$A$ first order reaction is given as $A \rightarrow \text{products}$. Its integrated rate equation is:

For a first order reaction,the rate constant is given as $\log_{10} K = 12 - \frac{6 \times 10^3}{T}$. What will be the value of temperature if its half-life period is $6.93 \times 10^{-3} \, \text{min}$ (in $, K$)?

Which of the following is correct regarding $t_{1/2}$ of a reaction if we double the initial concentration of a reactant in a first-order reaction?

If the time required for $90\%$ completion of a first-order reaction is '$t$', what is the time required for $99.9\%$ completion of the reaction at the same temperature?

Which one of the following formulas represents a first-order reaction?

Given below are two statements:
Statement $I$: The graph of $t_{1/2}$ versus initial concentration $[R]_0$ for a first-order reaction is a horizontal line.
Statement $II$: The graph of $\log \frac{[R]_0}{[R]}$ versus time $t$ for a first-order reaction is a straight line passing through the origin with a slope equal to $\frac{k}{2.303}$.
In the light of the above statements,choose the correct answer from the options given below: