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

Q.1 (i) If \(A = \{1, 2, 3, 4, 5\}\) then which of the following is not true?

Q.1 (ii) The value of \(\sin\left[\sin^{-1}\left(-\frac{1}{\sqrt{2}}\right) - 3\sin^{-1}\left(\frac{\sqrt{3}}{2}\right)\right] =\)

Q.1 (iii) If \(\hat{p}, \hat{q}, \hat{r}\) are mutually perpendicular unit vectors and form a right handed triplet then the value of \(\hat{p}\cdot(\hat{q}\times\hat{r}) + \hat{q}\cdot(\hat{p}\times\hat{r}) + \hat{r}\cdot(\hat{p}\times\hat{q}) =\) _____

Q.1 (iv) The Cartesian equation of the line passing through the points \(A(3, 4, -1)\) and \(B(2, -1, 3)\) is _____.

Q.1 (v) If \(y = \log_x a\) then \(\frac{dy}{dx} =\) _____.

Q.1 (vi) If \(f(x) = \frac{\sin^{-1}x}{\sqrt{1-x^2}}\) and \(g(x) = e^{\sin^{-1}x}\) then \(\int f(x)\cdot g(x)\,dx =\) _____.

Q.1 (vii) The area bounded by the line \(y = 2x\), X-axis and the lines \(x = -1\), \(x = 4\) is _____ sq. units.

Q.1 (viii) In a Binomial Distribution, \(E(X) = 6\), \(\mathrm{Var}(X) = 4.2\) then the value of \(n\) is _____

Q.2 (i) Write the dual of \(p \wedge t\).

Q.2 (ii) What is the distance of point \((3, -5, 6)\) from XZ plane?

Q.2 (iii) The displacement of a particle at time \(t\) is given by \(S = 3t^2 + 4t + 1\). Find its velocity at time \(t\).

Q.2 (iv) Write the order of differential equation \(\left(\frac{d^2y}{dx^2}\right)^{\frac{5}{2}} = \sqrt[3]{\frac{dy}{dx}}\)

Q.3 Construct the switching circuit for the statement \((\sim p \wedge q) \vee (p \wedge \sim r)\).

Q.4 Find the matrix of cofactors of matrix \(\begin{bmatrix} 5 & 2 \\ -1 & 6 \end{bmatrix}\).

Q.5 Find the general solution of equation \(\cos 5\theta = \frac{1}{2}\).

Q.6 Find the value of \(k\), if \(2x + y = 0\) is one of the lines represented by \(3x^2 + kxy + 2y^2 = 0\).

Q.7 If \(\bar{a}, \bar{b}, \bar{c}\) are the position vectors of points \(A, B, C\) respectively and \(10\bar{a} = 7\bar{b} + 3\bar{c}\) then find the ratio in which the point \(C\) divides the line segment \(AB\).

Q.8 Find the angle between the lines
\(\bar{r} = (\hat{i} + 2\hat{j} + 3\hat{k}) + \lambda(2\hat{i} - \hat{j} + \hat{k})\)
\(\bar{r} = (\hat{i} + 2\hat{j} + 3\hat{k}) + \mu(\hat{i} + 2\hat{j} + \hat{k})\)

Q.9 Test whether the function \(f(x) = x - \frac{1}{x},\ x \in R,\ x \ne 0\) is increasing or decreasing.

Q.10 Evaluate: \(\int_{-1}^{1} |x|\,dx\)

Q.11 Evaluate: \(\int x\log x\,dx\)

Q.12 Find the area bounded by curve \(x^2 + y = 0\), X-axis and lines \(x = 1\) and \(x = 4\).

Q.13 The probability distribution of r.v. \(X\) is as follows:

Find: (i) \(k\) (ii) \(F(2)\)

Q.14 Evaluate: \(\int\frac{e^x(1 + x)}{\cos^2(xe^x)}\,dx\)

Q.15 Using truth table, determine whether statement pattern \([(p \vee q) \wedge \sim p] \wedge \sim q\) is a tautology or a contradiction or a contingency.

