UPSC NDA 2 2019 Mathematics Solved Paper

Concept:The degree of a polynomial is the highest power of $x$ in its expression after simplification.Explanation:Given $f(x) = x^{2} + 2x - 5$ and $g(x) = 5x + 30$.First, compute $f[g(x)]$: substitute $g(x)$ into $f(x)$.$f[g(x)] = f(5x + 30) = (5x + 30)^{2} + 2(5x + 30) - 5$Expand: $(5x + 30)^{2} = 25x^{2} + 300x + 900$Then add $2(5x + 30) = 10x + 60$ and subtract $5$: $25x^{2} + 300x + 900 + 10x + 60 - 5 = 25x^{2} + 310x + 955$The highest power of $x$ is $2$, so $f[g(x)]$ is a polynomial of degree $2$.Thus, statement 1 (degree 3) is false.Now compute $g[g(x)]$: substitute $g(x)$ into itself.$g[g(x)] = g(5x + 30) = 5(5x + 30) + 30 = 25x + 150 + 30 = 25x + 180$The highest power of $x$ is $1$, so $g[g(x)]$ is a polynomial of degree $1$.Thus, statement 2 (degree 2) is false.Answer:D. Neither 1 nor 2