UPSC NDA 2 2019 Mathematics Solved Paper

Concept:The period of a sum of two functions equals the LCM of their individual periods.Explanation:First, find the period of $g(x)=\sin\left(\frac{x}{4}\right)$.The standard period of $\sin x$ is $2\pi$.Here, the coefficient of $x$ is $\frac{1}{4}$, so period becomes $\frac{2\pi}{1/4}=8\pi$.Next, find the period of $h(x)=\cos\left(\frac{4x}{5}\right)$.The standard period of $\cos x$ is $2\pi$.The coefficient of $x$ is $\frac{4}{5}$, so period is $\frac{2\pi}{4/5}=2\pi\times\frac{5}{4}=\frac{5\pi}{2}$.Now, period of $f(x)=g(x)+h(x)$ is LCM of $8\pi$ and $\frac{5\pi}{2}$.LCM of fractions: $\frac{\text{LCM of numerators}}{\text{HCF of denominators}}$.Numerators: $8\pi$ and $5\pi$; LCM = $40\pi$.Denominators: $1$ and $2$; HCF = $1$.Thus, LCM = $\frac{40\pi}{1}=40\pi$.Answer:$40\pi$, which corresponds to option C.