- [Maximum mark: 19]
(a) Solve (z^2 = -1 - \sqrt{3}i), giving your answers in the form (z = r(\cos \theta + i \sin \theta)). [4]
Let (z_1) and (z_2) be the square roots of (-1 - \sqrt{3}i), where (\text{Re}(z_1) > 0).
Let (z_3) and (z_4) be the square roots of (-1 + \sqrt{3}i), where (\text{Re}(z_3) > 0).
(b) Expressing your answers in the form (z = a + bi), where (a, b \in \mathbb{R}),
(i) find (z_1) and (z_2);
(ii) deduce (z_3) and (z_4). [4]
The four roots (z_1, z_2, z_3) and (z_4) are represented by the points A, B, C and D respectively on an Argand diagram.
(c) (i) Plot the points A, B, C and D on an Argand diagram.
(ii) Find the area of the polygon formed by these four points. [4]
The four roots (z_1, z_2, z_3) and (z_4) satisfy the equation (z^4 + 2z^2 + 4 = 0).
The four roots (\frac{1}{z_1}, \frac{1}{z_2}, \frac{1}{z_3}) and (\frac{1}{z_4}) satisfy the equation (pw^4 + qw^2 + r = 0) where (p, q, r \in \mathbb{Z}).
(d) Find the value of (p, q) and (r). [3]
The four roots (\frac{1}{z_1}, \frac{1}{z_2}, \frac{1}{z_3}) and (\frac{1}{z_4}) are represented by the points E, F, G and H respectively on an Argand diagram.
(e) (i) Find (\frac{1}{z_1}) in the form (z = a + bi), where (a, b \in \mathbb{R}).
(ii) Hence, deduce the area of the polygon formed by these four points. [4]