Question

Statement $I$ Period of 2 ${\sin }^{3}x+5{\cos }^{3}x$ is $2\pi$
Statement $II$. Period of $\tan \frac{\pi x}{3}+\cos ec\frac{\pi x}{5}$ is $10$
Which of the above statements is correct?

• A. only $I$
• B. only $II$
• C. Both $I$ and $II$
• D. Neither $I$ nor $II$

Answer 1

The correct option is A only $I$

Concept - 1

$\to$ [$\because$ By property of periodic function. If period of g(x) is $T_1$ and period of m(x) is $T_2$, then period of g(x)+m(x) is LCM of $(T_1,T_2)$]

I. Let $f\left(x\right)=2{\sin }^{3}x+5{\cos }^{3}x$

$=2\left[\frac{3\sin x-\sin 3A}{4}\right]+5\left[\frac{\cos 3x+3\cos x}{4}\right]$

$=\frac{3}{2}\sin x+\frac{15}{4}\cos x+\frac{5}{4}\cos 3x-\frac{\sin 3x}{2}$

$\therefore$ Period of $f\left(x\right)=LCMof(2\pi ,2\pi ,\frac{2\pi }{3},\frac{2\pi }{3})$ [concept -1]

$=2\pi$ True.

II. Period of $\tan \frac{\pi x}{3}+\cos ec\frac{\pi x}{5}is10$

$\Rightarrow LCMof(3,10)=30$ [Period of $\tan \frac{\pi x}{3}=\frac{\pi }{|\frac{\pi }{3}|}=3$ , Period of $cosec\frac{\pi x}{5}=\frac{\pi }{|\frac{\pi }{5}|}=10$]