First slide

Binomial theorem for positive integral Index

Question

If the constant term in the binomial expansion of ${(\sqrt{x} - \frac{k}{x^{2}})}^{10}$ is 405, then $| k |$ equal

Moderate

A

9
B

1
C

3
D

2

Solution

Since, the general term in the expansion of binomial
$\begin{aligned} {(\sqrt{x} - \frac{k}{x^{2}})}^{10} is T_{r + 1} =^{10} C_{r} x^{(\frac{10 - r}{2})} (- k)^{r} x^{- 2 r} \\ =^{10} C_{r} (- k)^{r} x^{\frac{10 - 5 r}{2}} \end{aligned}$
$∵$ the Term is constant,So r=2.
$\begin{array}{ll} ∴^{10} C_{2} (- k)^{2} = 405 \\ \Rightarrow \frac{10 \times 9}{2} k^{2} = 405 \Rightarrow k^{2} = 9 \Rightarrow | k | = 3 \end{array}$

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