The complex number 2 + yi is denoted by a, where y is a real number and y < 0. It is given that f(a) = a³ – a² – 2a.

Mark scheme

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

1. M1 Substitute 2 + yi in a³ – a² – 2a and attempt expansions of a² and a³ [a² = 4+4yi-y² a³ = 8+12yi−6y² – y³i. If using a(a²-a-2) must then expand fully. Must see working.]
2. M1 Use i² = -1 [Seen at least once (e.g. in squaring).]
3. A1 Obtain final answer −5y² + (6y – y³)i [Or simplified equivalent e.g. 6yi – 5y² – y³i . Do not ISW. SC B1 for the correct answer. No evidence of working for the square or the cube can score SC B1 for the correct answer.]
4. M1 Equate their -5y² to –20 and solve for y [Need to obtain a value for y. Available even if their y is not real.]
5. A1 Obtain y = -2 [From correct work. Allow after incorrect f(a) if the real part was correct. Condone ±2 with positive not rejected.]
6. A1 Correct only (must have rejected y positive). [OE e.g. -π/4 ± 2nπ . Accept –0.785, 5.50 . Allow after incorrect f(a) if the real part was correct. Accept degrees. Do not ISW.]