Question

The coefficient of the middle term in the binomial expansion in powers of \( x \) of \( (1 + \alpha x)^4 \) and \( (1 - \alpha x)^6 \) is the same, if \( \alpha \) is equal to

• A. \( -\frac{5}{3} \)
• B. \( \frac{10}{3} \)
• C. \( -\frac{3}{10} \)
• D. \( \frac{3}{5} \)

Hint:

Always check sign carefully in binomial with negative terms.

Answer 1

The Correct Option is A

Concept: Middle term: \[ T_{n/2+1} \]

Step 1: For \( (1+\alpha x)^4 \).
Middle term: \[ \binom{4}{2} \alpha^2 x^2 = 6\alpha^2 \]

Step 2: For \( (1-\alpha x)^6 \).
Middle term: \[ \binom{6}{3}(-\alpha)^3 x^3 = -20\alpha^3 \]

Step 3: Equate coefficients.
\[ 6\alpha^2 = -20\alpha^3 \Rightarrow \alpha = -\frac{5}{3} \]