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 [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 

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. [DCE 2002]

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 [2003]

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. $1\times {10}^{10}$