Question

The coefficient of the term independent of \( x \) in the expansion of \[ \left( \frac{x + 1}{x^{2/3} - x^{1/3} + 1} - \frac{x - 1}{x - x^{1/2}} \right)^{10} \] is:

• A. 210
• B. 105
• C. 70
• D. 112

Hint:

Simplify complex algebraic expressions using identities like sum of cubes and difference of squares.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
Simplify each fraction using algebraic identities.

Step 2: Detailed Explanation:
First term: \(\frac{x+1}{x^{2/3} - x^{1/3} + 1}\). Note that \(a^3 + b^3 = (a+b)(a^2 - ab + b^2)\). Let \(a = x^{1/3}, b=1\). Then \(x+1 = a^3 + 1^3 = (a+1)(a^2 - a + 1)\). So \(\frac{x+1}{x^{2/3} - x^{1/3} + 1} = a+1 = x^{1/3} + 1\). Second term: \(\frac{x-1}{x - x^{1/2}}\). Write \(x-1 = (\sqrt{x}-1)(\sqrt{x}+1)\). Denominator \(x - \sqrt{x} = \sqrt{x}(\sqrt{x}-1)\). So \(\frac{x-1}{x - \sqrt{x}} = \frac{(\sqrt{x}-1)(\sqrt{x}+1)}{\sqrt{x}(\sqrt{x}-1)} = \frac{\sqrt{x}+1}{\sqrt{x}} = 1 + x^{-1/2}\). Thus the expression becomes \((x^{1/3} + 1) - (1 + x^{-1/2}) = x^{1/3} - x^{-1/2}\). We need the term independent of \(x\) in \((x^{1/3} - x^{-1/2})^{10}\). General term: \(\binom{10}{r} (x^{1/3})^{10-r} (-x^{-1/2})^r = \binom{10}{r} (-1)^r x^{(10-r)/3 - r/2}\). Power = \(\frac{10-r}{3} - \frac{r}{2} = \frac{20 - 2r - 3r}{6} = \frac{20 - 5r}{6} = 0 \Rightarrow 20 - 5r = 0 \Rightarrow r = 4\). Coefficient = \(\binom{10}{4} (-1)^4 = 210\).

Step 3: Final Answer:
210, which corresponds to option (A).