Q.1 (A) (i) If \(y=\sqrt[3]{a^2+x^2}\) then \(\frac{dy}{dx}\) = _____.
Mathematics & Statistics (Commerce) Part 1
Every question from past board papers on Mathematics & Statistics (Commerce) Part 1, newest paper first. Open any question for its full worked answer.
Maharashtra HSC Mathematics and Statistics (Commerce) Board Paper March 2026 with Solutions
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Q.1 (A) (ii) If \(y=\log\left(\frac{e^x}{x^2}\right)\) then \(\frac{dy}{dx}\) = _____.
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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?
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Q.1 (A) (iv) The equation of normal to the curve \(y=3x^2-x+1\) at (1, 3) is _____.
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Q.1 (A) (v) The solution of \(\frac{dy}{dx}=1\) is :
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Q.1 (A) (vi) If \(\int_0^a 3x^2\,dx=8\) then a = _____.
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Q.1 (B) (i) State whether true or false: If \(y=20+15x+x^2\) then \(\frac{dx}{dy}=\frac{1}{15+2x}\).
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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.
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Q.1 (B) (iii) State whether true or false: Order and degree of a differential equation are always positive integers.
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Q.1 (C) (i) Statement \(p\leftrightarrow q\) is true, when p and q have _____ truth values.
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Q.1 (C) (ii) If \(y=e^{ax}\) then \(x\frac{dy}{dx}\) = _____.
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Q.1 (C) (iii) If \(f(x)=x\cdot\log x\) then its minimum value is _____.
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Q.2 (A) (i) Using truth table, examine whether the following statement pattern is a tautology, a contradiction or a contingency.
\((p\wedge\sim q)\to(\sim p\wedge\sim q)\)
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Q.2 (A) (ii) If \(e^x+e^y=e^{(x+y)}\) then show that \(\frac{dy}{dx}=-e^{y-x}\).
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Q.2 (A) (iii) Evaluate : \(\int\frac{1+x}{x+e^{-x}}\,dx\)
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Q.2 (B) (i) Find the inverse of \(\begin{bmatrix}1 & -1 & 1\\ 2 & 1 & -3\\ 1 & 1 & 1\end{bmatrix}\) by adjoint method.
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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.
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Q.2 (B) (iii) Evaluate : \(\int_2^7\frac{\sqrt{x}}{\sqrt{x}+\sqrt{9-x}}\,dx\)
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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.
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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 : \(\int\frac{1}{4x^2-20x+17}\,dx\)
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Q.3 (B) (i) Solve the following differential equation
\((x^2-y^2)\,dx+2xy\,dy=0\)
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Q.3 (B) (ii) Find the area of the region bounded by the curve \(x^2=16y\) and the line \(y=4\).
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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 \(x=\square,\ y=\square,\ z=\square\)
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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, \(\begin{vmatrix}x^2 & y^2 & 1\\ x & y\frac{dy}{dx} & 0\\ 1 & \square & 0\end{vmatrix}=0\)
\(\therefore \left\{x\left[y\frac{d^2y}{dx^2}+\left(\frac{dy}{dx}\right)^2\right]-y\frac{dy}{dx}\right\}=0\)
\(\therefore \square+x\left(\frac{dy}{dx}\right)^2-y\frac{dy}{dx}=0\)
Maharashtra HSC Mathematics and Statistics (Commerce) Board Paper June 2026 with Solutions
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Q.1 (A) (i) If \(p\) is any statement then \((p\vee\sim p)\) is a
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Q.1 (A) (ii) If \(A=\begin{bmatrix}4 & x\\6 & 3\end{bmatrix}\) is a singular matrix then \(x\) is
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Q.1 (A) (iii) If \(y=e^{\log x}\) then \(\frac{dy}{dx}\) is
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Q.1 (A) (iv) If \(0\lt\eta\lt 1\), then the demand is
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Q.1 (A) (v) \(\int x^2(3)^{x^3}\,dx\) = _____
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Q.1 (A) (vi) \(\int_{-9}^{9}\frac{x^3}{4-x^2}\,dx\) = _____.
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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.
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Q.1 (B) (ii) State whether true or false: \(\int\frac{1}{2x+3}\,dx=\log|2x+3|+c\)
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Q.1 (B) (iii) State whether true or false: Order and degree of a differential equation are always positive integers.
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Q.1 (C) (i) The dual of \((p\wedge\sim q)\vee t\) is _____.
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Q.1 (C) (ii) \(\int e^x\left(\frac{1}{x}-\frac{1}{x^2}\right)dx\) = _____.
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Q.1 (C) (iii) The integrating factor of the differential equation \(\frac{dy}{dx}-y=x\) is _____.
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Q.2 (A) (i) Write the converse, inverse and contrapositive of the following statement.
'If a man is rich, then he is happy.'
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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.
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Q.2 (A) (iii) Find the rate of change of demand (\(x\)) of a commodity with respect to its price (\(y\)) if \(y=\frac{5x+7}{2x-13}\)
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Q.2 (B) (i) Evaluate: \(\int\frac{e^x}{\sqrt{e^{2x}+4e^x+13}}\,dx\)
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Q.2 (B) (ii) Evaluate: \(\int_{2}^{5}\frac{\sqrt{x}}{\sqrt{x}+\sqrt{7-x}}\,dx\)
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Q.2 (B) (iii) Find the area of ellipse \(\frac{x^2}{25}+\frac{y^2}{4}=1\)
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Q.3 (A) (i) Determine whether the following statement pattern is tautology, contradiction or contingency.
