UPSC NDA I and II 2020 Math Solved Paper

Concept:For the function $y = \frac{1}{x-1}$, the domain excludes values that make the denominator zero, and the range excludes values that cannot be attained by $y$.Explanation:The denominator $x-1$ cannot be zero, so $x \neq 1$. Thus, domain = $\{x \in \mathbb{R} \mid x \neq 1\}$. To find the range, solve for $x$ in terms of $y$: $y = \frac{1}{x-1}$ implies $x-1 = \frac{1}{y}$, so $x = 1 + \frac{1}{y}$. For $x$ to be real, $y$ must not be zero (division by zero). Hence, all real $y$ except $y = 0$ are possible. So, range = $\{y \in \mathbb{R} \mid y \neq 0\}$, which is the set of points on the $y$-axis excluding $y = 0$. The graph is a rectangular hyperbola shifted right by 1 unit, with vertical asymptote $x = 1$ and horizontal asymptote $y = 0$. Thus, the correct statement matches option D. Answer:Option D: The domain is $\{x \in \mathbb{R} \mid x \neq 1\}$ and the range is the set of points on the $y$-axis except $y = 0$.