FP1 January 2011 Q7
7. \[z = -24 - 7\mathrm{i}\]
(a) Show \(z\) on an Argand diagram. (1)
(b) Calculate \(\arg z\), giving your answer in radians to 2 decimal places. (2)
It is given that \[w = a + b\mathrm{i}, \quad a \in \mathbb{R},\ b \in \mathbb{R}\]
Given also that \(|w| = 4\) and \(\arg w = \dfrac{5\pi}{6}\),
(c) find the values of \(a\) and \(b\), (3)
(d) find the value of \(|zw|\). (3)
| Scheme | Marks |
|---|---|
 | B1 |
| (1) |
| Scheme | Marks |
|---|---|
| \(\arg z = -\pi + \tan^{-1}\left(\tfrac{7}{24}\right)\) \(\tan^{-1}\left(\tfrac{7}{24}\right)\) or \(\tan^{-1}\left(\tfrac{24}{7}\right)\) | M1 |
| \(= -2.857798544\ldots = -2.86\) (2 dp) awrt −2.86 or awrt 3.43 | A1 |
| (2) |
| Scheme | Marks |
|---|---|
| \(|w| = 4,\ \arg w = \tfrac{5\pi}{6} \Rightarrow r = 4,\ \theta = \tfrac{5\pi}{6}\) \(w = r\cos\theta + \mathrm{i}r\sin\theta\) | |
| \(w = 4\cos\left(\tfrac{5\pi}{6}\right) + 4\mathrm{i}\sin\left(\tfrac{5\pi}{6}\right)\) Attempt to apply \(r\cos\theta + \mathrm{i}r\sin\theta\). | M1 |
| \(= 4\left(\tfrac{-\sqrt{3}}{2}\right) + 4\mathrm{i}\left(\tfrac{1}{2}\right)\) Correct expression for \(w\). | A1 |
| \(= -2\sqrt{3} + 2\mathrm{i}\) \(a = -2\sqrt{3},\ b = 2\) either \(-2\sqrt{3} + 2\mathrm{i}\) or awrt \(-3.5 + 2\mathrm{i}\) | A1 |
| (3) |
Notes
Aliter 7. (c) Way 2
| Scheme | Marks |
|---|---|
| \(|w| = 4,\ \arg w = \tfrac{5\pi}{6}\) and \(w = a + \mathrm{i}b\) | |
| \(|w| = 4 \Rightarrow a^2 + b^2 = 16\) Attempts to write down an equation in terms of \(a\) and \(b\) for either the modulus or the argument of \(w\). | M1 |
| \(\arg w = \tfrac{5\pi}{6} \Rightarrow \arctan\left(\tfrac{b}{a}\right) = \tfrac{5\pi}{6} \Rightarrow \tfrac{b}{a} = -\tfrac{1}{\sqrt{3}}\) Either \(a^2 + b^2 = 16\) or \(\tfrac{b}{a} = -\tfrac{1}{\sqrt{3}}\) | A1 |
| \(a = -\sqrt{3}\,b \Rightarrow a^2 = 3b^2\) So, \(3b^2 + b^2 = 16 \Rightarrow b^2 = 4\) \(\Rightarrow b = \pm 2\) and \(a = \mp 2\sqrt{3}\) As \(w\) is in the second quadrant | |
| \(w = -2\sqrt{3} + 2\mathrm{i}\) \(a = -2\sqrt{3},\ b = 2\) either \(-2\sqrt{3} + 2\mathrm{i}\) or awrt \(-3.5 + 2\mathrm{i}\) | A1 |
| (3) |
| Scheme | Marks |
|---|---|
| \(|z| = \sqrt{(-24)^2 + (-7)^2} = \underline{25}\) \(\underline{|z| = 25}\) or \(zw = (48\sqrt{3} + 14) + (14\sqrt{3} - 48)\mathrm{i}\) or awrt \(97.1 - 23.8\mathrm{i}\) | B1 |
| \(|zw| = |z| \times |w| = (25)(4)\) Applies \(|z| \times |w|\) or \(|zw|\) | M1 |
| \(= \underline{100}\) 100 | A1 |
| (3) | |
| [9] |