Maharashtra Board Class 12 Maths & Statistics (Commerce) 2025 Question Paper (Available): Download Question Paper with Answer Key And Solutions PDF

(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) \( \sim p \lor \sim q \)
• (B) \( \sim(p \land q) \)
• (C) \( \sim(p \lor q) \)
• (D) \( p \lor \sim q \)

(ii)
The value of \( \int \left(x + \frac{1}{x}\right)^3 dx \) is equal to:

• (A) \( \frac{1}{4}\left(x+\frac{1}{x}\right)^4 + c \)
• (B) \( \frac{x^4}{4} + \frac{3x^2}{2} + 3\log x - \frac{1}{2x^2} + c \)
• (C) \( \frac{x^4}{4} + \frac{3x^2}{2} + 3\log x + \frac{1}{x^2} + c \)
• (D) \( (x-x^{-1})^3 + c \)

(iii)
The value of the definite integral \( \int_2^7 \frac{\sqrt{x}}{\sqrt{x} + \sqrt{9-x}} dx \) is:

• (A) \( \frac{7}{2} \)
• (B) \( \frac{5}{2} \)
• (C) \( 7 \)
• (D) \( 2 \)

(iv)
The area of the region bounded by the curve \(y=x^2\) and the line \(y=4\) is:

• (A) \( \frac{32}{3} \) sq. units
• (B) \( \frac{64}{3} \) sq. units
• (C) \( \frac{16}{3} \) sq. units
• (D) \( 64 \) sq. units

(v)
The order and degree of the differential equation \( \left(\frac{d^2y}{dx^2}\right)^2 + \left(\frac{dy}{dx}\right)^2 = a^x \) are ________ respectively.

• (A) 1, 1
• (B) 1, 2
• (C) 2, 2
• (D) 2, 1

(vi)
The integrating factor of the differential equation \( \frac{dy}{dx} + \frac{y}{x} = x^3 - 3 \) is:

• (A) \( \log x \)
• (B) \( e^x \)
• (C) \( \frac{1}{x} \)
• (D) \( x \)

(i)
If A is a matrix and K is a constant, then \( (KA)^T = K A^T \).

(ii)
The value of \( \int \log x \, dx = x \log x + x + c \).

(iii)
The differential equation obtained by eliminating arbitrary constants from \( bx + ay = ab \) is \( \frac{d^2y}{dx^2} = 0 \).

(i)
The average revenue \( R_A \) is 50 and elasticity of demand \( \eta \) is 5, the marginal revenue \( R_M \) is ______.

(ii)
\( \int e^x \left(\frac{1}{x} - \frac{1}{x^2}\right) dx = \_\_\_\_\_\_ + c \)

(iii)
If \( f'(x) = x^2 + 5 \) and \( f(0) = -1 \) then \( f(x) = \_\_\_\_\_\_.\)

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

(ii)
Find x, y, z if 

(iii)
Evaluate: \( \int \frac{1}{x(x^6 + 1)} dx \)

(i)
Solve the following equations by the method of inversion: \( 2x-y+z=1 \), \( x+2y+3z=8 \), \( 3x+y-4z=1 \).

(ii)
Find MPC, MPS, APC and APS, if the expenditure \(E_c\) of a person with income I is given as \( E_c = (0.0003)I^2 + (0.075)I \); when \( I=1000 \).

(iii)
Evaluate: \( \int_1^2 \frac{dx}{x^2+6x+5} \)

(i)
Find \( \frac{dy}{dx} \) if \( y = x^x + a^x \).

(ii)
Find the area of the region bounded by the parabola \( y^2 = 25x \) and the line \( x=5 \).

(iii)
Find the differential equation by eliminating arbitrary constants from the relation \( y = Ae^{3x} + Be^{-3x} \).

(i)
Using the truth table, verify \( p \lor (q \land r) \equiv (p \lor q) \land (p \lor r) \).

(ii)
If \( x = \frac{4t}{1+t^2} \), \( y = 3\left(\frac{1-t^2}{1+t^2}\right) \), then show that \( \frac{dy}{dx} = -\frac{9x}{4y} \).

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

(ii)
Solve the following differential equation \( (x^2 - yx^2)dy + (y^2 + xy^2)dx = 0 \).

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

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

Moving averages are useful in identifying

• (a) Seasonal component
• (b) Irregular component
• (c) Trend component
• (d) Cyclical component

If \(P_{01}(L)=90\) and \(P_{01}(P)=40\), then \(P_{01}(D-B)\) is _____.

• (a) 65
• (b) 50
• (c) 25
• (d) 130

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

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

If \( b_{yx} + b_{xy} = 1.30 \) and \( r = 0.75 \) then the given data is inconsistent.

Cyclic variation can occur several times in a year.

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

The amount paid to the holder of the bill after deducting banker's discount is known as _______.

The simplest method of measuring trend of time series is _______.

Quantity index number by weighted aggregate method is given by _______.

Compute the appropriate regression equation for the following data :

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

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 INR 20 per kg and sand costs INR 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.

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

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

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 :

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.

A house valued at INR 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 commission is 9% of the premium, find agent's commission.

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

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 \(e^{-1} = 0.3678\))

The equations of two regression lines are \(10x-4y=80\) and \(10y-9x=-40\). Find

• (a) \(\bar{x}\) and \(\bar{y}\)
• (b) \(b_{yx}\) and \(b_{xy}\)
• (c) \(r\)
• (d) If Var(Y) = 36, obtain var (X).

Find x if the cost of living index is 150 :

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

Solve the following assignment problem for minimization :