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

# Find the Term Independent of X in the Expansion of the Expression: (Vi) ( X − 1 X 2 ) 3 N - Mathematics

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

(vi)  $\left( x - \frac{1}{x^2} \right)^{3n}$

#### Solution

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

$\left( x - \frac{1}{x^2} \right)^{3n}$
$T_{r + 1} = ^{3n}{}{C}_r x^{3n - r} \left( \frac{- 1}{x^2} \right)^r$
 = ( - 1 )^r "^(3n) C_r x^{3n - r - 2r}
$\text{ For this term to be independent of x, we must have}$
$3n - 3r = 0$
$\Rightarrow r = n$
$\text{ Hence, the required term is the (n + 1)th term .}$
$\text{ Now, we have}$
( - 1 )^n "^3n C_n

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#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 18 Binomial Theorem
Exercise 18.2 | Q 16.06 | Page 39