Maharashtra HSC Mathematics and Statistics (Commerce) Board Paper June 2026 with Solutions

Q.1 (A) (i) If \(p\) is any statement then \((p\vee\sim p)\) is a

Q.1 (A) (ii) If \(A=\begin{bmatrix}4 & x\\6 & 3\end{bmatrix}\) is a singular matrix then \(x\) is

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

Q.1 (A) (iv) If \(0\lt\eta\lt 1\), then the demand is

Q.1 (A) (v) \(\int x^2(3)^{x^3}\,dx\) = _____

Q.1 (A) (vi) \(\int_{-9}^{9}\frac{x^3}{4-x^2}\,dx\) = _____.

Q.1 (B) (i) State whether true or false: The slope of the tangent at any point (a, b) is called as gradient of the curve.

Q.1 (B) (ii) State whether true or false: \(\int\frac{1}{2x+3}\,dx=\log|2x+3|+c\)

Q.1 (B) (iii) State whether true or false: Order and degree of a differential equation are always positive integers.

Q.1 (C) (i) The dual of \((p\wedge\sim q)\vee t\) is _____.

Q.1 (C) (ii) \(\int e^x\left(\frac{1}{x}-\frac{1}{x^2}\right)dx\) = _____.

Q.1 (C) (iii) The integrating factor of the differential equation \(\frac{dy}{dx}-y=x\) is _____.

Q.2 (A) (i) Write the converse, inverse and contrapositive of the following statement.

'If a man is rich, then he is happy.'

Q.2 (A) (ii) Find the inverse of matrix \(\begin{bmatrix}1 & 2 & 3\\1 & 1 & 5\\2 & 4 & 7\end{bmatrix}\) by adjoint method.

Q.2 (A) (iii) Find the rate of change of demand (\(x\)) of a commodity with respect to its price (\(y\)) if \(y=\frac{5x+7}{2x-13}\)

Q.2 (B) (i) Evaluate: \(\int\frac{e^x}{\sqrt{e^{2x}+4e^x+13}}\,dx\)

Q.2 (B) (ii) Evaluate: \(\int_{2}^{5}\frac{\sqrt{x}}{\sqrt{x}+\sqrt{7-x}}\,dx\)

Q.2 (B) (iii) Find the area of ellipse \(\frac{x^2}{25}+\frac{y^2}{4}=1\)

Q.3 (A) (i) Determine whether the following statement pattern is tautology, contradiction or contingency.

\([p\wedge(\sim p\vee q)]\)

Q.3 (A) (ii) If \(x^y=e^{x-y}\) then prove that \(\frac{dy}{dx}=\frac{\log x}{(1+\log x)^2}\).

Q.3 (A) (iii) The manufacturing company produces \(x\) items at the total cost of ₹ (180+4\(x\)). The demand function for this product is \(P=(240-x)\). Find \(x\) for which profit is increasing.

Q.3 (B) (i) Solve the following equation by method of reduction:

\(x-y+z=4\)
\(2x+y-3z=0\)
\(x+y+z=2\)

Q.3 (B) (ii) In a certain culture of bacteria, the rate of increase is proportional to the number present. If it is found that the number doubles in 4 hours, find the number of times the bacteria increased in 12 hours by using the concept of application of differential equations.

Q.3 (C) (i) Demand function \(x\), for a certain commodity is given as \(x=200-4p\), where \(p\) is the unit price. Find

(a) elasticity of demand as a function of \(p\).
(b) elasticity of demand when \(p=10\). Interpret your result.

Solution:

(a) Here, \(x=200-4p\)
\(\therefore\) Elasticity of demand is
\(\eta=\frac{-p}{x}\cdot\)[ ]
\(=\frac{-p}{(200-4p)}\)[ ]
\(\therefore\eta=\frac{p}{50-p}\)

(b) When \(p=10\)
\(\eta=\frac{10}{[\ \ ]}\)
\(=0.25\)
\(\therefore\) Demand is [ ] for \(p=10\)

Q.3 (C) (ii) Solve the following differential equations \(\frac{dy}{dx}+y=5\)

Solution: \(\frac{dy}{dx}+y=5\)
Given equation is a linear differential equation.
Here \(P=\)[ ] \(Q=5\)
\(\therefore\) I.F. \(=e^{\int p\,dx}\)
\(\therefore\) I.F. \(=\)[ ]
The solution of given linear differential equation is
By formula [ ]
\(\therefore y\cdot e^x=\int 5\cdot e^x\,dx+c\)
\(\therefore\) [ ]

Q.4 (A) (i) The date on which period of the bill expires is called ____.

Q.4 (A) (ii) Loss = ____.

Q.4 (A) (iii) In regression analysis \(b_{yx}\cdot b_{xy}\) = _____.

