The term independent of x in the expansion of [ x 3 + 3 2 x 2 ] 10 is:

Question

The term independent of $x$ in the expansion of ${[\sqrt{\frac{x}{3}} + \frac{3}{2 x^{2}}]}^{10}$ is:

Smart Achievers · Accepted Answer

(1)

$Let {(r + 1)}^{th} term in the expansion of$

${[\sqrt{\frac{x}{3}} + \frac{3}{2 x^{2}}]}^{10} is independent of x$

$So T_{r + 1} = {}^{10}C_{r} {(\sqrt{\frac{x}{3}})}^{10 - r} {(\frac{3}{2 x^{2}})}^{r}$

$= {}^{10}C_{r} {(\frac{1}{\sqrt{3}})}^{10 - r} {(\frac{3}{2})}^{r} x^{\frac{10 - 5 r}{2}}$

$Putting x^{(\frac{10 - 5 r}{2})} equal to x^{0} \Rightarrow \frac{10 - 5 r}{2} = 0 \Rightarrow r = 2$

$∴$ $T_{2 + 1} = {}^{10}C_{2} {(\frac{1}{\sqrt{3}})}^{10 - 2} {(\frac{3}{2})}^{2} = \frac{5}{4}$

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