IB Mathematics Analysis and Approaches Topic Number…

The complex number $z$ is defined by $z = \frac{4-4i}{1+i}$ .

(a) Find an expression for $z$ in the form $re^{i\theta}$ , where $r>0$ and $-\pi < \theta \le \pi$ .

(b) Find the two square roots of $z$ , giving your answers in the form $re^{i\theta}$ , where $r>0$ and $-\pi < \theta \le \pi$ .

Prove that for any real number $x$ , $x(x+2) - (x+1)^2 \equiv -1$ .

The complex number $z$ is given in Euler form as $z = 4e^{i\frac{\pi}{3}}$ .

Express $z$ in the form $x+iy$ , where $x$ and $y$ are real.

Given expression $n^3 + (n+1)^3 + (n+2)^3$ , prove by induction that the expression is divisible by $9$ for all positive integers $n$ .

Given the complex number $w = 5(\cos \frac{\pi}{4} + i\sin \frac{\pi}{4})$ .

(a) Convert the complex number $w$ to its Cartesian form.

(b) Plot the complex number $w$ on an Argand diagram.

IB Mathematics Analysis and Approaches Number and Algebra Questionbank