Three metal rods of the same material and identical in all respects are joined as shown in the figure. The temperatures at the ends of these rods are maintained as indicated. Assuming no heat energy loss occurs through the curved surfaces of the rods,the temperature at the junction $X$ is (in $^{\circ} C$)

$A$ copper rod of length $18 \ cm$ and a steel rod of length $6 \ cm$ are joined together to form a composite rod of uniform cross-section. The temperature of the free end of the copper rod is $100 \ ^\circ C$ and the temperature of the free end of the steel rod is $0 \ ^\circ C$. What is the temperature at the junction in $^\circ C$? (Thermal conductivity of copper is $9$ times that of steel. The rod is in a steady state.)

An insulated container is filled with ice at $0\,^{\circ}\text{C}$,and another container is filled with water that is continuously boiling at $100\,^{\circ}\text{C}$. In a series of experiments,the containers are connected by various thick metal rods that pass through the walls of the container as shown in the figure.
In experiment $I$: a copper rod is used and all ice melts in $20$ minutes.
In experiment $II$: a steel rod of identical dimensions is used and all ice melts in $80$ minutes.
In experiment $III$: both the rods are used in series and all ice melts in $t_{10}$ minutes.
In experiment $IV$: both rods are used in parallel and all ice melts in $t_{20}$ minutes.

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 plates of equal area are placed in contact with each other. Their thicknesses are $2.0 \, cm$ and $5.0 \, cm$. The temperature of the outer surface of the first plate is $-20^{\circ}C$ and the temperature of the outer surface of the second plate is $20^{\circ}C$. If the plates are made of the same material,find the temperature of the contact surface in $^{\circ}C$.

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 . . . . . .