\([p\wedge(\sim p\vee q)]\)
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Q.3 (A) (ii) If \(x^y=e^{x-y}\) then prove that \(\frac{dy}{dx}=\frac{\log x}{(1+\log x)^2}\).
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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.
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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.
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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 [ ]
\(\therefore y\cdot e^x=\int 5\cdot e^x\,dx+c\)
\(\therefore\) [ ]
Maharashtra HSC Mathematics and Statistics (Commerce) Board Paper March 2025 with Solutions
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Q.1 (A) (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 :
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Q.1 (A) (ii) \(\int\left(x+\frac{1}{x}\right)^3dx=\)
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Q.1 (A) (iii) \(\int_2^7\frac{\sqrt{x}}{\sqrt{x}+\sqrt{9-x}}\,dx=\)
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Q.1 (A) (iv) The area of the region bounded by the curve \(y=x^2\) and the line \(y=4\) is
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Q.1 (A) (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.
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Q.1 (A) (vi) The integrating factor of the differential equation \(\frac{dy}{dx}+\frac{y}{x}=x^3-3\) is
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Q.1 (B) (i) State whether true or false: If A is a matrix and K is a constant, then \((KA)^T=KA^T\).
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Q.1 (B) (ii) State whether true or false: \(\int\log x\,dx=x\log x+x+c\).
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Q.1 (B) (iii) State whether true or false: The differential equation obtained by eliminating arbitrary constants from \(bx+ay=ab\) is \(\frac{d^2y}{dx^2}=0\).
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Q.1 (C) (i) The average revenue \(R_A\) is 50 and elasticity of demand \(\eta\) is 5, the marginal revenue \(R_M\) is _____.
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Q.1 (C) (ii) \(\int e^x\left(\frac{1}{x}-\frac{1}{x^2}\right)dx=\) _____ \(+c\)
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Q.1 (C) (iii) If \(f'(x)=x^2+5\) and \(f(0)=-1\) then \(f(x)=\) _____.
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Q.2 (A) (i) Write the converse, inverse and contrapositive of the statement "If a triangle is equilateral then it is equiangular".
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Q.2 (A) (ii) Find x, y, z if \(\left\{5\begin{bmatrix}0 & 1\\ 1 & 0\\ 1 & 1\end{bmatrix}-\begin{bmatrix}2 & 1\\ 3 & -2\\ 1 & 3\end{bmatrix}\right\}\begin{bmatrix}2\\ 1\end{bmatrix}=\begin{bmatrix}x-1\\ y+1\\ 2z\end{bmatrix}\)
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Q.2 (A) (iii) Evaluate : \(\int\frac{1}{x(x^6+1)}dx\)
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Q.2 (B) (i) Solve the following equations by the method of inversion :
\(2x-y+z=1\)
\(x+2y+3z=8\)
\(3x+y-4z=1\) -
Q.2 (B) (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 \(I=1000\)
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Q.2 (B) (iii) Evaluate : \(\int_1^2\frac{dx}{x^2+6x+5}\)
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Q.3 (A) (i) Find \(\frac{dy}{dx}\) if \(y=(x)^x+(a)^x\)
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Q.3 (A) (ii) Find the area of the region bounded by the parabola \(y^2=25x\) and the line \(x=5\).
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Q.3 (A) (iii) Find the differential equation by eliminating arbitrary constants from the relation \(y=Ae^{3x}+Be^{-3x}\).
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Q.3 (B) (i) Using the truth table, verify \(p\vee(q\wedge r)\equiv(p\vee q)\wedge(p\vee r)\)
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Q.3 (B) (ii) If \(x=\frac{4t}{1+t^2}\), \(y=3\left(\frac{1-t^2}{1+t^2}\right)\), then show that \(\frac{dy}{dx}=\frac{-9x}{4y}\).
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Q.3 (C) (i) Divide the number 84 into two parts such that the product of one part and square of the other is maximum.
Solution : Let one part be x then the other part will be \(84-x\).
\(\therefore f(x)=\square\)
\(\therefore f'(x)=168x-3x^2\)
For extreme values \(f'(x)=0\)
\(168x-3x^2=0\)
\(\therefore 3x(56-x)=0\)
\(\therefore x=\square\) OR \(\square\)
\(f''(x)=168-6x\)
If \(x=0,\ f''(0)=168-6(0)=168\gt0\)
\(\therefore\) function attains minimum at \(x=0\)
If \(x=56,\ f''(56)=\square\lt0\)
\(\therefore\) function attains maximum at \(x=56\)
\(\therefore\) Two parts of 84 are \(\square\) and \(\square\) -
Q.3 (C) (ii) Solve the following differential equation \((x^2-yx^2)dy+(y^2+xy^2)dx=0\)
Solution : Separating the variables, the given equation can be written as :
\(\square\,dy+\square\,dx=0\)
\(\therefore\left(y^{-2}-\frac{1}{y}\right)dy+\left(x^{-2}+\frac{1}{x}\right)dx=0\)
\(\square\,dy-\frac{1}{y}dy+x^{-2}dx+\square\,dx=0\)
Integrating we get,
\(\int y^{-2}dy-\int\frac{1}{y}dy+\int x^{-2}dx+\int\frac{1}{x}dx=0\)
\(\therefore\frac{y^{-1}}{-1}-\square+\frac{x^{-1}}{-1}+\square=c\)
\(-\frac{1}{y}-\frac{1}{x}+\log x-\log y=c\)
\(\log x-\log y=\square+c\)
is the required solution.