Question

$\text{Statement I}:\underset{0}{\overset{4\pi }{\int }}[|\sin x|+|\cos x|]dx$, (where $[\cdot ]$ denotes G.I.F.) equals $8\pi$.

$\text{Statement II}:$ If $f\left(x\right)=|\sin x|+|\cos x|$, then $1\le f\left(x\right)\le \sqrt{2}.$

• A. $\text{Statement I}$ is true, $\text{Statement II}$ is true. $\text{Statement II}$ is a correct explanation for $\text{Statement I}$
• B. $\text{Statement I}$ is true, $\text{Statement II}$ is true. $\text{Statement II}$ is not a correct explanation for $\text{Statement I}$
• C. $\text{Statement I}$ is true, $\text{Statement II}$ is false
• D. $\text{Statement I}$ is false, $\text{Statement II}$ is true.

Answer 1

The correct option is D $\text{Statement I}$ is false, $\text{Statement II}$ is true.

We know that $1\le |\sin x|+|\cos x|\le \sqrt{2}$
$\Rightarrow [|\sin x|+|\cos x|]=1$
$I=\underset{0}{\overset{4\pi }{\int }}[|\sin x|+|\cos x|]dx=\underset{0}{\overset{4\pi }{\int }}1\cdot dx=4\pi$
Thus, $\text{Statement I}$ is false, $\text{Statement II}$ is true.