A ball of mass \(m\) hits the floor with a speed \(v\) making an angle of incidence \(e\) with the normal. The coefficient of restitution is \(e.\) The speed of the reflected ball and the angle of reflection of the ball will be:

1. \(v'=v,~\theta=\theta'\)
2. \(v'=\frac v2,~\theta=2\theta'\)
3. \(v'=2v,~\theta=2\theta'\)
4. \(v'=\frac{3v}{2},~\theta=\frac{2\theta'}3\)

A particle slides on the surface of a fixed smooth sphere starting from the  topmost point. The angle rotated by the radius through the particle, when it leaves contact with the sphere, is:
1. \(\theta=cos^{-1}\Big(\frac13\Big)\)
2. \(\theta=cos^{-1}\Big(\frac23\Big)\)
3. \(\theta=tan^{-1}\Big(\frac13\Big)\)
4. \(\theta=sin^{-1}\Big(\frac43\Big)\)

What is the radius of curvature of the parabola traced out by the projectile in the previous problem at a point where the particle velocity makes an angle \(\frac{\theta}{2}\) with the horizontal?
1. \(r=\frac{v^2\cos^2\theta}{g\cos^2\theta}\)
2. \(r=\frac{2v\sin\theta}{g\tan\theta}\)
3. \(r=\frac{v\cos\theta}{g\sin^2\frac{\theta}{2}}\)
4. \(r=\frac{3v\cos\theta}{g\cot\theta}\)

A square loop is made by a uniform conductor wire as shown in the figure,     
The net magnetic field at the centre of the loop if the side length of the square is a:
1. \(\frac{\mu_{_0}i}{2a}\)
2. zero
3. \(\frac{\mu_{_0}i^2}{a^2}\)
4. None of these

The electron of an H-atom is revolving around the nucleus in circular orbit having radius \(\frac{h^2}{4\pi me^2}\) with \(\Big(\frac{2\pi e^2}{h}\Big).\) The current produced due to the motion of the electron is:
1. \(\frac{2\pi m^2e^2}{3h^2}\)
2. zero
3. \(\frac{2\pi^2me}{h^2}\)
4. \(\frac{4\pi^2me^5}{h^3}\)

Two small balls, each carrying a charge \(q\) are suspended by equal insulator strings of length 1 m from the hook of a stand. This arrangement is carried in a satellite in space. The tension in each string will be:
1. \(\frac{1}{4\pi \varepsilon_0}\frac{q}{I^2}\)
2. \(\frac{1}{4\pi\varepsilon_0}\frac{q^2}{4I^2}\)
3. \(\frac{1}{4\pi\varepsilon_0}\frac{q^2}{I^2}\)
4. \(\frac{1}{(4\pi~\varepsilon_0)}\frac{q}{I}\)

A vessel of depth \(t\) is half filled with a liquid having a refractive index \(n_1\) and the other half is filled with water having a refractive index \(n_2.\) The apparent depth of the vessel as viewed from the top is:
1. \(\frac{2t(n_1+n_2)}{n_1n_2}\)
2. \(\frac{tn_1n_2}{(n_1+n_2)}\)
3. \(\frac{t(n_1+n_2)}{2n_1n_2}\)
4. \(\frac{n_1n_2}{(n_1+n_2)t}\)