MBOSE Class 12 Question Paper 2025 for Maths

Meghalaya Board of School Education

QUESTION
PAPERS

Total No. of Printed Pages—12

HS/XII/A. Sc. Com/M/25

2025

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.

/226 [ P.T.O.

( 2 )

SECTION—A

-1
1. Find the principal value of cot (- 3 ). 1

Or

ìp æ 1 öü
Evaluate : sin í - sin -1 ç - ÷ý 1
î3 è 2 øþ

2. Write the elements a 23 and a 32 of a 3 ´ 3 matrix A = [a ij ],
|i - j|
whose elements a ij are given by a ij = . 1
2

3. Check the continuity of the function f given by
f (x ) = 2x + 3 at x = 1. 1

Or

Find the value of k, so that the function

ì kx 2 , if x £ 2
f (x ) = í
î 3 , if x > 2

is continuous at x = 2. 1

4. Define an equivalence relation. 1

Or

If R = {(1, - 1), (2, - 2), (3, - 1)} is a relation, then find the
domain and the range of R. 1

HS/XII/A. Sc. Com/M/25/226 [ Contd.

( 3 )

5. Evaluate : ò x log 2x dx 1

6. Find the equation of the line joining (1, 2) and (3, 6) using
determinants. 1

Or

é5-l l +1 ù
For what values of l the matrix ê is
ë 2 4 úû
invertible? 1

7. The random variable X has a probability distribution P (X )
of the following form, where k is some real number :

ì k, if X = 0
ï 2k , if X = 1
ï
P (X ) = í
ï 3k , if X = 2
ïî 0 , otherwise

Determine the value of k. 1

8. Find an anti-derivative of (ax + b )2 by the method of
inspection. 1

Or

Find the order and degree of the differential equation

2
d 2y æ dy ö
+ 5x ç ÷ - 6y = log x 1
dx 2 è dx ø

HS/XII/A. Sc. Com/M/25/226 [ P.T.O.

( 4 )

r r
9. If q is the angle between any two non-zero vectors a and b
r r r r
such that |a ´ b |= a × b , then find q. 1

Or

If a line makes angles 90º, 135º and 45º with the x-, y-
and z-axes respectively, then find its direction cosines. 1

10. Given two independent events A and B such that
P (A ) = 0 × 3 and P (B ) = 0 × 6. Find P (neither A nor B ). 1

11. Find the second-order derivative of x cos x w.r.t. x. 1

12. If f : R ® R is a function defined by f (x ) = x 2 , " x Î R,
then show that f is not one-one. 1

a
13. If ò 3x 2dx = 8, then write the value of a. 1
0

Or

Differentiate sin2 x w.r.t. e cos x . 1

14. Show that the function f (x ) = 3x + 17 is strictly increasing
on R. 1

HS/XII/A. Sc. Com/M/25/226 [ Contd.

( 5 )

15. Show that the relation R in R defined as R = {(a, b ) : a £ b }
is transitive. 1

dy
16. Find , if ax + by 2 = cos y. 1
dx

Or

log x
Find the integral ò dx. 1
x

Choose the correct answer :

17. The integrating factor of the differential equation
dy
x - y = 2x 2 is
dx

(a) e - x

(b) e - y

1
(c)
x

(d) x 1

18. The maximum value of the objective function Z = 3x + 4y
subject to the constraints x ³ 0, y ³ 0 and x + y £ 1 is
(a) 7
(b) 4
(c) 3
(d) 10 1

HS/XII/A. Sc. Com/M/25/226 [ P.T.O.

( 6 )

r
19. If a is a non-zero vector of magnitude ‘a’ and ‘l’ is a
r
non-zero scalar, then la is a unit vector if

(a) l = 1

(b) l = -1

(c) a =|l|

1
(d) a = 1
|l|

Or

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

(a) 0

(b) -1

(c) 1

(d) 3 1

ò e sec x (1 + tan x ) dx is equal to
20. x

(a) e x cos x + c

(b) e x sec x + c

(c) e x sin x + c

(d) e x tan x + c 1

HS/XII/A. Sc. Com/M/25/226 [ Contd.

( 7 )

SECTION—B

21. Find the vector and the Cartesian equations of the line
through the point (5, 2, - 4) and which is parallel to the
vector 3i$ + 2 $j - 8k$ . 2

22. Evaluate the integral : 2

p /2 sin x
ò0 1 + cos2 x
dx

Or

dy
Find , if x = a (cos q + q sin q) and y = a (sin q - q cos q ). 2
dx

—®
23. Find the unit vector in the direction of vector PQ ,
where P and Q are the points (1, 2, 3 ) and ( 4, 5, 6 )
respectively. 2

Or

r r r r r
Find |a - b |, if two vectors a and b are such that |a |= 2,
r r r
|b |= 3 and a × b = 4. 2

HS/XII/A. Sc. Com/M/25/226 [ P.T.O.

