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 $-\frac{3}{10}$

Middle term of $(1+\alpha x{)}^4$ is $T_3.$
Its coefficient is ${}^4{C}_2(\alpha {)}^2=6{\alpha }^2.$
Middle term of $(1-\alpha x{)}^6$ is $T_4.$
Its coefficient is ${}^6{C}_3(-\alpha {)}^3=-20{\alpha }^3.$
According to question,
$6{\alpha }^2=-20{\alpha }^3$
$\Rightarrow 3{\alpha }^2+10{\alpha }^3=0$
$\Rightarrow {\alpha }^2(3+10\alpha )=0$
$\Rightarrow \alpha =-\frac{3}{10}$