In the following circuit, the switch \(S\)  is closed at \(t = 0.\)  The charge on the capacitor  \(C_1\)  as a function of time will be given by \(\left( {{C_{eq}}\, = {\kern 1pt} \,\frac{{{C_1}{C_2}}}{{{C_1} + {C_2}}}} \right).\)

A bullet of \(10\, {g}\), moving with velocity \(v\), collides head-on with the stationary bob of a pendulum and recoils with velocity \(100 \, {m} / {s}\). The length of the pendulum is \(0.5\, {m}\) and mass of the bob is \(1\, {kg}\). The minimum value of \(v=\) \(....{m} / {s}\) so that the pendulum describes a circle. (Assume the string to be inextensible and \({g}=10\, {m} / {s}^{2}\) )

In an \(LCR \) circuit as shown below both switches are open initially. Now switch \(S_1\) is closed, \(S_2\) kept open. ( \(q\) is charge on the capacitor and \(\tau = RC\) is Capacitive time constant). Which of the following statement is correct?

For the given circuit the current \(i\) through the battery when the key in closed and the steady state has been reached is .....\(A\)

Shown in the figure is a hollow icecream cone (it is open at the top). If its mass is \(M,\) radius of its top, \(R\) and height, \(H\), then its moment of inertia about its axis is

Given below are two statements : one is labelled as Assertion \(A\) and the other is labelled as Reason \(R\). Assertion \(A\) : When a body is projected at an angle \(45^{\circ}\), it's range is maximum. Reason \(R\) : For maximum range, the value of \(\sin 2 \theta\) should be equal to one. In the light of the above statements, choose the correct answer from the options given below :

\(\mathrm{A}, \mathrm{B}\) and C are disc, solid sphere and spherical shell respectively with same radii and masses. These masses are placed as shown in figure.

The moment of inertia of the given system about \(P Q\) is \(\frac{X}{15} I\), where \(I\) is the moment of inertia of the disc about its diameter. The value of x is _______.

A prism of refractive index \(n_{1}\) and another prism of refractive index \(n_{2}\) are stuck together (as shown in the figure). \(n_{1}\) and \(n_{2}\) depend on \(\lambda\), the wavelength of light, according to the relation \({n}_{1}=1.2+\frac{10.8 \times 10^{-14}}{\lambda^{2}} \text { and } {n}_{2}=1.45+\frac{1.8 \times 10^{-14}}{\lambda^{2}}\) The wavelength for which rays incident at any angle on the interface \(B C\) pass through without bending at that interface will be \(....\,nm.\)

A particle moves in one dimension from rest under the influence of a force that varies with the distance traveled by the particle as shown in the figure. The kinetic energy of the particle after it has traveled \(3\, m\) is ................ \(\mathrm{J}\)