If \(x, y \in \mathrm{R}\) then match List-I with List-II
| List-I | List-II | ||
| (A) | \(|x| \lt |y|\) | (I) | iff \(x^2 \gt y^2\) |
| (B) | \(|x| \gt |y|\) | (II) | iff \(x^2 \le y^2\) |
| (C) | \(|x| \le |y|\) | (III) | iff \(x^2 \lt y^2\) |
| (D) | \(|x| \ge |y|\) | (IV) | iff \(x^2 \ge y^2\) |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A)-(III), (B)-(I), (C)-(II), (D)-(IV)
|x| and |y| are non-negative, and for non-negative numbers squaring keeps the order: if \(a, b \ge 0\), then \(a \lt b\) exactly when \(a^2 \lt b^2\) (because \(b^2 - a^2 = (b - a)(b + a)\)). Also \(|x|^2 = x^2\) and \(|y|^2 = y^2\). So each inequality between |x| and |y| holds exactly when the same inequality holds between \(x^2\) and \(y^2\).
(A) \(|x| \lt |y|\)
This matches (III) iff \(x^2 \lt y^2\).
(B) \(|x| \gt |y|\)
This matches (I) iff \(x^2 \gt y^2\).
(C) \(|x| \le |y|\)
This matches (II) iff \(x^2 \le y^2\).
(D) \(|x| \ge |y|\)
This matches (IV) iff \(x^2 \ge y^2\).
For example, x = −3, y = 2: \(|x| = 3 \gt 2 = |y|\) and \(x^2 = 9 \gt 4 = y^2\), although \(x \lt y\).
So the matching is (A)-(III), (B)-(I), (C)-(II), (D)-(IV).