Throughout this question the use of a calculator is not permitted. The complex number 2 – i is denoted by u.

Mark scheme

Marking points from the official mark scheme (May/June 2017). The mark or mark type is shown in brackets.

1. [M1] Substitute x = 2 – i (or x = 2+i ) in the equation and attempt expansions of x² and x³
2. [M1] Equate real and/or imaginary parts to zero
3. [A1] Obtain a = -2
4. [A1] Obtain b = 10
5. [M1] Substitute x = 2 – i in the equation and attempt expansions of X² and x³
6. [M1] Substitute x=2+i in the equation and add/subtract the two equations
7. [A1] Obtain a = -2
8. [A1] Obtain b = 10
9. [M1] Factorise to obtain (x−2+i)(x-2-1)(x-p) (= (x²-4x+5)(x-p))
10. [M1] Compare coefficients
11. [A1] Obtain a = -2
12. [A1] Obtain b = 10
13. [M1] Obtain the quadratic factor (x² - 4x+5)
14. [M1] Use algebraic division to obtain a real linear factor of the form x-p and set the remainder equal to zero
15. [A1] Obtain a = -2
16. [A1] Obtain b = 10
17. [M1] Use aβ = 5 and a + β =4 in αβ + βγ + γα = −3
18. [M1] Solve for γ and use in αβγ = −b and/or α + β + γ = -a
19. [A1] Obtain a = -2
20. [A1] Obtain b = 10
21. [M1] Factorise as (x-- (2-i))(x² + ex + g) and compare coefficients to form an equation in a and b
22. [M1] Equate real and/or imaginary parts to zero
23. [A1] Obtain a = -2
24. [A1] Obtain b = 10
25. [B1] Show a circle with centre 2 – i in a relatively correct position
26. [B1] Show a circle with radius 1 and centre not at the origin
27. [B1] Show the perpendicular bisector of the line segment joining 0 to – i
28. [B1] Shade the correct region