MH 12th Question Paper 2025 Maths (Arts & Science)

Maharashtra Board

Question Paper
2025
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QUESTION PAPER
Presented By

DAY — 09 SEAT NUMBER

2025 II 22 1100 (E)
J-312
MATHEMATICS & STATISTICS (40)
(ARTS & SCIENCE)

Time : 3 Hrs. ( 8 Pages ) Max. Marks : 80

General instructions :

The question paper is divided into FOUR sections.

(1) Section A : Q. 1 contains Eight multiple choice type
questions carrying Two marks each.

Q. 2 contains Four very short answer type
questions carrying One mark each.

(2) Section B : This section contains Twelve short answer type
questions carrying Two marks each.
(Attempt any Eight)

(3) Section C : This section contains Twelve short answer type
questions carrying Three marks each.
(Attempt any Eight)

(4) Section D : This section contains Eight long answer type
questions carrying Four marks each.
(Attempt any Five)

(5) Use of log table is allowed. Use of calculator is not allowed.

(6) Figures to the right indicate full marks.

0 3 1 2 Page 1 P.T.O

(7) Use of graph paper is not necessary. Only rough sketch of
graph is expected.

(8) For each multiple choice type of questions, only the first attempt
will be considered for evaluation.

(9) Start answer to each section on a new page.

SECTION – A

Q. 1. Select and write the correct answer of the following [16]
multiple choice type of questions :

(i) If A = {1, 2, 3, 4, 5} then which of the following is not
true ?
(a) x A such that x + 3 = 8
(b) x A such that x + 2 < 9
(c) x A, x 6 9
(d) x A such that x + 6 < 10 (2)

(ii) In 'ABC, (a + b)· cos C + (b + c)·cos A + (c + a)· cos B is
equal to _____.
(a) a–b+c
(b) a+b–c
(c) a+b+c
(d) a–b–c (2)

(iii) If a 5, b 13 and a b 25 then a b is equal to
_____.
(a) 30 (b) 60
(c) 40 (d) 45 (2)

0 3 1 2 Page 2

(iv) The vector equation of the line passing through the
point having position vector 4i j 2k and parallel to

vector 2i j k is given by _____.
(a) (4i j 2k ) ( 2i j k )
(b) (4i j 2k ) (2i j k)
(c) (4i j 2k ) ( 2i j k)
(d) (4i j 2k ) ( 2i j k) (2)

1 8
(v) Let f (1) 3, f (1) , g (1) 4 and g (1) 3
. The
3

derivative of [ f ( x )]2 [ g ( x )]2 w.r.t. x at x = 1 is _____.

29 7
(a) (b)
25 3
31 29
(c) (d) (2)
15 15

(vi) If the mean and variance of a binomial distribution are
18 and 12 respectively, then n is equal to _____.
(a) 36 (b) 54
(c) 18 (d) 27 (2)

(vii) The value of x x (1 log x ) dx is equal to _____.

1 2
(a) (1 log x ) c (b) x 2x c
2
(c) x x log x c (d) xx c (2)

(viii) The area bounded by the line y = x, X-axis and the lines
x = – 1 and x = 4 is equal to _____.
(in square units)
2
(a) (b) 8
17
17 1
(c) (d) 2 (2)
2

0 3 1 2 Page 3 P.T.O

Q. 2. Answer the following questions : [4]

(i) Write the negation of the statement : ‘ n N such
that n 8 11 ’ (1)

(ii) Write unit vector in the opposite direction to

u 8i 3 j k. (1)

(iii) Write the order of the differential equation
3
2 2 2
dy d y
1 (1)
dx dx 2

(iv) Write the condition for the function f (x), to be strictly
increasing, for all x R. (1)

SECTION – B

Attempt any EIGHT of the following questions : [16]

Q. 3. Using truth table, prove that the statement patterns p q
and ( p q ) (  p  q ) are logically equivalent. (2)

2 2
Q. 4. Find the adjoint of the matrix 4 3 . (2)

Q. 5. Find the general solution of tan 2 1. (2)

Q. 6. Find the co-ordinates of the points of intersection of the lines
represented by x 2 y 2 2x 1 0 . (2)

Q. 7. A line makes angles of measure 45º and 60º with the positive
directions of the Y and Z axes respectively. Find the angle
made by the line with the positive direction of the X-axis. (2)

