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

Q.1 (A) (i) If \(y=\sqrt[3]{a^2+x^2}\) then \(\frac{dy}{dx}\) = _____.

Q.1 (A) (ii) If \(y=\log\left(\frac{e^x}{x^2}\right)\) then \(\frac{dy}{dx}\) = _____.

Q.1 (A) (iii) [Venn diagram: a rectangle U contains a circle Q; inside Q is a smaller shaded circle I.] U : the set of all real numbers, Q : the set of all rational numbers, I : the set of all integers. The above Venn diagram represents the truth value of which of the following statements?

Q.1 (A) (iv) The equation of normal to the curve \(y=3x^2-x+1\) at (1, 3) is _____.

Q.1 (A) (v) The solution of \(\frac{dy}{dx}=1\) is :

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

Q.1 (B) (i) State whether true or false: If \(y=20+15x+x^2\) then \(\frac{dx}{dy}=\frac{1}{15+2x}\).

Q.1 (B) (ii) State whether true or false: If \(\int\frac{x}{(1+x)(2+x)}\,dx=\int\left(\frac{A}{1+x}+\frac{B}{2+x}\right)dx\) then A = 1, B = 2.

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

Q.1 (C) (i) Statement \(p\leftrightarrow q\) is true, when p and q have _____ truth values.

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

Q.1 (C) (iii) If \(f(x)=x\cdot\log x\) then its minimum value is _____.

Q.2 (A) (i) Using truth table, examine whether the following statement pattern is a tautology, a contradiction or a contingency.

\((p\wedge\sim q)\to(\sim p\wedge\sim q)\)

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

Q.2 (A) (iii) Evaluate : \(\int\frac{1+x}{x+e^{-x}}\,dx\)

Q.2 (B) (i) Find the inverse of \(\begin{bmatrix}1 & -1 & 1\\ 2 & 1 & -3\\ 1 & 1 & 1\end{bmatrix}\) by adjoint method.

Q.2 (B) (ii) The consumption expenditure \(E_c\) of a person with income x is given by \(E_c=0.0006x^2+0.003x\). Find average propensity to consume, marginal propensity to consume when his income is ₹ 200. Also find his marginal propensity to save and average propensity to save.

Q.2 (B) (iii) Evaluate : \(\int_2^7\frac{\sqrt{x}}{\sqrt{x}+\sqrt{9-x}}\,dx\)

Q.3 (A) (i) If the demand function is \(D=50-3p-p^2\), find the elasticity of demand at \(p=5\). Comment on the result.

Q.3 (A) (ii) If p : He swims, q : Water is warm. Give the verbal statements for the following symbolic statements :

(a) \(p\leftrightarrow\sim q\)
(b) \(\sim(p\vee q)\)
(c) \(q\to p\)

Q.3 (A) (iii) Evaluate : \(\int\frac{1}{4x^2-20x+17}\,dx\)

Q.3 (B) (i) Solve the following differential equation

\((x^2-y^2)\,dx+2xy\,dy=0\)

Q.3 (B) (ii) Find the area of the region bounded by the curve \(x^2=16y\) and the line \(y=4\).

Q.3 (C) (i) Express the following equations in matrix form and solve them by method of reduction.

\(x-y+z=1,\ 2x-y=1,\ 3x+3y-4z=2\)

Solution : The given equations can be written in matrix form as \(\begin{bmatrix}1 & -1 & 1\\ 2 & -1 & 0\\ 3 & 3 & -4\end{bmatrix}\begin{bmatrix}x\\ y\\ z\end{bmatrix}=\begin{bmatrix}1\\ 1\\ 2\end{bmatrix}\)

By \(R_2\to R_2-2R_1\): \(\begin{bmatrix}1 & -1 & 1\\ \square & \square & \square\\ 3 & 3 & -4\end{bmatrix}\begin{bmatrix}x\\ y\\ z\end{bmatrix}=\begin{bmatrix}1\\ -1\\ 2\end{bmatrix}\)

By \(R_3\to R_3-3R_1\): \(\begin{bmatrix}1 & -1 & 1\\ 0 & 1 & -2\\ \square & \square & \square\end{bmatrix}\begin{bmatrix}x\\ y\\ z\end{bmatrix}=\begin{bmatrix}1\\ -1\\ -1\end{bmatrix}\)

By \(R_3\to R_3-6R_2\): \(\begin{bmatrix}1 & -1 & 1\\ 0 & 1 & -2\\ 0 & 0 & 5\end{bmatrix}\begin{bmatrix}x\\ y\\ z\end{bmatrix}=\begin{bmatrix}1\\ -1\\ \square\end{bmatrix}\)

We write equations as \(x-y+z=1\) .....(I), \(y-2z=-1\) .....(II), \(5z=5\) .....(III). Solving equations (I), (II) and (III) we get \(x=\square,\ y=\square,\ z=\square\)

Q.3 (C) (ii) Obtain the differential equation from the relation \(Ax^2+By^2=1\), where A and B are constants.

