A2 June 2024 Paper 2 Q17
17 The Argand diagram below shows a circle \(C\)
(a) Write down the equation of the locus of \(C\) in the form\[|z - w| = a\]
where \(w\) is a complex number whose real and imaginary parts are integers, and \(a\) is an integer. [2 marks]
(b) It is given that \(z_1\) is a complex number representing a point on \(C\). Of all the complex numbers which represent points on \(C\), \(z_1\) has the least argument.
(i) Find \(|z_1|\)
Give your answer in an exact form. [3 marks]
(ii) Show that \(\arg z_1 = \arcsin\left(\dfrac{6\sqrt{3} - 2}{13}\right)\) [4 marks]
| Scheme | Marks | AO |
|---|---|---|
| Obtains \(4 + 6\mathrm{i}\) Allow \(|z - 4 - 6\mathrm{i}|\) | B1 | 1.2 |
| Obtains \(a = 2\) | B1 | 1.2 |
| (2) |
Typical solution
\[|z - (4 + 6\mathrm{i})| = 2\]| Scheme | Marks | AO |
|---|---|---|
| (i) Correctly identifies the point representing \(z_1\) PI by correct method. or Obtains \(m = \dfrac{6 - 2\sqrt{3}}{3}\) as the gradient of the tangent at \(z_1\) | B1 | 2.2a |
| (i) Uses Pythagoras or other correct method to obtain \(|z_1|\) | M1 | 3.1a |
| (i) Obtains \(4\sqrt{3}\) Accept any exact correct value eg \(\sqrt{48}\) | A1 | 1.1b |
| (3) | ||
| (ii) Deduces that \(\arg z_1 = P\hat{O}R - P\hat{O}Q\) or Uses \(\tan(\arg(z_1)) = \dfrac{6 - 2\sqrt{3}}{3}\) or Obtains \(x = \dfrac{48 + 12\sqrt{3}}{13}\) or \(y = \dfrac{72 - 8\sqrt{3}}{13}\) where \(z_1 = x + \mathrm{i}y\) | B1 | 2.2a |
| (ii) Uses a suitable trigonometric identity or Uses a correct method to obtain \(\sin(\arg(z_1))\) from \(\tan(\arg(z_1))\) or Obtains \(x = \dfrac{48 + 12\sqrt{3}}{13}\) and \(y = \dfrac{72 - 8\sqrt{3}}{13}\) where \(z_1 = x + \mathrm{i}y\) | M1 | 1.1a |
| (ii) Obtains sines and cosines of \(P\hat{O}R\) and \(P\hat{O}Q\) (at least three correct) or Obtains \(\sin^2(\arg(z_1))\) or \(\cos^2(\arg(z_1))\) or Obtains the values of the sides of a right-angled triangle with an angle equal to \(\arg(z_1)\) | M1 | 3.1a |
| (ii) Uses correct reasoning to obtain the required result. Condone omission of “\(\arg(z_1)\) is acute”. AG | R1 | 2.1 |
| (4) | ||
| (9 marks) |
Typical solution
(i)
\(Q\) is the point representing \(z_1\)
\[OP^2 = 52\]\[PQ^2 = 4\]\[OQ^2 = 48\]\[OQ = 4\sqrt{3}\]\[|z_1| = 4\sqrt{3}\](ii)
\(\arg(z_1)\) is acute, so
\[\arg(z_1) = \arcsin\left(\frac{6\sqrt{3} - 2}{13}\right)\]