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JEE Main 2018

JEE Main 2018 Mathematics question on Complex Numbers and Quadratic Equations

Let S={x∈R:x≥0S = \{x \in \mathbb{R} : x \geq 0 and 2∣x−3∣+x(x−6)+6=0}2|\sqrt{x} - 3| + \sqrt{x}(\sqrt{x} - 6) + 6 = 0\}. Then, SS
  1. A.
    is an empty set
  2. B.
    contains exactly one element
  3. C.
    contains exactly two elements
  4. D.
    contains exactly four elements
Show answer and solution

Correct option: C

Solution

Let t = √x ≥ 0. The equation becomes 2|t - 3| + t(t - 6) + 6 = 0, i.e., 2|t - 3| + t² - 6t + 6 = 0.

Note t² - 6t + 6 = (t-3)² - 3, so the equation is 2|t-3| + (t-3)² - 3 = 0. Let u = |t-3| ≥ 0: u² + 2u - 3 = 0 → (u+3)(u-1) = 0, giving u = 1 (since u ≥ 0).

So |t - 3| = 1, meaning t = 2 or t = 4. Both are non-negative, giving x = 4 or x = 16. S contains exactly two elements.

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