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  • The coefficient of the middle term in binomial expansion in power of x of <img alt="(1+\alpha x)^{4} \text { and }(1-\alpha x)^{6}" src="http://entrancecorner.oncodecogs.com/gif.late

The coefficient of the middle term in binomial expansion in power of x of (1+\alpha x)^{4} \text { and }(1-\alpha x)^{6} is the same if \alpha equals

Option: 1

= 3/10


Option: 2

= -3/5


Option: 3

= 3/10


Option: 4

= -3/10


Answers (1)

best_answer

 

 

Middle Term -

The middle term in the expansion (x + y)n, depends on the value of 'n'. 

 

Case 1 When 'n' is even

If n is even, and number of terms in the expansion is n + 1, so n +1 is odd number therefore only one middle term is obtained which is 

\left(\frac{\mathrm{n}}{2}+1\right)^{\mathrm{th}}term.

It is given by

\mathrm{T} _{\frac{\mathrm{n}}{2}+1}=\left(\begin{array}{c}{\mathrm{n}} \\ {\frac{\mathrm{n}}{2}}\end{array}\right) \mathrm{X}^{\frac{\mathrm{n}}{2}} \mathrm{y}^{\frac{\mathrm{n}}{2}} .

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We need to find T_3\; \text{and}\;T_4 terms of the respective expansion (1+\alpha x)^{4} \text { and }(1-\alpha x)^{6} and then equate them,

\\T_3 \text { of }(1+\alpha x)^{4} = T_4 \text { of }(1-\alpha x)^{6}\\ ^4C_2(\alpha )^2=^6C_3(-\alpha )^3\\ -\alpha=\frac{4C_2}{^6C_3}\\ -\alpha=\frac{4!}{2!2!}\cdot\frac{3!3!}{6!}\\\alpha=-\frac{3}{10}

Posted by

Nehul

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