Question

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

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

Answer 1

The correct option is C

$-\left(\frac{3}{10}\right)$

Explanation for correct option:

Finding the value of $\alpha$so that coefficient of the middle term of given expressions is equal.

Given expressions are${(1+\alpha x)}^4$and ${(1-\alpha x)}^6$

Now, the coefficient of the middle term in the expansion ${(1+\alpha x)}^4$ is

$T_3={}^{4}C_2{\left(\alpha \right)}^2$

And coefficient of the middle term in the expansion ${(1-\alpha x)}^6$ is

$T_4={}^{6}C_3{(-\alpha )}^3$

Now for middle terms to be equal-

${}^{4}C_2{\left(\alpha \right)}^2={}^{6}C_3{(-\alpha )}^3$

$$\begin{gathered} \Rightarrow 6{\alpha }^2=-20{\alpha }^3 \\ \Rightarrow 6{\alpha }^2+20{\alpha }^3=0 \\ \Rightarrow \alpha =\frac{-3}{10} \end{gathered}$$

Hence, Option(C) is correct.