| Exam Board | WJEC |
|---|---|
| Module | Further Unit 1 (Further Unit 1) |
| Session | Specimen |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Roots of polynomials |
| Type | Complex roots with real coefficients |
| Difficulty | Standard +0.8 This question requires knowledge of relationships between roots and coefficients (Vieta's formulas), manipulation of symmetric functions (using (α+β+γ)² = α²+β²+γ² + 2(αβ+βγ+γα)), and understanding that complex roots come in conjugate pairs. Part (a) involves algebraic manipulation and interpreting the negative sum of squares, while part (b) requires applying the conjugate root theorem and finding the third root via sum of roots. These are standard Further Maths techniques but require careful multi-step reasoning beyond typical A-level Core content. |
| Spec | 4.02g Conjugate pairs: real coefficient polynomials4.05a Roots and coefficients: symmetric functions |
The roots of the equation
$$x^3 - 4x^2 + 14x - 20 = 0$$
are denoted by $\alpha$, $\beta$, $\gamma$.
\begin{enumerate}[label=(\alph*)]
\item Show that
$$\alpha^2 + \beta^2 + \gamma^2 = -12.$$
Explain why this result shows that exactly one of the roots of the above cubic equation is real. [3]
\item Given that one of the roots is $1 + 3i$, find the other two roots. Explain your method for each root. [4]
\end{enumerate}
\hfill \mbox{\textit{WJEC Further Unit 1 Q4 [7]}}