Two radioactive substances A and B have decay constants $5\lambda$ and $\lambda$ respectively. At t=0 they have the same number of nuclei. The ratio of number of nuclei of A to those of B will be ${\left(\frac{1}{e}\right)}^2$ after a time interval

1. $\frac{1}{4\lambda }$   

2. $4\lambda$

3. $2\lambda$ 

4. $\frac{1}{2\lambda }$

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

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 

1. 172, 70 

2. 172, 69

3. 172, 68

4. 171, 69

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. 

1. 4$\nu$ / 206

2. 4$\nu$ / 214

3. $\nu$ / 206 

4. $\nu$ / 214

An atom of mass number 15 and atomic number 7 captures an $\alpha$-particle and then emits a proton. The mass number and atomic number of the resulting product will respectively be

1. 14 and 2

2. 15 and 3

3. 16 and 4 

4. 18 and 8

The wavelength of $K_{\alpha }$ X-rays for lead isotopes $P{b}^{208}, P{b}^{206}, P{b}^{204} are {\lambda }_1, {\lambda }_2 and {\lambda }_3$ respectively. Then

1. ${\lambda }_1={\lambda }_2>{\lambda }_3$   

2. ${\lambda }_1>{\lambda }_2>{\lambda }_3$

3. ${\lambda }_1<{\lambda }_2<{\lambda }_3$ 

4. ${\lambda }_2={\lambda }_1={\lambda }_3$

Solar energy is mainly caused due to 

1. fusion of protons during synthesis of heavier elements

2. gravitational contraction

3. burning of hydrogen in the oxygen

4. fission of uranium present in the sun

A sample of radioactive elements contains $4\times {10}^{10}$ active nuclei. If half-life of element is 10 days, then the number of decayed nuclei after 30 days is 

1. $0.5\times {10}^{10}$   

2. $2\times {10}^{10}$

3. $3.5\times {10}^{10}$

4. $1\times {10}^{10}$