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:
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:
Show Hint
To find the term independent of \( x \), equate the net power of \( x \) to 0 in the expanded expression.
Updated On: Mar 30, 2026
- 210
- 150
- 240
- 120
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The Correct Option is A
Solution and Explanation
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:
\[
T_r = {10 \choose r} \cdot x^{\frac{10 - 2r}{2}},\quad \text{set power of } x = 0
\Rightarrow \frac{10 - 2r}{2} = 0 \Rightarrow r = 5
\]
\[
T_5 = {10 \choose 5} = 210
\]
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