Question

The middle term of \(\left(\sqrt{x} - \frac{1}{\sqrt{x}}\right)^6\) is

• A. -20
• B. -1
• C. 1
• D. None of these

Hint:

In symmetric expressions like \(\sqrt{x}\) and \(1/\sqrt{x}\), powers cancel in the middle term.

Answer 1

The Correct Option is A

Concept: For \((a+b)^n\), middle term when \(n=6\) is: \[ T_{4} \quad \left(\text{since } \frac{6}{2}+1=4\right) \]

Step 1:General term.
\[ T_{r+1} = \binom{6}{r} (\sqrt{x})^{6-r}\left(-\frac{1}{\sqrt{x}}\right)^r \]

Step 2:Middle term (r = 3).
\[ T_4 = \binom{6}{3} (\sqrt{x})^{3}\left(-\frac{1}{\sqrt{x}}\right)^3 \]

Step 3:Simplify.
\[ = 20 \cdot x^{3/2} \cdot (-1)^3 x^{-3/2} \] \[ = 20 \cdot (-1) \cdot x^{0} = -20 \]