Question

The constant terms is the expansion of ${\left(\sqrt{x}-\frac{c}{x^2}\right)}^{10}$ is $180,$ then the value of $c$ equals to

• A. $\pm 2$
• B. $\pm 3$
• C. $\pm 4$
• D. $Noneofthese$

Answer 1

Answer: (A)

Given, ${\left(\sqrt{x}-\frac{C}{x^2}\right)}^{10}$
General term is $T_{r+1}{=}^{10(}{C}_r{(\sqrt{x})}^{10-r}{\left(-\frac{C}{x^2}\right)}^r$
${=}^{10}{C}_r{x}^{(5-r/2)}{(-C)}^r{x}^{-2r}$
${=}^{10}{C}_r{x}^{(5-r/2-2r)}{(-C)}^r$
${=}^{10}{C}_r{x}^{\left(5-\frac{5r}{2}\right)}{(-C)}^r$
For the constant term, Put $5-\frac{5r}{2}=0$
$\Rightarrow$ $(r=2)$
$\Rightarrow$ $T_{2+1}{=}^{10}{C}_2{x}^{(5-5)}{(-C)}^2{=}^{10}{C}_2.C^2$
Given, $T_3=180{=}^{10}{C}_2.C^2$
$\Rightarrow$ $C^2.\frac{10\times 9}{2}=180$
$\Rightarrow$ $C^2=4$
$\Rightarrow$ $C=\pm 2$