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Directions : In the following question more than one of the answers given may be correct. Select the correct answer and mark it according to the code:

The wavelengths and frequencies of photons in transitions 1, 2 and 3 for hydrogen like atom are\[{{\lambda }_{1}},{{\lambda }_{2}},{{\lambda }_{3}};{{v}_{1}},{{v}_{2}}\]and\[{{v}_{3}}\]respectively. Then (1) \[{{v}_{3}}={{v}_{1}}+{{v}_{2}}\]                         (2) \[{{\lambda }_{3}}={{\lambda }_{1}}+{{\lambda }_{2}}\] (3) \[{{\lambda }_{3}}=\frac{{{\lambda }_{1}}{{\lambda }_{2}}}{{{\lambda }_{1}}+{{\lambda }_{2}}}\]                       (4) \[{{v}_{3}}=\frac{{{v}_{1}}{{v}_{2}}}{{{v}_{1}}+{{v}_{2}}}\]

A)  1, 2 and 3 are correct

B)  1 and 2 are correct

C)   2 and 4 are correct

D)  1 and 3 are correct

Correct Answer: D

Solution :

                  From figure                 \[{{E}_{3}}={{E}_{1}}+{{E}_{2}}\]                 \[h{{v}_{3}}=h{{v}_{1}}+h{{v}_{2}}\] Again,   \[{{v}_{3}}={{v}_{1}}+{{v}_{2}}\]                 \[\frac{hc}{{{\lambda }_{3}}}=\frac{hc}{{{\lambda }_{1}}}+\frac{hc}{{{\lambda }_{2}}}\]                 \[\frac{1}{{{\lambda }_{3}}}=\frac{1}{{{\lambda }_{1}}}+\frac{1}{{{\lambda }_{2}}}\]                 \[\frac{1}{{{\lambda }_{3}}}=\frac{{{\lambda }_{2}}+{{\lambda }_{1}}}{{{\lambda }_{1}}{{\lambda }_{2}}}\]                 \[\lambda {{ & }_{3}}=\frac{{{\lambda }_{1}}{{\lambda }_{2}}}{{{\lambda }_{1}}+{{\lambda }_{2}}}\] Hence, option  is correct.