Showing all necessary working, express the complex number (2 + 3i) / (1 − 2i) in the form re^(iθ), where r > 0 and –π < θ < π. Give the values of r and θ correct to 3 significant figures.

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2018). The mark or mark type is shown in brackets.

1. [M1] (a) EITHER: Multiply numerator and denominator by 1 + 2i, or equivalent, or equate to x + iy, obtain two equations in x and y and solve for x or for y
2. [A1] (a) Obtain quotient -4/5 + 7/5 i, or equivalent
3. [M1] (a) Use correct method to find either r or θ
4. [A1] (a) Obtain r = 1.61
5. [A1] (a) Obtain θ = 2.09
6. [B1] (a) OR: Find modulus or argument of 2 + 3i or of 1 – 2i
7. [M1] (a) Use correct method to find r
8. [A1] (a) Obtain r = 1.61
9. [M1] (a) Use correct method to find θ
10. [A1] (a) Obtain θ = 2.09
11. [B1] (b) Show a circle with centre 3 – 2i
12. [B1ft] (b) Show a circle with radius 1
13. [M1] (b) Carry out a correct method for finding the least value of |z|
14. [A1] (b) Obtain answer √13 – 1