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

To simplify the expression, we first perform algebraic manipulation:\[\left( \frac{(x + 1)^2}{x^{3/2} + 1 - \sqrt{x}} \cdot \frac{1}{x - \sqrt{x}} \right)^{10}\Rightarrow \left( \sqrt{x} + \frac{1}{\sqrt{x}} \right)^{10}\]Next, apply the binomial expansion. The general term $T_r$ is given by:\[T_r = {10 \choose r} \cdot x^{\frac{10 - 2r}{2}}\]To find the constant term (where the power of $x$ is 0), we set the exponent of $x$ to zero:\[\frac{10 - 2r}{2} = 0 \Rightarrow r = 5\]The 5th term of the expansion, $T_5$, is:\[T_5 = {10 \choose 5} = 210\]