List of practice Questions

A convex lens has radius of curvature \( R = 40 \, \text{cm} \) for both surfaces and refractive index \( \mu = 1.5 \). Find the focal length.
Telescope has objective diameter 250 cm. For wavelength \( \lambda = 600 \, \text{nm} \), the limit of resolution is:
A particle initially at the mean position is executing simple harmonic motion with an angular frequency \( \frac{\pi}{4} \, \text{rad/s} \). The ratio of the distances travelled by the particle in the first second and second second is:
The time period of a simple pendulum on the surface of the Earth is \( T \). At what height above the surface will the time period become \( 2T \)?
(Radius of Earth = \( 6400 \, \text{km} \))
Which of the following statements are true about acceleration due to gravity \( g \)?
A. \( g \) is greater at poles.
B. \( g \) decreases with height.
C. \( g \) is same all over Earth.
D. \( g \) is maximum at centre of Earth.
A rectangular metallic block of size \( 40 \, \text{mm} \times 20 \, \text{mm} \) is pulled with a force of \( 50 \, \text{kN} \). The strain in the block is:
(Shear modulus \( = 40 \times 10^9 \, \text{Nm}^{-2} \))
A steel tape of 300 cm is graduated at \( 27^\circ \text{C} \). A steel rod measured to be 110 cm at \( 50^\circ \text{C} \) with this tape has actual length (given \( \alpha_{\text{steel}} = 1.2 \times 10^{-5} \, \text{K}^{-1} \)):
In a Carnot engine, if the temperatures of the source and sink are both decreased by 100 K, the efficiency:
A body moves with uniform speed of \( 20 \text{ ms}^{-1} \) in a horizontal circle. What is the change in velocity after half revolution?
Force is the mutual interaction between bodies according to:
A motor vehicle of mass 1000 kg is moving on a circular road having banking angle \( 30^\circ \) and coefficient of friction 0.2. Then the normal reaction force on the motor vehicle is about:
(Acceleration due to gravity = \( 10 \, \text{ms}^{-2} \))
If a body has a potential energy of \( (4x^2 + 2x) \, \text{J} \) at height 2 m, then the force acting on it is:
A freely falling body has attained a velocity of \( 2 \, \text{ms}^{-1} \). If it is opposed by air resistance, total distance before stopping is:
A spherical portion A of radius \( R \) is removed from a solid sphere B of radius \( 2R \), both centered. The ratio of moments of inertia of remaining part to original is:
A disc of moment of inertia \( 3.5 \, \text{kg m}^2 \) is rotating at \( 30 \, \text{rad/s} \). Torque to stop it in 5 seconds is:
If \( 5f(x) + 3f\left( \frac{1}{x} \right) = 2 - \frac{1}{x}, x \ne 0 \), then \( \int_1^2 f\left( \frac{1}{x} \right) dx = \)
Let \( c_1, c_2, c_3, c_4 \) be arbitrary constants. The order of the differential equation corresponding to \[ y = c_1 e^x + c_2 e^{\log_e x} + c_3 \sin^2 x - c_4 (\cos^5 x - 1) \] is:
The order and degree of the differential equation \[ 3x^2 \frac{d^2 y}{dx^2} - \sin\left( \frac{d^3 y}{dx^3} \right) + \cos(xy) = 0 \] are:
If \( y = y(x) \) is the solution of the differential equation \[ x \frac{dy}{dx} = y + xe^{-\frac{y}{x}}, \quad y(1) = \log e \] , then \( y(e) = \)
The dimensional formula of a physical quantity represented by \( \frac{e^2}{4 \varepsilon_0 h} \) is:
Where \( e \) is the charge of the electron, \( \varepsilon_0 \) is the permittivity of free space, and \( h \) is Planck's constant.
If a person moving along a straight line covers the first half of the distance with velocity \( V_1 \) and the next half with velocity \( V_2 \), then the average velocity is:
A 2 kg stone is tied to a 2 m long string and whirled in a circle. If the maximum tension is 64 N, what is the max number of revolutions per minute?
If \[ f(x) = \lim_{n \to \infty} n^2 \left( \frac{1}{x^n} - \frac{1}{x^{n+1}} \right), x>0, \] then \[ \int x f(x) \, dx = ? \]
Evaluate the following integral: \[ \int \frac{1 - \cos x}{\cos x(1 + \cos x)} \, dx \]
If \( g(x) \) is an antiderivative of \( f(x) = 1 + 2 \log 2 \) and the graph of \( y = g(x) \) passes through the point \( \left( -1, -\frac{1}{2} \right) \), then the curve meets the Y-axis at