The ratio of the diameters of two metallic rods of the same material is $2 : 1$ and their lengths are in the ratio $1 : 4$. If the temperature difference between their ends is equal,the rate of flow of heat in them will be in the ratio: (in $:1$)

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

Two identical containers are filled with the same amount of ice. The containers are made of different metals. If the ice in both containers melts in $20$ minutes and $35$ minutes respectively,find the ratio of their thermal conductivities.

Two metal cubes $A$ and $B$ of the same size are arranged as shown in the figure. The ends of the combination are maintained at the indicated temperatures. The arrangement is thermally insulated. If the thermal conductivities of $A$ and $B$ are $300 \, W/m \, ^\circ C$ and $200 \, W/m \, ^\circ C$ respectively,find the temperature $t$ at the junction when the steady state is reached.

Four rods of the same material with different radii $r$ and length $l$ are used to connect two reservoirs of heat at different temperatures. Which one will conduct the most heat?

In a steady state, the graph between temperature $(\theta)$ and distance $(x)$ from the hot end is: