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Question

The value of $$x$$ in the expression $${ \left( x+{ x }^{ \log _{ 10 }{ x }  } \right)  }^{ 5 }$$, if the third term in the expansion is $$1,000,000$$, is


A
10,103/2
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B
100 or 103/2
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C
10 or 105/2
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D
None of these
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Solution

The correct option is C $$10$$ or $${ 10 }^{ { -5 }/{ 2 } }$$
$$\log { x } $$ is defined only when $$x>0$$
Now, the $${ 3 }^{ rd }$$ term in the expansion
$${ T }_{ 2+1 }=^{ 5 }{ { C }_{ 2 } }\cdot { x }^{ 5-2 }\cdot { \left( { x }^{ \log _{ 10 }{ x }  } \right)  }^{ 2 }=1,000,000$$         ....(Given)
$$\Rightarrow { x }^{ 3+2\log _{ 10 }{ x }  }={ 10 }^{ 5 }$$
Taking logarithm of both sides, we get
$$\left( 3+2\log _{ 10 }{ x }  \right) \cdot \log _{ 10 }{ x } =5$$
$$\Rightarrow 2{ y }^{ 2 }+3y-5=0$$
where $$\log _{ 10 }{ x } =y$$
$$\Rightarrow \left( y-1 \right) \left( 2y+5 \right) =0$$
$$\Rightarrow y=1$$ or $${ -5 }/{ 2 }$$
$$\Rightarrow \log _{ 10 }{ x } =1$$ or $${ -5 }/{ 2 }$$
$$\Rightarrow x={ 10 }^{ 1 }=10$$ or $${ 10 }^{ { -5 }/{ 2 } }$$

Mathematics

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