Multiple choice

If $\alpha$ and $\beta$ are roots of the equation $x^{2} + 5|x| - 6 = 0$ then the value of $|\tan^{-1} \alpha - \tan^{-1}\beta|$ is

  1. $\dfrac {\pi}{2}$
  2. $0$
  3. $\pi$
  4. $\dfrac {\pi}{4}$
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A Correct answer
Explanation

x^2 + 5|x| - 6 = 0. Let |x| = t. t^2 + 5t - 6 = 0 => (t+6)(t-1) = 0. Since t=|x| >= 0, t=1. So |x|=1, x = 1 or -1. Roots are 1 and -1. |tan^-1(1) - tan^-1(-1)| = |pi/4 - (-pi/4)| = |pi/2| = pi/2.

AI explanation

The given equation is x^2 + 5|x| - 6 = 0. If x is positive, the equation becomes x^2 + 5x - 6 = 0, which factors to (x + 6)(x - 1) = 0, yielding the positive root x = 1. If x is negative, replacing |x| with -x gives x^2 - 5x - 6 = 0, which factors to (x - 6)(x + 1) = 0, yielding the negative root x = -1. Therefore, the roots are alpha = 1 and beta = -1. Substituting these into the inverse tangent expression gives |tan^-1(1) - tan^-1(-1)|. Since tan^-1(1) = pi/4 and tan^-1(-1) = -pi/4, the expression becomes |pi/4 - (-pi/4)|, which equals pi/2.

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