Question

The set of values of $k$ for which $x^2-kx+{\sin }^{-1}\left(\sin 4\right)>0$ for all real $x$ is

• A. $\varphi$
• B. $(-2,2)$
• C. $R$
• D. $noneofthese$

Answer 1

Answer: (A)

$\varphi$

Given, $x^2-kx+{\sin }^{-1}\left(\sin 4\right)>0$
Now, ${\sin }^{-1}\left(\sin 4\right)={\sin }^{-1}\left(\sin \right(\pi +(4-\pi )\left)\right)=\pi -4$
So, the given inequality is
$x^2-kx+\pi -4>0$
$\Rightarrow D<0$
$\Rightarrow k^2-4(\pi -4)<0$

$\Rightarrow k^2+16-4\pi <0$
which is false all the time.