Q.4 (A) (iv) Moving averages are useful in identifying ____.

Q.4 (A) (v) Price Index Number by Simple Aggregate Method is given by

Q.4 (A) (vi) Objective function of LPP is

Q.4 (B) (i) State whether true or false: The difference between the banker's discount and true discount is called the Banker's gain.

Q.4 (B) (ii) State whether true or false: One of the assumptions made while sequencing \(n\) jobs on 2 machines is: two jobs must be loaded at a time on any machine.

Q.4 (B) (iii) State whether true or false: Quantity Index Number by simple aggregate method is given by \(\frac{\Sigma q_1}{\Sigma q_0}\times 100\).

Q.4 (C) (i) If \(P_{01}(L)=225,\ P_{01}(P)=144\) then \(P_{01}(F)\) = _____.

Q.4 (C) (ii) Graphical solution set of the inequations \(x\ge 0,\ y\ge 0\) lies in _____ quadrant.

Q.4 (C) (iii) For solving an assignment problem the matrix should be a ____ matrix.

Q.5 (A) (i) Obtain the 4-yearly centered moving averages for the following data:

Q.5 (A) (ii) Find the sequence that minimizes the total elapsed time to complete the following jobs. Each job is processed in order \(AB\). Also find the idle time for machine \(B\).

Q.5 (A) (iii) A car firm has 2 cars, which are hired out day-by-day. The number of cars hired on a day follows Poisson distribution with mean 1.5. Find the probability that some demand is refused on a given day. [Given \(e^{-1.5}=0.2231\)]

Q.5 (B) (i) Minimize: \(z=4x+2y\)

Subject to: \(3x+y\ge 27\)
\(x+y\ge 21\)
\(x+2y\ge 30\)
\(x\ge 0,\ y\ge 0\)

Solve the above L.P.P. by graphical method.

Q.5 (B) (ii) Four different machines can do any of the three required jobs, with different profits resulting from each assignment as shown below:

Find the optimal assignment schedule.

Q.5 (B) (iii) A random variable \(X\) has the following probability distribution:

Determine (a) \(k\) (b) \(P(X\lt 3)\) (c) \(P(X\gt 6)\) (d) \(P(0\lt X\lt 1)\)

Q.6 (A) (i) Find the amount a company should set aside at the end of every year if it wants to buy a machine expected to cost ₹ 4,31,000 at the end of 4 years and interest rate is 5% p.a. compounded annually. [Given \((1.05)^4=1.2155\)]

Q.6 (A) (ii) For the following bivariate data obtain equation of regression line \(Y\) on \(X\):

Q.6 (A) (iii) Following table shows the number of traffic fatalities (in a state) resulting from drunken driving from years 1975 to 1983:

Fit a trend line by least square method.

Q.6 (B) (i) Given that \(\Sigma p_0q_0=120,\ \Sigma p_0q_1=160,\ \Sigma p_1q_1=140\) and \(\Sigma p_1q_0=200\). Find Laspeyre's, Paasche's, Dorbish-Bowley's and Marshall-Edgeworth's Price Index Numbers.

Q.6 (B) (ii) A die is thrown 4 times. If 'getting an odd number' is a success, find the probability of (a) 2 successes (b) at least 3 successes (c) at the most 2 successes.

Q.6 (C) (i) Rajesh gets a commission at 10% on cash sales and 8% on credit sales. If he receives ₹ 4,400 as commission on the total sales of ₹ 50,000, find the sales made by him on cash and on credit sales.

Solution:
Let cash sales be ₹ \(x\)
\(\therefore\) Credit sales = ₹ [ ]
Total commission = 10% on \(x\) + 8% on \((50{,}000-x)\)
\(\therefore\) [ ] \(=x\times\frac{10}{100}+(50{,}000-x)\times\frac{8}{100}\)
\(\therefore x=\) [ ]
\(\therefore\) Commission on cash sales is ₹ 20,000 and on credit sales is ₹ [ ]

Q.6 (C) (ii) The equations of two regression lines are \(16y+x=64\) and \(x+4y=5\). Find regression coefficients and correlation coefficient.

Solution:
Suppose that equation of regression line of \(Y\) on \(X\) be \(16y+x=64\) and equation of regression line of \(X\) on \(Y\) be \(x+4y=5\)
Now \(16y+x=64\) and \(x+4y=5\)
\(\therefore y=\frac{-1}{16}x+4\) and \(x=-4y+5\)
\(\therefore b_{yx}=\)[ ] and \(b_{xy}=\)[ ]
\(b_{yx}\times b_{xy}=\frac{1}{4}\)
\(\therefore b_{yx}\times b_{xy}\lt 1\)
\(\therefore\) Our assumption is correct.
\(\therefore r^2=\)[ ]
\(\therefore r=\)[ ]