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

Maharashtra State Board Sample Paper 2026

DAY — 08 SEAT NUMBER

2025 VII 03 1100 (E)
J-388
MATHEMATICS & STATISTICS (88)
(COMMERCE)
Time : 3 Hrs. ( 15 Pages ) Max. Marks : 80

General Instructions :
(i) All questions are compulsory.
(ii) There are six questions divided into
two sections.
(iii) Write answers of Section-I and
Section-II in the same answer book.
(iv) Use of logarithmic tables is allowed.
Use of calculator is not allowed.
(v) For L.P.P. and Time Series graph
paper is not necessary. Only rough
sketch of graph is expected.
(vi) Start answer to each question on a
new page.
(vii) For each objective type of question (i.e.
Q.1 and Q.4) only the first attempt
will be considered for evaluation.

0 3 8 8 Page 1 P.T.O

SECTION - I
Q. 1. (A) Select and write the correct answer of the following multiple
choice type of questions (1 mark each) : (6) [12]

0 3 1
(i) Matrix B 3 0 4 is skew-symmetric then value
P 4 0
of P is
(a) 1
(b) –1
(c) 0
(d) –3

dy
(ii) If y e log x then _____.
dx
1
(a)
x
1
(b)
2
log x
(c) e
x
(d) 0

3
(iii) (1 x ) dx _____.

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

0 3 8 8 Page 2

a
(iv) If 3x 2dx 8 then a = ____.
0

(a) 0 (b) 2
4
(c) 8 (d)
3
2
(v) Area of the region bounded by the curve y x , the
X-axis and the lines x = 1 and x = 3 is _____.
3
(a) sq. units (b) 3 sq. units
26
26
(c) 26 sq. units (d) sq. units
3

2 2
d2 y dy
(vi) The order and degree of 2
a x are
dx dx

respectively ____
(a) 1, 1 (b) 1, 2
(c) 2, 1 (d) 2, 2

(B) State whether the following statements are true or false
(1 mark each) : (3)

(i) Dual of ( p  q ) t is ( p  q) c

x 1 x x
(ii) For 3
e dx e f ( x ) c , where f ( x ) ( x 1)2
( x 1)

dy x
(iii) The integrating factor (I.F.) of y e is ex
dx

(C) Fill in the following blanks (1 mark each) : (3)

(i) Negation of "Some men are animals" is ____.

0 3 8 8 Page 3 P.T.O

(ii) If the average revenue is 45 and elasticity of demand
is 3 then marginal revenue is ____.

10 x 9 10 x log10
(iii) To find the value of dx , the proper
10 x x10
substitution is ____.

Q. 2. (A) Attempt any TWO of the following questions (3 marks [14]
each) : (6)

(i) Determine whether the following statement pattern
is tautology, contradiction or contingency.
[( p q ) (  p )] [ p (  q )]

dy
(ii) Find if y (log x )x x
5
dx

(iii) Find the area of the region bounded by the parabola
y2 4 x and line x = 3.

(B) Attempt any TWO of the following questions (4 marks
each) : (8)

(i) Find MPC, MPS, APC and APS, if the expenditure
Ec of a person with income I is given as Ec = (0.0003)
I2 + (0.075) I when I = 1000.

(ii) Solve the following differential equation :

x 2 y dx ( x 3 y3 ) dy 0

(iii) Express the following equations in matrix form and
solve them by method of reduction :
x + 2y + z = 8
2x + 3y – z = 11
3x – y – 2z = 5

0 3 8 8 Page 4

Q. 3. (A) Attempt any TWO of the following questions (3 marks [14]
each) : (6)

(i) Write the converse, inverse and contrapositive of the
following statement.
'If a man is bachelor, then he is happy.'

