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 $^{\circ}C$?

$A$ composite rod is made of three rods of equal length and cross-section as shown in the figure. The thermal conductivities of the materials of the rods are $K/2, 5K$ and $K$ respectively. The ends $A$ and $B$ are at constant temperatures. All heat entering the face $A$ goes out of the end $B$,with no loss of heat from the sides of the bar. The effective thermal conductivity of the bar is:

Six wires,each of cross-sectional area $A$ and length $l$,are combined as shown in the figure. The thermal conductivities of copper and iron are $K_1$ and $K_2$ respectively. The equivalent thermal resistance between points $A$ and $C$ is:

The temperature $\theta$ at the junction of two insulating sheets,having thermal resistances $R_{1}$ and $R_{2}$ as well as top and bottom temperatures $\theta_{1}$ and $\theta_{2}$ (as shown in figure) is given by

Six identical conducting rods are joined as shown. The ends $A$ and $D$ are maintained at $200\,^{\circ}C$ and $20\,^{\circ}C$ respectively. No heat is lost to the surroundings. The temperature of the junction $C$ will be ........ $^{\circ}C$.

Three rods $AB$, $BC$, and $BD$ made of the same material and having the same cross-section have been joined as shown in the figure. The ends $A$, $C$, and $D$ are held at temperatures of $20^{\circ} C$, $80^{\circ} C$, and $80^{\circ} C$ respectively. If each rod is of the same length, then the temperature at the junction $B$ of the three rods is: (in $^{\circ} C$)