Complex numbers: past paper questions

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• Question 4May/June 2026, paper 31Structured5 marks

  The square roots of $-71-(20\sqrt{3})\mathrm{i}$ can be expressed in the Cartesian form $x+\mathrm{i}y$, where $x$ and $y$ are real and exact. By first forming a quartic equation, find the square roots of $-71-(20\sqrt{3})\mathrm{i}$.

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• Question 5Oct/Nov 2025, paper 31Structured6 marksHas a diagram

  The shaded region on the Argand diagram shows points representing complex numbers $z$ defined by two inequalities. The shaded region is bounded by a circle and a line parallel to the imaginary axis. The boundaries of the region are included in the shaded region.

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• Question 8Oct/Nov 2024, paper 31Structured9 marks

  Given that $z=1+y\mathrm{i}$ and that $y$ is a real number, express $\frac{1}{z}$ in the form $a+b\mathrm{i}$, where $a$ and $b$ are functions of $y$.

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• Question 7May/June 2026, paper 31Structured6 marksHas a diagram

  The shaded region on the Argand diagram shows points representing the complex numbers, $z$, defined by two inequalities. The shaded region is bounded by the circle and the line parallel to the imaginary axis shown in the diagram and includes its boundaries.

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• Question 6Oct/Nov 2025, paper 31Structured5 marks

  Solve the quadratic equation $(2+\mathrm{i})w^2+4w+2-\mathrm{i}=0$.

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• Question 3Oct/Nov 2024, paper 32Structured5 marks

  The square roots of $6-8\mathrm{i}$ can be expressed in the Cartesian form $x+\mathrm{i}y$, where $x$ and $y$ are real and exact. By first forming a quartic equation in $x$ or $y$, find the square roots of $6-8\mathrm{i}$ in exact Cartesian form.

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• Question 4Oct/Nov 2023, paper 31Structured5 marks

  The complex number $u$ is defined by $u=\frac{3+2\mathrm{i}}{a-5\mathrm{i}}$, where $a$ is real. Express $u$ in the Cartesian form $x+\mathrm{i}y$, where $x$ and $y$ are in terms of $a$.

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• Question 5Oct/Nov 2024, paper 32Structured4 marks

  The complex number $u$ is given by $$u=\frac{(\cos \frac{1}{7}\pi +\mathrm{i}\sin \frac{1}{7}\pi {)}^4}{\cos \frac{1}{7}\pi -\mathrm{i}\sin \frac{1}{7}\pi }\mathrm{.}$$ Find the exact value of $\arg u$.

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• Question 4May/June 2026, paper 32Structured5 marks

  The square roots of $3-(6\sqrt{2})\mathrm{i}$ can be expressed in the Cartesian form $x+\mathrm{i}y$, where $x$ and $y$ are real and exact. By first forming a quartic equation, find the square roots of $3-(6\sqrt{2})\mathrm{i}$.

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• Question 8Oct/Nov 2023, paper 32Structured7 marks

  It is given that $\frac{2+3a\mathrm{i}}{a+2\mathrm{i}}=\lambda (2-\mathrm{i})$, where $a$ and $\lambda$ are real constants. Show that $3a^2+4a-4=0$.

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• Question 7Oct/Nov 2025, paper 33Structured6 marks

  Solve the equation $\frac{5z}{2-\mathrm{i}}-z{z}^{\ast }+20+8\mathrm{i}=0$.

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• Question 4Oct/Nov 2024, paper 33Structured5 marks

  Find the complex number $z$ satisfying the equation $$\frac{z-3\mathrm{i}}{z+3\mathrm{i}}=\frac{2-9\mathrm{i}}{5}\mathrm{.}$$ Give your answer in the form $x+\mathrm{i}y$, where $x$ and $y$ are real.

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• Question 7May/June 2026, paper 33Structured5 marks

  The square roots of $3-(20\sqrt{7})\mathrm{i}$ can be expressed in the Cartesian form $x+\mathrm{i}y$, where $x$ and $y$ are real and exact. By first forming a quartic equation, find the square roots of $3-(20\sqrt{7})\mathrm{i}$.

