$ABCDE$ is a regular pentagon made of uniform wire. Heat enters at $A$ and leaves at $C$. $T_B$ and $T_D$ are the temperatures of points $B$ and $D$ respectively. Find the temperature $T_C$.

Two rods of the same length and material transfer a given amount of heat in $12 \ s$ when they are joined end to end. But when they are joined lengthwise parallel to each other,they will transfer the same amount of heat under the same conditions in a time of: (in $s$)

$A$ wall consists of two layers $A$ and $B$ of equal thickness. Their thermal resistances are $R_1$ and $R_2$. Find the temperature of the interface.

$A$ composite slab is made of two different materials of equal thickness,having thermal conductivities $K$ and $2K$ respectively. The equivalent thermal conductivity of the slab is ........

Three identical rods have been joined at a junction to make a $Y$-shaped structure. If two free ends are maintained at $90\,^{\circ}C$ and the third end is at $30\,^{\circ}C$,what is the junction temperature $\theta$ in $^{\circ}C$?

Two rods of the same length and material transfer a given amount of heat in $12 \, s$ when they are joined in parallel. But when they are joined in series,then they will transfer the same amount of heat under the same conditions in ........ $s$.