Q110 / 160 · AP EAMCET 2023 (18 May Shift 2) · Mathematics

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 $\left[\frac{\pi}{2}, \frac{3 \pi}{2}\right]$

  • A
    Both $A$ and $R$ are true and $R$ is the correct explanation of $\mathrm{A}$
  • B
    Both $\mathrm{A}$ and $\mathrm{R}$ are true but $\mathrm{R}$ is not the correct explanation of $A$
  • C
    $\mathrm{A}$ is true, $\mathrm{R}$ is false
  • Correct answer
    D
    $\mathrm{A}$ is false, $\mathrm{R}$ is true

Solution

$\begin{aligned} & \text { (A) } \int_{\frac{\pi}{2}}^{\frac{3 \pi}{2}}[2 \sin x] d x=2 \int_{\frac{\pi}{2}}^{\frac{3 \pi}{2}}[\sin x] d x \\ & =2\left[\int_{\frac{\pi}{2}}^\pi 0 \mathrm{dx}+\int_\pi^{\frac{\pi}{2}}(-1) \mathrm{dx}\right] \\ & \end{aligned}$ $$ \begin{aligned} & =2\left([-\mathrm{x}]_\pi^{\frac{\pi}{2}}\right)=2\left[-\frac{\pi}{2}+\pi\right]=2 \times \frac{\pi}{2} \\ & \Rightarrow \int_{\frac{\pi}{2}}^{\frac{3 \pi}{2}}[2 \sin x] \mathrm{dx}=\pi \neq 0 \end{aligned} $$ $\therefore$ A is false (R) From the graph of $2 \sin x$, it is clear that $2 \sin x$ is decreasing function in the interval $\left[\frac{\pi}{2}, \frac{3 \pi}{2}\right]$

Hence, the correct answer is option: 'D'.

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