3 1 5
(ii) Find the inverse of 2 7 8 by adjoint method.
12 5

dy y
(iii) If then show that .
dx x

(B) Attempt any ONE of the following questions (4 marks
each): (4)

2 3x
(i) Evaluate : x e dx

4
x
(ii) Evaluate :

(C) Attempt any ONE of the following questions (Activity)
(4 marks each): (4)

(i) Find the values of x, such that f(x) is increasing
function
f(x) = 2x3 – 15x2 – 144x – 7
Solution :
Given : f ( x ) 2x 3 15x 2 144 x 7

f ( x ) 6x 2 30x 144
Now, f ( x ) 0 , as f(x) is increasing

6 x 2 30 x 144 0
2

0 3 8 8 Page 5 P.T.O

( x 8)( x 3) 0
Case (I) x 8 0 and x 3 0
x 8 and x 3

Case (II) x 8 0 and x 3 0
x 8 and x 3
x

f ( x ) 2x 3 15x 2 144 x 7 is increasing if and
only if x ( , ) or x ( , )

(ii) Solve the following differential equation, hence find
t h e par t i cu l ar sol u t i on w h en x = 0, y = 1

3 dy dy
y x
dx dx
Solution :

3 dy dy
y x
dx dx
3
y ( x 1)

( x 1) dy y3dx
Separating the variables, we get
1 1
3
dy
dx
y x 1
Now, integrating, we get
1 1
3
dy dx
y x 1

1 ....(I)
2
c
2y
which is required general solution
put x = 0 and y = 1 in (I)

0 3 8 8 Page 6

1
log|0 1| c
2(1)2
c

1 1
2 2
2y
is the particular solution.

SECTION - II
Q. 4. (A) Select and write the correct answer of the following multiple [12]
choice type of questions (1 mark each) : (6)

(i) The sum due is also called as ____
(a) True discount
(b) Face value
(c) Present value
(d) Cash value

(ii) Following are different types of insurance
(I) Life insurance
(II) Health insurance
(III) Liability insurance
(a) Only II
(b) Only III
(c) Only I
(d) All the three

(iii) b b ____
xy yx

(a) 2 |r|
(b) 2r
(c) r
(d) |r|

0 3 8 8 Page 7 P.T.O

(iv) If P ( L ) 120.4 and P ( P ) 130.6 then P ( D B)
01 01 01
is ____
(a) 25.1
(b) 60.2
(c) 125.5
(d) 65.3

(v) F(x) is c.d.f. of discrete r.v. X whose distribution is

Xi –2 –1 0 1 2
Pi 0.2 0.3 0.15 0.25 0.1

then F(–3) = ____
(a) 1
(b) 0.2
(c) 0.15
(d) 0

(vi) Given p.d.f. of a continuous r.v. X as

x 2 for
f (x ) , 1 x 2
3
= 0 , otherwise
then F(1) = ____
3
(a)
9
4
(b)
9
1
(c)
9
2
(d)
9

0 3 8 8 Page 8

(B) State whether the following statements are true or false
(1 mark each) : (3)

(i) The banker's discount is also called as commercial

discount.

(ii) The optimum value of the objective function of L.P.P.
occurs at the centre of the feasible region.

(iii) If E(X) > Var (X) then X follows Binomial distribution.

(C) Fill in the following blanks (1 mark each) : (3)

(i) The region represented by the inequality y 0 lies
in ____ quadrants.

(ii) The time required for printing of four books A, B, C
and D is 5, 8, 10 and 7 hours. While its data entry

requires 7, 4, 3 and 6 hours respectively. The

sequence that minimizes total elapsed time is ____.

(iii) If X has Poisson distribution with parameter m and

P(X = 3) = P(X = 4) then m = ____.

Q. 5. (A) Attempt any TWO of the following questions [14]
(3 marks each) : (6)

(i) A person wants to create a fund of 6, 96, 150 after 4
years at the time of his retirement. He decides to
invest a fixed amount at the end of every year in a

0 3 8 8 Page 9 P.T.O

bank that offers him interest of 10% p.a. compounded
annually. What amount should he invest every year?
[ Given (1.1)4 = 1.4641 ]

(ii) Following are given information about advertising
expenditure and sales.

