In the following figure,two insulating sheets with thermal resistances $R$ and $3R$ are shown. The temperature $\theta$ at the interface is ...... $^oC$.

$A$ hollow sphere of inner radius $R$ and outer radius $2R$ is made of a material of thermal conductivity $K$. It is surrounded by another hollow sphere of inner radius $2R$ and outer radius $3R$ made of the same material of thermal conductivity $K$. The inside of the smaller sphere is maintained at $0^oC$ and the outside of the bigger sphere at $100^oC$. The system is in steady state. The temperature of the interface will be ........ $^oC$.

For the figure shown,when arcs $ACD$ and $ADB$ are made of the same material,the total heat current carried between $A$ and $B$ is $H$. If $ADB$ is replaced with another material,the total heat current becomes $2H$. If the temperatures at $A$ and $B$ are fixed at $T_1$ and $T_2$,what is the ratio of the new thermal conductivity to the old thermal conductivity of the arc $ADB$?

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 ........ $^{\circ}C$.

Two identical conducting rods are first connected independently in parallel to two vessels,one containing water at $100^oC$ and the other containing ice at $0^oC$. In the second case,the rods are joined end to end (in series) and connected to the same vessels. Let $q_1$ and $q_2$ $g/s$ be the rate of melting of ice in the two cases respectively. The ratio of $q_1/q_2$ is

Two conducting rods $A$ and $B$ of same length and cross-sectional area are connected $(i)$ In series $(ii)$ In parallel as shown. In both combinations,a temperature difference of $100^{\circ}C$ is maintained. If the thermal conductivity of $A$ is $3K$ and that of $B$ is $K$,then the ratio of heat current flowing in the parallel combination to that flowing in the series combination is: