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  4. The term independent of x in the expansion of ( 9x - (1/3 √x) )18, x > 0 , is ‘a’ times the corresponding binomial coefficient. Then ‘a’ is

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Q. The term independent of x in the expansion of $\left( 9x - \frac{1}{3 \sqrt{x}} \right)^{18}, x > 0 $ , is ‘a’ times the corresponding binomial coefficient. Then ‘a’ is

Binomial Theorem

Solution:

$T_{r+1} = ^{18} C_{r} \left(9x\right)^{18-r} \left( - \frac{1}{3 \sqrt{x}}\right)^{r}$
$ = \left(-1\right)^{r} {^{18}C_{r}} 9^{18 - \frac{3r}{2} } x^{18 \frac{3r}{2}} $
is independent of x provided r = 12 and then a = 1.