Question

Find the term independent of x in the expansion of ${(2x^2-\frac{3}{x^3})}^{25}$.

• A. ${}^{25}{C}_{20}(2^5\times 3^{20})$
• B. ${}^{25}{C}_{11}(2^{16}\times 3^9)$
• C. ${}^{25}{C}_{10}(2^{15}\times 3^{10})$
• D. None of these

Answer 1

The correct option is C ${}^{25}{C}_{10}(2^{15}\times 3^{10})$

Given expression,

${(2x^2-\frac{3}{x^3})}^{25}$

$T_{r+1}{=}^{25}{C}_r(2x^2{)}^{25-r}{\left(\frac{-3}{x^3}\right)}^r$

${=}^{25}{C}_r\left(2x^2{)}^{25-r}\right(x{)}^{50-2r}(-3{)}^r(x{)}^{-3r}$

${=}^{25}{C}_r\left(2x^2{)}^{25-r}\right(-3{)}^r(x{)}^{50-5r}$

For the term independent of $x,50-5r=0\Rightarrow r=10$

$\therefore$ The term independent of $x$ is

$T_{10+1}{=}^{25}{C}_{10}\left(2{)}^{25-10}\right(-3{)}^{10}$

${=}^{25}{C}_{10}\left(\right(2{)}^{15}\left(3{)}^{10}\right)$