Thank you for reporting, we will resolve it shortly
Q.
Half-lives of a first order and a zero order reactions are same. Then, the ratio of the initial rates of first order reaction to that of the zero order reaction is
$ {{t}_{1/2}} $ for zero order reaction
$ =\frac{{{[A]}_{0}}}{2k} $
$ {{t}_{1/2}} $ for first order reaction
$ =\frac{0.693}{k} $ Rate of zero order reaction,
$ {{r}_{0}}=k\frac{{{[A]}_{0}}}{2.{{t}_{1/2}}} $ Rate of first order reaction,
$ {{r}_{1}}=k({{A}_{0}})=\frac{0.693}{{{t}_{1/2}}}{{[A]}_{0}} $
$ \Rightarrow $ $ \frac{{{r}_{1}}}{{{r}_{0}}}=\frac{0.693{{[A]}_{0}}}{{{t}_{1/2}}}\times \frac{2\times {{t}_{1/2}}}{{{[A]}_{0}}} $
$ =2\times 0.693 $