Advertisement Sales
expenditure ( in lakh)
( in lakh)
(X) (Y)
Arithmetic mean 10 90
Standard deviation 3 12

Correlation coefficient between X and Y is 0.8
(a) Obtain the regression of Y on X
(b) What is the likely sales when the advertising
budget is 15 lakh?

(iii) Find y if the cost of living index is 200 for the following
data :

Group Food Clothing Fuel and House Miscellaneous

Lighting Rent

I 180 120 160 300 200

W 4 5 3 y 2

0 3 8 8 Page 10

(B) Attempt any TWO of the following questions (4 marks
each) : (8)

(i) A bill of 5,475 drawn on 19th January 2015 for 8
months was discounted on 28th February 2015 at 8%
p.a. interest. What is the banker's discount? What is
the cash value of the bill?
(ii) Following table shows the all India Infant Mortality
Rates (per c000) for years 1980 to 1986 :

Years 1980 1981 1982 1983 1984 1985 1986

IMR 10 6 5 3 3 1 0

Fit a trend line to the above data by the method of
least squares.

(iii) Four new machines M1, M2, M3 and M4 are to be
installed in a machine shop. There are five vacant
places A, B, C, D and E available. Because of limited
space, machine M2 cannot be placed at C and M3
cannot be placed at A. The cost matrix is given below:

Places
Machines
A B C D E
M1 4 6 10 5 6
M2 7 4 – 5 4
M3 – 6 9 6 2
M4 9 3 7 2 3

Find the optimal assignment schedule.

0 3 8 8 Page 11 P.T.O

Q. 6. (A) Attempt any TWO of the following questions (3 marks [14]
each) : (6)

(i) Obtain 4-yearly centered moving averages for the
following time series :

Years 1981 1982 1983 1984 1985 1986
Number of
40 42 43 42 44 44
crimes (c000)
Years 1987 1988 1989 1990 1991
Number of
43 46 47 45 46
crimes (c000)

(ii) Calculate Walsh's Price Index Number for the
following data :

Base Year Current Year
Commodity
Price Quantity Price Quantity

L 4 8 3 2

M 6 16 8 9

N 8 18 7 32

(iii) A pair of dice is thrown 3 times. If getting a doublet is
considered a success, find the probability of two
successes.

0 3 8 8 Page 12

(B) Attempt any ONE of the following questions (4 marks
each) : (4)

(i) The equations of two regression lines are
8x 10 y 66 0 and 40x 18 y 214 . Find
(a) The mean values of X and Y
(b) Correlation coefficient between X and Y

(ii) Maximize : z 60 x 50 y
Subject to : x 2 y 40
3x 2 y 60

(C) Attempt any ONE of the following questions (Activity)
(4 marks each) : (4)

(i) Find the sequence that minimizes the total elapsed
time to complete the following jobs in the order A B.
Find the total elapsed time and idle times for both
the machines.

Jobs I II III IV V VI VII
Machine A 7 16 19 10 14 15 5
Machine B 12 14 14 10 16 5 7

Solution :

Using the optimal sequence algorithm, the
following optimal sequence can be obtained.

IV V III

0 3 8 8 Page 13 P.T.O

Total elapsed time is obtained as follows

Machine A Machine B
Jobs
Time in Time out Time in Time out
0 5 5 12
5 12 12 24
IV 12 22 24 34
V 22 36 36 52
III 36 55 55 69
55 71 71 85
71 86 86 91

? Total elapsed time T = 91 units

Idle time for machine A = units

Idle time for machine B = units

(ii) The probability distribution of X is as follows :

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

Find (a) k (b) P (X > 2) (c) P (1 < X d 4)

Solution :
The table gives a probability distribution and
therefore

P(X x) 1

P(X 3) P ( X 4) 1

0.1 k+ 2k+ 2k+ k = 1

0 3 8 8 Page 14

6k 1 – 0.1
6k 0.9

k

P( X 2) P ( X 3) P ( X 4)

