DT1143: Four Complex Numbers, z1, z2, z3, and z4 are Shown on an Argand Diagram: Maths 2 Assignment, AUA

Assignment Questions:

1. (a) Simplify each of the following:

• 3i² + 4i¹²
• i¹³
• 2i ⁸ + 3i ⁷

(b) Four complex numbers, z₁, z₂, z_3, and z₄ are shown on an Argand diagram. They satisfy the following conditions:

z₂ = iz₁

z₃ = kz₁ where k 2 R

z₄ = z₂ + z₃

• Copy the diagram and identify which point is which by labelling the points on the diagram.
• Write down the approximate value of k.

2. (a) Let =  Using de Moivre’s theorem,  nd all solutions to the equation z⁶ =  .
3. (a) Write P (z) = z⁵ 1 as a product of linear factors.

(b) The polynomial f(z) = 2z³          3z² + 18z + 10 has n roots.

• What is the value of n?
• If 1 3i is a root of f(z) = 0, write down another root.
• Find the real root of f(z).

(c) Find the roots of the polynomial g(z) = z³      2z² + (2 + i)z      (1 + i).

4. Let z = cos + i sin . Use de Moivre’s theorem to show that

• z  1/z = 2i sin
• z^n 1/z^n= 2i sin n
• sin³ = 1/4 (3 sin  sin 3).