Browse Complex numbers

1. 1
   Given $z_1 = 2 + 3i$ and $z_2 = 4 - i$, find
   (a) $z_1 + z_2$, (1 mark)
   (b) $z_1 - z_2$. (1 mark)
   Level 1 · 2 marksTry it→
2. 2
   Expand and simplify $(3 + 2i)(4 - 5i)$, giving your answer in the form $a + bi$. (2 marks)
   Level 2 · 2 marksTry it→
3. 3
   The complex number $z = 5 - 2i$, and $z^{*}$ denotes its conjugate.
   (a) Find $zz^{*}$. (2 marks)
   (b) Write down $z + z^{*}$ and $z - z^{*}$. (2 marks)
   Level 2 · 4 marksTry it→

All 17 complex numbers in cartesian form questions →

1. 1
   Find the modulus of the complex number $z = 8 + 6i$. (2 marks)
   Level 1 · 2 marksTry it→
2. 2
   Find the modulus and argument of $z = 1 + \sqrt{3}\,i$, giving the argument as an exact multiple of $\pi$. (3 marks)
   Level 2 · 3 marksTry it→
3. 3
   The complex number $z = 2 + 5i$ is shown on an Argand diagram.
   (a) Describe the geometric relationship between the points representing $z$ and $z^{*}$. (1 mark)
   (b) Describe the geometric relationship between the points representing $z$ and $-z$. (1 mark)
   (c) Find $|z|$, giving your answer as an exact surd. (2 marks)
   Level 2 · 4 marksTry it→

All 15 the argand diagram, modulus and argument questions →

1. 1
   Solve the equation $z^2 - 4z + 13 = 0$, giving your answers in the form $a + bi$ where $a$ and $b$ are real. (3 marks)
   Level 1 · 3 marksTry it→
2. 2
   Solve the equation $z^2 + 2z + 5 = 0$, giving your answers in the form $a + bi$. (3 marks)
   Level 2 · 3 marksTry it→
3. 3
   Solve the equation $z^{2} - 6z + 25 = 0$, giving your answers in the form $a + bi$. (3 marks)
   Level 2 · 3 marksTry it→

All 15 complex roots of polynomial equations questions →

1. 1
   Describe fully the locus of points in the Argand diagram given by $|z - (2 + i)| = 3$. (2 marks)
   Level 1 · 2 marksTry it→
2. 2
   Find the Cartesian equation of the locus $|z - 3i| = 2$, and describe it geometrically. (3 marks)
   Level 2 · 3 marksTry it→
3. 3
   The complex number $z$ satisfies $|z - 4| = 3$.
   (a) Describe the locus of $z$ fully. (2 marks)
   (b) Find the least and greatest values of $|z|$. (2 marks)
   Level 2 · 4 marksTry it→

All 16 loci in the argand diagram questions →

1. 1
   The complex number $z = 1 + \sqrt{3}\,i$.
   (a) Find $|z|$. (1 mark)
   (b) Find $\arg z$, and hence express $z$ in the form $r(\cos\theta + i\sin\theta)$, where $-\pi < \theta \le \pi$. (2 marks)
   Level 1 · 3 marksTry it→
2. 2
   Express each of the following in the form $r(\cos\theta + i\sin\theta)$, where $r > 0$ and $-\pi < \theta \le \pi$.
   (a) $\sqrt{3} + i$ (2 marks)
   (b) $-1 + i$ (2 marks)
   Level 2 · 4 marksTry it→
3. 3
   The complex number $z = 4\left(\cos\dfrac{2\pi}{3} + i\sin\dfrac{2\pi}{3}\right)$.
   
   Express $z$ in the form $a + bi$, giving $a$ and $b$ exactly. (2 marks)
   Level 2 · 2 marksTry it→

All 16 modulus–argument (polar) form questions →