0 3 1 2 Page 4

Q. 8. Find the vector equation of the plane passing through the point

having position vector 2i 3 j 4k and perpendicular to the
vector 2i j 2k . (2)

Q. 9. Divide the number 20 into two parts such that sum of their
squares is minimum. (2)

Q. 10. Evaluate : x 9 .sec2 ( x10 )dx (2)

1
Q. 11. Evaluate : dx (2)
25 9x 2

4
1
Q. 12. Evaluate : dx (2)
1 sin x
4

Q. 13. Find the area of the region bounded by the parabola y 2 16
and its latus rectum. (2)

Q. 14. Suppose that X is waiting time in minutes for a bus and its p.d.f.
is given by :

1
f (x ) , for 0 x 5
5

= 0, otherwise.

Find the probability that :

(i) waiting time is between 1 to 3 minutes.

(ii) waiting time is more than 4 minutes. (2)

0 3 1 2 Page 5 P.T.O

SECTION – C
Attempt any EIGHT of the following questions : [24]

Q. 15. Express the following switching circuit in the symbolic form of
logic. Construct the switching table and interpret it :

(3)

Q. 16. Prove that : . (3)

Q. 17. In if a = 13, b = 14, c = 15 then find the values of
A
(i) sec A (ii) cosec . (3)
2

Q. 18. A line passes through the points (6, –7, –1) and (2, –3, 1). Find
the direction ratios and the direction cosines of the line. Show
that the line does not pass through the origin. (3)

Q. 19. Find the cartesian and vector equations of the line passing
through A (1, 2, 3) and having direction ratios 2, 3, 7. (3)

Q. 20. Find the vector equation of the plane passing through points
A (1, 1, 2), B (0, 2, 3) and C (4, 5, 6). (3)

Q. 21. Find the nth order derivative of log x. (3)

Q. 22. The displacement of a particle at time t is given by
s t t t . Find the velocity and displacement at the
time when the acceleration is 14 ft/sec2. (3)

0 3 1 2 Page 6

Q. 23. Find the equations of tangent and normal to the curve

y = 2x3 – x2 + 2 at point . (3)

Q. 24. Three coins are tossed simultaneously, X is the number of heads.
Find the expected value and variance of X. (3)

dy y
Q. 25. Solve the differential equation : x x tan y. (3)
dx x

Q. 26. Five cards are drawn successively with replacement from a well-
shuffled deck of 52 cards. Find the probability that :
(i) all the five cards are spades.
(ii) none is spade. (3)

SECTION – D
Attempt any FIVE of the following questions : [20]

Q. 27. Find the inverse of by elementary row

transformations. (4)

Q. 28. Prove that homogeneous equation of degree two in x and y,

ax 2 2hxy by2 0 represents a pair of lines passing through
the origin if . Hence show that equation x 2 y2 0
does not represent a pair of lines. (4)

Q. 29. Let a and b be non-collinear vectors. If vector r is coplanar
with a and b then show that there exist unique scalars t1 and
t 2 such that For
find t1, t2. (4)

0 3 1 2 Page 7 P.T.O

Q. 30. Solve the linear programming problem graphically.
Maximize : z
Subject to : x 4 y 24,
3x y 21,
x y 9,
x 0, y 0
Also find the maximum value of z. (4)

Q. 31. If x f (t ) and y g (t ) are differentiable functions of t so that y
dx
is a function of x and if 0
dt
dy
dy dt
then prove that .
dx dx
dt
x 7
Hence find the derivative of 7 w.r.t. x . (4)

1 2
Q. 32. Evaluate : esin x x 1 x (4)
dx
2
1 x

b b
Q. 33. Prove that : f ( x )dx f ( a b x )dx
a a

3
x
Hence evaluate : dx (4)
0
x 3 x

Q. 34. If a body cools from 80ºC to 50ºC at room temperature of 25ºC
in 30 minutes, find the temperature of the body after 1 hour. (4)

‹‹‹

0 3 1 2 Page 8

DAY — 09 SEAT NUMBER

2025 II 22 1100 (M)
J-313
MATHEMATICS & STATISTICS (40)
(ARTS & SCIENCE )
Time : 3 Hrs. ( 8 Pages ) Max. Marks : 80

–

–

–

–

0 3 1 3 Page 1 P.T.O.

