A-LevelMathematicsComplex numbers9709/32 · Feb/Mar 2023 · Q25 Marks

On an Argand diagram, shade the region whose points represent complex numbers z satisfying the inequalities −⅓π ≤ arg(z − 1 − 2i) < π and Rez < 3.

(a)On an Argand diagram, shade the region whose points represent complex numbers z satisfying the inequalities −⅓π ≤ arg(z − 1 − 2i) < π and Rez < 3.[3]
(b)Calculate the least value of arg z for points in the region from (a). Give your answer in radians correct to 3 decimal places.[2]

This question uses a diagram from the original paper, which isn't shown here. Open the full paper from the past papers archive to see it.

Mark scheme

Marking points from the official mark scheme (Feb/Mar 2023). The mark or mark type is shown in brackets.

[B1] Show correct half-lines from 1 + 2i, symmetrical about y = 2i (drawn between π/4 and 5π/12).
[B1] Show the line x = 3 extending in both quadrants.
[B1 FT] Shade the correct region. Allow dashes on axes as scale. FT If only error is one of following: FULL lines or x ≠ 3 or one sign error in 1 + 2i or angle outside tolerance or scale missing on one axis. SC No scale on either axis allow B1 FT for otherwise correct figure in correct position.
[M1] Carry out a complete method for finding the least value of arg z e.g. -tan-¹((2√3-2)/3) or tan-¹((-2√3+2)/3)
[A1] Obtain answer -0.454 3dp SC B1 0.454.

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