Question

The greatest term (numerically) in the expansion of ${(3-5x)}^{11}$ when $x=\frac{1}{5}$ is

• A. $55\times 3^9$
• B. $46\times 3^9$
• C. $55\times 3^6$
• D. None of these

Answer 1

The correct option is A $55\times 3^9$

We have,

${(3-5x)}^{11}=3^{11}{(1-\frac{5x}{3})}^{11}=3^{11}{(1-\frac{1}{3})}^{11}(\because x=\frac{1}{5})$

$\therefore m=\frac{\left|x\right|(n+1)}{(\left|x\right|+1)}=\frac{\left(|-\frac{1}{3}|\right)(11+1)}{(|-\frac{1}{3}|+1)}=3$

The greatest term in the expansion are $T_3$ and $T_4$

$\therefore$ Greatest term $($ when $r=2)$ $=3^{11}\left|T_{2+1}\right|$

$=3^{11}\left|{}^{11}{C}_2{(-\frac{1}{3})}^2\right|=3^{11}|\frac{11.10}{1.2}\times \frac{1}{9}|=55\times 3^9$

and greatest term $($ when $r-3)$ $=3^{11}\left|T_{2+1}\right|$

$=3^{11}\left|{}^{11}{C}_3{(-\frac{1}{3})}^3\right|=3^{11}|\frac{\mathrm{11.10.9}}{\mathrm{1.2.3}}\times -\frac{1}{27}|=55\times 3^9$

From above we see that the values of both greatest term are equal