The decay constant of a radio isotope is $\mathrm{\lambda }.$ If ${\mathrm{A}}_1$ and ${\mathrm{A}}_2$ are its activities at times ${\mathrm{t}}_1$ and ${\mathrm{t}}_2$ respectively, the number of nuclei which have decayed during the time $({\mathrm{t}}_1-{\mathrm{t}}_2)\; is:$

1. ${\mathrm{A}}_1{\mathrm{t}}_1-{\mathrm{A}}_2{\mathrm{t}}_2$                                       

2. ${\mathrm{A}}_1-{\mathrm{A}}_2$

3. $\frac{({\mathrm{A}}_1-{\mathrm{A}}_2)}{\mathrm{\lambda }}$                                         

4. $\mathrm{\lambda }({\mathrm{A}}_1-{\mathrm{A}}_2)$ 

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 parent                                   

(2) isomer of parent

(3) isotone of parent                                 

(4) isotope of 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) ${\left(3\right)}^{1/3}:1$

(4) 1:1

The size of an atom is of the order of

(a) ${10}^{-8}$ m         (b) ${10}^{-10} \mathrm{m}$ 
(c) ${10}^{-12} \mathrm{m}$        (d) ${10}^{-14} \mathrm{m}$

The activity of a radioactive sample is measured as $N_0$ counts per minute at t=0 and $N_0/e$ counts per minute at t=5 min.The time (in minute) at which the activity reduces to half its value is 

(1) ${\log }_{e}2/5$                                                         

(2) $\frac{5}{{\log }_e 2}$

(3) $5 {\log }_{10 }2$                                                         

(4) $5 {\log }_e 2$

The mass of a ${}_3{}^{7}Li$ nucleus is 0.042u less than the sum of the masses of its nucleons.The binding energy per nucleon of ${}_3{}^{7}Li$ nucleus is nearly

(a) 46 MeV                              (b) 5.6 MeV

(c) 3.9 MeV                             (d) 23 MeV

Two radioactive nuclei P and Q, in a given sample decay into a stable nucleus R. At time t=0, the number of P species are $4N_0$ and that of Q is $N_0.$Half-life of P(for conversion to R) is 1 min whereas that of Q is 2 min. Initially there are no nuclei of R present in the sample. When number of nuclei of P and Q are equal, the number of nuclei of R present in the sample would be: 

1. $3N_0$                                         

2. $\frac{9N_0}{2}$

3. $\frac{5N_0}{2}$                                       

4. $2N_0$

Fusion reaction takes place at high temperature because

(1) atoms get ionised at high temperature 

(2) kinetic energy is high enough to overcome the Coulomb repulsion between nuclei

(3) molecules break up at high temperature

(4) nuclei break up at high temperature

The half-life of a radioactive nucleus is 50 days. The time interval $({\mathrm{t}}_2-{\mathrm{t}}_1)$ between the time ${\mathrm{t}}_2$ when $\frac{2}{3}$ of it has decayed and the time ${\mathrm{t}}_1$ when $\frac{1}{3}$ of it had decayed is:

1. 30 days

2. 50 days

3. 60 days

4. 15 days

A mixture consists of two radioactive materials $A_1$ and $A_2$ with half lives of 20 s and 10 s respectively.Initially the mixture has 40g of $A_1$ and 160g of $A_2$. The amount of the two in the mixture will become equal after

(a)60 s                                      (b)80 s

(c)20 s                                      (d) 40 s