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Three identical rods $A, B$ and $C$ are placed end to end. A temperature difference is maintained between the free ends of $A$ and $C$. The thermal conductivity of $B$ is thrice that of $C$ and half of that of $A$. The effective thermal conductivity of the system will be $\left(K_{A}\right.$ is the thermal conductivity of $\left.\operatorname{rod} A\right)$
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The correct answer is:
$\frac{1}{3} K_{A}$
$\begin{array}{ll}\text { Given, } & K_{B}=K_{A} / 2, \\ \text { and } & K_{B}=3 K_{C} \\ \therefore & K_{C}=K_{A} / 6\end{array}$
Rods are in series form so
$$
\begin{gathered}
\frac{L}{K}=\frac{l_{1}}{K_{A}}+\frac{l_{2}}{K_{B}}+\frac{l_{3}}{K_{C}} \\
\frac{3 l}{K}=\frac{l}{K_{A}}+\frac{l}{K_{A} / 2}+\frac{l}{K_{A} / 6} \\
\frac{3 l}{K}=\frac{9 l}{K_{A}} \\
K=\frac{K_{A}}{3}
\end{gathered}
$$
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