Question

The highest degree of ‘x’ in the expansion of ${(\frac{x+1}{x^{\frac{2}{3}}-x^{\frac{1}{3}}+1}-\frac{x-1}{x-x^{\frac{1}{2}}}+\frac{x-1}{x^{\frac{3}{2}}-x^{\frac{1}{2}}})}^9$ is

• A. 3
• B. 4
• C. 18
• D. 10

Answer 1

The correct option is A 3

${(\frac{x+1}{x^{\frac{2}{3}}-x^{\frac{1}{3}}+1}-\frac{x-1}{x-x^{\frac{1}{2}}}+\frac{x-1}{x^{\frac{3}{2}}-x^{\frac{1}{2}}})}^9={(\frac{(x^{1/3}{)}^3+1^3}{x^{\frac{2}{3}}-x^{\frac{1}{3}}+1}-\frac{x-1}{\sqrt{x}(\sqrt{x}-1)}+\frac{(x-1)}{\sqrt{x}})}^9$
${(\frac{(x^{\frac{1}{3}}+1)(x^{\frac{2}{3}}-x^{\frac{1}{3}}+1)}{x^{\frac{2}{3}}-x^{\frac{1}{3}}+1}-\frac{(\sqrt{x}+1)}{\sqrt{x}}+\frac{1}{\sqrt{x}})}^9={(x^{\frac{1}{3}}+1-1-\frac{1}{\sqrt{x}}+\frac{1}{\sqrt{x}})}^9={\left(x^{\frac{1}{3}}\right)}^9$
$=x^3$