The figure shows three different arrangements of materials $1, 2$,and $3$ to form a wall. The thermal conductivities are $k_1 > k_2 > k_3$. The left side of the wall is $20\,^{\circ}\text{C}$ higher than the right side. The temperature difference $\Delta T$ across material $1$ has the following relation in the three cases:

$A$,$B$ and $C$ are three identical conductors but made from different materials. They are kept in contact as shown. Their thermal conductivities are $K$,$2K$ and $K/2$. The free end of $A$ is at $100^{\circ} C$ and the free end of $C$ is at $0^{\circ} C$. During steady state,the temperature of the junction of $A$ and $B$ is nearly (in $^{\circ} C$)

For the figure shown,when arcs $ACD$ and $ADB$ are made of the same material,the total heat current carried between $A$ and $B$ is $H$. If $ADB$ is replaced with another material,the total heat current becomes $2H$. If the temperatures at $A$ and $B$ are fixed at $T_1$ and $T_2$,what is the ratio of the new thermal conductivity to the old thermal conductivity of the arc $ADB$?

Two rectangular blocks,having identical dimensions,can be arranged either in configuration $I$ or in configuration $II$ as shown in the figure. One of the blocks has thermal conductivity $k$ and the other $2k$. The temperature difference between the ends along the $x$-axis is the same in both the configurations. It takes $9 \ s$ to transport a certain amount of heat from the hot end to the cold end in the configuration $I$. The time to transport the same amount of heat in the configuration $II$ is: (in $s$)

$A$ rod $CD$ of thermal resistance $10.0 \; K/W$ is joined at the middle of an identical rod $AB$ as shown in the figure. The ends $A$,$B$,and $D$ are maintained at $200^{\circ}C$,$100^{\circ}C$,and $125^{\circ}C$ respectively. The heat current in $CD$ is $P$ watt. The value of $P$ is ... .

$A$ slab consists of two parallel layers of two different materials of same thickness having thermal conductivities $K_1$ and $K_2$. The equivalent conductivity of the combination is