Question

If in the expansion ${\left[3x-\frac{2}{x^2}\right]}^{15}$ $r$th term is independent of $x$, then the value of $r$ is

• A. $6$
• B. $10$
• C. $9$
• D. $12$

Answer 1

The correct option is A

$6$

Explanation for the correct option:

Find the value of $r$.

The $r^{th}$ term of the expansion of ${\left(a+b\right)}^n$ is given as $T_r={}^{n}C_{r-1}{a}^{n-r+1}{b}^{r-1}$.

Now, the $r^{th}$ term of the expansion of ${\left[3x-\frac{2}{x^2}\right]}^{15}$ is given as

$\begin{array}{rcl}{T}_r& =& {}^{15}C_{r-1}{\left(3x\right)}^{15-r+1}{\left(-\frac{2}{x^2}\right)}^{r-1}\\ & =& {}^{15}C_{r-1}·3^{16-r}·{\left(-2\right)}^{r-1}\frac{x^{16-r}}{x^{2r-2}}\\ & =& {}^{15}C_{r-1}·3^{16-r}·{\left(-2\right)}^{r-1}{x}^{16-r-2r+2}\\ & =& {}^{15}C_{r-1}·3^{16-r}·{\left(-2\right)}^{r-1}{x}^{18-3r}\end{array}$

As the $r^{th}$ term is independent of $x$, so the power of $x$ must be zero, so

$$\begin{gathered} 18-3r=0 \\ \Rightarrow -3r=-18 \\ \Rightarrow r=6 \end{gathered}$$

So the value of $r$ is $6$.

Hence, the correct option is A.