4.02a Complex numbers: real/imaginary parts, modulus, argument

$$z = \frac{a + 3i}{2 + ai}, \quad a \in \mathbb{R}.$$

1. Given that \(a = 4\), find \(|z|\). [3]
2. Show that there is only one value of \(a\) for which \(\arg z = \frac{\pi}{4}\), and find this value. [6]

Given that \(z = 2 - 2i\) and \(w = -\sqrt{3} + i\),

1. find the modulus and argument of \(wz^2\). [6]
2. Show on an Argand diagram the points \(A\), \(B\) and \(C\) which represent \(z\), \(w\) and \(wz^2\) respectively, and determine the size of angle \(BOC\). [4]

Given that \(z = 1 + \sqrt{3}i\) and that \(\frac{w}{z} = 2 + 2i\), find

1. \(w\) in the form \(a + ib\), where \(a, b \in \mathbb{R}\), [3]
2. the argument of \(w\), [2]
3. the exact value for the modulus of \(w\). [2]

On an Argand diagram, the point \(A\) represents \(z\) and the point \(B\) represents \(w\).

1. Draw the Argand diagram, showing the points \(A\) and \(B\). [2]
2. Find the distance \(AB\), giving your answer as a simplified surd. [2]

Given that \(z = -2\sqrt{2} + 2\sqrt{2}i\) and \(w = 1 - i\sqrt{3}\), find

1. \(\left|\frac{z}{w}\right|\), [3]
2. \(\arg \left( \frac{z}{w} \right)\). [3]

1. On an Argand diagram, plot points \(A\), \(B\), \(C\) and \(D\) representing the complex numbers \(z\), \(w\), \(\left( \frac{z}{w} \right)\) and 4, respectively. [3]
2. Show that \(\angle AOC = \angle DOB\). [2]
3. Find the area of triangle \(AOC\). [2]

$$z = -4 + 6i.$$

1. Calculate \(\arg z\), giving your answer in radians to 3 decimal places. [2]

The complex number \(w\) is given by \(w = \frac{A}{2 - i}\), where \(A\) is a positive constant. Given that \(|w| = \sqrt{20}\),

1. find \(w\) in the form \(a + ib\), where \(a\) and \(b\) are constants, [4]
2. calculate \(\arg \frac{w}{z}\). [3]

Given that \(\frac{z + 2i}{z - \lambda i} = i\), where \(\lambda\) is a positive, real constant,

1. show that \(z = \left( \frac{\lambda}{2} + 1 \right) + i \left( \frac{\lambda}{2} - 1 \right)\). [5]

Given also that \(\arg z = \arctan \frac{1}{3}\), calculate

1. the value of \(\lambda\), [3]
2. the value of \(|z|^2\). [2]

The complex numbers \(z\) and \(w\) satisfy the simultaneous equations $$2z + iw = -1,$$ $$z - w = 3 + 3i.$$

1. Use algebra to find \(z\), giving your answers in the form \(a + ib\), where \(a\) and \(b\) are real. [4]
2. Calculate \(\arg z\), giving your answer in radians to 2 decimal places. [2]

The complex numbers \(z_1\) and \(z_2\) are given by $$z_1 = 5 + 3i,$$ $$z_1 = 1 + pi,$$ where \(p\) is an integer.

1. Find \(\frac{z_2}{z_1}\), in the form \(a + ib\), where \(a\) and \(b\) are expressed in terms of \(p\). [3]

Given that \(\arg \left( \frac{z_2}{z_1} \right) = \frac{\pi}{4}\),

1. find the value of \(p\). [2]

$$z = \sqrt{3} - i.$$ \(z^*\) is the complex conjugate of \(z\).

1. Show that \(\frac{z}{z^*} = \frac{1}{2} - \frac{\sqrt{3}}{2} i\). [3]
2. Find the value of \(\left| \frac{z}{z^*} \right|\). [2]
3. Verify, for \(z = \sqrt{3} - i\), that \(\arg \frac{z}{z^*} = \arg z - \arg z^*\). [4]
4. Display \(z\), \(z^*\) and \(\frac{z}{z^*}\) on a single Argand diagram. [2]
5. Find a quadratic equation with roots \(z\) and \(z^*\) in the form \(ax^2 + bx + c = 0\), where \(a\), \(b\) and \(c\) are real constants to be found. [2]

$$z = -2 + i.$$

1. Express in the form \(a + ib\)
   1. \(\frac{1}{z}\)
   2. \(z^2\). [4]
2. Show that \(|z^2 - z| = 5\sqrt{2}\). [2]
3. Find \(\arg (z^2 - z)\). [2]
4. Display \(z\) and \(z^2 - z\) on a single Argand diagram. [2]

