As per the given figure,two plates $A$ and $B$ of thermal conductivity $K$ and $2K$ are joined together to form a compound plate. The thickness of the plates are $4.0 \,cm$ and $2.5 \,cm$ respectively,and the area of cross-section is $120 \,cm^{2}$ for each plate. If the equivalent thermal conductivity of the compound plate is $\left(1+\frac{5}{\alpha}\right) K$,then the value of $\alpha$ will be . . . . . .

$A$ cylinder of thermal conductivity $k_1$ and radius $r$ is surrounded by a cylindrical shell of inner radius $r$,outer radius $2r$,and thermal conductivity $k_2$. Both cylinders have the same length and the temperature difference across their ends is the same. The equivalent thermal conductivity of the system is:

Three rods of same dimensions are arranged as shown in the figure. They have thermal conductivities ${K_1}, {K_2}$ and ${K_3}$. The points $P$ and $Q$ are maintained at different temperatures. For the heat to flow at the same rate along the path $PRQ$ and the path $PQ$,which of the following options is correct?

$A$ slab consists of two parallel layers of copper and brass of equal thickness. The ratio of their thermal conductivities is $1:4$. If the temperature of the free side of the brass is $100^{\circ}C$ and that of the copper is $0^{\circ}C$,find the temperature of the interface in $^{\circ}C$.

$A$ cylinder of radius $R$ is made of a material with thermal conductivity $K_1$. It is surrounded by a cylindrical shell of a material with thermal conductivity $K_2$,having an inner radius $R$ and an outer radius $2R$. The two ends of this composite system are maintained at two different temperatures. There is no heat loss from the surface of the cylinder in the steady state. Find the equivalent thermal conductivity of the system.

$A$ large box is connected to an ice cube at $100^{\circ}C$ by two identical metal rods as shown in the figure. The ice melts at a rate of $Q_1 \, g/s$. When the rods are connected in series between the box and the ice cube,the new melting rate is $Q_2 \, g/s$. Then $Q_2/Q_1$ is: