The positive value of for which the coefficient of x 2 in the expression x 2 ( x + x 2 ) 10 is 720, is

Question

The positive value of $λ$ for which the coefficient of $x^{2}$ in the expression $x^{2} {(\sqrt{x} + \frac{λ}{x^{2}})}^{10}$ is 720, is

Smart Achievers · Accepted Answer

(2)

$T_{r + 1} term of {(\sqrt{x} + \frac{λ}{x^{2}})}^{10} is$

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

$For this to be independent of x,$ $5 - \frac{5 r}{2} = 0 \Rightarrow r = 2$

$∴$ ${}^{10}C_{2} λ^{2} = 720 \Rightarrow 45 λ^{2} = 720$

$\Rightarrow$ $λ^{2} = 16 \Rightarrow λ = \pm 4$

$∴$ $The positive value of λ for which the coefficient of x^{2} is 720 is 4 .$

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