List of top Questions asked in CBSE CLASS XII

Let \( A = \mathbb{R} - \{5\} \) and \( B = \mathbb{R} - \{1\} \). Consider the function \( f : A \to B \), defined by: \[ f(x) = \frac{x - 3}{x - 5} \] Show that \( f \) is one-one and onto.
Sketch the graph of \( y = x|x| \) and hence find the area bounded by this curve, the X-axis, and the ordinates \( x = -2 \) and \( x = 2 \), using integration.
A biased die is twice as likely to show an even number as an odd number. If such a die is thrown twice, find the probability distribution of the number of sixes. Also, find the mean of the distribution.
A card from a well-shuffled deck of 52 playing cards is lost. From the remaining cards of the pack, a card is drawn at random and is found to be a King. Find the probability of the lost card being a King.
Solve the following linear programming problem graphically: Maximise \( Z = 2x + 3y \), subject to the constraints: \( x + y \leq 6, \, x \geq 2, \, y \geq 3, \, x \geq 0, \, y \geq 0. \)
de Broglie wavelength \( \lambda \) as a function of \( \frac{1}{\sqrt{K}} \), for two particles of masses \( m_1 \) and \( m_2 \), is shown in the figure. Here, \( K \) is the energy of the moving particles. (a) What does the slope of a line represent? 
(b) Which of the two particles is heavier? 
(c) Is this graph also valid for a photon? Justify your answer in each case.

 
Find the particular solution of the differential equation \( \left( x (e)^y/^x + y \right) dx = x \, dy \), given that \( y = 1 \) when \( x = 1 \).
Find the particular solution of the differential equation: \[ \frac{dy}{dx} = y \cot 2x, \] given that \( y\left(\frac{\pi}{4}\right) = 2 \).
Two media A and B are separated by a plane boundary. The speed of light in medium A and B is \( 2 \times 10^8 \, \text{ms}^{-1} \) and \( 2.5 \times 10^8 \, \text{ms}^{-1} \), respectively. The critical angle for a ray of light going from medium A to medium B is:
Find: \[ \int \frac{3x+5}{\sqrt{x^2+2x+4}} \,dx. \]
Find: \[ \int e^x \left[ \frac{1}{(1+x^2)^{3/2}} + \frac{x}{\sqrt{1+x^2}} \right] dx. \]
A particle of mass \( m \) and charge \( q \) is moving with a velocity \( \vec{v} \) in a magnetic field \( \vec{B} \) as shown in the figure. Show that it follows a helical path. Hence, obtain its frequency of revolution.
charged particle in a magnetic field
 
Evaluate \( \int_0^{\pi/4} \frac{x}{1+\cos 2x + \sin 2x} \, dx \).
Given that \( y = (\sin x)^x \cdot x^{\sin x} + a^x \), find \( \frac{dy}{dx} \).
Draw a labelled ray diagram of a compound microscope showing image formation at least distance of distinct vision. Derive an expression for its magnifying power.
Evaluate \( \int \frac{dx}{x(x^2-1)} \).
The volume of a cube is increasing at the rate of \( 6 \, \text{cm}^3/\text{s} \). How fast is the surface area of the cube increasing, when the length of an edge is \( 8 \, \text{cm} \)?
If \( x^y = e^x - y \), prove that \( \frac{dy}{dx} = \frac{\log x (1 + \log x)^2}{1 + \log x} \).
In Young's double-slit experiment using monochromatic light of wavelength \( \lambda \), the intensity of light at a point on the screen, where path difference is \( \lambda \), is \( K \) units. Find the intensity of light at a point on the screen where the path difference is \( \lambda / 6 \).
A telescope consists of two lenses of focal length 100 cm and 5 cm. Find the magnifying power when the final image is formed at infinity.
Three point charges \( q \), \( 2q \) and \( nq \) are placed at the vertices of an equilateral triangle. If the potential energy of the system is zero, find the value of \( n \).
State Gauss's law in electrostatics. Apply this to obtain the electric field E at a point near a uniformly charged infinite plane sheet:
In a hydrogen atom, the electron moves in an orbit of radius \( 2 \, \text{Å} \) making \( 8 \times 10^{14} \) revolutions per second. Find the magnetic moment associated with the orbital motion of the electron.
What is current sensitivity of a galvanometer? Show how the current sensitivity of a galvanometer may be increased. Increasing the current sensitivity of a galvanometer may not necessarily increase its voltage sensitivity. Explain.
A moving coil galvanometer has a resistance of 15 Ω and takes 20 mA to produce full-scale deflection. How can this galvanometer be converted into a voltmeter of range 0 to 100 V?