Positive value of λ for which co-efficient of x^2 in expression {x^2}{left( {√ x + fr

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7.Binomial Theorem

hard

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

$3$

(JEE MAIN-2019)

Solution

$x^{2}\left(\sqrt{x}+\frac{\lambda}{x^{2}}\right)^{10}$

Consider constant term

$^{10} \mathrm{C}_{\mathrm{r}}(\sqrt{\mathrm{x}})^{10-\mathrm{r}}\left(\frac{\lambda}{\mathrm{x}^{2}}\right)^{r}$

$\frac{10-r}{2}-2 r=0$

$10-5 r=0$

$r=2$

$\Rightarrow^{10} \mathrm{C}_{2} \times \lambda^{2}=720$

$ \Rightarrow \lambda=4$

Standard 11

Mathematics

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