AS June 2022 Paper 1 Q7

OCR MEICurrent spec9 marksComplex Numbers

7 On an Argand diagram, the point A represents the complex number \(z\) with modulus 2 and argument \(\tfrac{1}{3}\pi\). The point B represents \(\dfrac{1}{z}\).

(a) Sketch an Argand diagram showing the origin O and the points A and B. [2]

(b) The point C is such that OACB is a parallelogram. C represents the complex number \(w\).

Determine each of the following.

The modulus of \(w\), giving your answer in exact form.
The argument of \(w\), giving your answer correct to 3 significant figures. [7]

Sketch: axes through O; A in the first quadrant at about 60 degrees to the real axis with OA = 2; B in the fourth quadrant at about 60 degrees below the real axis with OB = one half — Mark scheme (a)

Scheme	Marks	AO
	B1 B1	1.1 1.1
	[2]

Mark scheme (b)

Scheme	Marks	AO
\(z = 2\left(\cos\dfrac{\pi}{3} + \mathrm{i}\sin\dfrac{\pi}{3}\right)\)	M1	3.1a
\(= 1 + \sqrt{3}\mathrm{i}\)	A1	1.1
\(\dfrac{1}{z} = \dfrac{1}{2}\left(\cos\left(-\dfrac{\pi}{3}\right) + \mathrm{i}\sin\left(-\dfrac{\pi}{3}\right)\right) = \dfrac{1}{4}(1 - \sqrt{3}\mathrm{i})\)	B1	1.1
\(w = z + \dfrac{1}{z}\)	M1	3.1a
\(= \dfrac{5}{4} + \dfrac{3\sqrt{3}}{4}\mathrm{i}\)	A1	1.1ft
\(\|w\| = \sqrt{\dfrac{25}{16} + \dfrac{27}{16}} = \dfrac{\sqrt{13}}{2}\)	B1	3.2a
\(\arg(w) = \arctan\dfrac{3\sqrt{3}}{5} = 0.805\)	B1	3.2a
	[7]

Notes

M1: (1st) converting to \(a + b\mathrm{i}\) form

B1: (1st) NB these first three marks may be awarded if gained in part (a)

M1: (2nd) or equivalent methods (e.g. vector displacements)

A1: (2nd) or \(\left(\dfrac{5}{4}, \dfrac{3\sqrt{3}}{4}\right)\)

B1: (2nd) must be from correct \(w\)

B1: (3rd) 0.80 or better or \(46^\circ\) or better, must be from correct \(w\)

Alternative solution

Scheme	Marks
Equation of BC is \(y = \sqrt{3}x - \sqrt{3}/2\) Equation of AC is \(y = -\sqrt{3}x + 2\sqrt{3}\)	M1
Solving simultaneously \(x = 5/4,\ y = 3\sqrt{3}/4\)	A1
\(\|w\| = \sqrt{\dfrac{25}{16} + \dfrac{27}{16}} = \dfrac{\sqrt{13}}{2}\)	B1
\(\arg(w) = \arctan\dfrac{3\sqrt{3}}{5} = 0.805\)	B1

M1: equations of BC and AC both correctly calculated

B1: (1st) must be from correct \(w\)

B1: (2nd) 0.80 or better or \(46^\circ\) or better, must be from correct \(w\)