A launching vehicle carrying an artificial satellite of mass \(m\) is set for launch on the surface of the earth of mass \(M\) and radius \(R.\) If the satellite is intended to move in a circular orbit of radius \(7R,\) the minimum energy required to be spent by the launching vehicle on the satellite is:
(Gravitational constant= \(G\))
1. \(\frac{GMm}{R}\)
2. \(-\frac{13GMm}{14R}\)
3. \(\frac{GMm}{7R}\)
4. \(-\frac{GMm}{14R}\)

 

The two lenses of an achromatic doublet should have:

If the ratio of lengths, radii and Young's modulus of steel and brass wires shown in the figure are \(a,\) \(b\) and \(c\) respectively, the ratio between the increase in lengths of brass and steel wires would be:

1. \( \frac{{b}^{2}a}{2c}\)
2. \( \frac{bc}{2{a}^{3}}\)
3. \( \frac{{ba}^{2}}{2c}\)
4. \( \frac{a}{2{b}^{2}c}\)

A soap bubble of radius \(r\) is blown up to form a bubble of radius \(2r\) under isothermal conditions. If \(T\) is the surface tension of the soap solution, the energy spent in the blowing is:
1. \(3\pi Tr^2\)
2. \(6\pi Tr^2\)
3. \(12\pi Tr^2\)
4. \(24\pi Tr^2\)

A current of \(1.6\) A is passed through a solution of CuSO₄. How many Cu⁺⁺ ions are liberated in one minute? (Electronic charge= \(1.6\times10^{-19}\) C)
1. 3 × 10²⁰
2. 3 × 10¹⁰
3. 6 × 10²⁰
4. 6 × 10¹⁰

In an \(LCR\) circuit inductance is changed from \(L\) to \(L/2.\) To keep the same resonance frequency, \(C\) should be changed to:
1. \(2C\)
2. \(\frac C2\)
3. \(4C\)
4. \(\frac C4\)