The polynomial `p(z) = z^3 + alpha z^2 + beta z + gamma`, where `z ∈ C` and `alpha, beta, gamma ∈ R`, can also be written as `p(z) = (z - z_1)(z - z_2)(z - z_3)`, where `z_1 ∈ R` and `z_2, z_3 ∈ C`.
- i. State the relationship between `z_2` and `z_3`. (1 mark)
- ii. Determine the values of `alpha, beta` and `gamma`, given that `p(2) = -13, |z_2 + z_3| = 0` and `|z_2 - z_3| = 6`. (3 marks)
Consider the point `z_4 = sqrt3 + 1`.
- Sketch the ray given by `text(Arg)(z - z_4) = (5pi)/6` on the Argand diagram below. (2 marks)
- The ray `text(Arg)(z - z_4) = (5pi)/6` intersects the circle `|z - 3i| = 1`, dividing it into a major and a minor segment.
- i. Sketch the circle `|z - 3i| = 1` on the Argand diagram in part b. (1 mark)
- ii. Find the area of the minor segment. (2 marks)