A person of mass \(M\) is, sitting on a swing of length \(L\) and swinging with an angular amplitude \(\theta_0\). If the person stands up when the swing passes through its lowest point, the work done by him, assuming that his centre of mass moves by a distance \(l\, ( l < < L)\), is close to

In the below circuit, \(C\, = \,\frac{{\sqrt 3 }}{2}\,\mu F,\,R = 20\,\,\Omega ,\,L = \frac{{\sqrt 3 }}{{10}}H\) and \({R_1} = 10\,\,\Omega .\) Current in \(L-R_1\) path is \(I_1\) and \(C-R_2\) path is \(I_2.\) The voltage of \(A.C\) source is given by, \(V\, = \,200\sqrt 2 \,\sin \,(100\,t)\,volts.\) The phase difference between \(I_1\) and \(I_2\) is

When a resistance of \(5\,\Omega\) is shunted with a moving coil galvanometer, it shows a full scale deflection for a current of \(250\,mA\), however when \(1050\,\Omega\) resistance is connected with it in series, it gives full scale deflection for \(25\) volt. The resistance of galvanometer is \(......\,\Omega\).

Match Column\(-I\) with Column\(-II :\) Choose the correct answer from the options given below:

A modulating signal is a square wave, as shown in the figure.

If the carrier wave is given as \(c(t)=2 \sin (8 \pi t)\) volts, the modulation index is:

Which of the following are correct expression for torque acting on a body?
A. \(\vec{\tau}=\vec{r}\times\vec{L}\)
B. \(\vec{\tau}=\frac{\mathrm{d}}{\mathrm{dt}}(\overrightarrow{\mathrm{r}} \times \overrightarrow{\mathrm{p}})\)
C. \(\vec{\tau}=\overrightarrow{\mathrm{r}} \times \frac{\mathrm{d} \mathrm{p}}{\mathrm{dt}}\)
D. \(\vec{\tau}=\mathrm{I} \vec{\alpha}\)
E. \(\vec{\tau}=\overrightarrow{\mathrm{r}} \times \overrightarrow{\mathrm{F}}\)
( \(\overrightarrow{\mathrm{r}}=\) position vector; \(\overrightarrow{\mathrm{p}}=\) linear momentum;
\(\overrightarrow{\mathrm{L}}=\) angular momentum; \(\vec{\alpha}=\) angular acceleration;
\(I=\) moment of inertia; \(\vec{F}=\) force; \(t=\) time)
Choose the correct answer from the options given below :

A student measured the length of a rod and wrote it as \(3.50\;cm\). Which instrument did he use to measure it?

The number of numbers between \(2,000\)  and \(5,000\)  that can be formed with the digits \(0, 1, 2, 3, 4,\) (repetition of digits is not allowed) an are multiple of \(3\) is?

Let \(z\) be a complex number such that \(\left|\frac{z-2 i}{z+i}\right|=2, z \neq-i\). Then \(z\) lies on the circle of radius \(2\) and centre

A body of mass \(0.5\, kg\) travels on straight line path with velocity \(v =\left(3 x ^{2}+4\right) m / s\). The net workdone by the force during its displacement from \(x =0\) to \(x =2\, m\) is\(.......J\)

Let \(y=y(x)\) be the solution of the differential equation \(x d y=\left(y+x^{3} \cos x\right) d x\) with \(y(\pi)=0\) then \(y\left(\frac{\pi}{2}\right)\) is equal to :

The absolute difference of the coefficients of \(x^{10}\) and \(x^7\) in the expansion of \(\left(2 x^2+\frac{1}{2 x}\right)^{11}\) is equal to

Three concentric metal shells \(A, B\) and \(C\) of respective radii \(a, b\) and \(c (a < b < c)\) have surface charge densities \(+\sigma,-\sigma\) and \(+\sigma\) respectively. The potential of shell \(B\) is