Complex numbers – further maths test to download as PDF

Given \(z_1=2+3i\) and \(z_2=1-i\):

1. Find \(\dfrac{z_1}{z_2}\) in the form \(a+bi\). [2]
2. Find \(|z_1z_2|\). [1]

\(f(z)=z^3-3z^2+7z-5\).

1. Show that \(z=1\) is a root of \(f(z)=0\). [1]
2. Solve \(f(z)=0\) completely. [2]
3. Sketch the roots on an Argand diagram. [1]

\(u=-3+3\sqrt3\,i\) and \(v=2-2i\).

1. Find the modulus and argument of \(u\) and of \(v\). [2]
2. Find \(\left|\dfrac uv\right|\) and \(\arg\dfrac uv\), as exact values. [2]

Two sensors are at the points \(A=2\) and \(B=6i\) on an Argand diagram (distances in km). A boat is the same distance from both sensors and is also exactly 5 km from \(A\).

1. Write the first condition as an equation in \(z\) and show that its Cartesian equation is \(x-3y=-8\). [3]
2. Find the exact coordinates of the two positions where the boat could be. [3]

The quadratic equation \(z^2+pz+q=0\) has real coefficients and one root is \(2-i\).

1. Write down the other root. [1]
2. Find \(p\) and \(q\). [2]