| Exam Board | SPS |
|---|---|
| Module | SPS FM (SPS FM) |
| Year | 2024 |
| Session | November |
| Marks | 8 |
| Topic | Roots of polynomials |
| Type | Roots with special relationships |
| Difficulty | Standard +0.8 This is a Further Maths question requiring insight that if α and 1/α are roots, their product equals 1, which combined with Vieta's formulas allows systematic solution. It requires recognizing the special relationship, applying product of roots (α·(1/α)·β = 2), then sum/product formulas to find both k and the roots. More sophisticated than standard Vieta's applications but follows a clear logical path once the key insight is spotted. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem4.05a Roots and coefficients: symmetric functions |
3. In this question you must show detailed reasoning.
The roots of the equation $x ^ { 3 } - x ^ { 2 } + k x - 2 = 0$ are $\alpha , \frac { 1 } { \alpha }$ and $\beta$.
\begin{enumerate}[label=(\alph*)]
\item Evaluate, in exact form, the roots of the equation.
\item Find $k$.\\[0pt]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM 2024 Q3 [8]}}