Question 1414a

Complex Numbers
2014 Paper 1 Question 5 Variant

Question

It is given that z=3i.{z=3 - \mathrm{i}.}
(a)
Without using a calculator, find the values of z2{z^2} and 1z2{\displaystyle \frac{1}{z^{2}}} in cartesian form x+iy,{x + \mathrm{i}y,} showing your working.
[4]
(b)
The real numbers p{p} and q{q} are such that pz2+qz2{\displaystyle pz^2 + \frac{q}{z^{2}}} is real.
Find, in terms of p,{p,} the value of q{q} and the value of pz2+qz2.{\displaystyle pz^2 + \frac{q}{z^{2}}.}
[3]

Answer