2k k

P (1 X 4) P ( X ) P(X 3) P ( X 4)

2k k

0.75

‹‹‹

0 3 8 8 Page 15 P.T.O

Maharashtra State Board Sample Paper 2026

DAY — 08 SEAT NUMBER

2025 VII 03 1100 (M)
J-389
MATHEMATICS & STATISTICS (88)
(COMMERCE)
Time : 3 Hrs. ( 16 Pages ) Max. Marks : 80

(logarithmic table)
.
(L. P. P.)
(time series)

0 3 8 9 Page 1 P.T.O.

0 3 1
(i) (matrix) B 3 0 4 (skew-
P 4 0
symmetric) P
1
–1
0
–3
dy
(ii) y e log x .........
dx
1
x
1
2
log x
e
x
0
3
(iii) (1 x ) dx

1 2
(1 x ) c
2
1 2
(1 x ) c
2
1 2 x
(1 x ) c
2 2
1 2 x
(1 x ) c
2 2

0 3 8 9 Page 2

a
2
(iv) 3x dx 8 a = .........
0

0 2
4
8
3

(v) y x 2 , X- x=1 x=3
(bounded) (area)
3
26
3
26
26
3

2 2
d2 y dy
(vi) 2
ax (order) (degree)
dx dx

1, 1
1, 2
2, 1

2, 2

(i) (p  q) t (dual) ( p  q) c

x 1 x x 2
(ii) 3
e dx e f (x ) c f ( x ) ( x 1)
( x 1)

dy x
(iii) y e (Integratinge Factor)
dx
(I.F.) e

0 3 8 9 Page 3 P.T.O.

(i)

(negation)

(ii) (average revenue) 45
(elasticity of demand) 3 (marginal

revenue)

9 x
10 x 10 log10
(iii) x 10
dx
10 x
(proper substitution)

(i) (statement pattern) (tautology),
(contradiction) (contingency)

[( p q ) (  p )] [ p (  q )]

y (log x )x x5 dy
(ii)
dx
2
(iii) (parabola) y 4x (line) x = 3

(bounded)

(i) (MPC), (MPS)

(APC) (APS)

0 3 8 9 Page 4

(expenditure) Ec (income)

I Ec= (0.0003) I2 + (0.075) I

I = 1000.

(ii) x 2 y dx ( x 3 y3 ) dy 0

(differential equation)

(iii) (matrix)
(method of reduction)
x + 2y + z = 8, 2x + 3y – z = 11,
3x – y – 2z = 5

(i) (converse), (inverse)
(contrapositive)

3 1 5
(ii) (matrix) 2 7 8 (inverse)
1 2 5

(adjoint method)

dy y
(iii)
dx x

(i) x 2 e3x dx

0 3 8 9 Page 5 P.T.O.

4
x
(ii)

(i) x f(x) (increasing)
f(x) = 2x3 – 15x2 – 144x – 7

3 2
f ( x ) 2x 15x 144 x 7

f ( x ) 6x 2 30x 144

' fc(x) > 0, f (x) (increasing)

6 x 2 30 x 144 0
2

( x 8)( x 3) 0
I x 8 0 x 3 0
x 8 x 3

II x 8 0 x 3 0
x 8 x 3
x

f ( x ) 2x 3 15x 2 144 x 7

x ( , ) x ( , )

(ii) (differential equation)
(particular solution)
x = 0, y = 1

0 3 8 9 Page 6

3 dy dy
y x
dx dx

3 dy dy
y x
dx dx
y3 ( x 1)

( x 1) dy y3dx

(separating the variables)

1 1
3
dy dx
y x 1

(integrating)
1 1
3
dy dx
y x 1

1 ....(I)
2
c
2y

(I) x=0 y=1

1
2
log|0 1| c
2(1)

c

1 1
2 2
2y

0 3 8 9 Page 7 P.T.O.

