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

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Solution

The correct option is C

In $(1 + a x)^{4}$ and $(1 - a x)^{6}$ the power are even (4 and 6).So there will be only one middle term in both the expansion.The middle term in the expansion. The middle term in the expansion of $(1 + a x)^{4}$ is $(\frac{4 + 2}{2})^{t h}$ term or $3^{t h}$ term and $(\frac{6 + 2}{2})^{t h}$ term or $4^{t h}$ term in the expansion of $(1 - a x)^{6}$ .

$T_{3}$ in the expansion of $(1 + a x)^{4} =^{4} C_{2} (1)^{4 - 2} (a x)^{2}$

= $^{4} C_{2} a^{2} x^{2}$

$\Rightarrow$ The coefficient of the middle term = $^{4} C_{2} a^{2}$

$T_{4}$ in the expansion of $(1 - a x)^{6} =^{6} C_{3} (1)^{6 - 3} (- a x)^{3}$

= $-^{6} C_{3} a^{3} x^{3}$

$\Rightarrow$ The coefficient of the middle term = $-^{6} C_{3} a^{3}$

It is given that the coefficient are same

$\Rightarrow^{4} C_{2} a^{2} = -^{6} C_{3} a^{3}$

$\Rightarrow a = - (\frac{^{4} C_{2}}{^{6} C_{3}})$

= $- (\frac{3}{10})$

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