Solution : The given equation is \(Ax^2+By^2=1\) .....(I)

Differentiating equation (I) w.r.t. x, we get \(\square\,x+2By\frac{dy}{dx}=0\), i.e. \(Ax+By\frac{dy}{dx}=0\) .....(II)

Differentiating equation (II) w.r.t. x, we get \(A+B\,\square=0\) .....(III)

Since equations (I), (II) and (III) are consistent in A and B, \(\begin{vmatrix}x^2 & y^2 & 1\\ x & y\frac{dy}{dx} & 0\\ 1 & \square & 0\end{vmatrix}=0\)

\(\therefore \left\{x\left[y\frac{d^2y}{dx^2}+\left(\frac{dy}{dx}\right)^2\right]-y\frac{dy}{dx}\right\}=0\)

\(\therefore \square+x\left(\frac{dy}{dx}\right)^2-y\frac{dy}{dx}=0\)

Q.4 (A) (i) The difference between face value and present worth is called _____.

Q.4 (A) (ii) Insurance companies collect a fixed amount from their customers at a fixed interval of time. This amount is called _____.

Q.4 (A) (iii) If \(b_{YX}\gt 1\) then \(b_{XY}\) is _____.

Q.4 (A) (iv) Which of the following can't be a component of a time series?

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

Q.4 (A) (vi) If the corner points of the feasible region are (0, 10), (2, 2) and (4, 0) then the point of minimum \(z=3x+2y\) is _____.

Q.4 (B) (i) State whether true or false: The banker's discount is always lower than the true discount.

Q.4 (B) (ii) State whether true or false: The cost of living Index Number using weighted relative method is given by \(\frac{\Sigma IW}{\Sigma W}\).

Q.4 (B) (iii) State whether true or false: To convert an assignment problem into a minimization problem, the smallest element in the matrix is deducted from all other elements.

Q.4 (C) (i) The difference between the banker's discount and the true discount is called _____.

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

Q.4 (C) (iii) A dish washing machine holds up to 40 pieces of large crockery (x). This constraint is given by _____.

Q.5 (A) (i) A shop is valued at ₹ 3,60,000 and is insured for 75% of its value. If the rate of premium is 0.9%, find the premium paid by the owner of the shop. Also find the agent's commission if the agent gets commission at 15% of the premium.

Q.5 (A) (ii) Given the following information about the production (X) and demand (Y) of a commodity, obtain the regression line of X on Y.

Coefficient of correlation between X and Y is 0.6. Also estimate the production when demand is 100.

Q.5 (A) (iii) The following table shows the production of pig-iron and ferro-alloys ('000 metric tons)

Find trend values for the above data by using 5 yearly moving averages.

Q.5 (B) (i) Solve the following L.P.P. by graphical method.

Minimize : \(z=8x+10y\)

Subject to : \(2x+y\ge 7,\ 2x+3y\ge 15,\ y\ge 2,\ x\ge 0,\ y\ge 0\)

Q.5 (B) (ii) Solve the following assignment problem for minimization :

Q.5 (B) (iii) The probability distribution of a discrete r.v. X is as follows :

(a) Determine the value of k.
(b) Find \(P(X\lt 4),\ P(2\lt X\lt 4),\ P(X\gt 3)\)

Q.6 (A) (i) The true discount on a sum is \(\frac{3}{8}\) of the sum due at 12% p.a. Find the period of the bill.

Q.6 (A) (ii) Following data shows the number of bags of cereals sold in years 1977 to 1984.

Fit a trend line to the above data by graphical method.

Q.6 (A) (iii) The following is the p.d.f. of a r.v. X : \(f(x)=\frac{x}{8}\) for \(0\lt x\lt 4\); \(=0\), otherwise.

Find (a) \(P(x\lt 1.5)\) (b) \(P(1\lt x\lt 2)\) (c) \(P(x\gt 2)\)

Q.6 (B) (i) For the following bivariate data obtain the equation of regression line of Y on X.

Q.6 (B) (ii) Calculate (a) Laspeyre's and (b) Paasche's Price Index Numbers for the following data :

Q.6 (C) (i) Determine the optimal sequence of jobs that minimizes the total elapsed time for the data given below (processing time on machines is given in hours). Also find the total elapsed time and the idle time for three machines.

Solution : Here min A = 3, min C = 5, Max B = 5. Since Min \(C\ge\) max B is satisfied, the problem can be converted into a two machine problem. Let G and H be two fictitious machines \(\therefore G=A+B,\ H=B+C\). The above problem can be written as :

Using optimal sequence algorithm, the following sequence can be obtained : [ ] [ ] [ ] VI II [ ] [ ]

Work table :

∴ Total elapsed time is = 59 hrs. Idle time for machine A = [ ] hrs. Idle time for machine B = [ ] hrs. Idle time for machine C = 7 hrs.

Q.6 (C) (ii) In a town 10 accidents take place in the span of 50 days. Assuming that the number of accidents follows Poisson distribution, find the probability that there will be 3 or more accidents on a day. [ Given that \(e^{-0.2}=0.8187\) ]

Solution : Here \(m=\square\) and \(X\sim P(m)\) with parameter m. The p.m.f. of X is : \(P(X=x)=\frac{e^{-m}\cdot m^x}{x!},\ x=0,1,2,\ldots\)

\(P(X\ge 3)=1-P(X\lt 3)=1-\left[\square+\square+\square\right]=1-\left[\frac{e^{-0.2}(0.2)^0}{0!}+\frac{e^{-0.2}(0.2)^1}{1!}+\frac{e^{-0.2}(0.2)^2}{2!}\right]\)

\(=1-[0.8187(1+0.2+0.02)]=1-\square=\square\)