8 It is given that (2 + 3ai) / (a + 2i) = λ(2 – i), where a and λ are real constants. (a) Show that 3a² + 4a – 4 = 0. (b) Hence find the possible values of a and the corresponding values of λ.

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2023). The mark or mark type is shown in brackets.

1. *M1 Multiply both sides by a+2i and attempt expansion of right-hand side
2. DM1 Use of i² = −1 seen at least once (or implied) e.g. 2 + 3ai = λ(2a + 2) + λi(−a+4)
3. M1 Compare real and imaginary parts to obtain an equation in a only [2= λ(2a+2), 3a = λ(-a+4)]
4. A1 AG Obtain 3a² + 4a-4=0 from correct working
5. M1 Solve given quadratic to obtain a value of a and use this to form an equation in λ only (based on an equation seen in their working in (a) or (b))
6. A1 Obtain a = -2, λ=−1 or a = ⅔, λ = ⁶/₅ Allow ⅔ and 0.6.
7. A1 Obtain second correct pair of values