$A$ composite slab is prepared with two different materials $A$ and $B$. The relation between their coefficients of thermal conductivity and thickness is given as $K_A = \frac{K_B}{2}$ and $X_A = 2 X_B$, respectively. If the temperatures of the outer faces of $A$ and $B$ are $75^{\circ} C$ and $50^{\circ} C$ respectively, what will be the temperature of the common surface (in $^{\circ} C$)?

$A$ metal rod $AB$ of length $10x$ has its one end $A$ in ice at $0^{\circ}C$ and the other end $B$ in water at $100^{\circ}C$. If a point $P$ on the rod is maintained at $400^{\circ}C$,then it is found that equal amounts of water and ice evaporate and melt per unit time. The latent heat of evaporation of water is $540 \ cal/g$ and latent heat of melting of ice is $80 \ cal/g$. If the point $P$ is at a distance of $\lambda x$ from the ice end $A$,find the value of $\lambda$. (Neglect any heat loss to the surrounding.)

Three rods of the same dimensions have thermal conductivities $3k, 2k$ and $k$. They are arranged as shown,with their ends at $100\,^{\circ}C, 50\,^{\circ}C$ and $0\,^{\circ}C$. The temperature of their junction is

Rate of flow of heat through a cylindrical rod is $H_1$. The temperatures at the ends of the rod are $T_1$ and $T_2$. If all the dimensions of the rod become double and the temperature difference remains the same, and if the rate of flow of heat becomes $H_2$, then $H_2 =$

$Assertion :$ $A$ brass tumbler feels much colder than a wooden tray on a chilly day.
$Reason :$ The thermal conductivity of brass is more than the thermal conductivity of wood.

$A$ copper pipe of length $10 \,m$ carries steam at temperature $110^{\circ} C$. The outer surface of the pipe is maintained at a temperature $10^{\circ} C$. The inner and outer radii of the pipe are $2 \,cm$ and $4 \,cm$,respectively. The thermal conductivity of copper is $0.38 \,kW / m /^{\circ} C$. In the steady state,the rate at which heat flows radially outward through the pipe is closest to ............. $\,kW$.