Question

If $\theta \in[0,5 \pi]$ and $r \in R$ such that $2 \sin \theta= r ^{4}-2 r ^{2}+3$ then the maximum number of values of the pair $( r , \theta)$ is _______

Answer 1

$2 \sin \theta=r^{4}-2 r^{2}+3 $
$\Rightarrow 2 \sin \theta=\left(r^{2}-1\right)^{2}+2$
This is possible only when $\sin \theta=1$ and $r^{2}=1$
or $r =\pm 1$.
So, $\theta=\pi / 2,5 \pi / 2,9 \pi / 2$
$\therefore $ Number of values of the pair $(r, \theta)=6$