Multiple choice

The arithmetic mean of the roots of the equation $4\cos ^{ 2 }{ x } -4\cos ^{ 2 }{ x } -\cos { (315\pi +x) } =1$ in the interval $(0,315)$ is

  1. $50\pi $
  2. $51\pi $
  3. $100\pi $
  4. $315\pi $
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The two 4cos^2(x) terms cancel, leaving -cos(315*pi + x) = 1. Since 315 is odd, cos(315*pi + x) = -cos(x), so cos(x) = 1. Solutions in (0, 315) are x = 2*pi, 4*pi, ..., 100*pi (50 roots). Their mean is 2*pi(1+2+...+50)/50 = 2550*pi/50 = 51*pi.

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