List of practice Questions

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

Find the differential equation of the family of curves \[ y = a \sin(x + b), \] where \( a \) and \( b \) are arbitrary constants.

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.

The degree of differential equation \[ \frac{d^2y}{dx^2} = \left( y + \frac{dy}{dx} \right)^{\frac{1}{5}} \] will be

The value of \[ \int \cos^2 x \, dx \] will be

What are doped semiconductors? Explain the effect of (i) doping and (ii) temperature on the conductivity of semiconductors.

A proton (p) and an $\alpha$-particle are accelerated to the same potential difference. Find the ratio of de Broglie wavelengths associated with proton and $\alpha$-particle.

Find the resultant intensity of two waves having intensities $9I$ and $4I$ when waves are (i) coherent and (ii) non-coherent.

Draw an energy level diagram for hydrogen atom. Calculate the wavelengths of (i) H$_\alpha$, (ii) H$_\beta$, and (iii) Series limit of Balmer series.

Obtain an expression for (i) potential difference and (ii) capacitance of a parallel plate capacitor filled partly with dielectric material between plates.