List of practice Questions

The gravitational potential energy of a system of three masses $ m $, $ 2m $, and $ 3m $ placed at the three vertices of an equilateral triangle of side $ a $ is Options:
A metal wire with circular cross section and length one metre is pulled with tensile force of 1000 N on each side. For the wire to be stretched not more than 0.25 cm, the minimum diameter of the wire required is:
(Young's modulus of the metal $= 10^{11}$ \text{ Pa, take } $\sqrt{\pi} = 1.77$)
The temperature of the earth without the greenhouse effect will be: Options:
The change in internal energy when 20 g of a gas is heated from $ 25^\circ C $ to $ 35^\circ C $ at constant volume is: Options:
A body of mass 2 kg is on an inclined plane of inclination 30° and coefficient of friction is $ \frac{1}{\sqrt{3}} $. The minimum force required to move the body up the inclined plane is: \[ \text{(Acceleration due to gravity = 10 ms}^{-2}\text{)} \] Options:
The maximum acceleration with which a body of mass 200 kg is lowered into a well using a rope having a breaking force of 50 kg-wt is (Acceleration due to gravity = 10 $ ms^{-2} $)
A toy of mass 20 g at rest acquires a velocity $ (3\hat{i} - 2\hat{j}) $ ms$ ^{-1} $ in 2 seconds. Then the power of the toy is
A ball is dropped from some height and after first collision with the ground if it reaches $ \frac{3}{4} $th of its original height, then the % loss of its energy is: Options:
A solid sphere of mass 2 kg is rolling without slipping on a horizontal surface with a velocity 5 m/s. The rotational kinetic energy of the sphere is: Options:
A solid sphere and a solid cylinder roll down without slipping along an inclined plane. If they start from rest from the top of the inclined plane, the ratio of the velocities of the solid sphere and solid cylinder when they reach the bottom of the inclined plane is:
A pendulum of time period one second is losing its mechanical energy due to damping. Its mechanical energy at time $ t = 0 $ is 45 J. After completing 15 oscillations, its mechanical energy is 15 J. The ratio of the damping constant and the mass of the object making damped oscillations is:
Let $ a $ and $ b $ be arbitrary constants and $ C $ be a fixed constant. If $ y = ae^{2x} + bxe^{2x} + C $ is the general solution of a differential equation, then the order of that differential equation is:
The differential equation \[ y^2 \, dx + (3xy - 1) \, dy = 0 \] is:
The particular solution of the differential equation \[ \frac{dx}{dy} = \frac{\sin y(1 + y \cot y)}{x \log(x^2 e)}, \quad y(1) = 0 \] Options:
The potential difference across the ends of a conductor is $ (50 \pm 3) $ V and the current through it is $ (5 \pm 0.1) $ A. The percentage error in the measurement of the resistance of the conductor is: Options:
The velocity (v) of a particle starting from rest increases linearly with time (t) as $ v = 4t $, where v is in $ ms^{-1} $ and t is in second. The distance covered by the particle in the first 4 seconds is
The time taken by a boat to travel upstream to a certain distance and return back is 14 hours. If the velocity of boat in still water is 35 kmh$ ^{-1} $ and velocity of the stream is 5 kmh$ ^{-1} $, the distance travelled by the boat before it returns is
If the angle between two unit vectors $ \vec{A} $ and $ \vec{B} $ is $ \theta $, then \[ |\vec{A} + \vec{B}| = \]
$\int e^{-2x} \left( \frac{1 - \sin 2x}{1 + \cos 2x} \right) dx =$
If $ S_n = \int_0^{\frac{\pi}{2}} \frac{\sin((2n-1)x)}{\sin x} \, dx $, and $ n $ is an integer, then $ S_{n+1} - S_n = $:
If $ a = 2n $ and $ b = 2m+1 $ for all $ m, n \in \mathbb{N} $, then \[ \int_{-\pi}^{\pi} e^{\sin^2 x} \cot^{b}(2n+1) x \, dx = \]
If $ \int_{n}^{n+1} g(x) \, dx = n^2 $, $ \forall n \in Z $, then the value of $ \int_{-3}^{3} g(x) \, dx $ is
$ \int_{0}^{\frac{\pi}{2}} \frac{\sum_{n=0}^{4} \sin \left( \frac{n\pi}{4} + x \right)}{\cos x + \sin x} dx = $
The maximum value of $ a $ such that the second derivative of $ x^4 + ax^3 + \frac{3x^2}{2} + 1 $ is positive for all real $ x $ is
Let $ f : \mathbb{R} \to \mathbb{R} $ be a continuous function. If $ px + my + n = 0 $ is a tangent drawn to the curve $ y = f(x) $ at $ x = \alpha $, then at $ x = 0 $, the value of \[ \frac{d}{dx} \left( f(\alpha e^{2x}) \right) \]