Complex Numbers — practice questions

Simplify $\sqrt{98} + \sqrt{32}$, giving your answer in the form $a\sqrt{b}$ where $b$ is as small as possible.

Express $\dfrac{5}{2\sqrt{3} - 1}$ with a rational denominator, giving your answer in simplest form.

Solve $\sqrt{x + 7} - \sqrt{x} = 1$ for $x$, showing that your solution is valid.

Given $z = 4 - 3i$ and $w = -1 + 5i$, find $z + w$ and $zw$, giving each answer in the form $a + bi$.

Find the real number $k$ for which $(2 + ki)^2$ is purely imaginary.

The complex number $z = x + yi$, where $x$ and $y$ are real, satisfies $z^2 = \overline{z} \cdot 4i$ where $\overline{z} = x - yi$. Find all possible values of $z$.

Express $\dfrac{2 + 6i}{1 + i}$ in the form $a + bi$.

Find the real number $k$ such that $\dfrac{k + 3i}{2 - i}$ is purely real.

The complex number $z$ satisfies $\dfrac{z}{z + 2} = 1 - i$. Find $z$, and hence find the real number $m$ for which $mz + \overline{z}$ is purely imaginary.

Find the modulus and principal argument of $z = 5 + 5i$, giving the argument in exact radian form.

Find the modulus and principal argument of $z = -2 - 2\sqrt{3}\,i$, giving exact values and explaining how you determined the quadrant.

The complex number $z = k + 4i$, where $k$ is real and positive, has $|z| = 2\,\text{Re}(z)$. Find the exact value of $k$ and hence the exact principal argument of $z$.

Write $z = 4\,\text{cis}\,\dfrac{\pi}{6}$ in the form $a + bi$, giving exact values.

Write $z = -\sqrt{3} - i$ in polar form, giving both the modulus and the argument exactly, and explain why a calculator's $\tan^{-1}$ does not give the correct argument here.

The complex number $z = k - ki$, where $k$ is a non-zero real number, is written in polar form. Find the modulus and the principal argument of $z$ in terms of $k$, treating the two cases $k > 0$ and $k < 0$ separately.

Given $z = 5\,\text{cis}\,\dfrac{\pi}{4}$ and $w = 3\,\text{cis}\,\dfrac{\pi}{6}$, find $zw$ in polar form.

Given $z = 6\,\text{cis}\,\dfrac{5\pi}{6}$ and $w = 2\,\text{cis}\,\dfrac{3\pi}{4}$, find $\dfrac{z}{w}$ in polar form with a principal argument, and state the geometric effect of dividing by $w$.

The points $A$ and $B$ on an Argand diagram represent $z_1 = 3 + i$ and $z_2 = 1 + 4i$. The point $C$ is obtained by rotating $B$ anticlockwise about $A$ through $\dfrac{\pi}{2}$. Find the complex number represented by $C$, in the form $a + bi$.

Use De Moivre's theorem to evaluate $\left(3\,\text{cis}\,\dfrac{\pi}{9}\right)^3$, giving your answer in polar form.

Evaluate $(1 - i)^8$, giving your answer in the form $a + bi$. Show all conversion steps.

Use De Moivre's theorem to prove that $\cos 3\theta = 4\cos^3\theta - 3\cos\theta$.

Solve $z^3 = 27\,\text{cis}\,\dfrac{\pi}{2}$, giving all solutions in polar form.

Solve $z^4 = -16$, giving all solutions in the form $a + bi$ with exact values.

The equation $z^5 = 32$ has five solutions. Show that the sum of the five solutions is zero, and explain the result geometrically.

Solve $x^2 + 2x + 10 = 0$, giving the roots in the form $a + bi$.

A quadratic equation with real coefficients has $-2 + 5i$ as one of its roots and a leading coefficient of 3. Find the equation in the form $ax^2 + bx + c = 0$.

Prove that the equation $x^2 + kx + (k + 3) = 0$ has real roots for no values of $k$ in the interval $-2 < k < 6$, and determine the exact values of $k$ at which the nature of the roots changes.

Show that $(z - 2)$ is a factor of $P(z) = z^3 - 2z^2 + 9z - 18$, and find the remaining roots.

The cubic $z^3 + 2z^2 + az + b = 0$ has real coefficients and one root $z = -1 + 2i$. Find $a$, $b$, and the real root.

A cubic equation with real coefficients has a root $z = 3 + i$ and satisfies $P(0) = -30$. Given that the cubic is $P(z) = a(z^3 + bz^2 + cz + d)$ with $a$, $b$, $c$, $d$ real and the leading coefficient $a$ equal to 1, find the equation and all three roots. Justify why the third root must be real.

Describe fully the locus of points satisfying $|z + 2 - 4i| = 6$, stating the shape, its key point and its size.

Find the Cartesian equation of the locus $|z - 1| = |z + 3i|$, and describe the locus geometrically.

The locus $|z - 4| = 2|z - 1|$ is not a perpendicular bisector. Find its Cartesian equation, identify the curve exactly, and explain why the factor of 2 changes the shape.