question_answer

One end of a thermally insulated rod is kept at a temperature\[m{{y}^{2}}+(1-{{m}^{2}})xy-m{{x}^{2}}=0\]and the other at\[xy=0,\]. The rod is composed of two sections of lengths\[-\frac{1}{2}\]and\[-2\]and thermal conductivities\[\pm 1\]and\[F(x)=f(x)+f\left( \frac{1}{x} \right),\] respectively. The temperature at the interface of the two sections is                                                                                                                                       [AIEEE 2007]

A)  \[f(x)=\int_{1}^{x}{\frac{\log \,t}{1+t}}dt\]

B)  \[\frac{1}{2}\]

C)  \[f:R\to R\]

D)  \[f(x)=\]

Correct Answer: C

Solution :

[c] Let temperature at the interface be T.

For part AB,

Rate of heat transmission,

For part BC,

Rate of heat transmission,

Here, A is area of cross-sections.

At equilibrium,