Browse Complex numbers

1. 1
   Express $\dfrac{5 + i}{2 - 3i}$ in the form $a + bi$, where $a$ and $b$ are real. (3 marks)
   Level 3 · 3 marksTry it→
2. 2
   Find the real numbers $x$ and $y$ such that
   
   $(x + yi)(2 - i) = 5 + 5i$. (4 marks)
   Level 3 · 4 marksTry it→
3. 3
   The complex number $z = 3 - 4i$.
   (a) Find $z^2$, giving your answer in the form $a + bi$. (2 marks)
   (b) Express $\dfrac{1}{z}$ in the form $a + bi$. (2 marks)
   Level 3 · 4 marksTry it→

All 17 complex numbers in cartesian form questions →

1. 1
   The complex number $z = -2 + 2i$. Find $|z|$ and $\arg z$, giving the argument as an exact multiple of $\pi$. (3 marks)
   Level 3 · 3 marksTry it→
2. 2
   Find the modulus and argument of $z = -\sqrt3 - i$, giving the argument as an exact multiple of $\pi$ in the range $-\pi < \arg z \le \pi$. (4 marks)
   Level 3 · 4 marksTry it→
3. 3
   On an Argand diagram, points $A$ and $B$ represent the complex numbers $z_1 = 6 + 2i$ and $z_2 = 2 - i$. Find the distance $AB$. (3 marks)
   Level 3 · 3 marksTry it→

All 15 the argand diagram, modulus and argument questions →

1. 1
   The complex number $1 + 2i$ is a root of a polynomial equation that has real coefficients.
   (a) Write down another root of the equation. (1 mark)
   (b) Hence find a quadratic factor of the polynomial that has real coefficients. (2 marks)
   Level 3 · 3 marksTry it→
2. 2
   (a) Solve the equation $z^{2} + 4z + 29 = 0$, giving your answers in the form $a + bi$. (3 marks)
   (b) Verify that the product of your roots equals the constant term. (1 mark)
   Level 3 · 4 marksTry it→
3. 3
   Find a quadratic equation with real coefficients that has $4 + i$ as a root, giving your answer in the form $z^{2} + bz + c = 0$. (3 marks)
   Level 3 · 3 marksTry it→

All 15 complex roots of polynomial equations questions →

1. 1
   The locus of $z$ satisfies $|z + 2| = |z - 4|$.
   (a) Describe the locus geometrically. (1 mark)
   (b) Find its Cartesian equation. (3 marks)
   Level 3 · 4 marksTry it→
2. 2
   Show that the locus $|z - 1| = |z + i|$ has Cartesian equation $y = -x$. (3 marks)
   Level 3 · 3 marksTry it→
3. 3
   The complex number $z$ satisfies $|z - 3 - 4i| \leqslant 2$.
   (a) Describe the region represented by this inequality. (2 marks)
   (b) Find the least and greatest values of $|z|$. (2 marks)
   Level 3 · 4 marksTry it→

All 16 loci in the argand diagram questions →

1. 1
   The complex number $u$ is given by $u = 3 - 3\sqrt{3}\,\mathrm{i}$.
   (a) Express $u$ in the form $r(\cos\theta + \mathrm{i}\sin\theta)$, where $r > 0$ and $-\pi < \theta \leqslant \pi$. Give the exact values of $r$ and $\theta$.
   
   (2 marks)
   
   The complex number $v$ is given by $v = 2\left(\cos\tfrac{3}{4}\pi + \mathrm{i}\sin\tfrac{3}{4}\pi\right)$.
   (b) Express the complex number $\dfrac{u}{v}$ in the form $r\mathrm{e}^{\mathrm{i}\theta}$, where $r > 0$ and $-\pi < \theta \leqslant \pi$.
   
   (2 marks)
   Level 3 · 4 marksTry it→
2. 2
   Given $z_1 = 2\left(\cos\dfrac{\pi}{3} + i\sin\dfrac{\pi}{3}\right)$ and $z_2 = \sqrt{2}\left(\cos\dfrac{3\pi}{4} + i\sin\dfrac{3\pi}{4}\right)$, express $z_1 z_2$ in the form $r(\cos\theta + i\sin\theta)$, where $r > 0$ and $-\pi < \theta \le \pi$. (3 marks)
   Level 3 · 3 marksTry it→
3. 3
   Express $z = 6\left(\cos\dfrac{5\pi}{6} + i\sin\dfrac{5\pi}{6}\right)$ in the form $a + bi$, giving $a$ and $b$ exactly. (3 marks)
   Level 3 · 3 marksTry it→

All 16 modulus–argument (polar) form questions →