List of practice Questions

A block of mass \(5\,\text{kg}\) starts up a \(45^\circ\) incline with an initial kinetic energy of \(100\,\text{J}\). The coefficient of friction is \(0.5\). The distance it travels before coming to rest is:
A lift raises 50 passengers of average weight \(600\,\text{N}\) each to a height of \(100\,\text{m}\) in time \(T\). If average power used is \(15\,\text{kW}\), then the value of \(T\) is:
If the earth were to suddenly shrink to \( \dfrac{1}{64} \) of its present volume, keeping mass constant, then what would be the new duration of the day?
A wheel is rotating with angular speed \( \omega \). Another identical wheel with moment of inertia three times the first is initially at rest. It is suddenly coupled to the first. What fraction of the original kinetic energy is lost?
There is a planet which is 8 times massive and 27 times denser than the earth. If \( g' \) and \( g \) are the accelerations due to gravity on the surfaces of the planet and the earth respectively, then
The position of a \(3\,\text{kg}\) mass moving along the \(x\)-axis is given by \(x = 0.3 \cos (\omega t)\,\text{m}\). If \(K(t)\) denotes the kinetic energy at time \(t\), then the value of \( \dfrac{K\left( \dfrac{\pi}{60} \right)}{K\left( \dfrac{\pi}{30} \right)} \) is:
The differential equation of the family of circles passing through origin and having centers on x-axis is:
The number of significant figures in quantity \( 0.00005041 \; \text{J} \) is:
A biker travels \( \frac{1}{3} \) of the distance \( L \) with speed \( v_1 \) and \( \frac{2}{3} \) of the distance with speed \( v_2 \). Then the average speed is:
Three blocks \( M_1, M_2, M_3 \) (\( M_1>M_2>M_3 \)) lie on a plane. The angle of inclination \( \theta \) is increased. If the coefficient of friction is the same for all, then:
Three blocks M1,M2,M3
A bar of mass \(m\) resting on a smooth horizontal plane starts moving due to a constant force \(F\). In the process of its rectilinear motion the angle \(\theta\) between the direction of this force and the horizontal varies as \(\theta = kx\), where \(k\) is a constant and \(x\) is the distance traversed by the bar from its initial position. The velocity \(v(\theta)\) of the bar as a function of the angle \(\theta\) is:
If \( f(x) = \int \cos^{-1} \left( \frac{x}{\sqrt{a + x}} \right) dx + C \), then \( f'(a) = \)
Evaluate \( \lim_{n \to \infty} \left( \frac{1^2}{n^3 + 1^3} + \frac{2^2}{n^3 + 2^3} + \cdots + \frac{n^2}{n^3 + n^3} \right) \)
If \( \int f(x) dx = F(x) + C \), then \( \frac{d}{dt} \int_{g(t)}^{h(t)} f(x)\,dx = \)
If \( \int_{0}^{\pi/2} \sin^m x \cos^4 x \, dx = \frac{7\pi}{2048} \), then \( m = \)
Evaluate \( \int_{-\pi}^{\pi} \frac{\cos^{2022}x}{1 + (2022)^x} \, dx \).
If \( \frac{a}{a_1} = \frac{b}{b_1} \), then the substitution used to solve the differential equation \(\frac{dy}{dx} = \frac{ax + by + c}{a_1x + b_1y + c_1} \) using separation of variables is:
For the differential equation \( \frac{d^3 y}{dx^3} = 0 \), \( y = ax^2 + bx + c \) is:
The curves \( y = x^2 - 1 \) and \( y = 8x - x^2 - 9 \)
The perimeter of a sector is constant. If its area is to be maximum, the sectorial angle should be:
The set of all \( x \) for which \( \sin x \leq x \) is:
The function \( f(x) = 2x^3 - 9ax^2 + 12a^2x + 1 \ (a>0) \) attains its maximum and minimum at \( p \) and \( q \) respectively and \( p^2 = q \). Then the value of \( a \) is:
If \( f(x) \) is the antiderivative of \( g(x) \) and \[ \int f(x) g(x)(1 + f^2(x)) dx = F(x) \], then \( F(x) = \)
Evaluate the integral \( \int \frac{1 - k \cos^2 x}{\sin^k x \cos^2 x} dx \).
Evaluate \( \int \frac{\cos(\sec^2 x) - 2022}{\cos^{2022} x} dx = f(x) + C \Rightarrow f\left( \frac{\pi}{4} \right) = \)