MBOSE Class 12 Question Paper 2024 for Maths

Meghalaya Board of School Education

QUESTION
PAPERS

Total No. of Printed Pages—11

HS/XII/A. Sc. Com/M/24

2024

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 7 questions of Section—A, 3 questions
of Section—B, 5 questions of Section—C and 2 questions
of Section—D. You have to attempt only one of the
alternatives in all such questions.

(iv) Use of calculator is not permitted.

/26 [ P.T.O.

( 2 )

SECTION—A

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

2. Differentiate w.r.t. x : 1

y = log (log x ), x > 1

(i + 2 j )2
3. Construct a 2 ´ 2 matrix A = [a ij ], where a ij = . 1
2

Or

Find the values of x and y, if

é 3x + y -y ù é 1 2 ù
ê 2x - y =
ë 3 úû êë 4 3 úû 1

4. Find the unit vector in the direction of the sum of the
r r
vectors a = 2i$ - $j + 2k$ and b = -i$ + $j + 3k$ . 1

5. Find the Cartesian equation of the line which passes
through the point (-2, 4, - 5) and parallel to the line given
by

x + 3 y -4 z + 8
= = 1
3 5 6

HS/XII/A. Sc. Com/M/24/26 [ Contd.

( 3 )

6. Integrate the function sin x × sin (cos x ). 1

Or

1 dx
Evaluate the definite integral ò .
0 1
1 - x2

dy
7. Find for the following : 1
dx

2x + 3y = sin y

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

r
9. Find the direction cosines of the vector a = i$ + 2 $j + 3k$ . 1

Or

Find the scalar and vector components of the vector with
initial point ( 2, 1 ) and terminal point ( - 5, 7 ). 1

10. Find the value of k, if the function given by

ïì kx 2, x ³ 1
f (x ) = í
ïî 4 , x < 1

is continuous at x = 1. 1

HS/XII/A. Sc. Com/M/24/26 [ P.T.O.

( 4 )

x
11. Differentiate log (cos e ) w.r.t. x. 1

Or

dy
Find , if x = a cos q and y = a sin q. 1
dx

12. The radius of a circle is increasing at the rate of 0·7 cm/s.
What is the rate of increase in its circumference? 1

13. Find the second-order derivative of log x . 1

14. Examine the continuity of the function f (x ) = 5x - 3
at x = 5. 1

15. Evaluate : 1
dx
ò
9 - 25x 2

Or

Integrate w.r.t. x : 1

1
x + x log x

HS/XII/A. Sc. Com/M/24/26 [ Contd.

( 5 )

16. Verify that y = e x + 1 is a solution of the differential
equation y ¢¢ - y ¢ = 0 . 1

Or

Find the general solution of the differential equation
dy 1 + y 2
= . 1
dx 1 + x 2

Choose the correct answer :

17. Let R be the relation in the set n given by
R = { ( a, b ) : a = b - 2, b > 6 } . Then

(a) (2, 4) Î R

(b) (3, 8) Î R

(c) (6, 8) Î R

(d) (8, 7) Î R 1

18. The corner points of the feasible region determined by a
system of linear constraints are (0, 0), (0, 40), (20, 40),
(60, 20) and (60, 0). The maximum value of Z = 4x + 3y is

(a) 120

(b) 240

(c) 300

(d) 200 1

HS/XII/A. Sc. Com/M/24/26 [ P.T.O.

( 6 )

é5 0ù
19. The matrix ê ú is a
ë0 5û

(a) diagonal matrix

(b) row matrix

(c) column matrix

(d) unit matrix 1

Or

The square matrices A and B will be inverse of each
other, if

(a) AB = O, BA = I

(b) AB = BA = O

(c) AB = BA = I

(d) AB = I , BA = O 1

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

(a) 1

(b) 2

(c) 0

(d) -1 1

HS/XII/A. Sc. Com/M/24/26 [ Contd.

