A bimetallic strip consists of metals $X$ and $Y$. It is mounted rigidly at the base as shown. The metal $X$ has a higher coefficient of expansion compared to that for metal $Y$. When the bimetallic strip is placed in a cold bath:

A rod of length $10\ meter$ at $0\,^oC$ having expansion coefficient $\alpha  = (2x^2 + 1) \times  10^{-6}\,C^{-1}$ where $x$ is the distance from one end of rod. The length of rod at $10\,^oC$ is

The coefficient of apparent expansion of mercury in a glass vessel is $153 \times 10^{-6} /{ }^{\circ} C$ and in steel vessel is $144 \times 10^{-6} /{ }^{\circ} C$. If $\alpha$ for steel is $12 \times 10^{-6} /{ }^{\circ} C ,$ then that of glass is

Given below are two statement : one is labelled as Assertion $A$ and the other is labelled as Reason $R$.

Assertion $A$ : When a rod lying freely is heated, no thermal stress is developed in it.

Reason $R :$ On heating the length of the rod increases.

In the light of the above statements, choose the correct answer from the options given below

A beaker is filled with water at $4\,^oC$. At one time the temperature is increased by few degrees above $4\,^oC$ and at another time it is decreased by a few degrees below $4\,^oC$. One shall observe that:

Coefficient of real expansion of mercury is $ 0.18 \times 10^{-3}{°C^{-1}}$. If the density of mercury at $0°C$ is $13.6\, gm/cc$. its density at $473K$ is