Match List-I with List-II
Where \(\mathrm{R}\) is set of real numbers
| List-I | List-II | ||
| (A) | \(\sin x\) is continuous on: | (I) | \(\mathrm{R}-\{0\}\) |
| (B) | \(\tan x\) is continuous on: | (II) | \(\mathrm{R}\) |
| (C) | \(\cot x\) is continuous on: | (III) | \(\mathrm{R}-\{n\pi:n\in \mathrm{Z}\}\) |
| (D) | \(x^{-n}, n\in \mathrm{N}\) is continuous on: | (IV) | \(\mathrm{R}-\left\{(2n+1)\displaystyle\frac{\pi}{2}:n\in \mathrm{Z}\right\}\) |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A) - (II), (B) - (IV), (C) - (III), (D) - (I)
A quotient of continuous functions is continuous wherever the denominator is not zero.
(A) \(\sin x\) is continuous on:
This matches (II) R. \(\sin x\) is defined and continuous for every real x.
(B) \(\tan x\) is continuous on:
This matches (IV). \(\tan x = \frac{\sin x}{\cos x}\), and \(\cos x = 0\) exactly at
\( \displaystyle x = (2n + 1)\frac{\pi}{2},\quad n \in \mathrm{Z} \)
(C) \(\cot x\) is continuous on:
This matches (III). \(\cot x = \frac{\cos x}{\sin x}\), and \(\sin x = 0\) exactly at \(x = n\pi,\ n \in \mathrm{Z}\).
(D) \(x^{-n}, n\in \mathrm{N}\) is continuous on:
This matches (I) R − {0}. \(x^{-n} = \frac{1}{x^n}\), and \(x^n = 0\) only at x = 0.
So the matching is (A) - (II), (B) - (IV), (C) - (III), (D) - (I).