Q114 / 200 · Manipal (MET) 2013 · Mathematics
If $x, y \in[0,2 \pi]$ then the total number of ordered pairs $(x, y)$ satisfying $\sin x \cdot \cos y=1$ is equal to
- A$1$
- Correct answerB$3$
- C$5$
- D$7$
Solution
$\sin x \cdot \cos y=1$
$\Rightarrow \quad \sin x=1, \cos y=1$
or $\quad \sin x=-1, \cos y=-1$.
if $\quad \sin x=1, \cos y=1 \quad \Rightarrow x=\frac{\pi}{2}, y=0,2 \pi$
if $\quad \sin x=-1, \cos y=-1 \Rightarrow x=\frac{3 \pi}{2}, y=\pi$
Thus possible ordered pairs are
$\left(\frac{\pi}{2}, 0\right),\left(\frac{\pi}{2}, 2 \pi\right),\left(\frac{3 \pi}{2}, \pi\right)$
Hence, the correct answer is option: 'B'.