| Exam Board | SPS |
|---|---|
| Module | SPS FM Pure (SPS FM Pure) |
| Year | 2022 |
| Session | February |
| Marks | 9 |
| Topic | Roots of polynomials |
| Type | Finding specific root values |
| Difficulty | Challenging +1.2 This is a Further Maths polynomial question requiring use of Vieta's formulas and the special structure of roots (α, -α, β, 1/β). The root structure immediately gives α² · β · (1/β) = -9/4, so α = ±3/2, and the sum of roots yields β. While it requires careful algebraic manipulation across multiple steps (9 marks total), the approach is systematic once the root relationships are recognized—harder than standard A-level but not requiring deep insight beyond applying standard techniques to a structured problem. |
| Spec | 4.05a Roots and coefficients: symmetric functions4.05b Transform equations: substitution for new roots |
The equation $4x^4 - 4x^3 + px^2 + qx - 9 = 0$, where $p$ and $q$ are constants, has roots $\alpha$, $-\alpha$, $\beta$ and $\frac{1}{\beta}$.
\begin{enumerate}[label=(\alph*)]
\item Determine the exact roots of the equation. [5]
\item Determine the values of $p$ and $q$. [4]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM Pure 2022 Q9 [9]}}