List of top Mathematics & Statistics Questions asked in Maharashtra Board Class XII Exam

Evaluate: \[ \int \sin^{-1} x \left( \frac{x + \sqrt{1 - x^2}}{\sqrt{1 - x^2}} \right) dx \]

Prove that: \[ \int_a^b f(x) \, dx = \int_a^b f(a + b - x) \, dx \] Hence, evaluate: \[ \int_0^3 \frac{\sqrt{x}}{\sqrt{x + \sqrt{3 - x}}} \, dx \]

If \( x = f(t) \) and \( y = g(t) \) are differentiable functions of \( t \) so that \( y \) is a function of \( x \) and if \( \frac{dx}{dt} \neq 0 \), then prove that \[ \frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}. \] Hence, find the derivative of \( 7^x \) with respect to \( x^7 \).

If a body cools from 80°C to 50°C at room temperature of 25°C in 30 minutes, find the temperature of the body after 1 hour.

Find the inverse of the matrix \[ \begin{pmatrix} \cos \theta & -\sin \theta & 0 \\ \sin \theta & \cos \theta & 0 \\ 0 & 0 & 1 \end{pmatrix} \] by elementary row transformations.

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 \( h^2 - ab \geq 0 \). Hence, show that the equation \( x^2 + y^2 = 0 \) does not represent a pair of lines.

Let \( \vec{a} \) and \( \vec{b} \) be non-collinear vectors. If vector \( \vec{r} \) is coplanar with \( \vec{a} \) and \( \vec{b} \), then show that there exist unique scalars \( t_1 \) and \( t_2 \) such that \( \vec{r} = t_1 \vec{a} + t_2 \vec{b} \). For \( \vec{r} = 2\hat{i} + 7\hat{j} + 9\hat{k} \), \( \vec{a} = \hat{i} + 2\hat{j} \), \( \vec{b} = \hat{j} + 3\hat{k} \), find \( t_1, t_2 \).

Three coins are tossed simultaneously, \( X \) is the number of heads. Find the expected value and variance of \( X \).

Find the equations of the tangent and normal to the curve \( y = 2x^3 - x^2 + 2 \) at the point \( \left( \frac{1}{2}, 2 \right) \).

Solve the linear programming problem graphically. Maximize: \( z = 3x + 5y \) Subject to: \[ x + 4y \leq 24, 3x + y \leq 21, x + y \leq 9, x \geq 0, y \geq 0 \] Also, find the maximum value of \( z \).

Five cards are drawn successively with replacement from a well-shuffled deck of 52 cards. Find the probability that:
(i) all the five cards are spades.
(ii) none is spade.

Solve the differential equation: \( x \frac{dy}{dx} = x \cdot \tan \left( \frac{y}{x} \right) + y \).

The displacement of a particle at time \( t \) is given by \( s = 2t^3 - 5t^2 + 4t - 3 \). Find the velocity and displacement at the time when the acceleration is \( 14 \, \text{ft/sec}^2 \).

Find the \( n \)th order derivative of \( \log x \).

A line passes through the points \( (6, -7, -1) \) and \( (2, -3, 1) \). Find the direction ratios and the direction cosines of the line. Show that the line does not pass through the origin.

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

Suppose that \( X \) is the waiting time in minutes for a bus and its p.d.f. is given by: \[ f(x) = \frac{1}{5}, \text{for } 0 \leq x \leq 5, \text{and} f(x) = 0, \text{otherwise}. \] Find the probability that: (i) waiting time is between 1 to 3 minutes. (ii) waiting time is more than 4 minutes.

Express the following switching circuit in the symbolic form of logic. Construct the switching table and interpret it. \includegraphics[width=0.5\linewidth]{01.jpeg}

Find the cartesian and vector equations of the line passing through \( A(1, 2, 3) \) and having direction ratios \( 2, 3, 7 \).

Prove that: \( 2 \tan^{-1} \left( \frac{1}{3} \right) + \cos^{-1} \left( \frac{3}{5} \right) = \frac{\pi}{2} \).

In \( \triangle ABC \), if \( a = 13 \), \( b = 14 \), and \( c = 15 \), then find the values of:
(i) \( \sec A \)
(ii) \( \csc \frac{A}{2} \)

Find the coordinates of the points of intersection of the lines represented by \( x^2 - y^2 - 2x + 1 = 0 \).

Find the vector equation of the plane passing through the point having position vector \( 2\hat{i} + 3\hat{j} + 4\hat{k} \) and perpendicular to the vector \( 2\hat{i} + \hat{j} - 2\hat{k} \).

Evaluate: \( \int x^9 \cdot \sec^2(x^{10}) \, dx \).

Evaluate: \( \int \frac{1}{25 - 9x^2} \, dx \)