Q.16 Prove that: \(\sin^{-1}\left(\frac{3}{5}\right) + \cos^{-1}\left(\frac{12}{13}\right) = \sin^{-1}\left(\frac{56}{65}\right)\)

Q.17 In \(\triangle ABC\), if \(a = 10\), \(b = 8\), \(c = 6\) then find the value of (i) \(\sin\left(\frac{C}{2}\right)\) (ii) \(\cos\left(\frac{B}{2}\right)\)

Q.18 Are the four points \(A(1, 2, 1)\), \(B(2, -3, 4)\), \(C(3, 4, -5)\), \(D(2, 3, -2)\) coplanar? Justify your answer.

Q.19 Find the distance between parallel lines \(\frac{x}{2} = \frac{y}{-1} = \frac{z}{2}\) and \(\frac{x+1}{2} = \frac{y-1}{-1} = \frac{z+1}{2}\)

Q.20 Find the vector equation of the plane passing through the points \(A(2, 1, 1)\), \(B(0, 2, 3)\) and \(C(4, 5, 6)\).

Q.21 If \(y = \log\left[\frac{x + \sqrt{x^2 + a^2}}{\sqrt{x^2 + a^2} - x}\right]\), find \(\frac{dy}{dx}\).

Q.22 A wire of length 36 cm is bent to form a rectangle. Find its dimensions, if the area of the rectangle is maximum.

Q.23 Verify Rolle's theorem for the function \(f(x) = x^2 - 5x + 9,\ x \in [1, 4]\)

Q.24 Solve the differential equation \(x\frac{dy}{dx} - \sqrt{x^2 + y^2} = y\)

Q.25 The following is the p.d.f. of continuous r.v.:
\(f(x) = \frac{x}{8};\ 0 \lt x \lt 4\)
\(= 0\), otherwise
Find (i) expression for c.d.f. of \(X\). (ii) \(F(x)\) at 0.5 and 5

Q.26 If the p.m.f. of r.v. \(X\) be \(P(X = x) = {}^4C_x\left(\frac{5}{9}\right)^x\left(\frac{4}{9}\right)^{4-x}\) for \(x = 0, 1, 2, 3, 4\); \(= 0\) otherwise, then find \(E(X)\) and \(\mathrm{Var}(X)\).

Q.27 Find the inverse of matrix \(\begin{bmatrix} -4 & -3 & -3 \\ 1 & 0 & 1 \\ 4 & 4 & 3 \end{bmatrix}\)

Q.28 Prove that the homogeneous equation of degree two in \(x\) and \(y\), \(ax^2 + 2hxy + by^2 = 0\) represents a pair of lines passing through the origin if \(h^2 - ab \ge 0\).

Q.29 Prove that, if \(\bar{a}, \bar{b}, \bar{c}\) are three non coplanar vectors then any vector \(\bar{r}\) in the space can be uniquely expressed as a linear combination of \(\bar{a}, \bar{b}, \bar{c}\).

Q.30 Solve the linear programming problem (L.P.P.) by graphical method.
Minimize: \(z = 8x + 10y\)
Subject to, \(2x + y \ge 7\), \(2x + 3y \ge 15\), \(x \ge 0\), \(y \ge 2\)

Q.31 If \(x = f(t)\) and \(y = g(t)\) are differentiable functions of \(t\), so that \(y\) is differentiable function of \(x\) and \(\frac{dx}{dt} \ne 0\) then prove that \(\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}\).
Hence find \(\frac{dy}{dx}\), if \(x = \sin t,\ y = \cos t\).

Q.32 Evaluate: \(\int\frac{2}{3 + 2\sin^2 x + 5\cos^2 x}\,dx\)

Q.33 Prove that, \(\int_{-a}^{a} f(x)\,dx = 2\int_{0}^{a} f(x)\,dx\), if \(f(x)\) is even
\(= 0\), if \(f(x)\) is odd

Q.34 A body cools according to Newton's law from 100°C to 60°C in 20 minutes. The temperature of the surrounding being 20°C. How long will it take to cool down to 40°C?