Mathematics 9709 · AS & A Level
Complex numbers
100 practice questions on Complex numbers, with worked solutions and instant marking.
Determine the complex number $z$ that satisfies the equation $z^* + 1 = 2iz$, where $z^*$ represents the complex conjugate of $z$. Write your answer in the form $x + iy$, where $x$ and $y$ are real.
Feb/March 2016
For this question, calculators are not allowed. Let the polynomial $z^4 + 3z^2 + 6z + 10$ be written as $p(z)$. Let the complex number $-1 + i$ be written as $u$.
Feb/March 2017
The complex number $1 + 2i$ is represented by $u$.
Feb/March 2018
Show every step and, without a calculator, solve the equation $(1 + i)z^2 - (4 + 3i)z + 5 + i = 0$. Present your solutions in the form $x + iy$, where $x$ and $y$ are real.
Feb/March 2019
The complex numbers $v$ and $w$ are defined by the equations $v + iw = 5$ and $(1 + 2i)v - w = 3i$.
Feb/March 2020
The complex numbers $u$ and $v$ are given by $u = -4 + 2i$ and $v = 3 + i$.
Feb/March 2021
On a sketch of an Argand diagram, shade the set of points representing complex numbers $z$ that satisfy $|z + 2 - 3i| \leq 2$ and $\arg z \leq \frac{3}{4}\pi$.
Feb/March 2022
Find the complex numbers $w$ that satisfy the equation $w^2 + 2i w^* = 1$ while also meeting $\operatorname{Re} w \leq 0$. Write each answer in the form $x + iy$, where $x$ and $y$ are real.
Feb/March 2022
On an Argand diagram, shade the set of points representing complex numbers $z$ that satisfy the inequalities $-\frac{1}{3}\pi \leq \arg(z - 1 - 2i) \leq \frac{1}{3}\pi$ and $\Re z \leq 3$.
Feb/March 2023
Find the solutions of $\frac{5z}{1 + 2i} - zz^* + 30 + 10i = 0$, and give them in the form $x + iy$, with $x$ and $y$ real.
Feb/March 2023
You are told that $z = -\sqrt{3} + i$.
Feb/March 2024
On an Argand diagram sketch, shade the set of points corresponding to complex numbers $z$ that satisfy the inequalities $|z - 4 - 2i| \leq 3$ and $|z| \geq |10 - z|$.
Feb/March 2024
On the Argand diagram, the shaded area contains the points for complex numbers $z$ that meet two inequalities. A circle and a straight line parallel to the real axis form the boundary, and those boundary lines are included in the shaded area.
Feb/March 2025
The square roots of $-4 + 6\sqrt{5}i$ may be written in Cartesian form as $x + iy$, with $x$ and $y$ real and exact.
Feb/March 2025
The complex number $2 + 2i$ is represented by $u$.
May/June 2010
The complex variable z is defined by z = 1 + \cos 2\theta + i \sin 2\theta, and \theta can take any value in the interval -\frac{1}{2}\pi < \theta < \frac{1}{2}\pi.
May/June 2010
The equation $2x^3 - x^2 + 2x + 12 = 0$ has one real root and two complex roots. By showing your working, confirm that $1 + i\sqrt{3}$ is one of the complex roots. State the other complex root.
May/June 2010
The complex number $u$ is specified as $u = \dfrac{6 - 3i}{1 + 2i}$.
May/June 2011
The complex number $u$ is given by $u = \frac{5}{a + 2i}$, where $a$ is a real constant. Write $u$ in the form $x + iy$, with $x$ and $y$ both real.
May/June 2011
Find the roots of $z^2 + (2\sqrt{3})z + 4 = 0$, and give the answers in the form $x + iy$, where $x$ and $y$ are real.
May/June 2011
The complex number $u$ is specified by $u = \frac{(1 + 2i)^2}{2 + i}$.
May/June 2012
For this question, you must not use a calculator. The complex number $u$ is given by $u = \frac{1 + 2i}{1 - 3i}$.
May/June 2012
The complex numbers $u$ and $w$ satisfy the equations $u - w = 4i$ and $uw = 5$. Find $u$ and $w$, giving every answer in the form $x + iy$, where $x$ and $y$ are real.
