# Find the term independent of x in the expansion of the expression: (i) ( 3 2 x 2 − 1 3 x ) 9 - Mathematics

Short Note

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

(i) $\left( \frac{3}{2} x^2 - \frac{1}{3x} \right)^9$

#### Solution

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

$\left( \frac{3}{2} x^2 - \frac{1}{3x} \right)^9$
$T_{r + 1} =^{9}{}{C}_r \left( \frac{3}{2} x^2 \right)^{9 - r} \left( \frac{- 1}{3x} \right)^r$
$= ( - 1 )^r {9}{}{C}_r . \frac{3^{9 - 2r}}{2^{9 - r}} \times x^{18 - 2r - r}$
$\text{ For this term to be independent of x, we must have}$
$18 - 3r = 0$
$\Rightarrow 3r = 18$
$\Rightarrow r = 6$
$\text{ Hence, the required term is the 7th term } .$
$\text{ Now, we have }$
$^{9}{}{C}_6 \times \frac{3^{9 - 12}}{2^{9 - 6}}$
$= \frac{9 \times 8 \times 7}{3 \times 2} \times 3^{- 3} \times 2^{- 3}$
$= \frac{7}{18}$

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#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 18 Binomial Theorem
Exercise 18.2 | Q 16.01 | Page 39