Comprehensive assessment covering the arithmetic of complex numbers, conjugates and division, modulus and argument, the Argand diagram, polar (mod-arg) form, multiplication and division in polar form, and loci on the Argand plane.
The value of $i^2$ is:
Simplify $(3 + 2i) + (1 - 5i)$.
Expand and simplify $(2 + 3i)(1 - i)$.
The complex conjugate of $4 - 7i$ is:
The modulus $|3 + 4i|$ equals:
Simplify $\dfrac{1}{i}$.
The value of $i^{23}$ is:
The principal argument of $1 + i$ is:
Express $\dfrac{2 + i}{1 - i}$ in the form $a + bi$.
If $z_1 = r_1(\cos\theta_1 + i\sin\theta_1)$ and $z_2 = r_2(\cos\theta_2 + i\sin\theta_2)$, then the product $z_1 z_2$ has:
The modulus of $2\left(\cos\dfrac{\pi}{3} + i\sin\dfrac{\pi}{3}\right)$ is:
Written in Cartesian form, $2\left(\cos\dfrac{\pi}{2} + i\sin\dfrac{\pi}{2}\right)$ equals:
If $z = 3 + 4i$, then $z\bar{z}$ equals:
On the Argand plane, the equation $|z| = 3$ represents:
The locus of points satisfying $|z - 2| = |z + 2|$ is:
1. Simplify $(4 + 3i) - (2 - i)$. (1 mark)
2. Express $(1 + 2i)(3 - i)$ in the form $a + bi$. (2 marks)
3. Find the modulus and argument of $z = 1 + \sqrt{3}\,i$. (2 marks)
4. Express $\dfrac{5}{2 - i}$ in the form $a + bi$. (2 marks)
5. Write $z = -1 + i$ in polar (mod-arg) form. (2 marks)
6. If $z = 2\left(\cos\dfrac{\pi}{6} + i\sin\dfrac{\pi}{6}\right)$ and $w = 3\left(\cos\dfrac{\pi}{3} + i\sin\dfrac{\pi}{3}\right)$, find $zw$ in polar form. (2 marks)
7. Find the real numbers $x$ and $y$ such that $(x + yi)(1 - i) = 3 + i$. (2 marks)
8. Solve $z^2 = -9$ for $z$. (1 mark)
9. Describe geometrically the locus of points satisfying $|z - 3| = 2$. (1 mark)
10. Show that $\dfrac{1}{1 + i} + \dfrac{1}{1 - i}$ is real, and find its value. (2 marks)