Multiple choice

A quadratic equation with rational coefficients one of whose roots is $\displaystyle \tan \left| \frac{\pi}{12}\right|$ is

  1. $x^2 - 2x + 1 = 0$
  2. $x^2- 2x + 4 = 0$
  3. $x^2 - 4x + 1 = 0$
  4. $x^2 -4x - 1 = 0$
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C Correct answer
Explanation

The root is tan(pi/12) = tan(15 degrees). Using the half-angle formula, tan(15) = 2 - sqrt(3). For a quadratic with rational coefficients, the other root must be the conjugate, 2 + sqrt(3). The sum of roots is 4 and the product is (2-sqrt(3))(2+sqrt(3)) = 4 - 3 = 1. The equation is x^2 - (sum)x + (product) = 0, which is x^2 - 4x + 1 = 0.

AI explanation

First, evaluate tan(pi/12) using the half-angle formula for tangent: tan(theta/2) = sin(theta) / (1 + cos(theta)). With theta = pi/6, this becomes (1/2) / (1 + sqrt(3)/2), which simplifies to 1 / (2 + sqrt(3)) and rationalizes to 2 - sqrt(3). Because a quadratic equation with rational coefficients must have conjugate roots, the other root is 2 + sqrt(3). The sum of the roots is 4 and the product is 1, giving the equation x squared - 4x + 1 = 0.

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