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# Find the Term Independent of X in the Expansion of the Expression: (V) ( √ X 3 + 3 2 X 2 ) 10 - Mathematics

Find the term independent of x in the expansion of the expression:

(v)  $\left( \frac{\sqrt{x}}{3} + \frac{3}{2 x^2} \right)^{10}$

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#### Solution

(v) Suppose the (+ 1)th term in the given expression is independent of x.
Now,

$\left( \sqrt{\frac{x}{3}} + \frac{3}{2 x^2} \right)^{10}$
$T_{r + 1} =^{10}{}{C}_r \left( \sqrt{\frac{x}{3}} \right)^{10 - r} \left( \frac{3}{2 x^2} \right)^r$
$= ^{10}{}{C}_r . \frac{3^{r - \frac{10 - r}{2}}}{2^r} x^\frac{10 - r}{2} - 2r$
$\text{ For this term to be independent of x, we must have}$
$\frac{10 - r}{2} - 2r = 0$
$\Rightarrow 10 - 5r = 0$
$\Rightarrow r = 2$
$\text{ Hence, the required term is the 3rd term } .$
$\text{ Now, we have }$
$^{10}{}{C}_2 \times \frac{3^{2 - \frac{10 - 2}{2}}}{2^2}$
$= \frac{10 \times 9}{2 \times 4 \times 9}$
$= \frac{5}{4}$

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#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 18 Binomial Theorem
Exercise 18.2 | Q 16.05 | Page 39
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