List of practice Questions

Evaluate: \[ \int \frac{\sec^2(2x)}{(\cot x - \tan x)^2} \, dx. \]

If the coordinates of mid-points of the sides of a triangle are \( (1, 5, -1), (0, 4, -2) \) and \( (2, 3, 4) \), then find the coordinates of its vertices.

Find the value of \( \int_a^b x^2 \, dx \) with the help of definite integral as the limit of a sum.

Find the shortest distance between the lines \[ \vec{r} = (i + 2j + k) + \lambda(i - j + k) \] and \[ \vec{r} = (2i - j - k) + \mu(2i + j + 2k) \]

Find the coordinates of the point which divides the line joining the points \( (2, -5, 1) \) and \( (1, 4, -6) \) internally in the ratio 2 : 3.

Find the area of the triangle whose two sides are represented by the vectors \[ \vec{a} = 3\hat{i} - \hat{j} + 5\hat{k}, \quad \vec{b} = \hat{i} + 2\hat{j} - \hat{k}. \]

If \[ A = \begin{bmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{bmatrix}, \] prove that \[ A^3 = \begin{bmatrix} \cos 3\theta & \sin 3\theta \\ -\sin 3\theta & \cos 3\theta \end{bmatrix}. \]

Two integers among 1 to 11 are selected at random. If their sum is even, then find the probability that both integers are odd.

If \( f : \mathbb{R} \to \mathbb{R} \), where \( f(x) = \sin x \) and \( g : \mathbb{R} \to \mathbb{R} \), where \( g(x) = x^2 \), then find the range of \( f(x) \) and \( g(x) \).

If \( P(A) = \frac{1}{2} \), \( P(B) = \frac{1}{3} \), and \( P(A \cup B) = \frac{2}{3} \), prove that the events \( A \) and \( B \) are independent.

If \( \vec{a} = 2\hat{i} + 2\hat{j} + 3\hat{k} \), \( \vec{b} = -\hat{i} + 2\hat{j} + \hat{k} \), and \( \vec{c} = 3\hat{i} + \hat{j} \) are such that \( \vec{a} + \lambda \vec{b} \) is perpendicular to \( \vec{c} \), then find the value of \( \lambda \).

Solve the differential equation \( (x - y) \, dy - (x + y) \, dx = 0 \).

Prove: \[ \left| \begin{matrix} 1+\alpha & 1 & 1 \\ 1+\beta & 1 & 1 \\ 1 & 1 & 1+\gamma \\ \end{matrix} \right| = abc \left( \frac{1}{a} + \frac{1}{b} + \frac{1}{c} + 1 \right) \]

The angle between the vectors \[ \mathbf{A} = 2\hat{i} + \hat{j} + 3\hat{k} \quad \text{and} \quad \mathbf{B} = 3\hat{i} - 2\hat{j} + \hat{k} \] will be

If \( A = \{1, 2, 3\} \), \( B = \{2, 3, 4\} \), then the function from A to B will be

Prove that the function \( f: \mathbb{N} \to \mathbb{N} \) defined by \( f(x) = x - 1 \), when \( x>2 \), and \( f(1) = f(2) = 1 \), is onto but not one-to-one.

If \( P(A) = P(B) = \frac{5}{13} \) and \( P(A \cap B) = \frac{2}{5} \), then find \( P(A \cup B) \).

Solve the inequality \( 8x + 4<7x + 8 \).

Compare astronomical telescope with compound microscope. Can a telescope be used as a microscope and vice-versa? Explain with reason.

The refractive index for material of prism having refracting angle $60^\circ$ is $1.414$. For light incident on prism in minimum deviation position, calculate:
(i) Angle of minimum deviation,
(ii) Angle of incidence,
(iii) Angle of refraction,
(iv) Angle of emergence. Also draw ray diagram.

State, explain and compare features of
(i) Gauss’s law in electrostatics and
(ii) Ampere’s circuital law in magnetostatics.

Write the nature of path of a charged particle moving:
(i) along the electric field,
(ii) perpendicular to the electric field,
(iii) perpendicular to the magnetic field,
(iv) along the magnetic field,
(v) along the combination of electric field and magnetic field acting mutually perpendicular to each other.

A square loop of each side 10 cm and having resistance 0.5 $\Omega$ is kept vertical in east-west plane. The uniform magnetic flux density of 0.01 Tesla is established across the plane along north-east direction. After 0.70 sec, the magnetic field is reduced to zero with uniform rate. Calculate the following:
(i) the initial and final magnetic flux,
(ii) the induced e.m.f.,
(iii) the induced current,
(iv) the induced charge.

Establish the formula for the force acting between two parallel current carrying straight wires of infinite length. Explain when and why the force between both current carrying conductors becomes
(i) attractive and
(ii) repulsive.