Maharashtra 12th Std Model Question Paper Mathematics and Statistics (Art and Science)

Maharashtra State Board Sample Paper 2026

DAY — 08 SEAT NUMBER

2025 VII 03 1100 (E)
J-384
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 : Q. 3 to Q. 14 contain Twelve short answer type
questions, each carrying Two marks.
(Attempt any Eight)
(3) Section C : Q. 15 to Q. 26 contain Twelve short answer type
questions, each carrying Three marks.
(Attempt any Eight)
(4) Section D : Q. 27 to Q. 34 contain Eight long answer type
questions, each carrying Four marks.
(Attempt any Five)
(5) Use of log table is allowed. Use of calculator is not allowed.

(6) Figures to the right indicate full marks.

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

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(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) The inverse of statement pattern ( p q ) ( p q ) is ____
(a) ( p q) ( p q)
(b)  ( p q) ( p q)
(c) ( p  q) ( p  q)
(d) ( p  q) ( p  q) (2)

2
(ii) In ABC , if a 2, b 3 and sin A , then ____
3

(a) (b)
2 3

(c) (d) (2)
4 6

(iii) If AB 2i 4 j 7k and initial point A (1,5,0) then
terminal point B is ____
(a) (1, 3, 7) (b) (7, 3, 1)
(c) (1, 7, 3) (d) (3, 1, 7) (2)

(iv) The angle between the lines
r (i 2 j 3k ) (2i 2 j k ) and
r (i 2 j 3k ) (i 2 j 2k ) is ____

(a) (b)
4 2

(c) (d) 0 (2)
3

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(v) If y is a function of x and log(x + y) = xy then the value of
dy
dx at x = 0 is ____

(a) 1
(b) –1
(c) 2
(d) 0 (2)

(vi) If the displacement of a particle at time t is given
by S 2t3 5t2 4t 3, then its acceleration at time
t = 1 is ____
(a) 2 (b) 8
(c) 10 (d) 14 (2)

(vii) The solution of the D.E.
sec2 x tan y dx sec 2 y tan x dy 0 is ____
(a) tan x.cot y c
(b) cot x cot y c
(c) tan x .tan y c
(d) cot x tan y c (2)

(viii) If X is waiting time in minutes for a bus and its p.d.f. is
1
given by f (x ) , for 0 x 5,
5

= 0, otherwise
then the probability that waiting time is between 1 and
3 is ____
1 2
(a) (b)
5 5
3 4
(c) (d) (2)
5 5

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

(i) Find the general solution of tan 0. (1)

(ii) Find the magnitude of the vector a 3i j 7k (1)

dy
(iii) Find , if y sin(log x ) . (1)
dx

1
(iv) Evaluate : dx . (1)
x

SECTION – B
Attempt any EIGHT of the following questions : [16]

Q. 3. Using truth table, prove that  p q ( p q )  p (2)

2 3
Q. 4. Find the cofactors of the elements of the matrix 3 5 (2)

Q. 5. Find the polar coordinates of the point whose cartesian
coordinates are ( 2, 2). (2)

Q. 6. In ABC , if a = 2, b = 3, c = 4 , then prove that the triangle is
obtuse angled. (2)

Q. 7. If a 3i j 2k, b 2i j k, then find a b (2)

Q. 8. Find the cartesian equation of the plane passing through the
point A(–1, 2, 3), the direction ratios of whose normal are 0, 2, 5. (2)

dy
Q. 9. Find , if x x y y a a. (2)
dx

Q. 10. Show that the tangent to the curve y x 3 6x 2 x 3 at the
point (0, 3) is parallel to the line y x 5 . (2)

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Q. 11. Check whether the conditions of Rolle’s theorem are satisfied
2
by the function f ( x ) x 4 x 3, x [1,3] (2)

dx
Q. 12. Evaluate : 10 (2)
x x

1
x
Q. 13. Evaluate : e dx (2)
0

Q. 14. If X  B( n, p ) and E ( X ) 6, Var( X ) 4.2, then find n and p. (2)

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

Q. 15. In ABC , prove that a 2 b 2 c 2 2bc c o s A (3)

Q. 16. Find the combined equation of pair of lines passing through
(2, 3) and perpendicular to the lines 3 x 2y 1 0 and
. (3)

Q. 17. Show that the acute angle T between the lines represented by

ax 2 2hxy by2 0 is given by,

2
2 h ab (3)
tan
a b

Q. 18. Using vector method, prove that the medians of a triangle are
concurrent. (3)

Q. 19. Find the vector equation of the plane passing through the point
A (–1, 2, –5) and parallel to the vectors 4i j 3k and i j k. (3)

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x 1 y 2 z 3
Q. 20. Find the shortest distance between the lines,
2 3 4

x 2 y 4 z 5
and . (3)
3 4 5

Q. 21. Water is being poured at the rate of 27 m3/sec into a cylindrical
vessel of base radius 3 m. Find the rate at which the water
level is rising. (3)

sin( x a )
Q. 22. Evaluate : dx (3)
cos( x b)

Q. 23. Solve the D.E.

dy y
x3 3 (3)
dx x

Q. 24. Obtain the differential equation by eliminating the arbitrary
constants from y c1 cos(log x ) c2 sin(log x ). (3)

Q. 25. The probability distribution of X is as follows :

x 0 1 2 3 4
P[X=x] 0.1 k 2k 2k k

Find (i) k, (ii) P(X < 2), (iii) P [1 X 4] (3)

