10 (a) The complex number u is given by u = −3 – (2√10)i. Showing all necessary working and without
using a calculator, find the square roots of u. Give your answers in the form a + ib, where the
numbers a and b are real and exact.
[5]

(b) On a sketch of an Argand diagram shade the region whose points represent complex numbers
z satisfying the inequalities |z – 3 – i| ≤ 3, argz ¦π and Imz ≥ 2, where Im z denotes the
imaginary part of the complex number z.
[5]

Q: What topic does this question test?

This question tests Complex numbers in A-Level Mathematics (syllabus code 9709). It is worth 10 marks.

Q: Which exam paper is this question from?

This question appeared in the Cambridge A-Level Mathematics Oct/Nov 2019 examination, Paper 3 Variant 1 (9709_w19_qp_31.pdf).

Question

10 (a) The complex number u is given by u = −3 – (2√10)i. Showing all necessary working and without
using a calculator, find the square roots of u. Give your answers in the form a + ib, where the
numbers a and b are real and exact.
[5]

(b) On a sketch of an Argand diagram shade the region whose points represent complex numbers
z satisfying the inequalities |z – 3 – i| ≤ 3, argz > ¦π and Imz ≥ 2, where Im z denotes the
imaginary part of the complex number z.
[5]

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