Question

The function(s) which are not having $2\pi$ as it's period is/are

• A. $f\left(t\right)=\sin (2\pi t+\frac{\pi }{3})+2\sin (3\pi t+\frac{\pi }{4})+3\sin 5\pi t$
• B. $g\left(t\right)=\sin \frac{\pi }{3}t+\sin \frac{\pi }{4}t$
• C. $h\left(t\right)=\sin t+\cos 2t$
  
• D. $k\left(t\right)=\sin \frac{2\pi }{3}t+\sin \frac{\pi }{4}t$

Answer 1

Answer: (A), (B), (D)

$k\left(t\right)=\sin \frac{2\pi }{3}t+\sin \frac{\pi }{4}t$

As we know that if period of $\sin t=2\pi ,$ then period of $\sin (kt+p),k,p\in R=\frac{2\pi }{|k|}$

By using above property,
For $f\left(t\right)=\sin (2\pi t+\frac{\pi }{3})+2\sin (3\pi t+\frac{\pi }{4})+3\sin 5\pi t$
Period $=L.C.M.(\frac{2\pi }{2\pi },\frac{2\pi }{3\pi },\frac{2\pi }{5\pi })=2$

For $g\left(t\right)=\sin \frac{\pi }{3}t+\sin \frac{\pi }{4}t$
$\Rightarrow$ Period $=L.C.M.(\frac{2\pi }{\pi /3},\frac{2\pi }{\pi /4})=L.C.M.(6,8)=24$

For $h\left(t\right)=\sin t+\cos 2t$
Period $=L.C.M.(2\pi ,\frac{2\pi }{2})=2\pi$

For $k\left(t\right)=\sin \frac{2\pi }{3}t+\sin \frac{\pi }{4}t$
Period $=L.C.M.(\frac{2\pi }{2\pi /3},\frac{2\pi }{\pi /4})=L.C.M.(3,8)=24$