Question

Assertion : ${\int }_0^{100\pi }|\sin x|dx=200$
Reason: $|\sin x|$ is a periodic function of period $2\pi$ and ${\int }_0^{\pi }|\sin x|dx$ is 4

• A. Both $A$ and $R$ are correct and $R$ is correct explanation of $A$$
• B. Both $A$ and $R$ are correct but $R$ is NOT the correct explanation of $A$$
• C. $A$ is true but $R$ is false
• D. $A$ is false but $R$ is true

Answer 1

The correct option is C $A$ is true but $R$ is false

The assertion is true ${\int }_0^{100\pi }|\sin x|dx=200$

$\sin x$ is a periodic function with period $2\pi$

$|\sin x|$ is also a periodic function with period $\pi$.

It is because $\sin x$ is negative in $\pi$ to $2\pi$

and when we apply mod it becomes positive, it can be observed from the diagram.

So, as $\sin x$ period is $\pi$,

${\int }_0^{100\pi }|\sin x|dx=100\times {\int }_0^{\pi }|\sin x|dx$

Now, ${\int }_0^{\pi }|\sin x|dx=-(\cos \pi -\cos 0)=2$

Therefore, $100\times {\int }_0^{\pi }|\sin x|dx=200$

Hence, the assertion is true but reason is false.