Question

the coefficient of term independent of x in the expansion of
$( \frac{3 {x}^{2} }{2} - \frac{1}{3x} )^{9}$
is λ then 18 λ is ?
9
7
6
4 ​

Answer 1

The term independent of $x$ in the expansion of $\left(ax^p+\dfrac{b}{x^q}\right)^n$ is,

$\longrightarrow T_{r+1}=\,^n\!C_r\ a^{n-r}\ b^r$

where $r=\dfrac{pn}{p+q}.$

According to the question,

• $a=\dfrac{3}{2}$

• $p=2$

• $b=-\dfrac{1}{3}$

• $q=1$

• $n=9$

Then,

$\longrightarrow r=\dfrac{pn}{p+q}$

$\longrightarrow r=\dfrac{2\times9}{2+1}$

$\longrightarrow r=6$

Then, term independent of $x$ is,

$\longrightarrow T_7=\,^9\!C_6\ \left(\dfrac{3}{2}\right)^{9-6}\ \left(-\dfrac{1}{3}\right)^6$

$\longrightarrow\lambda=84\times\dfrac{3^3}{2^3}\times\dfrac{1}{3^6}$

$\longrightarrow\lambda=7\times\dfrac{1}{18}$

$\longrightarrow\underline{\underline{18\lambda=7}}$

Hence 7 is the answer.