Questions covered in Basic

Review the question prompts below before starting the timed test. Try each problem independently, then use the interactive test to record your answers and review the worked solutions.

1. The value of $i^{19}$ is
2. The conjugate of the complex number $z = 3 - 4i$ is
3. The roots of the equation $x^2 + 2 = 0$ are
4. The multiplicative inverse of $1+i$ is
5. If $4x + i(3x-y) = 3 - 6i$, where $x$ and $y$ are real numbers, then the values of $x$ and $y$ are
6. Express $(5 - 3i)^3$ in the form $a + ib$.
7. Solve the quadratic equation $x^2 + x + 1 = 0$.
8. Find the modulus of the complex number $z = \frac{1+i}{1-i}$.
9. Convert the complex number $\frac{1+3i}{1-2i}$ into the form $a+ib$.
10. Solve the equation $\sqrt{5}x^2 + x + \sqrt{5} = 0$.
11. If $(x+iy)^3 = u+iv$, then show that $\frac{u}{x} + \frac{v}{y} = 4(x^2 - y^2)$.
12. Case Study: Two complex numbers are given by $z_1 = 2+3i$ and $z_2 = 1-i$.
    (i) Find $z_1 + z_2$.
    (ii) Find $z_1 z_2$.
    (iii) Find $|z_1|$.