| Exam Board | SPS |
|---|---|
| Module | SPS FM Pure (SPS FM Pure) |
| Year | 2025 |
| Session | February |
| Marks | 8 |
| Topic | Roots of polynomials |
| Type | Finding specific root values |
| Difficulty | Challenging +1.2 This is a Further Maths cubic equation problem requiring use of Vieta's formulas and algebraic manipulation with the unusual root relationship. While the constraint α + 12/α - β looks complex, systematic application of sum/product of roots leads to a tractable system. The problem requires careful algebra and multiple steps but follows standard FM techniques without requiring deep insight or novel approaches. |
| Spec | 4.02j Cubic/quartic equations: conjugate pairs and factor theorem4.05a Roots and coefficients: symmetric functions |
$$f(z) = z^3 - 8z^2 + pz - 24$$
where $p$ is a real constant.
Given that the equation $f(z) = 0$ has distinct roots
$$\alpha, \beta \text{ and } \left(\alpha + \frac{12}{\alpha} - \beta\right)$$
\begin{enumerate}[label=(\alph*)]
\item solve completely the equation $f(z) = 0$ [6]
\item Hence find the value of $p$. [2]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM Pure 2025 Q9 [8]}}