May/June 2012
Without a calculator, solve $3w + 2i w^* = 17 + 8i$, where $w^*$ is the complex conjugate of $w$. Give your answer in the form $a + bi$.
May/June 2013
The complex number $w$ has $\Re w > 0$ and satisfies $w + 3w^* = iw^2$, where $w^*$ is the complex conjugate of $w$. Find $w$, giving the answer in the form $x + iy$, where $x$ and $y$ are real.
May/June 2013
The complex number $z$ is given by $z = a + ib$, where $a$ and $b$ are real numbers. Its complex conjugate is written as $z^*$.
May/June 2013
The complex number $z$ is specified by $z = \frac{9\sqrt{3} + 9i}{\sqrt{3} - i}$. Find, showing all your working,
May/June 2014
The equation $z^3 + 2z + a = 0$, where $a$ is real, has root $-1 + (\sqrt{5})i$. Show your working to find $a$, and then write down the other complex root of this equation.
May/June 2014
The complex number $\frac{3 - 5i}{1 + 4i}$ is labelled $u$. Show your working and express $u$ in the form $x + iy$, where $x$ and $y$ are real.
May/June 2014
The complex number $w$ is given by $w = \frac{22 + 4i}{(2 - i)^2}$.
May/June 2015
The complex number $u$ is defined as $u = -1 + (4\sqrt{3})i$.
May/June 2015
Let $u$ represent the complex number $1 - i$.
May/June 2015
Without using a calculator and showing all your working, determine the square roots of the complex number $7 - 6\sqrt{2}i$. Present your answers in the form $x + iy$, where $x$ and $y$ are real and exact.
May/June 2016
Show all working, and solve the equation $iz^2 + 2z - 3i = 0$, writing your answers in the form $x + iy$, where $x$ and $y$ are exact real numbers.
May/June 2016
For this question, calculators must not be used. Let the complex numbers $-1 + 3i$ and $2 - i$ be represented by $u$ and $v$ respectively. On an Argand diagram with origin $O$, points $A$, $B$ and $C$ stand for the numbers $u$, $v$ and $u + v$ respectively.
May/June 2016
Calculators are not allowed anywhere in this question. The complex numbers $u$ and $w$ are given by $u = -1 + 7i$ and $w = 3 + 4i$.
May/June 2017
For the whole question, calculator use is not allowed. Let the complex number $2 - i$ be called $u$.
May/June 2017
You must not use a calculator anywhere in this question.
May/June 2017
Show all working and, without using a calculator, solve the equation $z^2 + (2\sqrt{6})z + 8 = 0$, writing your answers in the form $x + iy$, where $x$ and $y$ are exact real values.
May/June 2018
For this question, a calculator must not be used. The complex numbers $-3\sqrt{3} + i$ and $\sqrt{3} + 2i$ are called $u$ and $v$ respectively.
May/June 2018
Determine the complex number $z$ that satisfies the equation $3z - iz^* = 1 + 5i$, where $z^*$ is the complex conjugate of $z$.
May/June 2018
No calculator may be used anywhere in this question. The complex number $(\sqrt{3}) + i$ is represented by $u$.
May/June 2019
For this question, calculator use is not allowed. The complex number $u$ is defined by $u = \frac{4i}{1 - (\sqrt{3})i}$.
May/June 2019
The complex number $u$ is given by $u = \dfrac{3i}{a + 2i}$, with $a$ taken to be real.
May/June 2020
Solve the equation $(1 + 2i)w + i w^* = 3 + 5i$. Express your answer in the form $x + iy$, with $x$ and $y$ real.
May/June 2020
The complex numbers $u$ and $w$ satisfy $u - w = 2i$ and $uw = 6$.
May/June 2020
On an Argand diagram with origin $O$, the roots of this equation are shown by the two distinct points $A$ and $B$.
May/June 2021
On an Argand diagram sketch, indicate by shading the set of points for complex numbers $z$ that satisfy the stated inequalities.
May/June 2021
The complex number $u$ is defined as $u = 10 - 4\sqrt{6}i$.
