List of top Mathematics & Statistics Questions

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 \( x^7 \).

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

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 \( z \).

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

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 \( t_1, t_2 \).

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.

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.

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

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.

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

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.

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

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 \( 2\hat{i} + \hat{j} - 2\hat{k} \).

Find the area of the region bounded by the parabola \( y^2 = 16x \) and its latus rectum.

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

Evaluate: \( \int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \frac{1}{1 - \sin x} \, dx \)

A line makes angles of measure 45° and 60° with the positive directions of the \( Y \) and \( Z \) axes respectively. Find the angle made by the line with the positive direction of the \( X \)-axis.

Find the general solution of \( \tan^2 \theta = 1 \).