TRB 2006 - Question 9

 Is $x (t) = cos \Big (\frac{1}{3} t \Big ) + sin \Big (\frac{1}{4} t \Big )$ periodic?

If so, its period is

 a) not periodic, can't find its period

b) periodic, $24 \pi$

c) periodic, $3/4 \pi$

d) periodic, $12 \pi $

Correct Answer: Option B

$x (t) = cos \Big (\frac{1}{3} t \Big ) + sin \Big (\frac{1}{4} t \Big )$

Frequency and time period be, 

$2\pi f_1=1/3$ and $2\pi f_2=1/4$

$f_1=1/6\pi$ and $f_2=1/8\pi$

$T_1=6\pi$ and $T_2=8\pi$

To find sum of sinusoids periodic or not, 

$$\frac{T_1}{T_2}=\frac{6\pi}{8\pi}=\frac{6}{8}$$

Since $6/8$ is a rational number, the given sum of sinusoids is periodic.

Fundamental period, $$T=lcm(6\pi, 8\pi)$$ $$T=24\pi$$