AS June 2018 Paper 1 Q3

OCR ACurrent spec9 marksComplex Numbers

3 In this question you must show detailed reasoning.

The complex numbers \(z_1\) and \(z_2\) are given by \(z_1 = 2 - 3\mathrm{i}\) and \(z_2 = a + 4\mathrm{i}\) where \(a\) is a real number.

(i) Express \(z_1\) in modulus-argument form, giving the modulus in exact form and the argument correct to 3 significant figures. [3]

(ii) Find \(z_1z_2\) in terms of \(a\), writing your answer in the form \(c + \mathrm{i}d\). [2]

(iii) The real and imaginary parts of a complex number on an Argand diagram are \(x\) and \(y\) respectively. Given that the point representing \(z_1z_2\) lies on the line \(y = x\), find the value of \(a\). [2]

(iv) Given instead that \(z_1z_2 = (z_1z_2)^*\) find the value of \(a\). [2]

Mark scheme (i)

Scheme	Marks	AO
\(\|z_1\| = \sqrt{13}\)	B1	1.1
\(\arg(z_1) = \arctan(-3/2)\)	M1	1.1
\(\sqrt{13}\operatorname{cis}(-0.983)\) \(\sqrt{13}\operatorname{cis}(5.30)\)	A1	1.1
	[3]

Notes

M1: Allow \(\arctan(3/2)\) for M1
M0 if no working i.e. just an angle of \(-0.983\) or \(-56.3^\circ\)
(Do not penalise here if degree sign is missing).
Allow finding another angle as long as part of a method to find a correct argument.
Need to see some evidence of use of arctan or \(\tan^{(-1)}\)
\(\tan\theta = \frac{3}{2}\) and \(\theta = 0.983\) is enough
E.g. \(\frac{3}{2}\pi + \arctan\left(\frac{2}{3}\right)\)

A1: Any equivalent mod-arg form (including exponential). Condone \(-56.3^\circ\) or \(304^\circ\) if degree symbol shown, or has been shown somewhere in the working.
Must not have a \(\pi\) attached to \(-0.983\)
A1 cannot be awarded if M0 awarded. Must have some evidence of use of arctan.
Condone \(-0.98\) or \(-56^\circ\) if correct angle to 3s.f. seen before
Condone 5.3
\([\sqrt{13}, -0.983]\)
\(\sqrt{13}e^{-0.983\mathrm{i}}\)
\(\sqrt{13}(\cos(-0.983) + \mathrm{i}\sin(-0.983))\)
Allow
\(\sqrt{13}(\cos(0.983) - \mathrm{i}\sin(0.983))\)

Mark scheme (ii)

Scheme	Marks	AO
\(2a + 8\mathrm{i} - 3a\mathrm{i} - 12\mathrm{i}^2\)	M1	1.1
\(2a + 12 + (8 - 3a)\mathrm{i}\)	A1	1.1
	[2]

Notes

M1: Expanding brackets; allow one error

A1: i terms must be collected

Mark scheme (iii)

Scheme	Marks	AO
\(2a + 12 = 8 - 3a\)	M1	2.2a
\(a = -\dfrac{4}{5}\)	A1ft	1.1
	[2]

Notes

M1: Equating their real and imaginary parts. No i

A1ft: or \(-0.8\)
Follow through provided that \(a\) appears in both the real and imaginary parts.

Mark scheme (iv)

Scheme	Marks	AO
\(8 - 3a = 0\)	M1	2.2a
\(a = \dfrac{8}{3}\)	A1ft	1.1
	[2]

Notes

M1: Or \(2a + 12 + (8 - 3a)\mathrm{i} = 2a + 12 - (8 - 3a)\mathrm{i}\)
Setting imaginary part of (ii) to 0.

A1ft: or awrt 2.67