List of top Mathematics & Statistics Questions

The differential equation obtained by eliminating arbitrary constants from \( bx + ay = ab \) is \( \frac{d^2y}{dx^2} = 0 \).

The value of \( \int \log x \, dx = x \log x + x + c \).

If A is a matrix and K is a constant, then \( (KA)^T = K A^T \).

The average revenue \( R_A \) is 50 and elasticity of demand \( \eta \) is 5, the marginal revenue \( R_M \) is ..............

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 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.

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 \( x^7 \).

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

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

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 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 \( z \).

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 \( t_1, t_2 \).

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.

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.

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

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