• # question_answer The positive value of $\lambda$ for which the co-efficient of ${{x}^{2}}$ in the expression ${{x}^{2}}{{\left( \sqrt{x}+\frac{\lambda }{{{x}^{2}}} \right)}^{10}}$ is $720,$ is: A) $4$ B) $2\sqrt{2}$ C) $\sqrt{5}$ D) $3$

 ${{x}^{2}}{{\left( \sqrt{x}+\frac{\lambda }{{{x}^{2}}} \right)}^{10}}$ Consider constant term $^{10}{{C}_{r}}{{(\sqrt{x})}^{10-r}}{{\left( \frac{\lambda }{{{x}^{2}}} \right)}^{r}}$ $\frac{10-r}{2}-2r=0$ $10-5r=0$ $r=2$ $\Rightarrow$$^{10}{{C}_{2}}\times {{\lambda }^{2}}=720$            $\Rightarrow$$\lambda =4.$