Sets, Relations & Functions
Functions
nta_abhyas_2025
Grade 11
Question:
$f(x) = \sin \frac{x}{6} + \cos \frac{7x}{10}$. Period of $\sin \frac{x}{6} = 6\pi$. Period of $\cos \frac{7x}{10} = \frac{20\pi}{7}$. $\sin a\theta, \cos a\theta$ are periodic with period $\frac{2\pi}{a}$. $\text{LCM}(6\pi, \frac{20\pi}{7}) = 60\pi$. Therefore, the period of $f(x) = 60\pi$. Hence, $n = 6$.
Step-by-Step Solution
Key Concept: The period of a sum of periodic functions is the LCM of their individual periods.
The period of $\sin \frac{x}{6}$ is $\frac{2\pi}{1/6} = 12\pi$. The period of $\cos \frac{7x}{10}$ is $\frac{2\pi}{7/10} = \frac{20\pi}{7}$. The period of $f(x)$ is the LCM of $12\pi$ and $\frac{20\pi}{7}$: $\text{LCM}(12\pi, \frac{20\pi}{7}) = \frac{\text{LCM}(12, 20)\pi}{\gcd(1, 7)} = 60\pi$. Since the period equals $n \times 10\pi$, we have $60\pi = n \times 10\pi$, giving $n = 6$.
Correct Answer: 6