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 half-life of a radioactive substance is 30 minutes. The time (in minutes) taken between 40% decay and 85% decay of the same radioactive substance is:

1. 15

2. 30

3. 45

4. 60

For radioactive material, the half-life is \(10\) minutes. If initially, there are \(600\) number of nuclei, the time taken (in minutes) for the disintegration of \(450\) nuclei is :
1. \(20\)
2. \(10\)
3. \(30\)
4. \(15\)

A nucleus of uranium decays at rest into nuclei of thorium and helium. Then:

If the radius of \(_{13}^{27}\mathrm{Al}\) nucleus is taken to be \({R}_{\mathrm{Al}},\) then the radius of \(_{53}^{125}\mathrm{Te}\) nucleus is near:
1. \(\left(\frac{53}{13}\right) ^{\frac{1}{3}}~{R_{Al}}\)
2. \(\frac{5}{3}~{R_{Al}}\)
3. \(\frac{3}{5}~{R_{Al}}\)
4. \(\left(\frac{13}{53}\right)~{R_{Al}}\)

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 radioisotope 'X' with a half-life 1.4 × 10⁹ years decays to 'Y' which is stable. A sample of the rock from a cave was found to contain 'X' and 'Y' in the ratio 1:7. The age of the rock is:

1. 1.96 x 10⁹ years

2. 3.92 x 10⁹ years

3. 4.20 x 10⁹ years

4. 8.40 x 10⁹ years

If the nuclear radius of \(^{27}\text{Al}\) is \(3.6\) Fermi, the approximate nuclear radius of \(^{64}\text{Cu}\) in Fermi is:
1. \(2.4\)
2. \(1.2\)
3. \(4.8\)
4. \(3.6\)