Three rods of same material, same area of crosssection but different lengths $10 \,cm , 20 \,cm$ and $30 \,cm$ are connected at a point as shown. What is temperature of junction $O$ is ......... $^{\circ} C$

Two metallic blocks $M_{1}$ and $M_{2}$ of same area of cross-section are connected to each other (as shown in figure). If the thermal conductivity of $M _{2}$ is $K$ then the thermal conductivity of $M _{1}$ will be ]...............$K$ [Assume steady state heat conduction]

A partition wall has two layers $A$ and $B$ in contact, each made of a different material. They have the same thickness but the thermal conductivity of layer $A$ is twice that of layer $B$. If the steady state temperature difference across the wall is $60K$, then the corresponding difference across the layer $A$ is ....... $K$

Three rods $A, B$ and $C$ of thermal conductivities $K, 2\,K$ and $4\,K$, cross-sectional  areas $A, 2\,A$ and $2\,A$ and lengths $2l, l$ and $l$ respectively are connected as shown  in the figure. If the ends of the rods are maintained at temperatures $100^o\,C, 50^o\,C$, and $0^o\,C$ respectively, then the temperature $\theta$ of the junction is ......... $^oC$

Two different rods $A$ and $B$ are kept as shown in figure. The ratio of thermal conductivities of $A$ and $B$ is

$A$ cylinder of radius $R$ made of a material of thermal conductivity ${K_1}$ is surrounded by a cylindrical shell of inner radius $R$ and outer radius $2R$ made of material of thermal conductivity ${K_2}$. The two ends of the combined system are maintained at two different temperatures. There is no loss of heat across the cylindrical surface and the system is in steady state. The effective thermal conductivity of the system is