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:
Correct Option: 1
Since, coefficient of $x^{2}$ in the expression $x^{2}\left(\sqrt{x}+\frac{\lambda}{x^{2}}\right)$
is a constant term, then
Coefficient of $x^{2}$ in $x^{2}\left(\sqrt{x}+\frac{\lambda}{x^{2}}\right)^{10}$
$=$ co-efficient of constant term in $\left(\sqrt{x}+\frac{\lambda}{x^{2}}\right)^{10}$
General term in $\left(\sqrt{x}+\frac{\lambda}{x^{2}}\right)^{10}={ }^{10} C_{r}(\sqrt{x})^{10-r}\left(\frac{\lambda}{x^{2}}\right)^{r}$
$={ }^{10} C_{r}(x)^{\frac{10-r}{2}-2 r} \cdot \lambda^{2}$
Then, for constant term,
$={ }^{10} C_{r}(x)^{\frac{10-r}{2}-2 r} \cdot \lambda^{2}$
Then, for constant term,
$\frac{10-r}{2}-2 r=0 \Rightarrow r=2$
Co-efficient is $x^{2}$ in expression $={ }^{10} C_{2} \lambda^{2}=720$
$\Rightarrow \lambda^{2}=\frac{720}{5 \times 9}=16$
$\lambda=4$]
Hence, required value of $\lambda$ is 4 .