| Exam Board | SPS |
|---|---|
| Module | SPS FM Pure (SPS FM Pure) |
| Year | 2026 |
| Session | November |
| Marks | 8 |
| Topic | Roots of polynomials |
| Type | Reciprocal sum of roots |
| Difficulty | Challenging +1.2 This is a Further Maths question on roots of polynomials requiring Vieta's formulas and symmetric function manipulation. Part (a) is straightforward application of sum of reciprocals formula. Part (b) requires using the identity (Σα)² = Σα² + 2Σαβ to find A, involving algebraic manipulation but following standard techniques. The 8 marks and multi-step nature elevate it above average, but it's a recognizable textbook-style problem without requiring novel insight. |
| Spec | 4.05a Roots and coefficients: symmetric functions |
The quartic equation
$$2x^4 + Ax^3 - Ax^2 - 5x + 6 = 0$$
where $A$ is a real constant, has roots $\alpha$, $\beta$, $\gamma$ and $\delta$
\begin{enumerate}[label=(\alph*)]
\item Determine the value of
$$\frac{3}{\alpha} + \frac{3}{\beta} + \frac{3}{\gamma} + \frac{3}{\delta}$$
[3]
\end{enumerate}
Given that $\alpha^2 + \beta^2 + \gamma^2 + \delta^2 = -\frac{3}{4}$
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item determine the possible values of $A$ [5]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM Pure 2026 Q10 [8]}}