If a radioactive element having a half-life of $30 \ min$ is undergoing beta decay,the fraction of the radioactive element that remains undecayed after $90 \ min$ will be:

Following statements related to radioactivity are given below:
$(A)$ Radioactivity is a random and spontaneous process and is dependent on physical and chemical conditions.
$(B)$ The number of undecayed nuclei in the radioactive sample decays exponentially with time.
$(C)$ Slope of the graph of $\log_{e}$ (no. of undecayed nuclei) $Vs.$ time represents the negative reciprocal of mean life time $(-\frac{1}{\tau})$.
$(D)$ Product of decay constant $(\lambda)$ and half-life time $(T_{1/2})$ is constant,equal to $\ln(2)$.
Choose the most appropriate answer from the options given below.

$A$ nucleus has a half-life of $30 \; min$. At $3 \; PM$, its decay rate was measured as $120000 \; cps$. What is the decay rate in $cps$ at $5 \; PM$?

Half-life of a substance is $10$ years. In what time,it becomes $\frac{1}{4}$ of the initial amount?

The half-life of a radioactive substance is $ 20 \text{ minutes} $. The time taken between $ 50\% $ decay and $ 87.5\% $ decay of the substance will be

Half-life period of a sample is $15$ years. How long will it take to decay $96.875\%$ of the sample?