Exam Questions & Step-by-Step Markschemes (4 Problems)
Question 1 • 2023
3 Marks
Find the modulus and principal argument of the complex number z = -1 - i\sqrt{3} , and express it in polar form.
Answer: |z| = 2, \quad \text{Arg}(z) = -\frac{2\pi}{3}, \quad z = 2\left[\cos\left(-\frac{2\pi}{3}\right) + i\sin\left(-\frac{2\pi}{3}\right)\right]
Question 2 • 2023
4 Marks
Convert the complex number z = \frac{1 + 7i}{(2 - i)^2} into standard polar form r(\cos\theta + i\sin\theta) .
Answer: z = \sqrt{2}\left(\cos\frac{3\pi}{4} + i\sin\frac{3\pi}{4}\right)
Question 3 • 2023
4 Marks
If (x + iy)^3 = u + iv , then prove that: \frac{u}{x} + \frac{v}{y} = 4(x^2 - y^2)
Answer: \text{Hence Proved, } \frac{u}{x} + \frac{v}{y} = 4(x^2 - y^2)
Question 4 • 2024
3 Marks
Convert the complex number z = \frac{-16}{1 + i\sqrt{3}} into polar form.
Answer: z = 8\left( \cos\frac{2\pi}{3} + i\sin\frac{2\pi}{3} \right)