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
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
Updated On: Jul 5, 2022
- $\frac{3}{5}$
- $\frac{10}{3}$
- $\frac{-3}{10}$
- $\frac{-5}{3}$
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The Correct Option is C
Solution and Explanation
The middle term in the expansion of
$\left(1+ \alpha x\right)^{4} =T_{3} = ^{4}C_{2} \left(\alpha x\right)^{2} = 6 \alpha^{2} x^{2}$
The middle term in the expansion of
$ \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
$ 6\alpha^{2} = - 20 \alpha^{3} \Rightarrow \alpha = - \frac{3}{10} $
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Concepts Used:
Binomial Theorem
The binomial theorem formula is used in the expansion of any power of a binomial in the form of a series. The binomial theorem formula is
Properties of Binomial Theorem
- The number of coefficients in the binomial expansion of (x + y)n is equal to (n + 1).
- There are (n+1) terms in the expansion of (x+y)n.
- The first and the last terms are xn and yn respectively.
- From the beginning of the expansion, the powers of x, decrease from n up to 0, and the powers of a, increase from 0 up to n.
- The binomial coefficients in the expansion are arranged in an array, which is called Pascal's triangle. This pattern developed is summed up by the binomial theorem formula.