Cells: EMF, Internal Resistance & Combinations question from JEE Main 2016 (9 Apr, Shift 1)

Answer: (b)

1. Heat in $r$ is maximum when $r$ equals the resistance of the rest of the circuit seen from its ends (ideal cell shorted).
2. Seen from $r$: series $R$ (through the cell) is in parallel with the other $R$, so $R_{th} = \dfrac{R \cdot R}{R + R} = \dfrac{R}{2}$.
3. $r = \dfrac{R}{2} \Rightarrow f = \dfrac{1}{2}$
4. So the answer is (b) $\dfrac{1}{2}$.