Question

The coefficient of the middle term in the binomial expansion in powers of x of $(1+\alpha x{)}^4$ and of $(1-\alpha x{)}^6$ is the same if $\alpha$ equals -

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

Answer 1

The correct option is C $-\frac{3}{10}$

The middle term in the expansion of ${(1+\alpha x)}^n$ is ${(\frac{n}{2}+1)}^{th}$ term.

General term in the expansion of ${(1+\alpha x)}^4$ is ${4_C}_r{\alpha }^r{x}^r$

General term in the expansion of ${(1-\alpha x)}^6$ is ${6_C}_r{\alpha }^r{x}^r$

The middle term in the expansion of ${(1+\alpha x)}^4$ is ${(\frac{4}{2}+1)}^{th}=3^{rd}$ term.

The middle term in the expansion of ${(1-\alpha x)}^6$ is ${(\frac{6}{2}+1)}^{th}=4^{th}$ term.

Given$:{4_C}_2{\alpha }^2=-{6_C}_3{\alpha }^3$

$\Rightarrow \alpha =\frac{-{4_C}_2}{{6_C}_3}$

$=\frac{-\frac{4\times 3}{2\times 1}}{\frac{6\times 5\times 4}{3\times 2\times 1}}$

$=\frac{-6}{20}=\frac{-3}{10}$