UPSC NDA I 2019 Mathematics Solved Paper

Concept:The absolute value function splits the equation into two separate cases, each restricted to a specific sign of $x$.Explanation:Case 1: $x \ge 0$.Here $\left|x\right| = x$.The equation becomes $x^2 + 9x + 20 = 0$.Factor: $(x+4)(x+5) = 0$ gives $x = -4$ or $x = -5$.Both solutions are negative, so they do not satisfy $x \ge 0$.Thus no real roots from this case.Case 2: $x < 0$.Here $\left|x\right| = -x$.The equation becomes $x^2 - 9x + 20 = 0$.Factor: $(x-4)(x-5) = 0$ gives $x = 4$ or $x = 5$.Both solutions are positive, so they do not satisfy $x < 0$.Thus no real roots from this case either.Since neither case yields a valid real root, the original equation has no real solutions.Answer:Zero real roots (Option A).