Question

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

• A. $8$
• B. $10$
• C. $6$
• D. $4$

Answer 1

Answer: (C)

$6$

$2\sin \theta =r^4-2r^2+3$

Maximum value of $\sin \theta$ can be 1.

$2\sin \theta =(r^2-1{)}^2+2$

So, right hand side of the equation can be $\ge 2$.

But as it is equal to $2\sin \theta$ and maximum of $\sin \theta$ is $1$. So, it's value is $2$.

$\therefore r^2-1=0$

or, $r=\pm 1$

$\sin \theta =2=\sin \frac{\pi }{2}$

$\theta =n\pi +(-1{)}^n\frac{\pi }{2}$

$\theta =\frac{\pi }{2},\frac{5\pi }{2},\frac{9\pi }{2}$( From the range).

Values of $r=2$

Values of $\theta =3$

$\therefore$ total number of possible pairs$=3\times 2=6$