# Find the Term Independent of X in the Expansion of the Expression: (Ii) ( 2 X + 1 3 X 2 ) 9 - Mathematics

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

(ii)  $\left( 2x + \frac{1}{3 x^2} \right)^9$

#### Solution

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

$\left( 2x + \frac{1}{3 x^2} \right)^9$
$T_{r + 1} = ^{9}{}{C}_r (2x )^{9 - r} \left( \frac{1}{3 x^2} \right)^r$
$= ^{9}{}{C}_r . \frac{2^{9 - r}}{3^r} x^{9 - r - 2r}$
$\text{ For this term to be independent of x, we must have}$
$9 - 3r = 0$
$\Rightarrow r = 3$
$\text{ Hence, the required term is the 4th term .}$
$\text{ Now, we have }$
$^{9}{}{C}_3 \frac{2^6}{3^3}$
$= ^{9}{}{C}_3 \times \frac{64}{27}$

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#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 18 Binomial Theorem
Exercise 18.2 | Q 16.02 | Page 39