First slide

Binomial theorem for positive integral Index

Question

If the term independent of x in the ${(\sqrt{x} - \frac{k}{x^{2}})}^{10}$ is 405, then k equals

Easy

A

2, -2
B

3, -3
C

4, -4
D

1, -1

Solution

$T_{r + 1} =^{10} C_{r} (\sqrt{x})^{10 - r} {(\frac{- k}{x^{2}})}^{r} =^{10} C_{r} x^{5 - 5 r / 2} (- k)^{r}$
For this to be independent of x, $5 - \frac{5 r}{2} = 0 \Rightarrow$ r must be 2, so that
$^{10} C_{2} k^{2} = 405 \Rightarrow 45 k^{2} = 405 \Rightarrow k^{2} = 9 \Rightarrow k = \pm 3$

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