Q230 / 269 · NDA 2025 (Phase 1) · Mathematics
Consider the following for the two (02) items that follow:
Let the function $y = (1 - \cos x)^{-1}$, where $x \neq 2n\pi$ and $n$ is an integer.
What is the range of the function?
Let the function $y = (1 - \cos x)^{-1}$, where $x \neq 2n\pi$ and $n$ is an integer.
What is the range of the function?
- A$[0, \infty)$
- Correct answerB$[0.5, \infty)$
- C$[1, \infty)$
- D$(-\infty, 0.5]$
Solution
Given function is $y = (1 - \cos x)^{-1} = \dfrac{1}{1 - \cos x}$
The range of $\cos x$ is $[-1, 1]$.
Since $x \neq 2n\pi$, $\cos x \neq 1$.
Therefore, $-1 \le \cos x < 1$
Multiplying by $-1$, we get $-1 < -\cos x \le 1$
Adding $1$, we get $0 < 1 - \cos x \le 2$
Taking the reciprocal, we get $\dfrac{1}{1 - \cos x} \ge \dfrac{1}{2}$
Thus, $y \ge 0.5$
The range of the function is $[0.5, \infty)$.
The range of $\cos x$ is $[-1, 1]$.
Since $x \neq 2n\pi$, $\cos x \neq 1$.
Therefore, $-1 \le \cos x < 1$
Multiplying by $-1$, we get $-1 < -\cos x \le 1$
Adding $1$, we get $0 < 1 - \cos x \le 2$
Taking the reciprocal, we get $\dfrac{1}{1 - \cos x} \ge \dfrac{1}{2}$
Thus, $y \ge 0.5$
The range of the function is $[0.5, \infty)$.
Hence, the correct answer is option: 'B'.