The complex number u is given by u = 8 – 15i. Showing all necessary working, find the two square roots of u. Give answers in the form a + ib, where the numbers a and b are real and exact.

Mark scheme

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

1. [M1] (a) Square x + iy and equate real and imaginary parts to 8 and -15
2. [A1] (a) Obtain x² – y² = 8 and 2xy = -15
3. [M1] (a) Eliminate one unknown and find a horizontal equation in the other
4. [A1] (a) Obtain 4x⁴ – 32x² – 225 = 0 or 4y⁴ + 32y² – 225 = 0, or three term equivalent
5. [A1] (a) Obtain answers ±½(5–3i) or equivalent
6. [B1] (b) Show a circle with centre 2+ i in a relatively correct position
7. [B1] (b) Show a circle with radius 2 and centre not at the origin
8. [B1] (b) Show line through i at an angle of ¼π to the real axis
9. [B1] (b) Shade the correct region