Question 1416

Complex Numbers
2016 Paper 1 Question 7 Variant

Question

Do not use a calculator in answering this question.
(a)
Verify that 4+4i{- 4 + 4 \mathrm{i}} is a root of the equation
w2+(6+i)w+(28+12i)=0.w^2 + \left( 6 + \mathrm{i} \right) w + \left( 28 + 12 \mathrm{i} \right) = 0.
Hence, or otherwise, find the second root of the equation in cartesian form, p+iq,{p+\mathrm{i}q,} showing your working.
[5]
(b)
The equation
z3+5z2+23z+k=0,z^3 + 5 z^2 + 23 z+k=0,
where k{k} is a real constant, has a root z=1+ai,{z=- 1 + a \mathrm{i},} where a{a} is a positive real constant.
Find the values of a{a} and k,{k,} showing your working.
[5]

Answer