Question

The independent term in the expansion of ${(2x^2+\frac{7}{x^5})}^{10}$ is

• A. ${}^{10}{C}_2{2}^8{7}^2$
• B. ${}^{10}{C}_3{2}^7{7}^3$
• C. ${}^{10}{C}_8{2}^8{7}^3$
• D. Does not exist.

Answer 1

The correct option is D Does not exist.

Given,

${(2x^2+\frac{7}{x^5})}^{10}$

General term

$T_{r+1}={}^n{C}_r{x}^{n-r}{y}^r$

$T_{r+1}={}^{10}{C}_r(2x^2{)}^{10-r}{\left(\frac{7}{x^5}\right)}^r$

$T_{r+1}={}^{10}{C}_r\left(2{)}^{10-r}\right(x{)}^{20-2r}\frac{(7{)}^r}{x^{5r}}$

$T_{r+1}={}^{10}{C}_r\left(2{)}^{10-r}\right(7{)}^r(x{)}^{20-7r}$.

To find independent terms in this expansion.

$20-7r=0$

$20=7r$

$r=\frac{20}{7}$

Which is not possible since r should be an integer.