Let f(x) = cos(ax) + sin(x) be a periodic fun | vedclass

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(B) function $f(x) = f_1(x) + f_2(x)$ is periodic if both $f_1(x)$ and $f_2(x)$ are periodic and the ratio of their periods is a rational number.Here,

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Rational

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(B) function $f(x) = f_1(x) + f_2(x)$ is periodic if both $f_1(x)$ and $f_2(x)$ are periodic and the ratio of their periods is a rational number.Here,$f_1(x) = \cos(ax)$ has a period $T_1 = \frac{2\pi}{|a|}$ and $f_2(x) = \sin(x)$ has a period $T_2 = 2\pi$.For $f(x)$ to be periodic,the ratio $\frac{T_1}{T_2} = \frac{2\pi/|a|}{2\pi} = \frac{1}{|a|}$ must be a rational number.If $\frac{1}{|a|}$ is a rational number,then $|a|$ must be a rational number,which implies $a$ is a rational number.Therefore,$a$ must be a rational number. Hence,option $B$ is correct.

Let $f(x) = \cos(ax) + \sin(x)$ be a periodic function. Then $a$ must be

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