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  • The constan term in<img alt="\left(x^{2 / 3}+4 x^{1 / 3}+4\right)^5\left(\frac{1}{x^{1 / 3}-1}+\frac{1}{x^{2 / 3}+x^{1 / 3}+1}\right)^{-9} is" src="https://entrancecorner.oncodecogs.com/gif.latex?%

The constan term in\left(x^{2 / 3}+4 x^{1 / 3}+4\right)^5\left(\frac{1}{x^{1 / 3}-1}+\frac{1}{x^{2 / 3}+x^{1 / 3}+1}\right)^{-9} is

Option: 1

152


Option: 2

160


Option: 3

168


Option: 4

None of these


Answers (1)

best_answer

\left(x^{2 / 3}+4 x^{1 / 3}+4\right)^5=\left(x^{1 / 3}+2\right)^{10}...(i)

\begin{aligned} & \left(\frac{1}{x^{1 / 3}-1}+\frac{1}{x^{2 / 3}+x^{1 / 3}+1}\right)^{-9}=\left(\frac{x^{2 / 3}+2 x^{1 / 3}}{x-1}\right)^{-9} \\ & =\frac{(x-1)^9}{x^3\left(x^{1 / 3}+2\right)^9} \end{aligned}....(ii)

The product of (i) and (ii) gives

\frac{\left(x^{1 / 3}+2\right)}{x^3}\left(x^9-\ldots\left(\begin{array}{l} 9 \\ 3 \end{array}\right) x^3-\left(\begin{array}{l} 9 \\ 2 \end{array}\right) x^2+9 x-1\right)\therefore \quad \text { The constant term is } 2\left(\begin{array}{l} 9 \\ 3 \end{array}\right)=2 \cdot 84=168

Posted by

Ajit Kumar Dubey

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