Question

Assertion($A$) : Period of $\frac{\sin x+\cos 2x}{\tan x+\tan 2x}$ is $2\pi$
Reason($R$) : Period of $f_1\left(x\right)+f_2\left(x\right)+f_3\left(x\right)+f_4\left(x\right)$ is $L.C.M$ of periods of $f_1\left(x\right),f_2\left(x\right),f_3\left(x\right)$, $f_4\left(x\right)$

• A. A is false, R is true.
• B. Both A and R are true and R is the correct explanation of A
• C. A is true, R is false
• D. A is false, R is false

Answer 1

Answer: (B)

Both A and R are true and R is the correct explanation of A

Period of $\sin x$ is $2\pi$

Period of $\cos 2x$ is $\pi$

Period of $\tan x$ is $\pi$

Period of $\tan 2x$ is $\frac{\pi }{2}$

The LCM of these periods is $2\pi$

So, each of the individual functions have a common multiple of $2\pi$ and thus the given function have the same value at $x$ and $x+2\pi$ respectively. Also, notice that the function $\sin x$ is periodic with at least $2\pi$ period.

The point is if a function is written in the form of periodic functions, then the LCM of all individual periods is the period of the whole function.

Thus the reason is correct in which $\frac{f_1\left(x\right)+f_2\left(x\right)}{f_3\left(x\right)+f_4\left(x\right)}$ where each function is periodic then the period of the total function is the L.C.M. of the individual periods.

Thus the correct answer is that both A and R are true and R is the correct explanation of A.