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  4. The coefficient of the middle term in the binomial expansion in powers of x of (1+α x)4 and of (1-α x)6 is the same if α equals

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Q. 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

Binomial Theorem

Solution:

$T _3$ in $(1+\alpha x )^4 \quad={ }^4 C _2 \alpha^2 x ^2 ; T _4$ in $(1-\alpha x )^6={ }^6 C _3(-\alpha x )^3=-{ }^6 C _3 \alpha^3 x ^3$
hence ${ }^4 C _2 \alpha^2=-{ }^6 C _3 \alpha^3$
$\alpha=-\frac{{ }^4 C _2}{{ }^6 C _3} ; \alpha=-\frac{6}{20} ; \alpha=-\frac{3}{10}$