In a radioactive material, the activity at time $t_1$ is $R_1$ and at a later time, $t_2$ is $R_2$. If the decay constant of the material is $\lambda$, then:

1. $R_1=R_2{e}^{-\lambda \left(t_1-t_2\right)}$   

2. $R_1=R_2{e}^{\lambda \left(t_1-t_2\right)}$

3. $R_1=R_2\left(\frac{t_2}{t_1}\right)$ 

4. $R_1=R_2$

Which of the following are suitable for the fusion process ? [2002]

1. Light nuclei

2. Heavy nuclei

3. Elements lying in the middle of periodic table

4. Elements lying in the middle of binding energy curve

An archeologist analyses the wood in a prehistoric structure and finds that ${\mathrm{C}}^{14}$ (Half - life=5700 years) to ${\mathrm{C}}^{12}$ is only one - fourth of that found in the cells buried plants. The age of the wood is about

1. 5700 yr

2. 2850 yr

3. 11,400 yr

4. 22,800 yr

In the nuclear reaction: $X{(n,}_{}$ $\alpha )3L{i}^7$, the term X will be:

1. ${}_{5}B_{10}$ 

2. ${}_{5}B_9$

3. ${}_{5}B_{11}$

4. ${}_{2}H{e}^4$

10 g of radioactive material of half-life 15 year is kept in store for 20 years. The disintegrated material mass is [Pb. PMT 2002]

1. 12.5 g

2. 10.5 g

3. 6.03 g

4. 4.03 g

If a radioactive nucleus decays according to the following reaction

${}_{72}X_{180}\stackrel{\alpha }{\to }{X}_1\stackrel{\beta }{\to }{X}_2\stackrel{\alpha }{\to }{X}_3\stackrel{\gamma }{\to }{X}_4$

then the mass number and the atomic number of $X_4$ then the mass number and the atomic number of $X_4$ will be, respectively [MP PET 2002]

1. 172, 70 

2. 172, 69

3. 172, 68

4. 171, 69

Two radioactive materials $X_1$ and $X_2$ have decay constants 10$\lambda$ and $\lambda$ respectively. If initially they have the same number of nuclei, then the ratio of the number of nuclei of $X_1$ to that of $X_2$ will be $\frac{1}{e}$ after a time

1. $\frac{1}{10 \lambda }$ 

2. $\frac{1}{11 \lambda }$

3. $\frac{11}{10 \lambda }$   

4. $\frac{1}{9 \lambda }$

A radioactive material has mean-lives of 1620 yr and 520 yr for $\alpha$ and $\beta$-emission. The material decays by simultaneous $\alpha$ and $\beta$-emission. The time in which 1/4th of the material remains intact is 

1. 4675 yr 

2. 720 yr

3. 545 yr 

4. 324 yr

The radioactivity of an element becomes $\frac{1}{64}th$ of its original value in 60 seconds. Then the half value period is

1. 5 sec 

2. 10 sec 

3. 20 sec

4. 30 sec

A nucleus of ${}_{84}{}^{210}Po$ originally at rest emits $\alpha$ particle with speed $\nu$. What will be the recoil speed of the daughter necleus. [DCE 2002]

1. 4$\nu$ / 206

2. 4$\nu$ / 214

3. $\nu$ / 206 

4. $\nu$ / 214