( 8 )

24. Find the values of x, y and z from the following equation : 2

éx+y 2 ù é6 2ù
ê 5+z =
ë xy úû êë 5 8 úû

Or

é 3 1ù 2
If A = ê ú , then show that A - 5A + 7I = 0. 2
ë - 1 2 û

25. Find the interval in which the function given by
f (x ) = x 2 + 2x - 5 is strictly increasing. 2

Or

ò e (sin x + cos x ) dx
x
Evaluate : 2

26. Find the values of p so that the lines

1 - x 7y - 14 z - 3 7 - 7x y - 5 6 - z
= = and = =
3 2p 2 3p 1 5

are at right angles. 2

Or

r r r
If a = 2i$ + 2 $j + 3k$ , b = -i$ + 2 $j + k$ and c = 3i$ + $j are
r r r
such that a + lb is perpendicular to c , then find the
value of l. 2

HS/XII/A. Sc. Com/M/25/226 [ Contd.

( 9 )

SECTION—C

27. By using the properties of definite integrals, evaluate the
integral

p /4
ò0 log (1 + tan x ) dx 4

Or

3x - 2
Find ò dx. 4
(x + 1)2 (x + 3)

28. A particle moves along the curve 6y = x 3 + 2. Find the
points on the curve at which the y-coordinate is changing
8 times as fast as the x-coordinate. 4

Or

Find the intervals in which the function f given
3 2
by f ( x ) = 2x - 3x - 36x + 7 is (a) increasing and
(b) decreasing. 2+2

29. Let f : X ® Y be a function. Define a relation R in X given
by R = { (a, b ) : f (a ) = f (b ) }. Show that R is an equivalence
relation. 4

Or

Show that the function f : R ® Rr
defined by f (x ) = 3 - 4x
is bijective. 4

HS/XII/A. Sc. Com/M/25/226 [ P.T.O.

( 10 )

dy x cos x x2 + 1
30. Find , if y = x + . 4
dx x2 -1

Or

If y = 3 cos (log x ) + 4 sin (log x ), then show that
x 2y 2 + xy 1 + y = 0. 4

31. 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

32. Find the equation of the curve passing through the point
æ pö
ç0, ÷ whose differential equation is
è 4ø

sin x cos y dx + cos x sin y dy = 0 4

Or

Solve the homogeneous differential equation

(x 2 + xy ) dy = (x 2 + y 2 ) dx 4

HS/XII/A. Sc. Com/M/25/226 [ Contd.

( 11 )

SECTION—D

33. Show that the height of the cylinder of maximum volume
2R
that can be inscribed in a sphere of radius R is . Also
3
find the maximum volume. 6

Or

Find the area of the region bounded by the ellipse
x2 y2
+ = 1 above x-axis. 6
16 9

é 2 -3 5 ù
34. If A = ê 3 2 -4 ú , then find A -1. Using A -1, solve the
ê ú
êë 1 1 -2 úû
following system of equations : 6

2x - 3y + 5z = 11
3x + 2y - 4z = -5
x + y - 2z = -3

35. Find the Cartesian equation of the line passing through
the point (1, 2, - 4) and perpendicular to the two lines

x - 8 y + 19 z - 10 x - 15 y - 29 z - 5
= = and = =
3 -16 7 3 8 -5

Also write the vector form of the line so obtained. 5+1

HS/XII/A. Sc. Com/M/25/226 [ P.T.O.

( 12 )

Or

Find the shortest distance between the lines whose vector
equations are
r
r = (1 - t ) i$ + (t - 2 ) $j + (3 - 2t ) k$ and
r
r = (s + 1 ) i$ + (2s - 1 ) $j - (2s + 1 ) k$ 6

36. State Bayes’ theorem on probability. Use this theorem to
solve any one of the following : 1+5
(a) Bag I contains 3 red and 4 black balls and Bag II
contains 4 red and 5 black balls. One ball is
transferred from Bag I to Bag II and then one ball is
drawn from Bag II. The ball so drawn is found to be
red in colour. Find the probability that the
transferred ball is black.

Or

(b) A man is known to speak truth 3 out of 4 times. He
throws a die and reports that it is a six. Find the
probability that it is actually a six.

HHH

HS/XII/A. Sc. Com/M/25/226 K25—6040

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