Radioactive material 'A' has decay constant '8\(\lambda\)' and material 'B' has a decay constant '\(\lambda\)'. Initially, they have the same number of nuclei. After what time, the ratio of the number of nuclei of material 'A' to that of 'B' will be \(\frac{1}{e}\)?

\(1 .   \frac{1}{7 \lambda}\)
\(2 .   \frac{1}{8 \lambda}\)
\(3 .   \frac{1}{9 \lambda}\)
\(4 .   \frac{1}{\lambda}\)

The Binding energy per nucleon of \(^{7}_{3}\mathrm{Li}\) and \(^{4}_{2}\mathrm{He}\) nucleon are \(5.60~\text{MeV}\) and \(7.06~\text{MeV}\), respectively. In the nuclear reaction \(^{7}_{3}\mathrm{Li} + ^{1}_{1}\mathrm{H} \rightarrow ^{4}_{2}\mathrm{He} + ^{4}_{2}\mathrm{He} +Q\), the value of energy \(Q\) released is:
1. \(19.6~\text{MeV}\)
 2. \(-2.4~\text{MeV}\)
 3. \(8.4~\text{MeV}\)
4. \(17.3~\text{MeV}\)

A nucleus \({ }_{{n}}^{{m}} \mathrm{X}\) emits one $\mathrm{}$\(\alpha\text -\text{particle}\) and two \(\beta\text- \text{particle}\) The resulting nucleus is:

The number of beta particles emitted by a radioactive substance is twice the number of alpha particles emitted by it. The resulting daughter is an:
1. Isobar of a parent.
2. Isomer of a parent.
3. Isotone of a parent.
4. Isotope of a parent.

Two nuclei have their mass numbers in the ratio of \(1:3.\) The ratio of their nuclear densities would be:
1. \(1:3\)
2. \(3:1\)
3. \((3)^{1/3}:1\)
4. \(1:1\)

An element \(\mathrm{X}\) decays, first by positron emission, and then two \(\alpha\text-\)particles are emitted in successive radioactive decay. If the product nuclei have a mass number \(229\) and atomic number \(89\), the mass number and the atomic number of element \(\mathrm{X}\) are:
1. \(237,~93\) 
2. \(237,~94\)
3. \(221,~84\)
4. \(237,~92\)

90% of a radioactive sample is left undecayed after time t has elapsed. What percentage of the initial sample will decay in a total time 2t?
1. 20% 

2. 19%

3. 40% 

4. 38%