Maharashtra HSC Mathematics and Statistics (Commerce) Board Paper July 2025 with Solutions

Q.1 (A) (i) Matrix \(B=\begin{bmatrix}0 & 3 & 1\\-3 & 0 & -4\\P & 4 & 0\end{bmatrix}\) is skew-symmetric then value of \(P\) is

Q.1 (A) (ii) If \(y=e^{\log x}\) then \(\frac{dy}{dx}=\) _____.

Q.1 (A) (iii) \(\int(1-x)^{-3}\,dx=\) _____.

Q.1 (A) (iv) If \(\int_0^a 3x^2\,dx=8\) then \(a=\) _____.

Q.1 (A) (v) Area of the region bounded by the curve \(y=x^2\), the X-axis and the lines \(x=1\) and \(x=3\) is _____.

Q.1 (A) (vi) The order and degree of \(\left(\frac{d^2y}{dx^2}\right)^2+\left(\frac{dy}{dx}\right)^2=a^x\) are respectively ____

Q.1 (B) (i) Dual of \((p\wedge\sim q)\vee t\) is \((p\vee\sim q)\vee c\).

Q.1 (B) (ii) For \(\int\frac{x-1}{(x+1)^3}e^x\,dx=e^x\cdot f(x)+c\), where \(f(x)=(x+1)^2\).

Q.1 (B) (iii) The integrating factor (I.F.) of \(\frac{dy}{dx}+y=e^{-x}\) is \(e^x\).

Q.1 (C) (i) Negation of \"Some men are animals\" is ____.

Q.1 (C) (ii) If the average revenue is 45 and elasticity of demand is 3 then marginal revenue is ____.

Q.1 (C) (iii) To find the value of \(\int\frac{10x^9+10^x\cdot\log 10}{10^x+x^{10}}\,dx\), the proper substitution is ____.

Q.2 (A) (i) Determine whether the following statement pattern is tautology, contradiction or contingency.
\([(p\wedge q)\vee(\sim p)]\vee[p\wedge(\sim q)]\)

Q.2 (A) (ii) Find \(\frac{dy}{dx}\) if \(y=(\log x)^x+x^5\)

Q.2 (A) (iii) Find the area of the region bounded by the parabola \(y^2=4x\) and line \(x=3\).

Q.2 (B) (i) 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 \(I=1000\).

Q.2 (B) (ii) Solve the following differential equation:
\(x^2\cdot y\,dx-(x^3+y^3)\cdot dy=0\)

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

Q.3 (A) (i) Write the converse, inverse and contrapositive of the following statement.
'If a man is bachelor, then he is happy.'

Q.3 (A) (ii) Find the inverse of \(\begin{bmatrix}3 & 1 & 5\\2 & 7 & 8\\1 & 2 & 5\end{bmatrix}\) by adjoint method.

Q.3 (A) (iii) If \(x^5\cdot y^7=(x+y)^{12}\) then show that \(\frac{dy}{dx}=\frac{y}{x}\).

Q.3 (B) (i) Evaluate: \(\int x^2\cdot e^{3x}\,dx\)

Q.3 (B) (ii) Evaluate: \(\int_1^4\frac{\sqrt[3]{x+6}}{\sqrt[3]{x+6}+\sqrt[3]{11-x}}\,dx\)

Q.3 (C) (i) Find the values of \(x\), such that \(f(x)\) is increasing function
\(f(x)=2x^3-15x^2-144x-7\)

Solution:
Given: \(f(x)=2x^3-15x^2-144x-7\)
\(\therefore f'(x)=6x^2-30x-144\)
Now, \(f'(x)\gt 0\), as \(f(x)\) is increasing
\(\therefore 6x^2-30x-144\gt 0\)
\(\therefore x^2-5x-24\gt 0\)
\(\therefore (x-8)(x+3)\gt 0\)
Case (I) \(x-8\gt 0\) and \(x+3\gt 0\)
\(x\gt 8\) and \(x\gt -3\)
\(\therefore x\gt\)[ ]
Case (II) \(x-8\lt 0\) and \(x+3\lt 0\)
\(x\lt 8\) and \(x\lt -3\)
\(\therefore x\lt\)[ ]
\(\therefore f(x)=2x^3-15x^2-144x-7\) is increasing if and only if \(x\in(-\infty,\)[ ]\()\) or \(x\in(\)[ ]\(,\infty)\)

Q.3 (C) (ii) Solve the following differential equation, hence find the particular solution when \(x=0,\ y=1\)
\(y^3-\frac{dy}{dx}=x\frac{dy}{dx}\)

