In figure, battery of emf E has internal resistance r and a variable resistor. At an instant, current flowing through the circuit is i, potential difference between the terminals of, cells is V, thermal power developed in external circuit is P, and thermal power developed in the cell is equal to fraction $\eta$ of total electrical power generated in it. Which of the following graphs is/are correct?

Potential difference across the terminals of the cell is given by $\mathrm{V}=\mathrm{E}-\mathrm{ir}$. Hence, the graph between V and i will be a straight line having negative slope and positive intercept. Hence, (a) is correct.
Total electrical power generated is Ei and thermal power generated in the cell is ${\mathrm{i}}^2\mathrm{r}$. Hence, thermal power generated in the external circuit is $\mathrm{P}=\mathrm{Ei}-{\mathrm{i}}^2\mathrm{r}$. So, the graph between P and i is a parabola passing through the origin. Hence, choice (b) is correct. 

When terminal potential difference across the cell is V, current flowing through the circuit will be i = E-V/R and thermal power generated in the external circuit will be equal to $\mathrm{P}=\mathrm{Vi}=\mathrm{V}\left(\frac{\mathrm{E}-\mathrm{V}}{\mathrm{r}}\right)=\frac{\mathrm{EV}}{\mathrm{r}}-\frac{{\mathrm{V}}^2}{\mathrm{r}}$
Hence, the graph between P and V will be a parabola passing through the origin and have a maximum value of P at V = E/2.
Hence, choice (d) is correct.
At an instant, thermal power generated in the cell is equal to ${\mathrm{i}}^2\mathrm{r}$ and total electrical power generated in the cell is equal to Ei. Hence, the fraction $\mathrm{\eta }={\mathrm{i}}^2\mathrm{r}/\mathrm{Ei}=(\mathrm{r}/\mathrm{E})\mathrm{i}.$ Hence, the graph between $\eta$ and i will be a straight line passing through the origin and have a positive slope. Hence, (c) is correct.