| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/1 (Pre-U Further Mathematics Paper 1) |
| Year | 2013 |
| Session | November |
| Marks | 8 |
| Topic | Roots of polynomials |
| Type | Sum of powers of roots |
| Difficulty | Standard +0.8 This is a standard symmetric functions question requiring Vieta's formulas and Newton's identities. Part (i) uses the identity (Σα)² - 2Σαβ, part (ii) is straightforward using Σα/αβγδ, and part (iii) requires either Newton's identity or factoring (Σα)³. While systematic, it demands careful algebraic manipulation across multiple steps and knowledge of power sum techniques beyond basic A-level, typical of Further Maths content. |
| Spec | 4.05a Roots and coefficients: symmetric functions |
The roots of the equation $x^4 - 2x^3 + 2x^2 + x - 3 = 0$ are $\alpha$, $\beta$, $\gamma$ and $\delta$. Determine the values of
\begin{enumerate}[label=(\roman*)]
\item $\alpha^2 + \beta^2 + \gamma^2 + \delta^2$, [2]
\item $\frac{1}{\alpha} + \frac{1}{\beta} + \frac{1}{\gamma} + \frac{1}{\delta}$, [2]
\item $\alpha^3 + \beta^3 + \gamma^3 + \delta^3$. [4]
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9795/1 2013 Q10 [8]}}