Three rods of the same material and same cross-sectional area,but different lengths $10 \, cm$,$20 \, cm$,and $30 \, cm$,are connected at a junction $O$ as shown in the figure. What is the temperature of the junction $O$ in $^{\circ} C$?

Six identical conducting rods are joined as shown in the figure. Points $A$ and $D$ are maintained at $100^{\circ} C$ and $10^{\circ} C$ respectively. The temperature of junction $C$ will be $.... ^{\circ} C$.

Two rods $A$ and $B$ of different materials but same cross-section are joined as shown in the figure. The free end of $A$ is maintained at $100^{\circ}C$ and the free end of $B$ is maintained at $0^{\circ}C$. If $l_2 = 2l_1$,$K_1 = 2K_2$ and the rods are thermally insulated from the sides to prevent heat losses,then the temperature $\theta$ of the junction of the two rods is ........ $^{\circ}C$.

$A$ wall has two layers $A$ and $B$,each made of different materials. Both layers are of the same thickness. The thermal conductivity of material $A$ is twice that of $B$. If in the steady state,the temperature difference across the wall is $24^{\circ} C$,then the temperature difference across the layer $B$ is (in $^{\circ} C$)

Two rectangular blocks,having identical dimensions,can be arranged either in configuration $I$ or in configuration $II$ as shown in the figure. One of the blocks has thermal conductivity $k$ and the other $2k$. The temperature difference between the ends along the $x$-axis is the same in both the configurations. It takes $9 \ s$ to transport a certain amount of heat from the hot end to the cold end in the configuration $I$. The time to transport the same amount of heat in the configuration $II$ is: (in $s$)

Two rods of same length, radius and material transfer a given amount of heat in '$t$' seconds when they are joined as shown in fig $(1)$. But when they are joined as shown in fig $(2)$, then they will transfer the same amount of heat under the same conditions in time: (in $t$)