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

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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