First slide

Binomial theorem for positive integral Index

Question

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

Moderate

A

$- \frac{3}{10}$
B

$\frac{10}{3}$
C

$- \frac{5}{3}$
D

$\frac{3}{5}$

Solution

Middle term in the expansion of ${(1 + a x)}^{4}$ is $^{4} C_{2} (α x)^{2} = 6 α^{2} x^{2}$
and the middle term in the expansion of $(1 - α x)^{6}$ is $^{6} C_{3} (- α x)^{3} = - 20 α^{3} x^{3}$ .
We are given $6 α^{2} = - 20 α^{3} \Rightarrow α = 0$ or $α = - \frac{3}{10}$

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