The number of x [ ] [0, 2[ ] ] for which [| 2 ^4x + 18 ^2x - 2 ^4x + 18 ^2x | = 1] is :

Question

The number of x $$ \in $$ [0,   2$$\pi $$ ] for  which

$$\left| {\sqrt {2{{\sin }^4}x + 18{{\cos }^2}x}  - \sqrt {2{{\cos }^4}x + 18{{\sin }^2}x} } \right| = 1$$ is :

Accepted Answer

We have,

$$
\left|\sqrt{2 \sin ^4 x+18 \cos ^2 x}-\sqrt{2 \cos ^4 x+18 \sin ^2 x}\right|=1
$$

That is,

$$
f(x)=\left|\sqrt{2 \sin ^4 x+8 \cos ^2 x}-\sqrt{2 \cos ^4 x+18 \sin ^2 x}\right|
$$

Now, $f(x)=f\left(\frac{\pi}{2}-x\right) $

$\Rightarrow f(x)$ is symmetric about $x=\frac{\pi}{4}$.

If $f(x)$ has solution in $(0, \pi / 4)$, then in $(0,2 \pi)$, there are eight solutions.

Solutions · 1