( 7 )

SECTION—B
1
21. Show that the function f : r * ® r * defined by f (x ) =
2x
is onto. Here, r * is the set of non-zero real numbers. 2

Or

Let A = r - { 3} and B = r - {1} . Consider the function
x -2
f : A ® B defined by f (x ) = . Show that f is one-one. 2
x -3

22. Prove that the function f (x ) = log x is strictly increasing
on the interval ( 0, ¥ ). 2

23. Evaluate : 2

dx
ò (x + 1)(x + 2)
Or
dy
Find , if xy + y 2 = tan x + y.
dx 2

24. If y = 5 cos x - 3 sin x, then prove that

d 2y
+y =0 2
dx 2

Or
dy
Find , if x = a (q + sin q ) and y = a (1 - cos q ). 2
dx

HS/XII/A. Sc. Com/M/24/26 [ P.T.O.

( 8 )

1
25. Given that the events A and B are such that P (A ) = ,
2
3
P (B ) = p and P (A È B ) = . Find p, if A and B are
5
independent. 2

26. Using determinants, find the value of k so that the points
(3, - 2), (k, 2) and (8, 8) lie on the same line. 2

SECTION—C

27. Using integrals, find the area enclosed by the circle
x 2 + y 2 = 4. 4

Or

Evaluate : 4

2x + 1
ò 2
dx
x + 4x + 3

28. Differentiate (log x )cos x w.r.t. x. 4

Or

Using the properties of definite integral, evaluate

p /2
ò0 (2 log sin x - log sin 2x ) dx
4

HS/XII/A. Sc. Com/M/24/26 [ Contd.

( 9 )

29. Show that the relation R on z given by
R = { (x, y ) : x - y is divisible by 2 } is an equivalence
relation. 4

Or

Let A and B be two sets. Show that f : A ´ B ® B ´ A
such that f ( a, b ) = ( b, a ) is one-one and onto. 4

30. Solve the following LPP graphically : 4

Minimize Z = 200x + 500y

subject to
x + 2y ³ 10
3x + 4y £ 24
x, y ³ 0

31. Find the angle between the two lines whose equations are

x + 3 y -1 z + 3 x +1 y - 4 z - 5
= = and = = 4
3 5 4 1 1 2

Or
r
Find l and m, if ( 2i$ + 6 $j + 27k$ ) ´ ( i$ + l$j + mk$ ) = 0. 4

32. Find the intervals in which the function f given by
f (x ) = 2x 3 - 3x 2 - 36x + 7 is—

(a) strictly increasing;

(b) strictly decreasing. 4

HS/XII/A. Sc. Com/M/24/26 [ P.T.O.

( 10 )

Or

-1
If y = e a cos x
, -1 £ x £ 1, then show that

(1 - x 2 ) y 2 - xy 1 - a 2y = 0 4

SECTION—D
33. Solve the following system of linear equations : 6

x+y+z =6
x -y + z = 2
2x + y - z = 1

34. Find the shortest distance between the lines l1 and l2 ,
whose vector equations are
r
r = ( i$ + $j ) + l ( 2i$ - $j + k$ )
r
r = ( 2i$ + $j - k$ ) + m ( 3i$ - 5 $j + 2k$ ) 6

35. Show that a closed right circular cylinder of given total
surface area and maximum volume is such that its height
is equal to its diameter. 6

Or

Solve the linear differential equation

dy æ pö
cos2 x + y = tan x ç0 £ x £ ÷
dx è 4ø

given that y = 1, when x = 0. 6

HS/XII/A. Sc. Com/M/24/26 [ Contd.

( 11 )

36. State Bayes’ theorem and use it to solve the following :
2+4=6

In answering a question on a multiple choice test, a
student either knows the answer or guesses. Let 3 / 4 be
the probability that he knows the answer and 1 / 4 be the
probability that he guesses. Assuming that a student who
guesses at the answer will be correct with probability 1 / 4,
what is the probability that the student knows the answer
given that he answered it correctly?

Or

A random variable X has the following probability
distribution :

X 0 1 2 3 4 5 6 7
P (X ) 0 k 2k 2k 3k k2 2k 2 7k 2 + k

Determine the following : 6
(i) k
(ii) P (X < 3 )
(iii) P (X > 6 )
(iv) P (0 < X < 3 )

HHH

HS/XII/A. Sc. Com/M/24/26 24K—5660

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