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Question

If the third term in the expansion of
$${ \left[ (1/x)+{ x }^{ \log _{ 10 }{ x }  } \right]  }^{ 5 }\quad $$ is $$1000$$, then $$x$$ is equal to


A
100
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B
10
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C
1
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D
110
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Solution

The correct options are
A $$100$$
D $$\dfrac {1}{\sqrt {10}}$$
$$T_{3}=\:^{5}C_{2}x^{-3+2log_{10}x}$$

$$=10x^{-3+2log_{10}x}$$

$$=10^{3}$$

Hence
$$x^{-3+2\log_{10}x}=10^{2}$$

Taking $$log_{10}$$ on both sides we get

$$\Rightarrow \log(x)(2\log(x)-3)=2$$

$$\Rightarrow 2\log(x)^{2}-3\log(x)-2=0$$

$$\Rightarrow 2\log(x)^{2}-4\log(x)+\log(x)-2=0$$

$$\Rightarrow (\log(x)-2)(2\log(x)+1)=0$$

$$\Rightarrow \log(x)=2$$  and $$\log(x)=\dfrac{-1}{2}$$

$$\therefore x=100$$ and $$x=\dfrac{1}{\sqrt{10}}$$

Mathematics

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