MH 12th Question Paper 2025 Maths (Commerce)

Maharashtra Board

Question Paper
2025
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DAY — 09 SEAT NUMBER

2025 II 22 1100 (E)
J-316
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 1 6 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]

(i) If p : He is intelligent
q : He is strong
Then, symbolic form of statement "It is wrong that,
he is intelligent or strong" is :
(a)  
(b) 
(c) 
(d) 

3
1
(ii) x dx
x
4
1 1
(a) x c
4 x
4 2
x 3x 1
(b) 3 log x 2
c
4 2 2x
4 2
x 3x 1
(c) 3log x 2
c
4 2 x

(d) 1 3
x x c

7
x
(iii) dx
2
x 9 x

7 5
(a) (b)
2 2
(c) 7 (d) 2

0 3 1 6 Page 2

(iv) The area of the region bounded by the curve y x 2
and the line y 4 is
32 64
(a) sq. units (b) sq. units
3 3
16
(c) sq. units (d) 64 sq. units
3

(v) The order and degree of the differential equation
2 2
d2 y dy x
2
a are _____ respectively.
dx dx

(a) 1, 1 (b) 1, 2
(c) 2, 2 (d) 2, 1
(vi) The integrating factor of the differential equation
dy y 3
x 3 is
dx x
(a) log x (b) ex
1
(c) (d) x
x

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

(i) If A is a matrix and K is a constant, then

(ii) log x dx x log x x c

(iii) The differential equation obtained by eliminating
2
arbitrary constants from bx ay ab is d y 0.
2
dx

0 3 1 6 Page 3 P.T.O

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

(i) The average revenue RA is 50 and elasticity of
demand K is 5, the marginal revenue RM is _____.

x 1 1
(ii) e dx _____ c
x x2

2
(iii) If f ( x ) x 5 and f (0) 1 then f ( x ) _____.

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

(i) Write the converse, inverse and contrapositive of the
statement "If a triangle is equilateral then it is
equiangular".

01 2 1 x 1
(ii) Find x, y, z if 5 1 0 3 2 2 y 1
1 1 1 3 1 2z

1
(iii) Evaluate : 6
dx
x(x 1)

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

(i) Solve the following equations by the method of
inversion :
2x y z 1
x 2 y 3z 8
3x y 4z 1

(ii) Find MPC, MPS, APC and APS, if the expenditure
Ec of a person with income I is given as :
2
E (0.0003) I (0.075) I ; when I
c

0 3 1 6 Page 4

2
dx
(iii) Evaluate : 2
1x 6x 5

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

dy x
(i) Find if y ( x ) ( a )x
dx
(ii) Find the area of the region bounded by the parabola

y2 25x and the line x 5 .

(iii) Find the differential equation by eliminating
arbitrary constants from the relation
3x 3x .
y Ae Be

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

(i) Using the truth table, verify
p (q r ) ( p q ) ( p r )

2
4t 1 t
(ii) If x 2
, y 3 2
, then show that
1 t 1 t

dy 9x .
dx 4y

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

(i) Divide the number 84 into two parts such that the
product of one part and square of the other is
maximum.

0 3 1 6 Page 5 P.T.O

Solution :
Let one part be x then the other part will be 84 – x.
f (x )

f ( x ) 168x 3x 2
For extreme values f ( x ) 0

168 x 3x 2 0
3x (56 x ) 0

x OR

f ( x ) 168 6 x
If x 0, f (0) 168 6(0) 168 0
function attains minimum at x = 0

If x 56, f (56) 0
function attains maximum at x = 56

Two parts of 84 are and

(ii) Solve the following differential equation

(x2 2
yx 2 )dy ( y2 xy ) dx 0
Solution :
Separating the variables, the given equation can be
written as :
dy dx 0

1 1
y 2 dy x 2 dx 0
y x
1
dy x 2dx
dy dx 0
y
Integrating we get,
1 1
y 2dy dy x 2dx dx 0
y x

0 3 1 6 Page 6

y 1 x 1
c
1 1
1 1
log x log y c
y x
log x log y c
is the required solution.

SECTION - II

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

(i) An agent who gives guarantee to his principal that
the party will pay the sale price of goods is called –
(a) Auctioneer
(b) Del credere agent
(c) Factor
(d) Broker

(ii) In an ordinary annuity, payments or receipts
occur at
(a) Beginning of each period
(b) End of each period
(c) Mid of each period
(d) Quarterly basis

(iii) Moving averages are useful in identifying
(a) Seasonal component
(b) Irregular component
(c) Trend component
(d) Cyclical component

0 3 1 6 Page 7 P.T.O

(iv) If P ( L ) 90 and P ( P ) 40 , then P ( D B ) is
01 01 01
_____.
(a) 65
(b) 50
(c) 25
(d) 130

(v) The objective of an assignment problem is to assign
(a) Number of jobs to equal number of persons at
maximum cost
(b) Number of jobs to equal number of persons at
minimum cost
(c) Only to maximize the cost
(d) Only to minimize the cost

(vi) The expected value of the sum of two numbers
obtained when two fair dice are rolled is _____.
(a) 5
(b) 6
(c) 7
(d) 8

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

(i) If b b 1.30 and r 0.75 then the given data is
yx xy

inconsistent.

