If the term independent of x in the expansion of ( x - x 2 ) 10 is 405, then the value of | 4 | is

Question

If the term independent of $x$ in the expansion of ${(\sqrt{x} - \frac{λ}{x^{2}})}^{10}$ is 405, then the value of $| 4 λ |$ is _________.

Smart Achievers · Accepted Answer

(12)

$Let {(r + 1)}^{th} term is independent of x .$

$We have,$ $T_{r + 1} = {}^{10}C_{r} {(\sqrt{x})}^{10 - r} \cdot {(- \frac{λ}{x^{2}})}^{r}$

$= {}^{10}C_{r} \cdot x^{5 - \frac{r}{2} - 2 r} \cdot {(- λ)}^{r}$

$The term will be independent of x if 5 - \frac{r}{2} - 2 r = 0$

$\Rightarrow$ $r = 2$

$Then,$ $T_{2 + 1} = {}^{10}C_{2} \cdot {(- λ)}^{2} = 45 λ^{2} = 405$ $(Given)$

$∴$ $λ^{2} = 9 \Rightarrow λ = \pm 3 \Rightarrow$ $| 4 λ |$ $= 12$

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