The three rods shown in figure have identical dimensions. Heat flows from the hot end at a rate of $40 \,W$ in the arrangement $(a)$. Find the rates of heat flow when the rods are joined as in arrangement $(b)$ is ......... $W$ (Assume $K_al=200 \,W / m ^{\circ} C$ and $\left.K_{c u}=400 \,W / m ^{\circ} C \right)$

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The end $A$ of a rod $AB$ of length $1\,m$ is maintained at $80\,^oC$ and the end $B$ at $0\,^oC.$ The temperature at a distance of  $60\,\,c.m.$  from the end $A$  is......... $^oC$

Three rods made of the same material and having same cross-sectional area but different lengths $10\, cm, 20\, cm$ and $30\, cm$ are joined as shown. The temperature of the junction is......... $^oC$ 

The ends $\mathrm{Q}$ and $\mathrm{R}$ of two thin wires, $\mathrm{PQ}$ and $RS$, are soldered (joined) togetker. Initially each of the wires has a length of $1 \mathrm{~m}$ at $10^{\circ} \mathrm{C}$. Now the end $\mathrm{P}$ is maintained at $10^{\circ} \mathrm{C}$, while the end $\mathrm{S}$ is heated and maintained at $400^{\circ} \mathrm{C}$. The system is thermally insulated from its surroundings. If the thermal conductivity of wire $\mathrm{PQ}$ is twice that of the wire $RS$ and the coefficient of linear thermal expansion of $P Q$ is $1.2 \times 10^{-5} \mathrm{~K}^{-1}$, the change in length of the wire $\mathrm{PQ}$ is

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