A2 June 2025 Paper 1 Q12
12 In this question you must show detailed reasoning.
The roots of the equation \(z^4 - z^3 + cz^2 + dz + 18 = 0\) are \(\alpha\), \(\dfrac{2}{\alpha}\), \(\beta\) and \(-\beta\).
Determine, in any order, the exact values of the following.
- The four roots of the equation
- The value of \(c\)
- The value of \(d\) [8]
| Scheme | Marks | AO |
|---|---|---|
| DR \(\alpha + \dfrac{2}{\alpha} + \beta + (-\beta) = 1\) | M1 | 3.1a |
| \(\Rightarrow \alpha^2 - \alpha + 2 = 0\) | A1 | 1.1 |
| \(\frac{1}{2} + \mathrm{i}\frac{\sqrt{7}}{2},\; \frac{1}{2} - \mathrm{i}\frac{\sqrt{7}}{2}\) | A1 | 2.2a |
| \(\alpha \cdot \dfrac{2}{\alpha} \cdot \beta \cdot (-\beta) = 18\) | M1 | 3.1a |
| \(\Rightarrow 3\mathrm{i}, -3\mathrm{i}\) | A1 | 1.1 |
| \(c = 2 + \alpha\beta - \alpha\beta + \dfrac{2\beta}{\alpha} - \dfrac{2\beta}{\alpha} - \beta^2\) | M1 | 3.1a |
| \(-d = 2\beta - 2\beta - \alpha\beta^2 - \frac{2\beta^2}{\alpha}\) | M1 | 1.1 |
| so \(c = 11\) and \(d = -9\). | A1 | 2.1 |
| [8] |
Notes
M1: sum of roots, allow a sign error. No missing terms except for those which have cancelled.
A1: any correct quadratic equation e.g. \(\alpha^2 + 2 = \alpha\); soi by correct \(\alpha\)
M1: product of roots, allow a sign error. No missing terms except for those which have cancelled.
M1: using product of roots two at a time, soi by correct numerical work. Allow only a sign error on RHS, not on \(c\). No missing terms except for those which have cancelled. Condone missing brackets. Or for substituting a root correctly into equation if \(d\) already found.
M1: using product of roots three at a time, soi by correct numerical work. Allow only a sign error on RHS, not on \(d\). No missing terms except for those which have cancelled. Condone missing brackets. Or for substituting a root correctly into equation if \(c\) already found.
Alternative method 1
| Scheme | Marks |
|---|---|
| \((z - \alpha)\left(z - \dfrac{2}{\alpha}\right)(z - \beta)(z + \beta) = 0\) \(\Rightarrow \left(z^2 + \left(-\dfrac{2}{\alpha} - \alpha\right)z + 2\right)\left(z^2 - \beta^2\right) = 0\) | M1 M1 |
| \(\Rightarrow (z^2 - z + 2)(z^2 + 9) = 0\) so \(c = 11\) and \(d = -9\) | A1 A1,A1 |
| \(z^2 - z + 2 = 0 \Rightarrow z = \frac{1 \pm \sqrt{-7}}{2} = \frac{1}{2} \pm \frac{\sqrt{7}}{2}\mathrm{i}\) | M1A1 |
| \(z^2 + 9 = 0 \Rightarrow z = \pm 3\mathrm{i}\) | B1 |
M1 expanding; M1 comparing coefft of \(x^3\) and constants
(corrected from the printed mark scheme: the roots of \(z^2 - z + 2 = 0\) are printed as \(\frac{1}{2} \pm \frac{7}{2}\mathrm{i}\); they are \(\frac{1}{2} \pm \frac{\sqrt{7}}{2}\mathrm{i}\), as in the main scheme)
Alternative method 2
| Scheme | Marks |
|---|---|
| \(\alpha + \dfrac{2}{\alpha} + \beta + (-\beta) = 1\) | M1 |
| \(\Rightarrow \alpha^2 - \alpha + 2 = 0\) | A1 |
| \(\alpha = \frac{1}{2} + \mathrm{i}\frac{\sqrt{7}}{2},\; \frac{2}{\alpha} = \frac{1}{2} - \mathrm{i}\frac{\sqrt{7}}{2}\) | A1 |
| \(\alpha \cdot \dfrac{2}{\alpha} \cdot \beta \cdot (-\beta) = 18\) | M1 |
| \(\Rightarrow \beta = 3\mathrm{i},\; -\beta = -3\mathrm{i}\) | A1 |
| \((z^2 - z + 2)(z^2 + 9) = 0\) | M1 |
| \(z^4 - z^3 + 11z^2 - 9z + 18 = 0\) | M1 |
| so \(c = 11\) and \(d = -9\). | A1 |
M1: sum of roots, allow a sign error
A1: any correct quadratic equation e.g. \(\alpha^2 + 2 = \alpha\)
M1: product of roots, allow a sign error
M1: at least one quadratic factor identified
M1: multiplying their quadratic factors
Alternative method 3
| Scheme | Marks |
|---|---|
| \(\alpha + \dfrac{2}{\alpha} + \beta + (-\beta) = 1\) | M1 |
| \(\Rightarrow \alpha^2 - \alpha + 2 = 0\) | A1 |
| \(\frac{1}{2} + \mathrm{i}\frac{\sqrt{7}}{2},\; \frac{1}{2} - \mathrm{i}\frac{\sqrt{7}}{2}\) | A1 |
| \(\alpha \cdot \dfrac{2}{\alpha} \cdot \beta \cdot (-\beta) = 18\) | M1 |
| \(\Rightarrow 3\mathrm{i}, -3\mathrm{i}\) | A1 |
| e.g. \((3\mathrm{i})^4 - (3\mathrm{i})^3 + c(3\mathrm{i})^2 + d(3\mathrm{i}) + 18 = 0\) | M1 |
| \(81 - 9c + 18 = 0\) and \(27 + 3d = 0\) | M1 |
| so \(c = 11\) and \(d = -9\). | A1 |
M1: sum of roots, allow a sign error
A1: Any correct quadratic equation e.g. \(\alpha^2 + 2 = \alpha\)
M1: product of roots, allow a sign error
M1: substituting any root into equation correctly
M1: equating real and imaginary parts leading to two equations