List of practice Questions

The ratio of the time periods of a simple pendulum at heights $2R_E$ and $3R_E$ from the surface of the earth is ($R_E$ is radius of the earth)

Two wires A and B made of same material and areas of cross-section in the ratio $1:2$ are stretched by same force. If the masses of the wires A and B are in the ratio $2:3$, then the ratio of the elongations of the wires A and B is

Water is filled in a tank up to a height of 20 cm from the bottom of the tank. Water flows through a hole of area $1\text{ mm}^2$ at its bottom. The mass of the water coming out from the hole in a time of 0.6 s is
(Density of water $= 1000\text{ kg m}^{-3}$ and acceleration due to gravity $= 10\text{ ms}^{-2}$)

For which of the following Reynold's number, a flow is streamlined?

If the system of blocks shown in the figure is released from rest, the ratio of the tensions $T_1$ and $T_2$ is (Neglect the mass of the string shown in the figure)

If the component of the vector $\vec{A}$ along the vector $\vec{B}$ is twice the component of $\vec{B}$ along $\vec{A}$, then the ratio of magnitudes of vectors $\vec{A}$ and $\vec{B}$ is

A body projected vertically up with an initial speed of $10\text{ ms}^{-1}$ reaches the point of projection after sometime with a speed of $8\text{ ms}^{-1}$. The maximum height reached by the body is (Acceleration due to gravity $= 10\text{ ms}^{-2}$)

Due to global warming, if the ice in the polar region melts and some of this water flows to the equatorial region, then

If the moment of inertia of a thin circular ring about an axis passing through its edge and perpendicular to its plane is I, then the moment of inertia of the ring about its diameter is

A particle is executing simple harmonic motion. If the force acting on the particle at a position is 86.6% of the maximum force on it, then the ratio of its velocity at that point and its maximum velocity is

The substitution required to reduce the differential equation \(t^2 dx + (x^2 - tx + t^2) dt = 0\) to a differential equation which can be solved by variables separable method is

The equation which represents the system of parabolas whose axis is parallel to y-axis satisfies the differential equation

Bose-Einstein statistics is applicable to particles with

If L and C are inductance and capacitance respectively, then the dimensional formula of $(LC)^{-\frac{1}{2}}$ is

Two bodies are projected from the same point with the same initial velocity 'u' making angles '$\theta$' and ($90^\circ-\theta$) with the horizontal in opposite directions. The horizontal distance between their positions when the bodies are at their maximum heights is

If \(I_n = \int \frac{1}{(x^2+1)^n} dx\), then \(2n I_{n+1} - (2n-1) I_n =\)

\(\int \frac{x^3}{x^4 + 3x^2 + 2} dx =\)

If \(\int \frac{dx}{(x^2+9)\sqrt{x^2+16}} = \frac{1}{3\sqrt{7}} \tan^{-1} \left( K \frac{x}{\sqrt{16+x^2}} \right) + c\), then \(K =\)

\(\lim_{n \to \infty} \frac{1}{n^2} \left[ e^{1/n} + 2e^{2/n} + 3e^{3/n} + \dots + 2n e^{2n/n} \right] =\)

Let m, n, p, q be four positive integers. If \(\int_0^{2\pi} \sin^m x \cos^n x dx = 4 \int_0^{\pi/2} \sin^m x \cos^n x dx\), \(\int_0^{2\pi} \sin^p x \cos^q x dx = 0\), \(a = m+n+p\) and \(b = m+n+q\), then

The area of the region bounded by the curves \(y=x^3\), \(y=x^2\) and the lines \(x=0\) and \(x=2\) is

If \(y = (\sin^{-1}x)^2\), then \((1-x^2)\frac{d^2y}{dx^2} - x\frac{dy}{dx} =\)

The radius of a cone of height 9 units is changed from 2 units to 2.12 units. The exact change and approximate change in the volume of the cone are respectively

The local maximum value \(l\) and local minimum value \(m\) of \(f(x) = \frac{x^2+2x+2}{x+1}\) in \(\mathbb{R} - \{-1\}\) exist at \(\alpha, \beta\) respectively, then \(\frac{l+m}{\alpha+\beta} =\)

\(P(5,2)\) is a point on the curve \(y=f(x)\) and \(\frac{7}{2}\) is the slope of the tangent to the curve at P. The area of the triangle formed by the tangent and the normal to the curve at P with x-axis is