Match List-I with List-II
| List-I | List-II | ||
| (A) | \( y=x^{2}+1 \) | (I) | Increasing on \( (0,\infty) \) |
| (B) | \( y=|x-1| \) | (II) | Decreasing on \( (-\infty,1) \) |
| (C) | \( y=1-|x| \) | (III) | Increasing on \( (-\infty,0) \) |
| (D) | \( y=1-2x \) | (IV) | Decreasing on \( (-\infty,\infty) \) |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A) - (I), (B) - (II), (C) - (III), (D) - (IV)
(A) \( y'=2x\gt 0 \) for x > 0, so it is increasing on \( (0,\infty) \). This matches (I).
(B) For x < 1, \( y=1-x \), so it is decreasing on \( (-\infty,1) \). This matches (II).
(C) For x < 0, \( y=1+x \), so it is increasing on \( (-\infty,0) \). This matches (III).
(D) The slope is −2 everywhere, so it is decreasing on \( (-\infty,\infty) \). This matches (IV).
So the matching is (A) - (I), (B) - (II), (C) - (III), (D) - (IV).
Note: Some List-I items fit more than one List-II entry. The function \( y=1-2x \) in (D) is decreasing on \( (-\infty,\infty) \), so it is also decreasing on \( (-\infty,1) \): (D) fits both (II) and (IV). The question can still be solved because (A) fits only (I), (B) fits only (II) and (C) fits only (III), and Option (3) is the only option containing (A)-(I), (B)-(II) and (C)-(III).