A solid cylinder of mass 'M' is suspended by two strings wrapped around it as shown. The acceleration 'a' and the tension T when the cylinder falls and the string unwinds itself are, respectively,
             
1. \(a=g,~T=\dfrac{Mg}{2}\)
2. \(a=\dfrac{g}{2},~T=\dfrac{Mg}{2}\)
3. \(a=\dfrac{g}{3},~T=\dfrac{Mg}{3}\)
4. \(a=\dfrac{2g}{3},~T=\dfrac{Mg}{6}\)

A sphere can not roll on:

1.  a smooth horizontal surface 

2.  a rough horizontal surface 

3.  a smooth inclined surface 

4.  a rough inclined surface 

A solid cylinder and a solid sphere, both having the same mass and radius, are released from a rough inclined plane. Both roll without slipping. Then,

1.  the force of friction that acts on the two is the same

2.  the force of friction is greater in the case of a sphere than in a cylinder

3.  the force of friction is greater in the case of a cylinder than in a sphere

4.  the force of friction will depend on the nature of the surface of the body that is moving and that of the inclined surface and is independent of the shape and size of the moving body

The speed of a uniform spherical shell after rolling down an inclined plane of vertical height h from rest is:

1.  \(\sqrt{\frac{10   g h}{7}}\)

2.    \(\sqrt{\frac{6   g h}{5}}\)

3.  \(\sqrt{\frac{4   g h}{5}}\)

4.  \(\sqrt{2   g h}\)

In a radioactive substance at t = 0, the number of atoms is 8$\times {10}^4$, its half-life period is 3 yr. The number of atoms equal to $1\times {10}^4$ will remain after an interval of:

1. 9 yr

2. 8 yr 

3. 6 yr 

4. 24 yr

In a radioactive sample the fraction of initial number of radioactive nuclei, which remains undecayed after n mean lives is:

1. $\frac{1}{e^n}$ 

2.  $e^n$   

3. $1-\frac{1}{e^n}$ 

4. ${\left(\frac{1}{e-1}\right)}^n$

At time t = 0, N₁ nuclei of decay constant λ₁ and N₂ nuclei of decay constant λ₂ are mixed. The decay rate of the mixture is:

1. $-N_1{N}_2{e}^{-\left({\lambda }_1+{\lambda }_2\right)t}$ $$

2. $-\left(\frac{N_1}{N_2}\right)$ $e^{{}^{-\left({\lambda }_1+{\lambda }_2\right)t}}$ $$

3. $-\left(N_1{\lambda }_1{e}^{-{\lambda }_{1}t}\right)$ $+$ $N_2{\lambda }_2{e}^{-{\lambda }_{2}t}$ $$

4. $-N_1{\lambda }_1{N}_2{\lambda }_2{e}^{{}^{-\left({\lambda }_1+{\lambda }_2\right)t}}$

The half-life period of a radioactive substance is 6 h.  If after 24 h, activity is 0.01 $\mu$Ci, what was the initial activity? 

1. 0.04 $\mu$Ci 

2. 0.08 $\mu$Ci

3. 0.24 $\mu$Ci 

4. 0.16 $\mu$Ci

The rate of disintegration of a fixed quantity of a radioactive substance can be increased by:

1. increasing the temperature.

2. increasing the pressure.

3. chemical reaction.

4. it is not possible.

The counting rate observed from a radioactive source at t = 0 second was 1600 counts per second and at t = 8 seconds it was 100 counts per second. The counting rate observed, as counts per second, at t = 6 seconds will be:

1. 400

2. 300

3. 200

4. 150