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• Question 3May/June 2025, paper 31Structured6 marks

  Find the complex numbers $z$ for which $\frac{z+5\mathrm{i}}{z-5}$ is real and $\mathrm{\mid }z\mathrm{\mid }=\sqrt{17}$.

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• Question 4May/June 2024, paper 31Structured4 marks

  The complex number $u$ is given by $u=-1-\mathrm{i}\sqrt{3}$. Express $u$ in the form $r(\cos \theta +\mathrm{i}\sin \theta )$, where $r>0$ and $-\pi <\theta ⩽\pi$.

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• Question 4Oct/Nov 2023, paper 33Structured5 marks

  Solve the quadratic equation $(3+\mathrm{i})w^2-2w+3-\mathrm{i}=0$, giving your answers in the form $x+\mathrm{i}y$, where $x$ and $y$ are real.

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• Question 7May/June 2024, paper 31Structured6 marks

  On a single Argand diagram sketch the loci given by the equations $\mathrm{\mid }z-3+2\mathrm{i}\mathrm{\mid }=2$ and $\mathrm{\mid }w-3+2\mathrm{i}\mathrm{\mid }=\mathrm{\mid }w+3-4\mathrm{i}\mathrm{\mid }$ where $z$ and $w$ are complex numbers.

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• Question 10May/June 2023, paper 31Structured12 marks

  The polynomial $x^3+5x^2+31x+75$ is denoted by $\mathrm{p}(x)$. Show that $(x+3)$ is a factor of $\mathrm{p}(x)$.

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• Question 9May/June 2024, paper 32Structured10 marks

  The complex numbers $z$ and $\omega$ are defined by $z=1-\mathrm{i}$ and $\omega =-3+3\sqrt{3}\mathrm{i}$. Express $z\omega$ in the form $a+b\mathrm{i}$, where $a$ and $b$ are real and in exact surd form.

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• Question 3May/June 2023, paper 32Structured4 marks

  On an Argand diagram, sketch the locus of points representing complex numbers $z$ satisfying $\mathrm{\mid }z+3-2\mathrm{i}\mathrm{\mid }=2$.

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• Question 5May/June 2025, paper 32Structured5 marks

  The square roots of $-1-4\sqrt{5}\mathrm{i}$ can be expressed in the Cartesian form $x+\mathrm{i}y$, where $x$ and $y$ are real and exact. By first forming a quartic equation in $x$ or $y$, find the square roots of $-1-4\sqrt{5}\mathrm{i}$ in exact Cartesian form.

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• Question 3May/June 2024, paper 33Structured5 marks

  The square roots of $24-7\mathrm{i}$ can be expressed in the Cartesian form $x+\mathrm{i}y$, where $x$ and $y$ are real and exact. By first forming a quartic equation in $x$ or $y$, find the square roots of $24-7\mathrm{i}$ in exact Cartesian form.

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• Question 5May/June 2023, paper 32Structured6 marks

  The complex number $2+y\mathrm{i}$ is denoted by $a$, where $y$ is a real number and $y<0$. It is given that $\mathrm{f}(a)=a^3-a^2-2a$. Find a simplified expression for $\mathrm{f}(a)$ in terms of $y$.

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• Question 4May/June 2025, paper 33Structured6 marks

  It is given that $z_1=r_1{\mathrm{e}}^{\mathrm{i}{\theta }_1}$ and $z_2=r_2{\mathrm{e}}^{\mathrm{i}{\theta }_2}$. Show that $(z_1{z}_2{)}^{\ast }=z_1^{\ast }{z}_2^{\ast }$.

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• Question 11May/June 2023, paper 33Structured9 marks

  The complex number $z$ is defined by $z=\frac{5a-2\mathrm{i}}{3+a\mathrm{i}}$, where $a$ is an integer. It is given that $\arg z=-\frac{1}{4}\pi$. Find the value of $a$ and hence express $z$ in the form $x+\mathrm{i}y$, where $x$ and $y$ are real.

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