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Question: If the middle term in the expansion of \(\left( \frac{1}{x} + x^{\log_{2}^{x}} \right)^{5}\) is 40, ...

If the middle term in the expansion of (1x+xlog⁡2x)5\left( \frac{1}{x} + x^{\log_{2}^{x}} \right)^{5} is 40, then x equals-

A

1/2,21/\sqrt{2},2

B

2,4\sqrt{2},4

C

1/2,41/\sqrt{2},4

D

2,1/2\sqrt{2},1/\sqrt{2}

Answer

1/2,41/\sqrt{2},4

Explanation

Solution

Middle term 5C2(1x)3(xlog⁡2x)2=10x2log⁡2x−3=405C_{2}\left( \frac{1}{x} \right)^{3}\left( x^{\log_{2}^{x}} \right)^{2} = 10x^{2\log_{2}^{x - 3}} = 40

Ž Taking log of base 2 on both sides,

let log⁡2x=y\log_{2}^{x} = y

Ž (2y – 3) y = 2

Ž y = – 1/2 or 2

Ž x=12x = \frac{1}{\sqrt{2}}, 4