If z₁ = (1 + i) and z₂ = (–2 + 4i), prove that Im

If z₁ is a complex number other than –1 such that |z₁| = 1 and z₂ = then show that z2 is purely imaginary.

For all z C, prove that

(i)

(ii) .

(iii) = |z|²

(iv) is real

(v) is 0 or imaginary.

If a and b are real numbers such that a² + b² = 1 then show that a real value of x satisfies the equation,

Find the modulus of each of the following complex numbers and hence express each of them in polar form: 4