Q. 26. In a multiple choice examination with three possible answers
for each of the five questions, what is the probability that a
candidate would get four correct answers just by guessing? (3)

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

Q. 27. Give an alternative equivalent simple circuit for the following
switching circuit :

(4)

Q. 28. The sum of three numbers is 2. If twice of the second number is
added to the sum of first and third number we get 0. Adding
five times the first number to twice the sum of second and third
number we get 7. Find the numbers using matrix method. (4)

Q. 29. Using properties of scalar triple product, prove that
[a b b c c a ] 2 [a b c] (4)

Q. 30. Solve the following L.P.P. using graphical method :
Maximize, z 9x 13 y
Subject to, 2x 3 y 18,
2x y 10
x 0, y 0 (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 0, then prove that
dt
dy
dy dt
dx dx
dt

dy
Hence find , if y at 2 and x 2at. (4)
dx

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x a2 x
Q. 32. Prove that : a 2 x 2 dx a2 x 2 sin 1 c (4)
2 2 a

1
2
dx
Q. 33. Evaluate : (4)
2 2
0 (1 2x ) 1 x

2
Q. 34. Find the area of the region lying between the parabolas y 4x
2
and x 4y . (4)

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Maharashtra State Board Sample Paper 2026

DAY — 08 SEAT NUMBER

2025 VII 03 1100 (M)
J-385
MATHEMATICS & STATISTICS (40)
(ARTS & SCIENCE )
Time : 3 Hrs. ( 8 Pages ) Max. Marks : 80

–

–

–

–

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(i) ( p q) ( p q) (statement pattern)
(inverse)

(a) ( p q) ( p q)
(b)  ( p q) ( p q)
(c) ( p  q) ( p  q)
(d) ( p  q) ( p  q)

2
(ii) a 2, b 3 sin A , .......
3
(a) (b)
2 3

(c) (d)
4 6

(iii) AB 2i 4 j 7k A (1,5,0)
B

(a) (1, 3, 7) (b) (7, 3, 1)

(c) (1, 7, 3) (d) (3, 1, 7)

(iv) r (i 2 j 3k ) (2i 2 j k )
r (i 2 j 3k ) (i 2 j 2k )

(a) (b)
4 2

(c) (d) 0
3

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(v) y x log(x + y) = xy x=0
dy
dx
(a) 1 (b) –1
(c) 2 (d) 0

(vi) t (particle) (displacement)
S = 2t3– 5t2 + 4t–3 t=1

(a) 2 (b) 8
(c) 10 (d) 14

(vii) 2 2
sec x tan y dx sec y tan x dy 0
(differential equation)

(a) tan x.cot y c
(b) cot x cot y c
(c) tan x .tan y c
(d) cot x tan y c

(viii) X:
(p.d.f.)
1
f (x ) ; 0 x 5,
5
=0;
1 3
1 2
(a) (b)
5 5
3 4
(c) (d)
5 5

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(i) tan 0 (general solution)

(ii) (vector) a 3i j 7k (magnitude)

dy
(iii) y sin(log x ) ,
dx

1
(iv) dx
x

(truth-table)
 p q ( p q)  p

2 3
3 5 (matrix) (elements)

(cofactors)

(cartesian co-ordinates)
(polar co-ordinates)

ABC a = 2, b = 3, c = 4 ABC

a 3i j 2k, b 2i j k, a b

A(–1, 2, 3) (normal)
(direction ratios) 0, 2, 5 (plane)
(cartesian equation)

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dy
x x y y a a
dx

y x 3 6x 2 x 3 (0, 3)

(tangent) y x 5

x

(Rolle’s theorem)

dx
10
x x

1
x
e dx
0

X  B ( n, p ) E ( X ) 6, Var( X ) 4.2, n p

ABC

a2 b2 c2 2bc cos A

(2, 3) 3x 2 y 1 0 x 3y 2 0

ax 2 2hxy by2 0
T
2
2 h ab
tan
a b

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(medians) (concurrent)

A (–1, 2, –5) (vector) 4i j 3k

i j k (parallel) (plane)
(vector equation)

x 1 y 2 z 3 x 2 y 4 z 5
2 3 4 3 4 5
(shortest distance)

(cylindrical vessel) (base radius)
3
3 27 m
(rising)

sin( x a )
dx
cos( x b)

dy y 3
x 3 (differential equation)
dx x

y c x c x

(arbitrary constants) (eliminating)
(differential equation)

X

x 0 1 2 3 4
P[X=x] 0.1 k 2k 2k k

(i) k (ii) P(X < 2) (iii) P [1 X 4]

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(switching circuit)
(alternative equivalent) (simple circuit)

2
0
7
(matrix method)

(scalar triple product)
(properties)

[ a b b c c a ] 2[ a b c ]

(L.P.P.)
(maximize) z 9x 13 y
2x 3 y 18,
2x y 10
x 0, y 0

x f (t ) y g (t ) t (differentiable
dx
function) y 0,
dt
dy
dy dt ,
dx dx
dt
dy
y = at2 x = 2at ,
dx

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2
2 2 x 2 2 a x
a x dx a x sin 1 c
2 2 a

1
2
dx
0 (1 2x 2 ) 1 x 2

y2 = 4 x2 = 4y (parabolas)
(region) (area)
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