Department of Pre-University Education, KarnatakaPUC Karnataka Science Class 11

# Find the Coefficient Of: (Iv) X 9 in the Expansion of ( X 2 − 1 3 X ) 9 - Mathematics

Find the coefficient of:

(iv)  $x^9$  in the expansion of  $\left( x^2 - \frac{1}{3x} \right)^9$

#### Solution

(iv) Suppose x9 occurs at the (+ 1)th term in the above expression.

Then, we have:

$T_{r + 1} = ^{9}{}{C}_r ( x^2 )^{9 - r} \left( \frac{- 1}{3x} \right)^r$
$= ( - 1 )^r {9}{}{C}_r \left( x^{18 - 2r - r} \right) \left( \frac{1}{3^r} \right)$
$\text{ For this term to contain } x^9 , \text{ we must have: }$
$18 - 3r = 9$
$\Rightarrow 3r = 9$
$\Rightarrow r = 3$
$\therefore \text{ Coefficient of } x^9 = ( - 1 )^3 {9}{}{C}_3 \frac{1}{3^3} = - \frac{9 \times 8 \times 7}{2 \times 9 \times 9} = \frac{- 28}{9}$

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#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 18 Binomial Theorem
Exercise 18.2 | Q 9.4 | Page 37