Correct optlon (c)
Explanation:
Given, \(Q-a t-b t^{2}\)
$$
\therefore \quad I=\frac{d Q}{d t}=a-2 h t
$$
$$
\begin{aligned}
&\text { At } t=0, Q=0 \Rightarrow J=0 \\
&\text { Also, } I=0 \text { at } t=a / 2 b
\end{aligned}
$$
\(\therefore\) Total heat produced in resistance \(\mathrm{R}\),
$$
\begin{aligned}
H &=\int_{0}^{a / 2 b} 1^{2} R d t=R \int_{0}^{a / 2 b}(a-2 b t)^{2} d t \\
&=R \int_{0}^{a / 2 b}\left(a^{2}+4 b^{2, t^{2}}-4 a b r\right) d t \\
&=R\left[a^{2} t+4 b^{2} \frac{r^{3}}{3}-4 a b \frac{t^{2}}{2}-a \cdot 2 b\right.\\
&=R\left[a^{2} \times \frac{a}{2 b}+\frac{4 b^{2}}{3} \times \frac{a^{3}}{3 b^{3}}-\frac{4 a b}{2} \times \frac{a^{2}}{4 b^{2}}\right] \\
&=\frac{a^{3} R}{b}\left[\frac{1}{2}+\frac{1}{6}-\frac{1}{2}\right]=\frac{a^{3} R}{6 b}
\end{aligned}
$$