(ii) Cyclic variation can occur several times in a year.

(iii) Cost of living index number is used in calculating
purchasing power of money.

0 3 1 6 Page 8

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

(i) The amount paid to the holder of the bill after
deducting banker's discount is known as _____.

(ii) The simplest method of measuring trend of time
series is _____.

(iii) Quantity index number by weighted aggregate
method is given by _____.

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

(i) Compute the appropriate regression equation for the
following data :

X 1 2 3 4 5
Y 5 7 9 11 13

X is the independent variable and Y is the
dependent variable.

(ii) A company makes concrete bricks made up of cement
and sand. The weight of a concrete brick has to be at
least 5 kg. Cement costs ` 20 per kg and sand costs
6 per kg. Strength consideration dictate that a
concrete brick should contain minimum 4 kg of
cement and not more than 2 kg of sand. Formulate
the L.P.P. for the cost to be minimum.

(iii) Find the mean of number of heads in three tosses of
a fair coin.

0 3 1 6 Page 9 P.T.O

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

(i) Obtain the trend value for the following data using
4-yearly centered moving averages :

Years 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985
Index 0 2 3 3 2 4 5 6 7 10

(ii) Find the sequence that minimizes the total elapsed
time to complete the following jobs in the order AB.
Find the total elapsed time and idle time for machine
B:

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

(iii) Five cards are drawn successively with replacement
from a well shuffled deck of 52 cards. Find the
probability that :
(a) all the five cards are spades
(b) only 3 cards are spades.

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

(i) A house valued at ` 8,00,000 is insured at 75% of its
value. If the rate of premium is 0.80%, find the
premium paid by the owner of the house. If agent's

0 3 1 6 Page 10

commission is 9% of the premium, find agent's
commission.

(ii) Solve the following L.P.P. by graphical method.
Maximize : z 4x 6 y
Subject to : 3x 2 y 12
x y 4
x, y 0

(iii) Defects on plywood sheet occur at random with the
average of one defect per 50 sq.ft. Find the probability
that such a sheet has :
(a) no defect
(b) at least one defect

(use e 1 0.3678 )

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

(i) The equations of two regression lines are
10 x 4 y 80 and 10 y 9x 40 . Find
(a) x and y
(b) b and b
YX XY

(c) r
(d) If var(Y) = 36, obtain var (X).

(ii) Find x if the cost of living index is 150 :

Group Food Clothing Fuel and House Miscellaneous
electricity Rent
I 180 120 300 100 160
W 4 5 6 x 3

0 3 1 6 Page 11 P.T.O

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

(i) A bill of ` 18,000 was discounted for ` 17,568 at a
bank on 25th October 2017. If the rate of interest was
12% p.a. what is the legal due date?

Solution :

Given SD = 18,000; CV = 17,568

r = 12% p.a.

Now, BD

= 18,000 – 17,568

= 432

Also BD

18,000 n 12
432
100

432 100
n
18,000 12

1
n years = days
5

The period for which the discount is deducted is 73
days, which is counted from the date of discounting
i.e. 25th October 2017 :

October November December January Total
6 30 31 6 73

Hence legal due date is

0 3 1 6 Page 12

(ii) Solve the following assignment problem for
minimization :

I II III IV V
1 18 24 19 20 23
2 19 21 20 18 22
3 22 23 20 21 23
4 20 18 21 19 19
5 18 22 23 22 21

Solution :
Step-I
Subtract the smallest element of each row from every
element of that row

I II III IV V

Step-II
Subtract the smallest element of each column from
every element of that column :

I II III IV V

0 3 1 6 Page 13 P.T.O

Step-III

Draw minimum number of lines covering all zeros.

I II III IV V

Here minimum number of lines (4) < order of matrix
(5)
Step-IV
The smallest uncovered element is 1, which is to be
subtracted from all uncovered elements and add it to
all elements which lie at the intersection of two
lines :

I II III IV V
3

3 3 0 2
4 0 3 1 0
5 3 4 3

Step-V
Draw minimum number of lines that are required to
cover all zeros :

I II III IV V

Here minimum number of lines z order of matrix.

