m and n are real numbers;

m and n are conjugate complex numbers.

attempt to equate real and imaginary parts     M1

equate real parts: $4m + 4n = 16$; equate imaginary parts: $-5m = 15$     A1

$\Rightarrow m = -3,{\text{ }}n = 7$     A1

[3 marks]

let $m = x + {\text{i}}y,{\text{ }}n = x - {\text{i}}y$     M1

$\Rightarrow (4 - 5{\text{i}})(x + {\text{i}}y) + 4(x - {\text{i}}y) = 16 + 15{\text{i}}$

$\Rightarrow 4x - 5{\text{i}}x + 4{\text{i}}y + 5y + 4x - 4{\text{i}}y = 16 + 15{\text{i}}$

attempt to equate real and imaginary parts     M1

$8x + 5y = 16,{\text{ }} -5x = 15$     A1

$\Rightarrow x = -3,{\text{ }}y = 8$     A1

$( \Rightarrow m = -3 + 8{\text{i}},{\text{ }}n = -3 - 8{\text{i}})$

[4 marks]

Part (a) was generally well answered. In (b), however, some candidates put $m = a + {\text{i}}b$ and $n = c + {\text{i}}d$ which gave four equations for two unknowns so that no further progress could be made.

Part (a) was generally well answered. In (b), however, some candidates put $m = a + {\text{i}}b$ and $n = c + {\text{i}}d$ which gave four equations for two unknowns so that no further progress could be made.