$A$ slab consists of two parallel layers of copper and brass of equal thickness. The ratio of their thermal conductivities is $1:4$. If the temperature of the free side of the brass is $100^{\circ}C$ and that of the copper is $0^{\circ}C$,find the temperature of the interface in $^{\circ}C$.

$A$ large box is connected to an ice cube at $100^{\circ}C$ by two identical metal rods as shown in the figure. The ice melts at a rate of $Q_1 \, g/s$. When the rods are connected in series between the box and the ice cube,the new melting rate is $Q_2 \, g/s$. Then $Q_2/Q_1$ is:

Two identical metal rods are welded end-to-end as shown in figure $(1)$. It takes $4 \ min$ for $20 \ cal$ of heat to flow through them. If these rods are now welded side-by-side as shown in figure $(2)$,the time taken for the same amount of heat to flow through them is .......... $\min$.

Three conducting rods of the same material and cross-section are shown in the figure. The temperatures at $A$, $D$, and $C$ are maintained at $20 ^oC$, $90 ^oC$, and $0 ^oC$ respectively. If there is no heat flow in $AB$, find the ratio of the lengths of $BD$ and $BC$.

Three identical rods $A, B$ and $C$ are placed end to end. $A$ temperature difference is maintained between the free ends of $A$ and $C$. The thermal conductivity of $B$ is thrice that of $C$ and half of that of $A$. The effective thermal conductivity of the system will be ($K_{A}$ is the thermal conductivity of rod $A$).

$A$ metallic prong consists of $4$ rods made of the same material,with the same cross-sections and the same lengths as shown below. The three forked ends are kept at $100^{\circ} C$ and the handle end is at $0^{\circ} C$. The temperature of the junction is ............. $^{\circ} C$.