(i) A = {1, 2, 3, 4, 5}
(a) x A x+3=8
(b) x A x+2<9
(c) x A, x 6 9
(d) x A x + 6 < 10

(ii) (a + b)· cos C + (b + c)·cos A + (c + a)· cos B

(a) a–b+c
(b) a+b–c
(c) a+b+c
(d) a–b–c

(iii) a 5, b 13 a b 25 a b
(a) 30
(b) 60
(c) 40
(d) 45

(iv) (position vector) i  k
2i j k
(vector equation)
(a) (4i j 2k ) ( 2i j k )
(b) (4i j 2k ) (2i j k)
(c) (4i j 2k ) ( 2i j k)
(d) (4i j 2k ) ( 2i j k)

0 3 1 3 Page 2

1 8
(v) f (1) 3, f (1) , g (1) 4 g (1)
3 3
[ f ( x )]2 [ g ( x )]2 x
(derivative) x =1
29 7
(a) (b)
25 3
31 29
(c) (d)
15 15

(vi) (binomial distribution) (mean)
(variance)
n
(a) 36 (b) 54
(c) 18 (d) 27

(vii) x x (1 log x ) dx
1 2
(a) (1 log x ) c
2
(b) x 2x c
(c) x x log x c
(d) xx c

(viii) y = x, X- x=–1 x=4

2
(a) (b) 8
17
17 1
(c) (d) 2
2

(i) ‘ n N (negation)
n 8 11
(ii) u 8i 3 j k (opposite) (unit vector)

0 3 1 3 Page 3 P.T.O.

3
2 2 2
dy d y
(iii) 1 (differential
dx dx 2
equation) (order)

(iv) f (x ) x R (strictly
increasing) (condition)

(truth-table) p q
( p q ) ( p q) (statement pattern)
(logically equivalent)

2 2
4 3 (adjoint matrix)

tan 2 1 (general solution)

x2 y 2 2x 1 0 (point

of intersection)

Y Z 45º 60º
X

2i 3 j 4k (vector) 2i j 2k
(perpendicular) (plane) (vector
equation)

(minimum)

x 9 .sec2 ( x10 )dx

0 3 1 3 Page 4

1
dx
25 9x 2

4
1
dx
1 sin x
4

y 2 16 (latus rectum)

(waiting time)
(probability distribution function)

f x for 0 x 5

=0
(i)
(ii) (probability)

(switching circuit)
(symbolic form) (switching table)

0 3 1 3 Page 5 P.T.O.

a = 13, b = 14, c = 15
A
(i) sec A (ii) cosec
2

(6, –7, –1) (2, –3, 1)
(direction cosines and direction ratios)
(origin)

A (1, 2, 3) (direction
ratios) 2, 3, 7 (cartesian
and vector equation)

A(1, 1, 2), B (0, 2, 3) C (4, 5, 6)
(vector equation)

Log x nth (derivative)

(particle) t s 2t 3 5t 2 4t 3
(acceleration) 14 ft/sec2 (time) (velocity)
(displacement)

y = 2x3 – x2 + 2 (tangent)

(normal)

(head)
X (expected value) (variance)

dy y
x x tan y (differential
dx x
equation)

0 3 1 3 Page 6

(with
replacement) (cards)

(i) (cards) (spade)
(ii) (spade)

(elementary row transformation)

(inverse)

x y (degree 2) (homogeneous)

ax 2 2hxy by2 0

x2 y2 0

a b (non-collinear vectors)
r a b (coplanar)
t t r t1a t 2 b
1 2
b j 3k t1 t2

(linear programing problem)

(maximize) z
(subject to) x 4 y 24,
3x y 21,
x y 9,
x 0, y 0
(z) (maximize)

0 3 1 3 Page 7 P.T.O.

x f (t ) y g (t ) t (differentiable
dx
function) y 0,
dt

dy
dy dt , 7x x7 (derivative)
dx dx
dt

sin
1
x x 1 x2
e dx
2
1 x

b b
f ( x )dx f ( a b x )dx
a a

3
x
dx
0
x 3 x

25ºC 80ºC 50ºC
30

‹‹‹

0 3 1 3 Page 8

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