Question

The arrangement of following functions in descending order of their periods:
A) $3\cos \frac{x}{2}+\sin \frac{x}{2}$
B) $3\tan \frac{x}{3}-\cot \frac{x}{3}$
C) $3{\sin }^{3}x+4{\cos }^{3}x$
D) ${\sin }^{2}x-2{\cos }^{2}x$

• A. $C,A,B,D$
• B. $D,C,A,B$
• C. $A,B,C,D$
• D. $B,D,C,A$

Answer 1

Answer: (C)

$A,B,C,D$

(A) Period of $\cos \frac{x}{2}$ is $\frac{2\pi }{\frac{1}{2}}=4\pi$

Period of$\sin \frac{x}{2}$ is $\frac{2\pi }{\frac{1}{2}}=4\pi$

$\therefore$ Period of $A$ is $4\pi$

(B) Period of $\tan \frac{x}{3}$ is $\frac{\pi }{\frac{1}{3}}=3\pi$

Period of $\cot \frac{x}{3}$ is $\frac{\pi }{\frac{1}{3}}=3\pi$

Hence, period $B$ is $3\pi$

(C) period of ${\sin }^{3}x$ is $2\pi$ as period of ${\left(\sin x\right)}^{2m}$ is $\pi$ and ${\left(\sin x\right)}^{2m+1}$ is $2\pi$

Period of ${\cos }^{3}x$ is $2\pi$ as period of ${\left(\cos x\right)}^{2m}$ is $\pi$ and ${\left(\cos x\right)}^{2m+1}$ is $2\pi$

$\therefore$ period of $C$ is $2\pi$

(D) Period of ${\sin }^{2}x$ is $\pi$

Period of ${\cos }^{2}x$ is $\pi$

$\therefore$ period of $D$ is $\pi$

Hence, sequired ordes is $ABCD.$