Questions covered in Hard

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 $\sum_{n=1}^{13} (i^n + i^{n+1})$ is
2. If $(x+iy)^{1/3} = a+ib$, then $\frac{x}{a} + \frac{y}{b}$ is equal to
3. The smallest positive integer $n$ for which $(1+i)^{2n} = (1-i)^{2n}$ is
4. If $z = x+iy$ and $|z+6| = |2z+3|$, then the locus of $z$ is
5. If $z = 1 + i\sqrt{3}$, then $|\text{arg}(z)| + |\text{arg}(\bar{z})|$ is
6. Solve the equation $x^2 - (3\sqrt{2} - 2i)x - 6\sqrt{2}i = 0$.
7. Find the modulus and argument of $z = \frac{1+2i}{1-3i}$.
8. If $|z|=1$ and $z \neq -1$, prove that $\frac{z-1}{z+1}$ is purely imaginary.
9. If $\alpha$ and $\beta$ are roots of $x^2 - 2x + 4 = 0$, find the value of $\alpha^3 + \beta^3$.
10. Find the number of non-zero integral solutions of the equation $|1-i|^x = 2^x$.
11. If $z_1, z_2$ are complex numbers such that $|\frac{1-z_1 \bar{z}_2}{z_1-z_2}| = 1$, prove that $|z_1|=1$ or $|z_2|=1$.
12. Case Study: Let $z_1 = 1+i$ be a complex number represented by point P in the Argand plane.
    (i) Find the modulus and argument of $z_1$.
    (ii) If $z_1$ is rotated by $90^\circ$ in the anti-clockwise direction about the origin to get $z_2$, find $z_2$.
    (iii) Find the distance between $z_1$ and $z_2$.