May/June 2021
Verify that $-1 + \sqrt{2}i$ is a root of the equation $z^{4} + 3z^{2} + 2z + 12 = 0$.
May/June 2021
The complex number $u$ is defined as $u = \dfrac{\sqrt{2} - a\sqrt{2} i}{1 + 2i}$, where $a$ is a positive integer.
May/June 2022
Let $\mu$ stand for the complex number $-1 + \sqrt{7}i$. It is stated that $\mu$ is a root of $2x^3 + 3x^2 + 14x + k = 0$, where $k$ is a real constant.
May/June 2022
Let $u$ stand for the complex number $3 - i$.
May/June 2022
The polynomial $x^3 + 5x^2 + 31x + 75$ is represented by $p(x)$.
May/June 2023
Using an Argand diagram, sketch the locus of points corresponding to complex numbers $z$ for which $|z + 3 - 2i| = 2$.
May/June 2023
The complex number $2 + yi$ is represented by $a$, with $y$ being a real number and $y < 0$. It is stated that $f(a) = a^3 - a^2 - 2a$.
May/June 2023
The complex number $z$ is specified by $z = \dfrac{5a - 2i}{3 + ai}$, where $a$ is an integer. It is also given that $\arg z = -\dfrac{\pi}{4}$.
May/June 2023
On an Argand diagram sketch, shade the set of points corresponding to complex numbers $z$ that satisfy the stated inequalities.
May/June 2023
The complex number $u$ is defined by $u = -1 - i\sqrt{3}$.
May/June 2024
On a single Argand diagram, sketch the loci described by the equations $|z - 3 + 2i| = 2$ and $|w - 3 + 2i| = |w + 3 - 4i|$ where $z$ and $w$ are complex numbers.
May/June 2024
The complex numbers $z$ and $\omega$ are given by $z = 1 - i$ and $\omega = -3 + 3\sqrt{3}i$.
May/June 2024
The square roots of $24 - 7i$ may be written in Cartesian form as $x + iy$, with $x$ and $y$ both real and exact.
May/June 2024
On an Argand diagram, shade the set of points representing complex numbers $z$ that satisfy both $|z - 4 - 3i| \leq 2$ and $\arg(z - 2 - i) \geq \frac{1}{3}\pi$.
May/June 2024
Determine the complex numbers $z$ for which $\frac{z + 5i}{z - 5}$ is real and $|z| = \sqrt{17}$. Give your answers in the form $z = x + iy$, where $x$ and $y$ are real.
May/June 2025
Given that $z_1 = 3e^{\frac{\pi i}{4}}$, $z_2 = \tfrac{3}{2}e^{\frac{3\pi i}{4}}$ and $\omega = 2e^{\frac{\pi i}{2}}$.
May/June 2025
On an Argand diagram, shade the set of points representing complex numbers $z$ that satisfy both inequalities.
May/June 2025
The square roots of $-1 - 4\sqrt{5}i$ may be written in Cartesian form as $x + iy$, with $x$ and $y$ both real and exact.
May/June 2025
You are given $z_1 = r_1 e^{i\theta_1}$ and $z_2 = r_2 e^{i\theta_2}$. Show that $(z_1 z_2)^* = z_1^* z_2^*$.
May/June 2025
Find all complex numbers $z$ such that $\displaystyle \frac{z + 4}{z + 4i}$ is real and $|z| = \sqrt{10}$. Give your answers in the form $z = x + iy$, where $x$ and $y$ are real.
May/June 2025
The complex numbers $s$ and $t$ are specified by $s = 5(\cos 0.25 + i\sin 0.25)$ and $t = 6e^{3i}$.
May/June 2025
The diagram displays the locus of points for the complex numbers $z$ that satisfy $|z + 5 - 4i| = 3$.
May/June 2025
The complex number $z$ is defined by $z = (\sqrt{3}) + i.$
Oct/Nov 2010
The complex number $z$ is defined as $z = \sqrt{3} + i$.
Oct/Nov 2010
The polynomial $p(z)$ is given by $p(z) = z^3 + mz^2 + 24z + 32$, with $m$ a constant. You are told that $(z + 2)$ divides $p(z)$.
Oct/Nov 2010
The complex number $w$ is given by $w = 2 + i$.
