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
B
10
C
1
D
110

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

Suggest Corrections

0

Similar questions
View More

People also searched for
View More