The position-time graphs for two students \(A\) and \(B\) returning from the school to their homes are shown in figure \((A)\) \(A\) lives closer to the school \((B)\) \(B\) lives closer to the school \((C)\) \(A\) takes lesser time to reach home \((D)\) \(A\) travels faster than \(B\) \((E)\) \(B\) travels faster than \(A\) Choose the correct answer from the options given below :

The energy density associated with electric field \(\overrightarrow{ E }\) and magnetic field \(B\) of an electromagnetic wave in free space is given by ( \(\epsilon_0-\) permittivity of free space, \(\mu_0\) - permeability of free space)

The plot given below is of the average power delivered to an \(LRC\) circuit versus frequency. The quality factor of the circuit is

A charge \(Q\) is moving \(\overrightarrow{dl}\) distance in the magnetic field \(\overrightarrow{ B }\). Find the value of work done by \(\overrightarrow{ B }\).

The driver sitting inside a parked car is watching vehicles approaching from behind with the help of his side view mirror, which is a convex mirror with radius of curvature \(\mathrm{R}=2 \mathrm{~m}\). Another car approaches him from behind with a uniform speed of \(90 \mathrm{~km} / \mathrm{hr}\). When the car is at a distance of 24 m from him, the magnitude of the acceleration of the image of the car in the side view mirror is ' a '. The value of 100 a is _______ \(\mathrm{m} / \mathrm{s}^2\).

In a compound microscope, the magnified virtual image is formed at a distance of \(25\, cm\) from the eye-piece. The focal length of its objective lens is \(1\, cm .\) If the magnification is \(100\) and the tube length of the microscope is \(20\, cm ,\) then the focal length of the eye-piece lens (in \(cm\) ) is

The equation of a particle executing simple harmonic motion is given by \(x =\sin \pi\left( t +\frac{1}{3}\right) m\). At \(t =1 \,s\), the speed of particle will be .......... \(cm s ^{-1}\) (Given : \(\pi=3.14\) )

From a solid sphere of mass \(M\) and radius \(R\), a spherical portion of radius \(R/2\) is removed, as shown in the figure. Taking gravitational potential \(V = 0\) at \(r = \infty\), the potential at the centre of the cavity thus formed is 
( \(G =\) gravitational constant)

The charge flowing in a conductor changes with time as \(Q ( t )=\alpha t -\beta t ^2+\gamma t ^3\). Where \(\alpha, \beta\) and \(\gamma\) are constants. Minimum value of current is :

Given below are two statements. One is labelled as Assertion (A) and the other is labelled as Reason (R).
Assertion (A) :


Three identical spheres of same mass undergo one dimensional motion as shown in figure with initial velocities \(v_{\mathrm{A}}=5 \mathrm{~m} / \mathrm{s}, v_{\mathrm{B}}=2 \mathrm{~m} / \mathrm{s}, v_{\mathrm{C}}=4 \mathrm{~m} / \mathrm{s}\). If we wait sufficiently long for elastic collision to happen, then \(v_{\mathrm{A}}=4 \mathrm{~m} / \mathrm{s}, v_{\mathrm{B}}=2 \mathrm{~m} / \mathrm{s}\), \(v_{\mathrm{C}}=5 \mathrm{~m} / \mathrm{s}\) will be the final velocities.

Reason (R): In an elastic collision between identical masses, two objects exchange their velocities.
In the light of the above statements, choose the correct answer from the options given below :

One main scale division of a vernier callipers is \(a\) \(cm\) and \(n ^{\text {th }}\) division of the vernier scale coincide with \(( n -1)^{\text {th }}\) division of the main scale. The least count of the callipers in \(mm\) is

Let \(y = y(x)\) be the solution curve of the differential equation \((1 + \sin x)\dfrac{dy}{dx} + (y+1)\cos x = 0\), \(y(0) = 0\). If the curve \(y = y(x)\) passes through the point \(\left(\alpha, \dfrac{-1}{2}\right)\), then a value of \(\alpha\) is :

If the data \(x_1, x_2, ...., x_{10}\) is such that the mean of first four of these is \(11\), the mean of the remaining six is \(16\) and the sum of squares of all of these is \(2,000\); then the standard deviation of this data is