Question

If $f\left(x\right)=cos\left(ax\right)+\sin \left(x\right)$ is periodic, then $a$ must be

• A. Irrational
• B. Rational
• C. Positive real number
• D. None of these

Answer 1

The correct option is B

Rational

Explanation for the correct option:

Step 1: Write the period of the trigonometric functions :

All trigonometric functions are periodic functions. The time period is the length of a complete sine wave or one complete cycle of the curve.

Therefore, the period of the functions

$\therefore \sin \left(x\right)=\frac{2\pi }{1}\text{and}\cos \left(ax\right)=\frac{2\pi }{a}$

Step 2: Take Least common multiple (LCM)

$\begin{array}{l}=\frac{2\pi }{1},\frac{2\pi }{a}\\ =\frac{\text{LCM of}\{2\pi ,2\pi \}}{\text{HCF of}\{1,a\}}\\ =\frac{2\pi }{k}\end{array}$

Where the Highest common factor (HCF) $\frac{1}{k}$

$$\begin{gathered} \therefore \frac{1}{k}=\mathrm{integer}=\mathrm{q}\left(\mathrm{say}\right)\left[q\ne 0\right] \\ \mathrm{And},\frac{\mathrm{a}}{\mathrm{k}}=\mathrm{integer}=\mathrm{p}\left(\mathrm{say}\right) \end{gathered}$$

Now, $\frac{p}{q}=\frac{\frac{a}{k}}{\frac{1}{k}}$

$\begin{array}{l}\Rightarrow \frac{p}{q}=\frac{a}{k}\times \frac{k}{1}\\ \Rightarrow \frac{p}{q}=a\\ \therefore a=\text{rational number}\end{array}$

Hence, the correct option is (B).