Question

Statement $1$: The value of ${\cos }^{-1}\left(\cos 30\right)$ is $30-9\pi$.

Statement $2:30-9\pi \in [0,\pi ]$.

• A. STATEMENT $1$ is TRUE and STATEMENT $2$ is FALSE.
• B. Both the statements are TRUE, and STATEMENT $2$ is NOT the correct explanation of STATEMENT $1$.
• C. Both the statements are TRUE, and STATEMENT $2$ is the correct explanation of STATEMENT $1$.
• D. STATEMENT $1$ is FALSE and STATEMENT $2$ is TRUE.

Answer 1

The correct option is D STATEMENT $1$ is FALSE and STATEMENT $2$ is TRUE.

${\cos }^{-1}\left(\cos 30\right)={\cos }^{-1}\left[\cos (10\pi -30)\right]=10\pi -30(\because {\cos }^{-1}(\cos x)=x,x\in [0,\pi \left]\right)$
Hence, statement $1$ is False.

Clearly, $30-9\pi \in [0,\pi ]$
Hence, statement $2$ is True.