KVPY Sample Paper KVPY Stream-SX Model Paper-29

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) \[3\]

Correct Answer: A

Solution :

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

Consider constant term

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

\[\frac{10-r}{2}-2r=0\]

\[10-5r=0\]

\[r=2\]

\[\Rightarrow \]\[^{10}{{C}_{2}}\times {{\lambda }^{2}}=720\] \[\Rightarrow \]\[\lambda =4.\]