L3 Complex Numbers
Question Bank
The complex numbers $u$ and $v$ are shown on the Argand diagram below.
Find $w = u + \bar{v}$, and show $w$ on the Argand diagram.
The complex number $z$ is shown on the Argand diagram below.
Find $iz$ in the form $a + bi$, and show $iz$ on the Argand diagram.
The complex numbers $u$ and $w$ are shown on the Argand diagram below.
Find $z = 2u - 3w$ in the form $a + bi$, and show $z$ on the Argand diagram.
The complex number $w$ is shown on the Argand diagram below.
Find $w^2$ in the form $a + bi$, and show $w^2$ on the Argand diagram.
The complex numbers $p$ and $q$ are shown on the Argand diagram below.
Find $\dfrac{p}{q}$ in the form $a + bi$, and show it on the Argand diagram.
The complex number $z = \sqrt{3} + i$ is shown on the Argand diagram below.
Find the exact value of $z^3$, and show $z^3$ on the Argand diagram.
The complex number $z$ is shown on the Argand diagram below.
(a) Find $w = (1 + i)z$ in the form $a + bi$, and show $w$ on the Argand diagram.
(b) Show that the triangle with vertices $0$, $z$ and $w$ is isosceles, with a right angle at $z$.
The complex numbers $p$ and $q$ are shown on the Argand diagram below.
(a) Find $p - q$, and show it on the Argand diagram.
(b) Find $pq$ in the form $a + bi$.
(c) Find the value of the real number $k$ for which $p + kq$ lies on the line $y = x$.
The complex numbers $u$, $v$, and $w$ are shown on the Argand diagram below.
(a) Calculate the value of $u - 2v$.
(b) Calculate the value of $\displaystyle \frac{u(v + w)}{v}$. Give your answer in the form $a + bi$.
(c) Find the value of the real constant $k$ such that $u + kw$ is a purely imaginary number.
The complex number $z$ is shown on the Argand diagram below.
(a) Find $z^2$ and $\bar{z}$, and show both on the Argand diagram.
(b) Find $iz$, and describe how multiplying by $i$ moves the point $z$ on the Argand diagram.
(c) The points $0$, $z$, $w$ and $iz$ are the four corners of a square, with $w$ opposite $0$. Find $w$.
The complex number $u = 4 + 2i$ is shown on the Argand diagram below.
The point $v$ is found by rotating $u$ anticlockwise about the origin through an angle of $\dfrac{\pi}{3}$.
(a) Explain why $v = u\,\text{cis}\dfrac{\pi}{3}$, and find $v$ exactly in the form $a + bi$.
(b) Find the exact area of the triangle with corners $0$, $u$ and $v$.