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Question

Mathematics Question on Binomial theorem

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

A

35\frac{3}{5}

B

103\frac{10}{3}

C

−310\frac{-3}{10}

D

−53\frac{-5}{3}

Answer

−310\frac{-3}{10}

Explanation

Solution

The middle term in the expansion of (1+αx)4=T3=4C2(αx)2=6α2x2\left(1+ \alpha x\right)^{4} =T_{3} = ^{4}C_{2} \left(\alpha x\right)^{2} = 6 \alpha^{2} x^{2} The middle term in the expansion of (1−αx)6=T4=6C3(−αx)3=−20α3x3 \left(1-\alpha x\right)^{6} =T_{4} = ^{6}C_{3} \left(-\alpha x\right)^{3} = - 20 \alpha^{3} x^{3} According to the question 6α2=−20α3⇒α=−3106\alpha^{2} = - 20 \alpha^{3} \Rightarrow \alpha = - \frac{3}{10}