Q50 / 120 · VITEEE 2011 · Mathematics
If $\mathrm{D}$ is the set of all real $\mathrm{x}$ such that $1-e^{(1 / x)-1}$ is positive, then $\mathrm{D}$ is equal to
- A$(-\infty, 1]$
- B$(-\infty, 0)$
- C$(1, \infty)$
- Correct answerD$(-\infty, 0) \cup(1, \infty)$
Solution
$$
\begin{array}{l}
1-\mathrm{e}^{\frac{1}{x}-1}>0 \\
\Rightarrow \mathrm{e}^{\frac{1}{x}-1} < 1 \Rightarrow \frac{1}{x}-1 < \log 1 \\
\Rightarrow \frac{1}{x}-1 < 0 \Rightarrow \frac{1}{x} < 1 \\
\Rightarrow x \in(-\infty, 0) \cup(1, \infty)
\end{array}
$$
Hence, the correct answer is option: 'D'.