When is thermal conductivity said to be constant?

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?

Two rods with thermal conductivities $K$ and $3K$ and equal cross-sectional areas are joined as shown in the figure. Their lengths are $1 \ cm$ and $2 \ cm$,respectively. If the temperatures of the two ends of this composite rod are $0^{\circ}C$ and $100^{\circ}C$ (see figure),then the temperature $\phi$ of their interface is:

Two spheres of different materials,one with double the radius and one-fourth wall thickness of the other,are filled with ice. If the time taken for complete melting of ice in the large radius sphere is $25 \ minutes$ and that for the smaller one is $16 \ minutes$,the ratio of thermal conductivities of the materials of the larger sphere to the smaller sphere is:

What is the temperature (in $^oC$) of the steel-copper junction in the steady state of the system shown in the figure? Length of the steel rod $= 15.0 \; cm,$ length of the copper rod $= 10.0 \; cm,$ temperature of the furnace $= 300^{\circ} C,$ temperature of the other end $= 0^{\circ} C.$ The area of cross-section of the steel rod is twice that of the copper rod. (Thermal conductivity of steel $= 50.2 \; J s^{-1} m^{-1} K^{-1};$ and of copper $= 385 \; J s^{-1} m^{-1} K^{-1}$)

Two materials having coefficients of thermal conductivity $3K$ and $K$ and thickness $d$ and $3d$,respectively,are joined to form a slab as shown in the figure. The temperatures of the outer surfaces are $\theta_2$ and $\theta_1$ respectively $(\theta_2 > \theta_1)$. The temperature at the interface is