(i) (sum due)
(true discount)
(face value)
(present value)
(cash value)

(ii)
I. (Life insurance)
II. (Health insurance)
III. (Liability insurance)
II
III
I

(iii) b b
xy yx

2 |r|
2r
r
|r|

(iv) P ( L ) 120.4 P ( P ) 130.6 P ( D B) =
01 01 01

25.1 60.2
125.5 65.3

0 3 8 9 Page 8

(v) F(x) (discrete r.v.) X
(c.d.f)

Xi –2 –1 0 1 2
Pi 0.2 0.3 0.15 0.25 0.1

F(–3) =
1

0.2
0.15

0

(vi) X (continuous r.v.)

(p.d.f.)
x2
f (x ) , 1 x 2
3
=0

F(1) = .........
3
9
4
9
1
9
2
9

(i) (banker’s discount)
(commercial discount)

(ii) LPP (objective function)
(optimum value) (feasible region)

(centre)

0 3 8 9 Page 9 P.T.O.

(iii) E(X) > Var (X) X (binomial

distribution)

(i) y 0 (inequality)

(ii) A, B, C, D 5,

8, 10 7

7, 4, 3 6

(iii) X (Poisson distribution)

(parameter) m P(X = 3) = P(X = 4)

m=

(i) 4 ` 6,96,150

10

(1.1)4 = 1.4641

0 3 8 9 Page 10

(ii)

( ) ( )
(X) (Y)

10 90
arithmetic mean

3 12

0.8

X Y

(budget) 15
(likely sales)

(iii) (cost of living
index) 200 y

I 180 120 160 300 200

W 4 5 3 y 2

(i) 19 2015 ` 5,475 8
28 2015 8

(discounted)

0 3 8 9 Page 11 P.T.O.

(ii) 1980 1986
c000

1980 1981 1982 1983 1984 1985 1986

IMR 10 6 5 3 3 1 0

(trend line)

(iii) M1, M2, M3 M4
A, B, C, D E
M2 C M3 A
(cost matrix)

A B C D E
M1 4 6 10 5 6
M2 7 4 – 5 4
M3 – 6 9 6 2
M4 9 3 7 2 3

(optimal) (assignment)
(schedule)

(i) (time series) 4
(4-yearly centred moving averages)

0 3 8 9 Page 12

1981 1982 1983 1984 1985 1986

40 42 43 42 44 44
(cc000)
1987 1988 1989 1990 1991

43 46 47 45 46
(cc000)

(ii) (Walsch’s price index
number)

(Base Year) (Current Year)

L 4 8 3 2

M 6 16 8 9

N 8 18 7 32

(iii) 3 (success)

(i) (regression)

8 x 10 y 66 0 40 x 18 y 214

Y (mean values)

Y (correlation coefficient)

0 3 8 9 Page 13 P.T.O.

(ii) (maximize)

z 60 x 50 y

x 2 y 40

3x 2 y 60

(i) 7 A B

(optimal sequence)

(total elapsed time)

(idle time)

I II III IV V VI VII
A 7 16 19 10 14 15 5
B 12 14 14 10 16 5 7

IV V III

0 3 8 9 Page 14

A B

0 5 5 12
5 12 12 24
IV 12 22 24 34
V 22 36 36 52
III 36 55 55 69
55 71 71 85
71 86 86 91

T = 91
A =
B =

(ii)
x 0 1 2 3 4
P[X = x] 0.1 2 2

k P (X > 2) P (1 < X d 4)

P(X x) 1

P(X 3) P ( X 4) 1

0.1 k+ 2k+ 2k+ k = 1
6k 1 – 0.1
6k 0.9
k

0 3 8 9 Page 15 P.T.O.

P( X 2) P ( X 3) P ( X 4)

2k k

P (1 X 4) P ( X ) P(X 3) P ( X 4)

2k k

0.75

‹‹‹

0 3 8 9 Page 16