Question

The term independent of $ x $ in the expansion of $$ \left( \frac{x + 1}{x^{3/2} + 1 - \sqrt{x}} \cdot \frac{x + 1}{x - \sqrt{x}} \right)^{10} $$ for $ x>1 $ is:

• A. 210
• B. 150
• C. 240
• D. 120

Hint:

To find the term independent of \( x \), equate the net power of \( x \) to 0 in the expanded expression.

Answer 1

The Correct Option is A

Simplify the given expression: \[ \left( \frac{(x + 1)^2}{x^{3/2} + 1 - \sqrt{x}} \cdot \frac{1}{x - \sqrt{x}} \right)^{10} \Rightarrow \left( \sqrt{x} + \frac{1}{\sqrt{x}} \right)^{10} \] Now use binomial expansion: \[ T_r = {10 \choose r} \cdot x^{\frac{10 - 2r}{2}},\quad \text{set power of } x = 0 \Rightarrow \frac{10 - 2r}{2} = 0 \Rightarrow r = 5 \] \[ T_5 = {10 \choose 5} = 210 \]