Answer : (b) $\;\lambda_{1}=\large\frac{\lambda_{2} \lambda_{3}}{\lambda_{2} + \lambda _{3} }$

Explanation :

Conservation of momentum yields

$\large\frac{h}{\lambda_{1}}+0=\large\frac{h}{\lambda_{2}}+mV$

$\large\frac{h}{\lambda_{1}}-\large\frac{h}{\lambda_{2}}=mV$

$\large\frac{1}{\lambda_{1}}-\large\frac{1}{\lambda_{2}}=\large\frac{mV}{h}-----(1)$

$\large\frac{h}{mV}=\lambda_{3}$

$\large\frac{1}{\lambda_{1}}-\large\frac{1}{\lambda_{2}}=\large\frac{1}{\lambda_{3}}$

$\large\frac{1}{\lambda_{1}}=\large\frac{1}{\lambda_{2}}+\large\frac{1}{\lambda_{3}}$

$\lambda_{1}=\large\frac{\lambda_{2} \lambda_{3}}{\lambda_{2} + \lambda _{3} }\;.$