Question

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)

$2$

Given equation is $x^3-(1+cos\theta +\sin \theta )x^2+(\cos \theta .\sin \theta +\cos \theta +\sin \theta )x-\sin \theta \cos \theta =0$

$⟹x_1=1,x_2=\cos \theta ,x_3=\sin \theta$

$x_1-x_2=1-\cos \theta =2{\sin }^2\frac{\theta }{2}\le 2$

Similarly $x_1-x_3\le 2,x_2-x_3=\cos \theta -\sin \theta \le \sqrt{2}$

Greatest possible difference is $2$