UPSC NDA 2 2019 Mathematics Solved Paper

Concept:The absolute value $|x|$ changes the equation into two separate quadratic equations depending on whether $x \ge 0$ or $x < 0$.Explanation:Given equation: $x^{2} + 3|x| + 2 = 0$.Case 1: $x \ge 0$, then $|x| = x$.Equation becomes $x^{2} + 3x + 2 = 0$.Factor: $(x + 1)(x + 2) = 0$, so $x = -1$ or $x = -2$.Both roots are negative, but $x \ge 0$. Hence no solution in this case.Case 2: $x < 0$, then $|x| = -x$.Equation becomes $x^{2} - 3x + 2 = 0$.Factor: $(x - 1)(x - 2) = 0$, so $x = 1$ or $x = 2$.Both roots are positive, but $x < 0$. Hence no solution in this case.Since neither case gives a valid root, the equation has zero real roots.Answer:A. Zero