Question

The number of ordered pairs $(x,y),$ satisfying $|x|+|y|=3$ and $\sin \left(\frac{\pi x^2}{3}\right)=1$ is equal to

• A. $4$
• B. $12$
• C. $8$
• D. $16$

Answer 1

The correct option is C $8$

Given : $\sin \left(\frac{\pi }{3}{x}^2\right)=1$
$\Rightarrow \frac{\pi }{3}{x}^2=2n\pi +\frac{\pi }{2};n\in Z$
$\Rightarrow \frac{\pi }{3}{x}^2=(4n+1)\frac{\pi }{2};n\in Z$
$\Rightarrow x^2=\frac{3}{2}(4n+1);n\in Z$
Only $n=0$ and $n=1$ is possible. $(\because |x|+|y|=3)$
$x^2=\frac{3}{2}$ or $x^2=\frac{15}{2}$
$\therefore x=\pm \sqrt{\frac{3}{2}}$ or $x=\pm \sqrt{\frac{15}{2}}$
For each value of $x,$ we will get $2$ values of $y$.
Hence, $8$ ordered pairs.