dx

Or
dy
If 2x + 3y = sin x, find . 1
dx

HS/XII/A. Sc. Com/M/23/52 [ P.T.O.

( 4 )

6 5 7
11. If P (A) = , P (B ) = and P (A È B ) = , find P (A Ç B ). 1
11 11 11

3 3
12. Let E and F be events with P (E ) = , P (F ) = and
5 10
1
P (E Ç F ) = . Are E and F independent? 1
5

13. Evaluate : 1

2æ 1 ö
ò x çè1 - x 2 ÷ø dx
Or

Evaluate : 1

ò tan x dx
2

14. Compute the magnitude of the following vector : 1
r 1 $ 1 $ 1 $
a = i + j- k
3 3 3

15. Evaluate : 1
p /2
ò- p/2 sin x dx
7

16. Prove that the function

f (x ) = x 3 - 3x 2 + 3x - 100

is increasing in R, where R is the set of real numbers. 1

HS/XII/A. Sc. Com/M/23/52 [ Contd.

( 5 )

Choose the correct answer :

17. If sin -1 x = y, then

(a) 0 £ y £ p

p p
(b) - £y £
2 2

(c) 0<y <p

p p
(d) - <y < 1
2 2

écos a - sin a ù
18. If A = ê ú, then A + A ¢ = I , if the value of a is
ë sin a cos a û

p
(a)
6

p
(b)
3

(c) p

3p
(d) 1
2

HS/XII/A. Sc. Com/M/23/52 [ P.T.O.

( 6 )

æ 1 ö
19. The anti-derivative of ç x + ÷ is equal to
è xø

1 13 1
(a) x + 2x 2 + c
3

2 23 1 2
(b) x + x +c
3 2

2 32 1
(c) x + 2x 2 + c
3

3 32 1 12
(d) x + x +c 1
2 2
Or
The rate of change of the area of a circle with respect to
its radius r at r = 6 cm is

(a) 10p

(b) 12p

(c) 8p

(d) 11p 1

20. The value of i$ × ( $j ´ k$ ) + $j × (i$ ´ k$ ) + k$ × (i$ ´ $j ) is

(a) 0

(b) -1

(c) 1

(d) 3 1

HS/XII/A. Sc. Com/M/23/52 [ Contd.

( 7 )

SECTION—B

21. Find x and y, if

é1 3 ù éy 0ù é5 6ù
2ê ú+ê ú=ê ú 2
ë0 x û ë1 2û ë1 8û

Or

é1 2ù
If A = ê ú, show that |2A| = 4|A|. 2
ë4 2 û

22. Find the value of k, if the function defined by

ìkx 2 , if x £ 2
f (x ) = í
î 3 , if x > 2
is continuous at x = 2. 2

Or

Show that f (x ) = 5x - 3 is continuous at x = 5. 2

23. Evaluate : 2

ò x e dx
x

Or

Evaluate : 2
x æ1 1 ö
ò e çè x - x 2 ÷ø dx

HS/XII/A. Sc. Com/M/23/52 [ P.T.O.

( 8 )

24. If y = 500e 7x + 600e - 7x , show that

d 2y
= 49y 2
dx 2

r r r r
25. Find |a ´ b |, if a = 2i$ + $j + 3k$ and b = 3i$ + 5 $j - 2k$ . 2

Or

Show that the points A (2 , 3 , 4), B (-1, - 2 , 1) and
C (5 , 8 , 7) are collinear. 2

dy
26. Find , if
dx

æ1 - x 2 ö
y = cos -1 çç ÷÷ , 0 < x < 1 2
è1 + x 2 ø

Or

Evaluate : 2

x2
ò 1 - x 6 dx
HS/XII/A. Sc. Com/M/23/52 [ Contd.

( 9 )

SECTION—C

27. Show that the function f : R ® R defined by f (x ) = 3 - 4x
is one-one and onto, where R is the set of real numbers. 4

Or

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 intervals in which the function f given by

f (x ) = 4x 3 - 6x 2 - 72x + 30

is (a) strictly increasing and (b) strictly decreasing. 4

Or

Sand is pouring from a pipe at the rate of 12 cm 3 / s. The
falling sand forms a cone on the ground in such a way
that the height of the cone is always one-sixth of the
radius of the base. How fast is the height of the sand cone
increasing when the height is 4 cm? 4

29. Using the properties of definite integrals, evaluate

p /4
ò0 log (1 + tan x ) dx 4

Or

Using partial fraction, evaluate

1
ò x 4 - 1 dx 4

HS/XII/A. Sc. Com/M/23/52 [ P.T.O.

( 10 )

30. Solve the following homogeneous differential equation : 4

xdy - ydx = x 2 + y 2 dx

Or

Solve the following linear differential equation :

dy 1
(1 + x 2 ) + 2xy =
dx 1 + x2
given y = 0, when x = 1. 4

r p p
31. If a unit vector a makes angle with i$, with $j and
3 4
an acute anglerq with k,$ then find q and hence the
components of a . 4

Or
r r r r r r r
If a , b , c are unit vectors and a + b + c = 0, find the
r r r r r r
value of a × b + b × c + c × a . 4

32. Solve the following linear programming problem
graphically : 4
Maximize Z = 4x + y
subject to the constraints
x + y £ 50
3x + y £ 90
x ³ 0, y ³ 0

HS/XII/A. Sc. Com/M/23/52 [ Contd.

( 11 )

SECTION—D

33. Verify A (adj A) = (adj A) A = |A| I for the matrix
é1 -1 2 ù
A = ê3 0 - 2 ú
ê ú 6
êë1 0 3 úû

Or
Solve the following system of linear equations using matrix
method : 6
x - y + 2z = 7
3x + 4y - 5z = -5
2x - y + 3z = 12

34. Prove that the volume of the largest cone that can be
8
inscribed in a sphere of radius R is of the volume of
27
the sphere. 6
Or
Using definite integral, find the area of the region
bounded by the ellipse
x2 y2
+ =1 6
16 9

35. Find the shortest distance between the lines whose vector
equations are
r
r = (i$ + 2 $j + 3k$ ) + l (i$ - 3 $j + 2k$ )
and
r
r = (4i$ + 5 $j + 6k$ ) + m (2i$ + 3 $j + k$ ) 6

HS/XII/A. Sc. Com/M/23/52 [ P.T.O.

( 12 )

36. A manufacturer has three machine operators A, B and C.
The first operator A produces 1% defective items whereas
the other two operators B and C produce 5% and 7%
defective items respectively. A is on the job for 50% of the
time, B is on the job for 30% of the time and C is on the
job for 20% of the time. A defective item is produced.
What is the probability that it was produced by A? 6

Or

Let X denote the number of hours you study during a
randomly selected school day. The probability that X can
take the values x has the following form, where k is a
constant :
ì 0 ×1 , if x = 0
ïkx if x = 1 or 2
ï ,
P (X = x ) = í
ïk (5 - x ) , if x = 3 or 4
ïî0 , otherwise

Find the value of k. What is the probability that you study
at least 2 hours, exactly 2 hours and at most 2 hours? 6

HHH

HS/XII/A. Sc. Com/M/23/52 K23—5730