| Exam Board | SPS |
|---|---|
| Module | SPS FM (SPS FM) |
| Year | 2025 |
| Session | October |
| Marks | 8 |
| Topic | Factor & Remainder Theorem |
| Type | Multiple unknowns with derivative condition |
| Difficulty | Moderate -0.3 This is a straightforward multi-part question testing standard A-level techniques: differentiation and substitution (part a), factor theorem application (part b), and factorization with discriminant analysis (part c). Each part follows directly from the previous with no novel insight required, making it slightly easier than average but still requiring competent execution of multiple techniques. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.07i Differentiate x^n: for rational n and sums |
The cubic polynomial $2x^3 - kx^2 + 4x + k$, where $k$ is a constant, is denoted by f(x). It is given that f'(2) = 16.
\begin{enumerate}[label=(\alph*)]
\item Show that $k = 3$. [3]
\end{enumerate}
For the remainder of the question, you should use this value of $k$.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Use the factor theorem to show that $(2x + 1)$ is a factor of f(x). [2]
\item Hence show that the equation f(x) = 0 has only \textbf{one} real root. [3]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM 2025 Q4 [8]}}