The universal gravitational constant is dimensionally represented as:
1. \(\left[ML^2T^{-1}\right]\)
2. \(\left[M^{-2}L^3T^{-2}\right]\)
3. \(\left[M^{-2}L^2T^{-1}\right]\)
4. \(\left[M^{-1}L^3T^{-2}\right]\)

In India, electricity is supplied for domestic use at \(220~\text{V}.\) It is supplied at \(110~\text{V}\) in the USA. If the resistance of a \(60~\text{W}\) bulb for use in India is \(R\), the resistance of a \(60~\text{W}\) bulb for use in the USA will be:

The magnetic flux through a circuit of resistance \(R\) changes by an amount \(\Delta \phi\) in a time \(\Delta t\). Then the total quantity of electric charge \(Q\) that passes any point in the circuit during the time \(\Delta t\) is represented by:
1. \(Q= \frac{\Delta \phi}{R}\)
2. \(Q= \frac{\Delta \phi}{\Delta t}\)
3. \(Q=R\cdot \frac{\Delta \phi}{\Delta t}\)
4. \(Q=\frac{1}{R}\cdot \frac{\Delta \phi}{\Delta t}\)

A bullet of mass \(2~\text {gm}\) has a charge of \(2~\mu\text{C}.\) Through what potential difference must it be accelerated, starting from rest, to acquire a speed of \(10~\text{m/s}?\)
1. \(50~\text {kV}\)
2. \(5~\text {V}\)
3. \(50~\text {V}\)
4. \(5~\text {kV}\)

The equation of state for 5g of oxygen at a pressure P and temperature T, when occupying a volume V, will be: (where R is the gas constant)
1. PV = 5 RT
2. PV = (5/2) RT
3. PV = (5/16) RT
4. PV = (5/32) RT

If \(\lambda_m\) denotes the wavelength at which the radioactive emission from a black body at a temperature \(T\) K is maximum, then:
1. \(\lambda_m\) is independent of \(T\)

2. \(\lambda_m \propto T\)

3. \(\lambda_m \propto T^{-1}\)

4. \(\lambda_m \propto T^{-4}\)

The ratio of the radii of gyration of a circular disc about a tangential axis in the plane of the disc and of a circular ring of the same radius about a tangential axis in the plane of the ring will be:
1. \(2:1\)
2. $\sqrt{5}$ : $\sqrt{6}$
3. \(2:3\)
4. \(1:\) $\sqrt{2}$

A round disc of the moment inertia ${\mathrm{I}}_2$ about its axis perpendicular to its plane and passing through its centre is placed over another disc of moment of inertia ${\mathrm{I}}_1$ rotating with an angular velocity ω about the same axis. The final angular velocity of the combination of discs is:

1. ω

2. $\frac{{\mathrm{I}}_1\omega }{{\mathrm{I}}_1+{\mathrm{I}}_2}$

3. $\frac{({\mathrm{I}}_1+{\mathrm{I}}_2)\omega }{{\mathrm{I}}_1}$

4. $\frac{{\mathrm{I}}_2\omega }{{\mathrm{I}}_1+{\mathrm{I}}_2}$

A ball of mass 2 kg and another of mass 4 kg are dropped together from a 60 feet tall building. After a fall of 30 feet each towards the earth, their respective kinetic energies will be in the ratio of:

1. 1: 4

2. 1: 2

3. 1: $\sqrt{2}$

4. $\sqrt{2}$ :1

The half-life of radium is about 1600 years. Out of 100 g of radium existing now, 25 g will remain undecayed after:

1. 6400 years

2. 2400 years

3. 3200 years

4. 4800 years