Oct/Nov 2010
Showing your working, determine the two square roots of the complex number $1 - 2\sqrt{6}i$. Present your answers in the form $x + iy$, with $x$ and $y$ exact.
Oct/Nov 2011
With your working shown, determine the two square roots of the complex number $1 - (2\sqrt{6})i$. Write each answer in the form $x + iy$, with $x$ and $y$ exact.
Oct/Nov 2011
The complex number $w$ is given by $w = -1 + i$.
Oct/Nov 2011
Let the complex number $1 + (\sqrt{2})i$ be represented by $u$. Let the polynomial $x^4 + x^2 + 2x + 6$ be represented by $p(x)$.
Oct/Nov 2012
Use $u$ to denote the complex number $1 + (\sqrt{2})i$. Write $x^4 + x^2 + 2x + 6$ as $p(x)$.
Oct/Nov 2012
Solve the equation $iw^2=(2-2i)^2$ without using a calculator.
Oct/Nov 2012
Calculators are not allowed anywhere in this question.
Oct/Nov 2013
A calculator must not be used anywhere in this question.
Oct/Nov 2013
Without a calculator, use the quadratic formula to solve $(2 - i)z^2 + 2z + 2 + i = 0$. Write the solutions in the form $a + bi$.
Oct/Nov 2013
For this question, calculator use is not allowed. The complex numbers $w$ and $z$ satisfy the relation $w = \frac{z + i}{iz + 2}$.
Oct/Nov 2014
For this question, you must not use a calculator. The complex numbers $w$ and $z$ satisfy the relation $w = \frac{z + i}{iz + 2}$.
Oct/Nov 2014
The complex numbers $w$ and $z$ are given by $w = 5 + 3i$ and $z = 4 + i$.
Oct/Nov 2014
Let the complex number $3 - i$ be written as $u$. Write its complex conjugate as $u^*$.
Oct/Nov 2015
The complex number $3 - i$ is represented by $u$. Its complex conjugate is represented by $u^*$.
Oct/Nov 2015
It is given that $(1 + 3i)w = 2 + 4i$. Show all the working needed to prove that the exact value of $|w^2|$ equals $2$ and determine $\arg(w^2)$ correct to $3$ significant figures.
Oct/Nov 2015
For this question, calculator use is not allowed.
Oct/Nov 2016
No calculator may be used anywhere in this question.
Oct/Nov 2016
You must not use a calculator anywhere in this question. The complex number $z$ is given by $z = (\sqrt{2}) - (\sqrt{6})i$. We denote the complex conjugate of $z$ by $z^*$.
Oct/Nov 2016
The complex number $u$ is $u = 8 - 15i$. With full working shown, determine the two square roots of $u$. Express your answers in the form $a + ib$, where $a$ and $b$ are exact real numbers.
Oct/Nov 2017
For this question, a calculator must not be used. Write the complex number $1 - \sqrt{3}i$ as $u$.
Oct/Nov 2017
The complex number $u$ is $8 - 15i$. With full working shown, determine the two square roots of $u$. Present your answers in the form $a + ib$, with $a$ and $b$ being exact real values.
Oct/Nov 2017
Show all working, and write the complex number $\frac{2 + 3i}{1 - 2i}$ in the form $r e^{i\theta}$, with $r > 0$ and $-\pi < \theta \leq \pi$. State $r$ and $\theta$ correct to 3 significant figures.
Oct/Nov 2018
Without a calculator, rewrite the complex number $\frac{2 + 6i}{1 - 2i}$ in the form $x + iy$, where $x$ and $y$ are real.
Oct/Nov 2018
Show all working needed, and write the complex number $\frac{2 + 3i}{1 - 2i}$ in the form $re^{i\theta}$, where $r > 0$ and $-\pi < \theta \leq \pi$. State $r$ and $\theta$ correct to $3$ significant figures.
Oct/Nov 2018
The complex number $u$ is defined by $u = -3 - (2\sqrt{10})i$. With all working shown and no calculator used, determine the square roots of $u$. Present your answers in the form $a + ib$, where $a$ and $b$ are exact real numbers.
Oct/Nov 2019