The complex number \(z\) is defined by $$z = \frac{a + 2i}{a - 1}, \quad a \in \mathbb{R}, a > 0 .$$ Given that the real part of \(z\) is \(\frac{1}{2}\) , find

1. the value of \(a\), [4]
2. the argument of \(z\), giving your answer in radians to 2 decimal places. [3]

Solve the equation $$z^2 = 4\sqrt{2} - 4\sqrt{2}i,$$ giving your answers in the form \(r(\cos \theta + i \sin \theta)\), where \(-\pi < \theta \leq \pi\). [6]

A transformation \(T\) from the \(z\)-plane to the \(w\)-plane is given by $$w = \frac{z}{z-i}, \quad z \neq i.$$ The circle with equation \(|z| = 3\) is mapped by \(T\) onto the curve \(C\).

1. Show that \(C\) is a circle and find its centre and radius. [8]

The region \(|z| < 3\) in the \(z\)-plane is mapped by \(T\) onto the region \(R\) in the \(w\)-plane.

1. Shade the region \(R\) on an Argand diagram. [2]

\(z = -8 + (8\sqrt{3})i\)

1. Find the modulus of \(z\) and the argument of \(z\). [3]

Using de Moivre's theorem,

1. find \(z^3\). [2]
2. find the values of \(w\) such that \(w^4 = z\), giving your answers in the form \(a + ib\), where \(a, b \in \mathbb{R}\). [5]

A transformation \(T\) from the \(z\)-plane to the \(w\)-plane is given by $$w = \frac{z + 2i}{iz}$$ The transformation maps points on the real axis in the \(z\)-plane onto a line in the \(w\)-plane. Find an equation of this line. [4]

\(z = 5\sqrt{3} - 5i\) Find

1. \(|z|\), [1]
2. \(\arg(z)\), in terms of \(\pi\). [2]

$$w = 2\left[\cos \frac{\pi}{4} + i\sin \frac{\pi}{4}\right]$$ Find

1. \(\left|\frac{w}{z}\right|\), [1]
2. \(\arg\left(\frac{w}{z}\right)\), in terms of \(\pi\). [2]

A complex number \(z\) is represented by the point \(P\) in the Argand diagram. Given that $$|z - 3i| = 3,$$

1. sketch the locus of \(P\). [2]
2. Find the complex number \(z\) which satisfies both \(|z - 3i| = 3\) and \(\arg (z - 3i) = \frac{3}{4}\pi\). [4] The transformation \(T\) from the \(z\)-plane to the \(w\)-plane is given by $$w = \frac{2i}{w}.$$
3. Show that \(T\) maps \(|z - 3i| = 3\) to a line in the \(w\)-plane, and give the cartesian equation of this line. [5]

The complex number \(2 - i\) is denoted by \(z\).

1. Find \(|z|\) and \(\arg z\). [2]
2. Given that \(az + bz^* = 4 - 8i\), find the values of the real constants \(a\) and \(b\). [5]

The complex numbers \(a\) and \(b\) are given by \(a = 7 + 6\text{i}\) and \(b = 1 - 3\text{i}\). Showing clearly how you obtain your answers, find

1. \(|a - 2b|\) and \(\arg(a - 2b)\), [4]
2. \(\frac{b}{a}\), giving your answer in the form \(x + \text{i}y\). [3]

The roots of the equation \(z^3 - 1 = 0\) are denoted by \(1, \omega\) and \(\omega^2\).

1. Sketch an Argand diagram to show these roots. [1]
2. Show that \(1 + \omega + \omega^2 = 0\). [2]
3. Hence evaluate
   1. \((2 + \omega)(2 + \omega^2)\), [2]
   2. \(\frac{1}{2 + \omega} + \frac{1}{2 + \omega^2}\). [2]
4. Hence find a cubic equation, with integer coefficients, which has roots \(2, \frac{1}{2 + \omega}\) and \(\frac{1}{2 + \omega^2}\). [4]

Find the cube roots of \(\frac{1}{2}\sqrt{3} + \frac{1}{2}i\), giving your answers in the form \(\cos \theta + i \sin \theta\), where \(0 \leqslant \theta < 2\pi\). [4]