Complex arithmetic

The complex numbers z and w are defined by

By evaluating the product zw, show that

The complex number z is such that

where k is a real number.

Find the real part of z and the imaginary part of z, giving your answers in terms of k

Find the imaginary part of

Circle your answer.

Given that z 1 3i is one root of the equation z² pz r 0, where p and r are real, find the value of r.

Circle your answer.

Show that the equation

has exactly four solutions and state these solutions.

Plot the four solutions to the equation in part (a) on the Argand diagram below and join them together to form a quadrilateral with one line of symmetry.

The complex number z satisfies the equations

and

where a is real.

numbers.

Find the value of p and the value of r.

Solve the equation 2z 5iz* 12

Given that z is a complex number, and that z * is the complex conjugate of z, which of the following statements is not always true?

Circle your answer.

Abel and Bonnie are trying to solve this mathematical problem:

Abel says he has solved the problem.

Bonnie says there is not enough information to solve the problem.

Abel’s solution begins as follows:

State one extra piece of information about m and p which could be added to the problem to make the beginning of Abel’s solution correct.

Prove that Bonnie is right.