0 3 1 6 Page 14

Step-VI
Find the smallest uncovered element (1). Subtract
this number from all uncovered elements and add it
to all elements which lie at the intersection of
two lines :

I II III IV V
1
3 3 0 3

4 0 4 0 0
2 0

Now minimum number of lines = order of matrix.
The optimal assignment can be made.
Optimal solution is

3
4

Minimum value =

‹‹‹

0 3 1 6 Page 15 P.T.O

DAY — 09 SEAT NUMBER

2025 II 22 1100 (M)
J-317
MATHEMATICS & STATISTICS (88)
(COMMERCE)
Time : 3 Hrs. ( 15 Pages ) Max. Marks : 80

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

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

(i) p:
q:

p q
 ( p q)
 ( p q)
p q
3
1
(ii) x dx
x

4
1 1
x c
4 x

4 2
x 3x 1
3 log x 2
c
4 2 2x
4 2
x 3x 1
3log x 2
c
4 2 x
3
x x 1 c
7
x
(iii) dx
2
x 9 x

7 5
2 2
7 2

0 3 1 7 Page 2

(iv) y x2 y 4

32
3
64
3
16
3
64

2 2
d2 y dy x
(v) 2
a
dx dx

1, 1 1, 2

2, 2 2, 1

dy y
(vi) x3 3 (I.F.)
dx x

log x ex
1
x
x

(i) K T T
( KA ) KA

(ii) log x dx x log x x c

(iii) bx ay ab
2
d y
2
0
dx

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

(i) RA = 50

K=5 RM

x 1 1
(ii) e dx _____ c
x x2

(iii) f (x ) x2 5 f (0) 1 f (x )

(i)

(converse) (inverse)

(contrapositive)

01 2 1 x 1
5 1 0 3 2 2 y 1
(ii) 1 x, y, z
1 1 1 3 2z

1
(iii) 6
dx
x(x 1)

(i) (method in inversion)

2x y z 1

x 2 y 3z 8

3x y 4z 1

0 3 1 7 Page 4

(ii) Ec I
2
E (0.0003) I (0.075) I I 1000
c

MPC, MPS, APC APS

2
dx
(iii) 2
1x 6x 5

dy
(i) y ( x )x ( a )x
dx
2
(ii) (parabola) y 25x x 5

3x 3x
(iii) y Ae Be

(i)

p (q r ) ( p q ) ( p r )

2
4t 1 t
(ii) x , y 3 ,
1 t2 1 t
2

dy 9x
dx 4y

0 3 1 7 Page 5 P.T.O.

(i) 84

x 84 – x

f (x )
2
f ( x ) 168x 3x
f (x ) 0

168 x 3x 2 0

3x (56 x ) 0

x

f ( x ) 168 6 x

x 0, f (0) 168 6(0) 168 0

x =0

x 56, f (56) 0

x = 56

84

(ii)

(x2 2
yx 2 )dy ( y2 xy ) dx 0

dy dx 0

1 1
y 2 dy x 2 dx 0
y x

0 3 1 7 Page 6

1
dy dy x 2dx dx 0
y

1 1
y 2dy dy x 2dx dx 0
y x

y 1 x 1
c
1 1
1 1
log x log y c
y x
log x log y c

(i)

(Auctioneer)

(Del credere agent)

(Factor)

(Broker)

(ii)

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

(iii)

(iv) P ( L ) 90 P ( P ) 40 P ( D B)
01 01 01

65
50
25
130

(v)

(vi)

5
6
7
8

(i) b b 1.30 r 0.75
yx xy

0 3 1 7 Page 8

(ii)

(iii)

(i)

(ii)

(iii)

(i)

X 1 2 3 4 5

Y 5 7 9 11 13

X

(ii)

5 20 6

4

2

(L.P.P.)

(iii)

(mean)

0 3 1 7 Page 9 P.T.O.

(i) 4-

(trend values)

1976 1977 1978 1979 1980 1981 1982 1983 1984 1985
0 2 3 3 2 4 5 6 7 10

(ii) AB

B

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

(iii) 52 5

3

(i) ` 8,00,000 75%

0.80%

9%

0 3 1 7 Page 10

(ii) (L.P.P.)

z 4x 6 y

3x 2 y 12

x y 4

x, y 0

(iii)

50

e 1 0.3678

(i) 10 x 4 y 80

10 y 9x 40

x y

b b
YX XY
r
var(Y) = 36 var (X)

(ii) 150 x

I 180 120 300 100 160

W 4 5 6 x 3

0 3 1 7 Page 11 P.T.O.

(i) 25 2017 `18,000 `17,568

12%

SD = 18,000; CV = 17,568,

r = 12%

BD

= 18,000 – 17,568

= 432

BD

18,000 n 12
432
100

432 100
n
18,000 12

1
n =
5

73

25 2017

6 30 31 6 73

0 3 1 7 Page 12

(ii)

I II III IV V
1 18 24 19 20 23
2 19 21 20 18 22
3 22 23 20 21 23
4 20 18 21 19 19
5 18 22 23 22 21

-I

I II III IV V

-II

I II III IV V

0 3 1 7 Page 13 P.T.O.

-III

I II III IV V

(4) < 5

1

I II III IV V
3

3 3 0 2
4 0 3 1 0
5 3 4 3

I II III IV V

z

0 3 1 7 Page 14

-VI

(1)

I II III IV V
1
3 3 0 3

4 0 4 0 0
2 0

3
4

=

‹‹‹

0 3 1 7 Page 15 P.T.O.

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