Browse Complex numbers

1. 1
   The complex numbers $7 + \mathrm{i}$ and $3 + 4\mathrm{i}$ are denoted by $u$ and $v$ respectively.
   (a) Find $\dfrac{u}{v}$ in the form $x + \mathrm{i}y$, where $x$ and $y$ are real.
   
   (3 marks)
   (b) State the argument of $\dfrac{u}{v}$.
   
   (1 mark)
   
   In an Argand diagram, with origin $O$, the points $A$, $B$ and $C$ represent the complex numbers $u$, $v$ and $u + v$ respectively.
   (c) State fully the geometrical relationship between $AC$ and $OB$.
   
   (2 marks)
   (d) Show that angle $OAC = \tfrac{3}{4}\pi$ radians.
   
   (2 marks)
   Level 4 · 8 marksTry it→
2. 2
   Solve the equation $(2 + i)z = 3 - i$, giving your answer for $z$ in the form $a + bi$. (3 marks)
   Level 4 · 3 marksTry it→
3. 3
   Find the two square roots of $3 - 4i$, giving your answers in the form $a + bi$ where $a$ and $b$ are real. (5 marks)
   Level 4 · 5 marksTry it→

All 17 complex numbers in cartesian form questions →

1. 1
   A complex number $z$ has modulus $\sqrt2$ and argument $-\dfrac{\pi}{4}$. Express $z$ in the form $a + bi$. (3 marks)
   Level 4 · 3 marksTry it→
2. 2
   On an Argand diagram, the points $O$, $A$ and $B$ represent $0$, $z_{1} = 4 + 2i$ and $z_{2} = -1 + 3i$ respectively.
   
   Find the area of triangle $OAB$. (4 marks)
   Level 4 · 4 marksTry it→
3. 3
   Given $z_{1} = 3 + 4i$ and $z_{2} = 5 - 12i$:
   (a) Find $|z_{1} + z_{2}|$, giving your answer as a simplified surd. (3 marks)
   (b) Verify that $|z_{1} + z_{2}| < |z_{1}| + |z_{2}|$, and state the geometric fact this illustrates. (2 marks)
   Level 4 · 5 marksTry it→

All 15 the argand diagram, modulus and argument questions →

1. 1
   It is given that $3 + 2i$ is a root of the equation $z^3 - 8z^2 + 25z - 26 = 0$, whose coefficients are all real.
   (a) Write down a second root of the equation. (1 mark)
   (b) Hence find the real root of the equation and factorise $z^3 - 8z^2 + 25z - 26$ completely. (4 marks)
   Level 4 · 5 marksTry it→
2. 2
   Find the two square roots of $3 + 4i$, giving each in the form $a + bi$ where $a$ and $b$ are real. (5 marks)
   Level 4 · 5 marksTry it→
3. 3
   It is given that $1 + 2i$ is a root of the equation
   
   $$z^{3} - 3z^{2} + 7z - 5 = 0$$
   (a) Explain why $1 - 2i$ must also be a root. (1 mark)
   (b) Find all three roots of the equation. (4 marks)
   Level 4 · 5 marksTry it→

All 15 complex roots of polynomial equations questions →

1. 1
   (a) On a single Argand diagram sketch the loci given by the equations $|z - 5 - 2\mathrm{i}| = 2$ and $\arg(w + 1) = \tfrac{1}{3}\pi$, where $z$ and $w$ are complex numbers.
   
   (4 marks)
   (b) Hence find the least value of $|z - w|$ for points on these loci. Give your answer in an exact form.
   
   (2 marks)
   Level 4 · 6 marksTry it→
2. 2
   Describe the locus $\arg(z - 2) = \dfrac{\pi}{4}$, and give the equation of the line on which it lies together with any restriction. (3 marks)
   Level 4 · 3 marksTry it→
3. 3
   Find the Cartesian equation of the locus $|z - 6i| = |z - 2|$. (4 marks)
   Level 4 · 4 marksTry it→

All 16 loci in the argand diagram questions →

1. 1
   Using $z_1 = 2\left(\cos\dfrac{\pi}{3} + i\sin\dfrac{\pi}{3}\right)$ and $z_2 = \sqrt{2}\left(\cos\dfrac{3\pi}{4} + i\sin\dfrac{3\pi}{4}\right)$, express $\dfrac{z_1}{z_2}$ in the form $r(\cos\theta + i\sin\theta)$, where $r > 0$ and $-\pi < \theta \le \pi$. (3 marks)
   Level 4 · 3 marksTry it→
2. 2
   The complex number $z = 3\left(\cos\dfrac{\pi}{5} + i\sin\dfrac{\pi}{5}\right)$.
   (a) Write down $z^{2}$ in polar form. (2 marks)
   (b) Find the smallest positive integer $n$ for which $z^{n}$ is real, and state the value of $z^{n}$ for this $n$. (3 marks)
   Level 4 · 5 marksTry it→