Question

Assertion :The function $f\left(x\right)=\cos x+\cos ax$ is a periodic function and $a\in R$. Reason: The number $a$ is a rational number and sum of two periodic functions is periodic.

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

The period of $\cos x$ & $\cos ax$ are $2\pi$ and $\frac{2\pi }{\left|a\right|}$ respectively

$\therefore$ Period of $\cos x+\cos ax=L.C.Mof(\frac{2\pi }{1},\frac{2\pi }{\left|a\right|})$

$=\frac{L.C.Mof(2\pi ,2\pi )}{H.C.Fof(1,\left|a\right|)}$

$=\frac{2\pi }{\lambda }$ (where $\lambda$ is H.C.F of 1,$\left|a\right|$)

Since
$\lambda$ is H.C.F of 1 & $\left|a\right|$ so
$\frac{1}{\lambda }$ & $\frac{\left|a\right|}{\lambda }$ should be an integer.

Say $\frac{1}{\lambda }=m$ & $\frac{\left|a\right|}{\lambda }=n$ then we have

$\left|a\right|=\frac{n}{m}wheren,m\in I$ & $H.C.F(m,n)=1$

$\therefore a=a$ rational number.