AS June 2023 Paper 1 Q3
3 In this question you must show detailed reasoning.
In this question the principal argument of a complex number lies in the interval \([0, 2\pi)\).
Complex numbers \(z_1\) and \(z_2\) are defined by \(z_1 = 3 + 4\mathrm{i}\) and \(z_2 = -5 + 12\mathrm{i}\).
| Scheme | Marks | AO |
|---|---|---|
| DR \(z_1z_2 = (3 + 4\mathrm{i})(-5 + 12\mathrm{i}) = -15 + 36\mathrm{i} - 20\mathrm{i} - 48\) | M1 | 1.1 |
| \(= -63 + 16\mathrm{i}\) | A1 | 1.1 |
| [2] |
Notes
M1: Attempt at expansion (4 terms soi) using \(\mathrm{i}^2 = -1\)
DR – Need to see at least one line of expanded terms before answer
| Scheme | Marks | AO |
|---|---|---|
| DR \(|z_2| = \left(\sqrt{(-5)^2 + 12^2} = \sqrt{169}\right) = 13\) | B1 | 1.1 |
| \(\tan^{-1}\left(\dfrac{12}{-5}\right)\) | M1 | 1.1 |
| \(\therefore z_2 = 13(\cos 1.97 + \mathrm{i}\sin 1.97)\) (3 sf) | A1 | 2.5 |
| [3] |
Notes
B1: Not \(\pm\) unless later corrected. Allow modulus of 13 for the B1 as long as no incorrect working
Treat attempt to write \(z_1\) or \(z_1z_2\) in mod/arg form as MR so B0M1A1 available
M1: Evidence of using trigonometry towards finding the correct angle, perhaps by finding a related angle.
Treat \(\tan^{-1}\left(\dfrac{12}{5}\right)\) as such evidence for M1 but not \(\tan^{-1}\left(-\dfrac{5}{12}\right)\) or \(\tan^{-1}\left(\dfrac{5}{12}\right)\) unless supported eg by a diagram or by working leading to correct answer.
A1: For argument accept awrt 1.97 only. Do not accept answers not written correctly in mod-arg form. Do not accept \(-1.18\) or \(-4.32\) as argument.
Answer must be in radians for A1.
Accept \([r, \theta]\) or \(r\,\mathrm{cis}\,\theta\)
Is \(1.965587446\ldots\)
eg do not accept the following
\(13\cos 1.97 + 13\mathrm{i}\sin 1.97\)
\(13(\cos 4.32 - \mathrm{i}\sin 4.32)\)
NB
\(z_1 = 5(\cos 0.927 + \mathrm{i}\sin 0.927)\)
\(z_1z_2 = 65(\cos 2.89 + \mathrm{i}\sin 2.89)\)
| Scheme | Marks | AO |
|---|---|---|
| DR \((\arg(z_1z_2) =)\tan^{-1}\left(\dfrac{16}{-63}\right)\) | M1 | 1.1 |
| \(\arg(z_1) + \arg(z_2) = \tan^{-1}\left(\dfrac{4}{3}\right) + 1.965587\ldots\) \(= 0.927295\ldots + 1.965587\ldots\) \(= 2.89288\ldots\) | M1 | 2.1 |
| \(\arg(z_1z_2) = -0.2487099\ldots + \pi = 2.892882\ldots\) so they are equal | A1 | 2.2a |
| [3] |
Notes
M1: Using trigonometry to find the argument. Do not accept any other form unless supported by clear evidence (eg diagram)
This mark may be awarded if \(z_1z_2\) was incorrect from (a).
M1: Attempt to calculate RHS using their values (either value could have been found earlier but both must be in \([0, 2\pi)\)).
Could accept \(0.927\ldots\) as evidence of \(\arctan(4/3)\)
A1: Accept rounding to 3 sf or better but rounding must be correct (e.g. \(0.927 + 1.96 = 2.89\) would score A0).
Answer must be in radians for A1.
If MR \(z_1\) or \(z_1z_2\) in part (b) then full credit available for a correct solution here.