Question

Consider the cubic equation $x^3-(1+\cos \theta +\sin \theta )x^2+(\cos \theta \sin \theta +\cos \theta +\sin \theta )x-\sin \theta \cos \theta =0$
whose roots are $x_1,x_2,x_3$
The greatest possible difference between two of the roots if $\theta ϵ[0,2\pi ]$ is

• A. $2$
• B. $1$
• C. $\sqrt{2}$
• D. $2\sqrt{2}$

Answer 1

Answer: (A)

$2$

Roots are $\sin \theta ,\cos \theta$ and $1$
$|\sin \theta -\cos \theta |\ge \sqrt{2}$
But for $\theta =\pi$
$1-\cos \theta =2$