Solution:
\(y^3=x\frac{dy}{dx}+\frac{dy}{dx}\)
\(\therefore y^3=(x+1)\)[ ]
\(\therefore (x+1)\,dy=y^3dx\)
Separating the variables, we get
\(\frac{1}{y^3}dy=\frac{1}{x+1}dx\)
Now, integrating, we get
\(\therefore\int\frac{1}{y^3}dy=\int\frac{1}{x+1}dx\)
\(\therefore -\frac{1}{2y^2}=\)[ ]\(+c\) ....(I)
which is required general solution
put \(x=0\) and \(y=1\) in (I)
\(-\frac{1}{2(1)^2}=\log|0+1|+c\)
\(\therefore\)[ ]\(=c\)
\(\therefore -\frac{1}{2y^2}=\)[ ]\(-\frac{1}{2}\)
is the particular solution.

Q.4 (A) (i) The sum due is also called as ____

Q.4 (A) (ii) Following are different types of insurance
(I) Life insurance
(II) Health insurance
(III) Liability insurance

Q.4 (A) (iii) \(\left|b_{xy}+b_{yx}\right|\ge\) ____

Q.4 (A) (iv) If \(P_{01}(L)=120.4\) and \(P_{01}(P)=130.6\) then \(P_{01}(D\text{-}B)\) is ____

Q.4 (A) (v) \(F(x)\) is c.d.f. of discrete r.v. \(X\) whose distribution is

then \(F(-3)=\) ____

Q.4 (A) (vi) Given p.d.f. of a continuous r.v. \(X\) as
\(f(x)=\frac{x^2}{3}\), for \(-1\lt x\lt 2\)
\(=0\), otherwise
then \(F(1)=\) ____

Q.4 (B) (i) The banker's discount is also called as commercial discount.

Q.4 (B) (ii) The optimum value of the objective function of L.P.P. occurs at the centre of the feasible region.

Q.4 (B) (iii) If \(E(X)\gt\text{Var}(X)\) then \(X\) follows Binomial distribution.

Q.4 (C) (i) The region represented by the inequality \(y\le 0\) lies in ____ quadrants.

Q.4 (C) (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 ____.

Q.4 (C) (iii) If \(X\) has Poisson distribution with parameter m and \(P(X=3)=P(X=4)\) then \(m=\) ____.

Q.5 (A) (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 bank that offers him interest of 10% p.a. compounded annually. What amount should he invest every year? [Given \((1.1)^4=1.4641\)]

Q.5 (A) (ii) Following are given information about advertising expenditure and sales.

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?

Q.5 (A) (iii) Find \(y\) if the cost of living index is 200 for the following data:

Q.5 (B) (i) A bill of ₹ 5,475 drawn on 19^th January 2015 for 8 months was discounted on 28^th February 2015 at 8% p.a. interest. What is the banker's discount? What is the cash value of the bill?

Q.5 (B) (ii) Following table shows the all India Infant Mortality Rates (per '000) for years 1980 to 1986:

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

Q.5 (B) (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:

Find the optimal assignment schedule.

Q.6 (A) (i) Obtain 4-yearly centered moving averages for the following time series:

Q.6 (A) (ii) Calculate Walsh's Price Index Number for the following data:

Q.6 (A) (iii) A pair of dice is thrown 3 times. If getting a doublet is considered a success, find the probability of two successes.

Q.6 (B) (i) The equations of two regression lines are \(8x-10y+66=0\) and \(40x-18y=214\). Find
(a) The mean values of \(X\) and \(Y\)
(b) Correlation coefficient between \(X\) and \(Y\)

Q.6 (B) (ii) Maximize: \(z=60x+50y\)
Subject to: \(x+2y\le 40\)
\(3x+2y\le 60\)
\(x\ge 0,\ y\ge 0\)

Q.6 (C) (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.

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

Total elapsed time is obtained as follows

\(\therefore\) Total elapsed time T = 91 units
Idle time for machine A = [ ] units
Idle time for machine B = [ ] units

Q.6 (C) (ii) The probability distribution of \(X\) is as follows:

Find (a) \(k\) (b) \(P(X\gt 2)\) (c) \(P(1\lt X\le 4)\)

Solution:
The table gives a probability distribution and therefore
\(\Sigma P(X=x)=1\)
\(P(X=0)+P(X=1)+P(X=2)+P(X=3)+P(X=4)=1\)
\(0.1+k+2k+2k+k=1\)
\(6k=1-0.1\)
\(6k=0.9\)
\(k=\)[ ]
\(P(X\gt 2)=P(X=3)+P(X=4)\)
\(=2k+k\)
\(=\)[ ]
\(P(1\lt X\le 4)=P(X=\)[ ]\()+P(X=3)+P(X=4)\)
\(=\)[ ]\(+2k+k\)
\(=0.75\)