Question

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

• A. $7$
• B. $8$
• C. $9$
• D. $10$

Answer 1

Answer: (B), (C), (D)

The correct options are
B $8$
C $9$
D $10$

$\sin \left(\frac{\pi }{3}{x}^2\right)=1\frac{\pi }{3}{x}^2=2n\pi +\frac{\pi }{2};n\in I\frac{\pi }{3}{x}^2=(4n+1)\frac{\pi }{2}{x}^2=\frac{3}{2}(4n+1);n\in I$
only $n=0$ and $n=1$ is possible.
($\because |x|+|y|=3$)
$x^2=\frac{3}{2}$ or $x^2=\frac{15}{2}$
$\therefore x=\pm \sqrt{\frac{3}{2}}\text{or}x=\pm \sqrt{\frac{15}{2}}$
for each value of $x$ will get $2$ values of $y$, hence $8$ ordered pairs.