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

Q.1 (i) The converse of contrapositive of \(\sim p \rightarrow q\) is _____.

Q.1 (ii) If \(A = \begin{bmatrix} 2 & -4 \\ 3 & 1 \end{bmatrix}\), then the adjoint of matrix \(A\) is _____.

Q.1 (iii) If \(\tan^{-1}(2x) + \tan^{-1}(3x) = \dfrac{\pi}{4}\), then \(x =\) _____.

Q.1 (iv) The angle between the line \(\bar{r} = (\hat{i} + 2\hat{j} + \hat{k}) + \lambda(\hat{i} + \hat{j} + \hat{k})\) and the plane \(\bar{r}\cdot(2\hat{i} - \hat{j} + \hat{k}) = 8\) is _____.

Q.1 (v) If \(y = \sec(\tan^{-1} x)\), then \(\dfrac{dy}{dx}\) at \(x = 1\) is _____.

Q.1 (vi) The approximate value of the function \(f(x) = x^3 - 3x + 5\) at \(x = 1.99\) is _____.

Q.1 (vii) \(\displaystyle\int_1^2 \frac{1}{x^2}\cdot e^{\frac{1}{x}}\,dx =\) _____.

Q.1 (viii) If the p.d.f. of a continuous r.v. \(X\) is
\(f(x) = \dfrac{x + 2}{18}\), for \(-2 \lt x \lt 4\)
\(\quad = 0\), otherwise
then \(P(|X| \lt 1) =\) _____

Q.2 (i) Write the dual of \((p \vee q) \vee r \equiv p \vee (q \vee r)\)

Q.2 (ii) Evaluate : \(\cos^{-1}\left(\dfrac{1}{2}\right) + 2\sin^{-1}\left(\dfrac{1}{2}\right)\)

Q.2 (iii) Evaluate : \(\displaystyle\int \frac{2x}{1 + x^2}dx\)

Q.2 (iv) Write the degree of the differential equation \((y''')^2 + 3(y'')^3 + 3xy' + 5y = 0\)

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

Q.4 In \(\triangle ABC\), prove that \(a(b\cos C - c\cos B) = b^2 - c^2\).

Q.5 Find the general solution of \(4\cos^2\theta = 3\).

Q.6 Find \(k\), if the sum of the slopes of the lines represented by \(x^2 + kxy - 3y^2 = 0\) is twice their product.

Q.7 Find the value of \(p\), for which the vectors \(\bar{a} = 3\hat{i} + 2\hat{j} + 9\hat{k}\) and \(\bar{b} = \hat{i} + p\hat{j} + 3\hat{k}\) are perpendicular to each other.

Q.8 Find the vector equation of the line passing through the points \(A(1, 2, 3)\) and \(B(2, 3, 4)\).

Q.9 Find \(\dfrac{dy}{dx}\), if \(\sqrt{x} + \sqrt{y} = \sqrt{a}\).

Q.10 Find \(\dfrac{d^2y}{dx^2}\), if \(y = x^3 + 7x^2 - 2x - 9\).

Q.11 Test whether the function \(f(x) = x^3 + 6x^2 + 12x - 7\) is increasing or decreasing for all \(x \in R\).

Q.12 A stone is dropped into a quiet lake and waves in the form of circles are generated. Radius of the circular wave increases at the rate of 3 cm/sec. How fast the area enclosed is increasing when the radius is 8 cm?

Q.13 Evaluate : \(\displaystyle\int\sqrt{1 + \sin 2x}\cdot dx\)

Q.14 Given that, \(X \sim B(n, p)\), if \(n = 10,\ E(X) = 8\) then find \(\text{Var}(X)\).

Q.15 Examine whether the statement pattern \((p \wedge q) \wedge (\sim p \vee \sim q)\) is a tautology or contradiction or contingency.

Q.16 In \(\triangle ABC\), if \(A = 45^\circ,\ B = 60^\circ\) then find the ratio of its sides.

Q.17 If two vertices of a triangle are \(A(3, 1, 4)\) and \(B(-4, 5, -3)\) and the centroid of the triangle is \(G(-1, 2, 1)\), then find the coordinates of the third vertex \(C\) of the triangle.

Q.18 If \(D, E, F\) are the mid-points of the sides \(BC, CA, AB\) respectively of \(\triangle ABC\), then prove that \(\overline{AD} + \overline{BE} + \overline{CF} = \bar{0}\).

Q.19 Show that the lines \(\bar{r} = (\hat{i} + \hat{j} - \hat{k}) + \lambda(2\hat{i} - 2\hat{j} + \hat{k})\) and \(\bar{r} = (4\hat{i} - 3\hat{j} + 2\hat{k}) + \mu(\hat{i} - 2\hat{j} + 2\hat{k})\) intersect each other.

Q.20 Find the cartesian equation of the plane \(\bar{r} = (\hat{i} - \hat{j}) + \lambda(\hat{i} + \hat{j} + \hat{k}) + \mu(\hat{i} - 2\hat{j} + 3\hat{k})\).

Q.21 If \(y = f(u)\) is a differentiable function of \(u\) and \(u = g(x)\) is a differentiable function of \(x\) then prove that \(y\) is a differentiable function of \(x\) and \(\dfrac{dy}{dx} = \dfrac{dy}{du}\times\dfrac{du}{dx}\).

Q.22 Verify LMVT for the function \(f(x) = \log x\), on \([1, e]\).

Q.23 Evaluate : \(\displaystyle\int\frac{\sin x}{\sin 3x}dx\)

Q.24 Solve the D.E. \(3e^x\tan y\,dx + (1 + e^x)\sec^2 y\,dy = 0\).

Q.25 Find \(E(X)\) and \(V(X)\), where X is the number obtained on uppermost face, when a fair die is thrown.

Q.26 A pair of dice is thrown 4 times. If getting a doublet is considered as success, find the probability of two successes.

Q.27 If \(A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}\), prove that \(A\cdot(\operatorname{adj} A) = (\operatorname{adj} A)\cdot A = |A|\cdot I\)

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

Q.29 If \(A(\bar{a})\) and \(B(\bar{b})\) are any two points in space and \(R(\bar{r})\) be a point on the line segment \(AB\) dividing internally in the ratio \(m : n\) then prove that \(\bar{r} = \dfrac{m\bar{b} + n\bar{a}}{m + n}\).

Q.30 Solve the L.P.P. graphically :
Minimize : \(z = 5x + 2y\),
Subject to, \(5x + y \ge 10,\ x + y \ge 6,\)
\(x \ge 0,\ y \ge 0\)

Q.31 Evaluate : \(\displaystyle\int\frac{3x^2 + 4x - 5}{(x^2 - 1)(x + 2)}\cdot dx\)

Q.32 Prove that : \(\displaystyle\int_a^b f(x)\,dx = \int_a^b f(a + b - x)\cdot dx\)
Hence, find \(\displaystyle\int_{\pi/6}^{\pi/3}\sin^2 x\cdot dx\)

Q.33 Find the area enclosed between the circle \(x^2 + y^2 = 1\) and the line \(x + y = 1\) lying in the first quadrant.

Q.34 Solve the differential equation \(x^2\dfrac{dy}{dx} = x^2 + xy + y^2\).