Two thin metallic spherical shells of radii $r_1$ and $r_2\left(r_1<r_2\right)$ are placed with their centres coinciding. A material of thermal conductivity $K$ is filled in the space between the shells. The inner shell is maintained at temperature ${\theta }_1$ and the outer shell at temperature ${\theta }_2\left({\theta }_1<{\theta }_2\right).$ The rate at which heat flows radially through the material is :

Due to cold weather, a $1 \mathrm{m}$ water pipe of cross-sectional area $1{ \mathrm{cm}}^2$ is filled with ice at $-10 °\mathrm{C}.$ Resistive heating is used to melt the ice. Current of $0.5 \mathrm{A}$ is passed through $4 \mathrm{k\Omega }$ resistance. Assuming that all the heat produced is used for melting, what is the minimum time required?

(Given latent heat of fusion for water/ice$=3.33\times {10}^5 \mathrm{J}{ \mathrm{kg}}^{-1},$ specific heat of ice $=2\times {10}^3 \mathrm{J}{ \mathrm{kg}}^{-1} °{\mathrm{C}}^{-1}$ and density of ice $={10}^3 \mathrm{kg} {\mathrm{m}}^{-3} )$

The area of cross-section of a railway track is $0.01{ \mathrm{m}}^2.$ The temperature variation is $10 °\mathrm{C}.$ Coefficient of linear expansion of material of track is ${10}^{-5} {}^{\circ }\mathrm{C}^{-1}.$ The energy stored per meter in the track is $\mathrm{J} {\mathrm{m}}^{-1}.$ (Young's modulus of material of track is ${10}^{11} \mathrm{N} {\mathrm{m}}^{-2}$)

In $5 \mathrm{minutes}$, a body cools from $75 °\mathrm{C}$ to $65 °\mathrm{C}$ at room temperature of $25 °\mathrm{C}.$ The temperature of body at the end of next $5 \mathrm{minutes}$ is $\underline{ }$ ${}^{\circ }\mathrm{C}.$

A bimetallic strip consists of metals $A$ and $B$. It is mounted rigidly as shown. The metal $A$ has higher coefficient of expansion compared to that of metal $B$. When the bimetallic strip is placed in a cold both, it will :