Multiple choice

Consider the cubic equation ${ x }^{ 3 }-\left( 1+\cos { \theta } +\sin { \theta } \right) { x }^{ 2 }+\left( \cos { \theta } \sin { \theta } +\cos { \theta } +\sin { \theta } \right) x-\sin { \theta } \cos { \theta } =0$ whose roots are ${ x }{ 1 },{ x }{ 2 },{ x }_{ 3 }$ The greatest possible difference between two of the roots if $\displaystyle \theta \epsilon \left [ 0,2\pi \right ]$ is

  1. $2$
  2. $1$
  3. $\displaystyle\sqrt{2}$
  4. $\displaystyle 2\sqrt{2} $
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A Correct answer
Explanation

The cubic equation is (x - 1)(x - cos(theta))(x - sin(theta)) = 0. The roots are 1, cos(theta), and sin(theta). We want to maximize the difference between any two roots. The range of sin and cos is [-1, 1]. The maximum difference occurs between 1 and -1, which is 2.

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