Question:
If $(5.55)^{x}=(0.555)^{y}=1000,$ then the value of $\frac{1}{x}-\frac{1}{y}$ is
- 3
- 1
- $\frac{1}{3}$
- $\frac{2}{3}$
Correct Answer: Option: 3
We have ,
$\begin{align} & {{(5.55)}^{x}}=1000 \\ & \Rightarrow {{(5.55)}^{x}}={{10}^{3}} \\ \end{align}$Taking log both the side we get
$\begin{align} & x{{\log }_{10}}(5.55)=3 \\ & \Rightarrow {{\log }_{10}}(5.55)=\frac{3}{x} \\ & \Rightarrow {{\log }_{10}}(10\times 0.555)=\frac{3}{x} \\ & \Rightarrow {{\log }_{10}}(0.555)+1=\frac{3}{x}...(1) \\ \end{align}$Also, we have been given
${{(0.555)}^{y}}=1000$
Taking log both the side
$\begin{align} & y{{\log }_{10}}(0.555)=3 \\ & \Rightarrow {{\log }_{10}}(5.55)=\frac{3}{y}...(2) \\ \end{align}$From (1) and (2)
$\begin{align} & \frac{3}{y}+1=\frac{3}{x} \\ & \Rightarrow \frac{1}{x}-\frac{1}{y}=\frac{1}{3} \\ \end{align}$CAT Online Course
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