# Question 1:

## Part (a)

| Answer/Working | Marks | Guidance Notes |
|---|---|---|
| $\beta = 3 + 2\sqrt{2}\text{i}$ is also a root | B1 | Identifies the correct complex conjugate as another root |
| $\alpha\beta = 17,\ \alpha + \beta = 6$ | B1 | Correct values for the sum and product for the conjugate pair |
| $\alpha\beta + \alpha\gamma + \beta\gamma = \frac{57}{3}$ | M1 | Correct application of the pair sum |
| $\alpha\gamma + \beta\gamma = \frac{57}{3} - 17 = \gamma(\alpha+\beta) = 6\gamma \Rightarrow \gamma = \ldots$ | M1 | Identifies a complete and correct strategy for identifying the third root |
| $\gamma = \frac{1}{3}$ | A1 | Deduces the correct third root |
| $3 \pm 2\sqrt{2}\text{i}$ plotted correctly in quadrants 1 and 4 (reflections in real axis) | B1 | Do not be concerned about labelling or scaling |
| Real root plotted correctly in correct relative position to the two complex roots | B1ft | Scales not needed but if correct, real root must be close to origin compared to complex roots |

### Part (a) Alternative

| Answer/Working | Marks | Guidance Notes |
|---|---|---|
| $\beta = 3 + 2\sqrt{2}\text{i}$ is also a root | B1 | Identifies the correct complex conjugate |
| $\left(z-(3+2\sqrt{2}\text{i})\right)\left(z-(3-2\sqrt{2}\text{i})\right) = z^2 - 6z + 17$ | B1 | Correct quadratic factor obtained |
| $f(z) = (z^2-6z+17)(3z+a) = 3z^3 + az^2 - 18z^2 - 6az + 51z + 17a$ | M1 | Expands quadratic $\times (3z + a)$ or attempts to factor/long division, leading to factor $(3z + a)$ |
| $\Rightarrow 51 - 6a = 57 \Rightarrow a = -1 \Rightarrow \gamma = \ldots$ | M1 | Proceeds to extract root from third factor $(3z + a)$ |
| $\gamma = \frac{1}{3}$ | A1 | Deduces the correct third root |

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## Part (b)

| Answer/Working | Marks | Guidance Notes |
|---|---|---|
| $3-2\sqrt{2}\text{i}+3+2\sqrt{2}\text{i}+\frac{1}{3} = -\frac{p}{3} \Rightarrow p = \ldots$ **or** $(3-2\sqrt{2}\text{i})(3+2\sqrt{2}\text{i})\times\frac{1}{3} = -\frac{q}{3} \Rightarrow q = \ldots$ | M1 | Correct strategy used for identifying at least one of $p$ or $q$ |
| $p = -19$ **or** $q = -17$ | A1 | At least one value correct |
| $p = -19$ **and** $q = -17$ | A1 | Both values correct |

### Part (b) Alternative

| Answer/Working | Marks | Guidance Notes |
|---|---|---|
| $f(z) = (z^2-6z+17)(3z-1) = 3z^3 + pz^2 + 57z + q$ | M1 | Correct strategy by expanding quadratic and linear factors to identify at least one of $p$ or $q$ |
| $p = -19$ **or** $q = -17$ | A1 | At least one value correct |
| $p = -19$ **and** $q = -17$ | A1 | Both values correct |

### Part (b) Factor Theorem Note

| Answer/Working | Marks | Guidance Notes |
|---|---|---|
| $f(3-2\text{i}\sqrt{2}) = 36+p+q+\text{i}(-228\sqrt{2}-12\sqrt{2}p) = 0$ | — | Some may attempt factor theorem with complex root |
| $36+p+q=0,\ -228\sqrt{2}-12\sqrt{2}p=0$ | B1 (2nd) | Equate real and imaginary components to 0 to get correct equations |
| $\Rightarrow p=-19,\ q=-17$ | M1 (1st) | Solves their equations |

**Total: 10 marks**