NDA 1 2025 Maths, question 81

Topic: Differential calculus

Passage · Q81 to Q82

Let the function $y = \left( 1 - \cos x \right)^{- 1}$ , where $x \neq 2n\pi$ and $n$ is an integer.

Q81

What is the range of the function?

(a) $\lbrack 0,\infty)$
(b) $\lbrack 0.5,\infty)$ Correct answer
(c) $\lbrack 1,\infty)$
(d) $( - \infty,0.5\rbrack$

Solution.

Answer: (b) $\lbrack 0.5,\infty)$

Because $- 1 \leq \cos x < 1$ on the allowed domain, $0 < 1 - \cos x \leq 2$ .
Taking reciprocals gives $y \geq 1/2$ .
The minimum $1/2$ occurs at $x = (2n + 1)\pi$ .
As $x$ approaches an excluded multiple of $2\pi$ , $y$ becomes arbitrarily large; every value between is attained.