Complex number arithmetic and simplification

A question is this type if and only if it asks to perform operations (multiplication, division, powers) on complex numbers and simplify to a specific form, without solving equations.

27 questions · Standard +0.3

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OCR Further Pure Core 2 2018 December Q9
5 marks Standard +0.8
  1. By using Euler's formula show that \(\cosh(\text{iz}) = \cos z\). [3]
  2. Hence, find, in logarithmic form, a root of the equation \(\cos z = 2\). [You may assume that \(\cos z = 2\) has complex roots.] [2]
Pre-U Pre-U 9795/1 2011 June Q12
10 marks Challenging +1.2
The complex number \(z_1\) is such that \(z_1 = a + ib\), where \(a\) and \(b\) are positive real numbers.
  1. Given that \(z_1^2 = 2 + 2i\), show that \(a = \sqrt{\sqrt{2} + 1}\) and find the exact value of \(b\) in a similar form. [5]
The complex number \(z_2\) is such that \(z_2 = -a + ib\).
    1. Determine \(\arg z_2\) as a rational multiple of \(\pi\). [You may use the result \(\tan(\frac{1}{8}\pi) = \sqrt{2} - 1\).] [2]
    2. The point \(P_n\) in an Argand diagram represents the complex number \(z_2^n\), for positive integers \(n\). Find the least value of \(n\) for which \(P_n\) lies on the half-line with equation $$\arg(z) = \frac{1}{4}\pi.$$ [3]