| Exam Board | SPS |
|---|---|
| Module | SPS FM (SPS FM) |
| Year | 2025 |
| Session | October |
| Marks | 7 |
| Topic | Curve Sketching |
| Type | Solutions from graphical analysis |
| Difficulty | Standard +0.8 This question requires understanding the relationship between a function and its derivative, constructing a quartic from given conditions (including using the y-intercept to find a scaling constant), and analyzing how vertical translations affect the number of x-intercepts. While systematic, it demands multiple conceptual connections and careful algebraic manipulation beyond routine exercises. |
| Spec | 1.02n Sketch curves: simple equations including polynomials1.07n Stationary points: find maxima, minima using derivatives |
\includegraphics{figure_1}
Figure 1 shows a sketch of a curve C with equation $y = \text{f}(x)$, where f(x) is a quartic expression in $x$.
The curve
• has maximum turning points at $(-1, 0)$ and $(5, 0)$
• crosses the $y$-axis at $(0, -75)$
• has a minimum turning point at $x = 2$
\begin{enumerate}[label=(\alph*)]
\item Find the set of values of $x$ for which
$$\text{f}'(x) \geq 0$$
writing your answer in set notation. [2]
\item Find the equation of C. You may leave your answer in factorised form. [3]
\end{enumerate}
The curve $C_1$ has equation $y = \text{f}(x) + k$, where $k$ is a constant.
Given that the graph of $C_1$ intersects the $x$-axis at exactly four places,
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item find the range of possible values for $k$. [2]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM 2025 Q9 [7]}}