4.02b Express complex numbers: cartesian and modulus-argument forms

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SPS SPS FM Pure 2023 June Q10
6 marks Standard +0.3
The complex number \(z\) is given by \(z = k + 3i\), where \(k\) is a negative real number. Given that \(z + \frac{12}{z}\) is real, find \(k\) and express \(z\) in exact modulus-argument form. [6]
SPS SPS FM Pure 2025 January Q6
12 marks Standard +0.3
You are given the complex number \(w = 2 + 2\sqrt{3}i\).
  1. Express \(w\) in modulus-argument form. [3]
  2. Indicate on an Argand diagram the set of points, \(z\), which satisfy both of the following inequalities. $$-\frac{\pi}{2} \leq \arg z \leq \frac{\pi}{3} \text{ and } |z| \leq 4$$ Mark \(w\) on your Argand diagram and find the greatest value of \(|z - w|\). [9]
OCR FP1 AS 2021 June Q2
9 marks Standard +0.3
In this question you must show detailed reasoning. The complex numbers \(z_1\) and \(z_2\) are given by \(z_1 = 2 - 3i\) and \(z_2 = a + 4i\) where \(a\) is a real number.
  1. Express \(z_1\) in modulus-argument form, giving the modulus in exact form and the argument correct to 3 significant figures. [3]
  2. Find \(z_1z_2\) in terms of \(a\), writing your answer in the form \(c + id\). [2]
  3. The real and imaginary parts of a complex number on an Argand diagram are \(x\) and \(y\) respectively. Given that the point representing \(z_1z_2\) lies on the line \(y = x\), find the value of \(a\). [2]
  4. Given instead that \(z_1z_2 = (z_1z_2)^*\) find the value of \(a\). [2]
Pre-U Pre-U 9794/1 2011 June Q10
9 marks Moderate -0.3
  1. The complex number \(z\) is such that \(|z| = 2\) and \(\arg z = -\frac{3}{4}\pi\). Find the exact value of the real part of \(z\) and of the imaginary part of \(z\). [2]
  2. The complex numbers \(u\) and \(v\) are such that $$u = 1 + ia \quad \text{and} \quad v = b - i,$$ where \(a\) and \(b\) are real and \(a < b\). Given that \(uv = 7 + 9i\), find the values of \(a\) and \(b\). [7]