FP1 June 2016 Q4
4. \[z = \frac{4}{1 + \mathrm{i}}\]
Find, in the form \(a + \mathrm{i}b\) where \(a, b \in \mathbb{R}\)
Given that \(z\) is a complex root of the quadratic equation \(x^2 + px + q = 0\), where \(p\) and \(q\) are real integers,
| Scheme | Marks |
|---|---|
| \(z = \dfrac{4(1 - \mathrm{i})}{(1 + \mathrm{i})(1 - \mathrm{i})}\) | M1 |
| \(z = 2(1 - \mathrm{i})\) or \(2 - 2\mathrm{i}\) or exact equivalent. | A1 |
| (2) |
Notes
M1: Multiplies numerator and denominator by \(1 - \mathrm{i}\) or by \(-1 + \mathrm{i}\)
A1: cao
| Scheme | Marks |
|---|---|
| \(z^2 = (2 - 2\mathrm{i})(2 - 2\mathrm{i}) = 4 - 8\mathrm{i} + 4\mathrm{i}^2\) | M1 |
| \(= -8\mathrm{i}\) | A1 cao |
| (2) |
Notes
M1: Squares their \(z\), or the given \(z = \dfrac{4}{1 + \mathrm{i}}\), to produce at least 3 terms which can be implied by the correct answer.
A1: \(-8\mathrm{i}\) or \(0 - 8\mathrm{i}\) only
| Scheme | Marks |
|---|---|
| If \(z\) is a root so is \(z^*\) So \((x - 2 + 2\mathrm{i})(x - 2 - 2\mathrm{i})\) (or \(x^2 - 2\mathrm{Re}(z) \cdot x + |z|^2\)) | M1 |
| So \((x - 2 + 2\mathrm{i})(x - 2 - 2\mathrm{i}) = 0\) (or \(x^2 - 2\mathrm{Re}(z) \cdot x + |z|^2 = 0\)) and so \(p = q =\) | M1 |
| Equation is \(x^2 - 4x + 8\ (= 0)\) or \(p = -4\) and \(q = 8\) | A1 |
| ALT 1 (c) Substitutes \(z = 2 - 2\mathrm{i}\) and \(z^2 = -8\mathrm{i}\) into quadratic and equates real and imaginary parts to obtain \(2p + q = 0\) and \(-2p - 8 = 0\) | M1 |
| Attempts to solve simultaneous equations to obtain \(p = -4\) and \(q = 8\) | M1A1 |
| ALT 2 (c) Attempts to obtain \(p = -\) sum of roots | M1 |
| Attempts product of roots to obtain \(q =\) | M1 |
| Equation is \(x^2 - 4x + 8\ (= 0)\) or \(p = -4\) and \(q = 8\) | A1 |
| ALT 3 (c) \(x - 2 = \pm 2\mathrm{i}\) either sign acceptable | M1 |
| \((x - 2)^2 = -4 \Rightarrow x^2 - 4x + 4 = -4\) i.e square and attempt to expand to give 3-term quadratic | M1 |
| Equation is \(x^2 - 4x + 8\ (= 0)\) or \(p = -4\) and \(q = 8\) | A1 |
| (3) | |
| (7 marks) |
Notes
M1: Uses their \(z\) and \(z^*\) in \((x - z)(x - z^*)\)
M1: Multiplies two factors and obtains \(p =\) or \(q =\)
A1: Both correct required – can be implied by \(x^2 - 4x + 8\)
ALT 1
M1: Substitutes their \(z\) and their \(z^2\) into the quadratic and equates real and imaginary parts to obtain two equations in \(p\) and \(q\)
M1: Attempts to solve for one unknown to obtain \(p =\) or \(q =\)
A1: Both correct required – can be implied by \(x^2 - 4x + 8\ (= 0)\)