A-LevelMathematicsComplex numbers9709/31 · Oct/Nov 2019 · Q1010 Marks

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]

(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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Mark scheme

Marking points from the official mark scheme (Oct/Nov 2019). The mark or mark type is shown in brackets.

[*M1] Square a + ib and equate real and imaginary parts to – 3 and -2√10 respectively
[A1] Obtain a²-b² =−3 and 2ab =-2√10
[DM1] Eliminate one unknown and find an equation in the other
[A1] Obtain a⁴ + 3a² −10 = 0, or b⁴ – 3b² – 10 = 0, or horizontal 3-term equivalent
[A1] Obtain answers ±(√2-√5i), or exact equivalent
[B1] Show point representing 3 + i in relatively correct position
[B1] Show a circle with radius 3 and centre not at the origin
[B1] Show correct half line from the origin at π/4 to the real axis
[B1] Show horizontal line y = 2
[B1] Shade the correct region

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