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Assertion (A): $\int_{\frac{\pi}{2}}^{\frac{3 \pi}{2}}[2 \sin x] d x=0$, where [.] denotes the greatest integer function
Reason (R) : $2 \sin x$ is a decreasing function in $\left[\frac{\pi}{2}, \frac{3 \pi}{2}\right]$
Options:
Reason (R) : $2 \sin x$ is a decreasing function in $\left[\frac{\pi}{2}, \frac{3 \pi}{2}\right]$
Solution:
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Verified Answer
The correct answer is:
$\mathrm{A}$ is false, $\mathrm{R}$ is true
$\begin{aligned} & \text { (A) } \int_{\frac{\pi}{2}}^{\frac{3 \pi}{2}}[2 \sin x] d x=2 \int_{\frac{\pi}{2}}^{\frac{3 \pi}{2}}[\sin x] d x \\ & =2\left[\int_{\frac{\pi}{2}}^\pi 0 \mathrm{dx}+\int_\pi^{\frac{\pi}{2}}(-1) \mathrm{dx}\right] \\ & \end{aligned}$
$$
\begin{aligned}
& =2\left([-\mathrm{x}]_\pi^{\frac{\pi}{2}}\right)=2\left[-\frac{\pi}{2}+\pi\right]=2 \times \frac{\pi}{2} \\
& \Rightarrow \int_{\frac{\pi}{2}}^{\frac{3 \pi}{2}}[2 \sin x] \mathrm{dx}=\pi \neq 0
\end{aligned}
$$
$\therefore$ A is false
(R) From the graph of $2 \sin x$, it is clear that $2 \sin x$ is decreasing function in the interval $\left[\frac{\pi}{2}, \frac{3 \pi}{2}\right]$
$$
\begin{aligned}
& =2\left([-\mathrm{x}]_\pi^{\frac{\pi}{2}}\right)=2\left[-\frac{\pi}{2}+\pi\right]=2 \times \frac{\pi}{2} \\
& \Rightarrow \int_{\frac{\pi}{2}}^{\frac{3 \pi}{2}}[2 \sin x] \mathrm{dx}=\pi \neq 0
\end{aligned}
$$
$\therefore$ A is false
(R) From the graph of $2 \sin x$, it is clear that $2 \sin x$ is decreasing function in the interval $\left[\frac{\pi}{2}, \frac{3 \pi}{2}\right]$
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