For a first order reaction,the graph between $\log \frac{a}{(a-x)}$ (on $y$-axis) and time (in $min$,on $x$-axis) gave a straight line passing through origin. The slope is $2 \times 10^{-3} \ min^{-1}$. What is the rate constant (in $min^{-1}$)?

$A$ first-order reaction starts with a decimolar solution of the reactant. After $8$ minutes,its concentration becomes $M/100$. Calculate the rate constant of the reaction.

$T_{50}$ of a first-order reaction is $10 \ min$. Starting with $10 \ mol \ L^{-1}$,the rate after $20 \ min$ is:

Consider the following first order gas phase reaction at constant temperature:
$A_{(g)} \rightarrow 2B_{(g)} + C_{(g)}$
If the total pressure of the gases is found to be $200 \ torr$ after $23 \ s$ and $300 \ torr$ upon the complete decomposition of $A$ after a very long time,then the rate constant of the given reaction is . . . . . . $\times 10^{-2} \ s^{-1}$ (nearest integer).
[Given: $\log_{10}(2) = 0.301$]

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:

The thermal decomposition of a compound is of first order. If a sample of the compound decomposes $50\%$ in $120 \, minutes$,in what time will it undergo $90\%$ decomposition?