$A$ radioactive sample of half-life $ 10 $ days contains $ 1000x $ nuclei. The number of original nuclei present after $ 5 $ days is: (in $x$)

At time $t=0$ some radioactive gas is injected into a sealed vessel. At time $T$ some more of the gas is injected into the vessel. Which one of the following graphs best represents the logarithm of the activity $A$ of the gas with time $t$?

$1 \, mg$ of gold undergoes decay with a $2.7 \, \text{days}$ half-life period. The amount left after $8.1 \, \text{days}$ is ......... $mg$.

The half-life of a radioactive substance is $5$ years. After $x$ years,a given sample of the radioactive substance gets reduced to $6.25 \%$ of its initial value. The value of $x$ is ...............

The half-life of a radioactive sample is $5 \,s$. If the initial mass of the sample is $60 \,g$, then the time required to reduce the sample to $7.5 \,g$ is (in $\,s$)

The half-life of a radioactive substance is $100 \ s$. How many grams of the radioactive substance will remain after $5 \ min$ if the initial amount is $8 \ g$ (in $g$)?