UPSC NDA 2 2019 Mathematics Solved Paper

Concept:If $f(x)$ and $g(x)$ are given, then $g[f(x)] = g(f(x))$. We set it to zero and solve for $x$ to find the roots.Explanation:Given $f(x) = x^{2} + 2x - 5$ and $g(x) = 5x + 30$.First, substitute $f(x)$ into $g(x)$: $g[f(x)] = g(x^{2} + 2x - 5) = 5(x^{2} + 2x - 5) + 30$.Simplify: $5x^{2} + 10x - 25 + 30 = 5x^{2} + 10x + 5$.Set equal to zero: $5x^{2} + 10x + 5 = 0$.Divide the entire equation by $5$: $x^{2} + 2x + 1 = 0$.Factor the perfect square: $(x + 1)^{2} = 0$.Take square root: $x + 1 = 0$ → $x = -1$ (double root).Since the square yields two identical factors, the two roots are both $-1$.Answer:The roots are $-1$ and $-1$. Thus, option B is correct.