An example of an autocatalytic reaction is:

Given below are two statements: $R = 8.314 \text{ J K}^{-1} \text{ mol}^{-1}$ and $1 \text{ cal} = 4.2 \text{ J}$.
Statement $I$: When $E_a = 12.6 \text{ kcal/mol}$, the room temperature rate constant is doubled by a $10 \text{ }^\circ\text{C}$ increase in temperature ($298 \text{ K}$ to $308 \text{ K}$).
Statement $II$: For a first-order reaction $A \to B$, the graph of half-life $(t_{1/2})$ versus initial concentration $[A]_o$ is a straight line passing through the origin.
In the light of the above statements, choose the correct answer from the options given below:

Fill in the blanks:
$(1)$ The time required to complete $99.9\%$ of a first-order reaction is ............ times the ${t_{1/2}}$.
$(2)$ The reaction in which the rate is given by $-\frac{d[R]}{dt} = k$ has an order of reaction equal to ........
$(3)$ For the reaction $[R] = [R]_0 e^{-kt}$,the order is .......

Half-lives of a first order and a zero order reaction are same. Then the ratio of the initial rates of first order reaction to that of the zero order reaction is

Which of the following graphs is incorrect for a first-order reaction? $A \rightarrow B$

For a chemical reaction, half-life period $(t_{1/2})$ is $10 \ minutes$. How much reactant will be left after $20 \ minutes$ if one starts with $100 \ moles$ of reactant and the order of the reaction is $(i)$ zero, $(ii)$ one, and $(iii)$ two?