Question

If $\sin \alpha$ is a root of the equation $25x^2+5x-12=0$ and $x$ belongs to the range of the function $f\left(x\right)=-|\cos x|,$ then the value of $25|\sin 2\alpha |$ is

Answer 1

$25x^2+5x-12=0$
$\Rightarrow x=\frac{-5\pm \sqrt{1225}}{50}$
$\Rightarrow x=\frac{-5\pm 35}{50}=\frac{3}{5}$ or $-\frac{4}{5}$
Range of $f$ is $[-1,0]$
$\therefore \sin \alpha =-\frac{4}{5}$
$|\sin 2\alpha |=2|\sin \alpha ||\cos \alpha |=2\left(\frac{4}{5}\right)\left(\frac{3}{5}\right)$
$\therefore |\sin 2\alpha |=\frac{24}{25}$