| Exam Board | SPS |
|---|---|
| Module | SPS SM Pure (SPS SM Pure) |
| Year | 2024 |
| Session | September |
| Marks | 7 |
| Topic | Curve Sketching |
| Type | Finding quadratic from vertex information |
| Difficulty | Moderate -0.8 This is a straightforward multi-part question requiring standard techniques: using vertex form to find a quadratic (with one additional constraint), sketching a parabola, and solving a quartic equation that factors nicely. All steps are routine A-level procedures with no novel problem-solving required, making it easier than average. |
| Spec | 1.02d Quadratic functions: graphs and discriminant conditions1.02e Complete the square: quadratic polynomials and turning points1.02f Solve quadratic equations: including in a function of unknown1.02n Sketch curves: simple equations including polynomials |
\begin{enumerate}
\item The curve $C _ { 1 }$ has equation $y = \mathrm { f } ( x )$.
\end{enumerate}
Given that
\begin{itemize}
\item $\mathrm { f } ( x )$ is a quadratic expression
\item $C _ { 1 }$ has a maximum turning point at $( 2,20 )$
\item $C _ { 1 }$ passes through the origin\\
(a) sketch a graph of $C _ { 1 }$ showing the coordinates of any points where $C _ { 1 }$ cuts the coordinate axes,\\
(b) find an expression for $\mathrm { f } ( x )$.
\end{itemize}
The curve $C _ { 2 }$ has equation $y = x \left( x ^ { 2 } - 4 \right)$\\
Curve $C _ { 1 }$ and $C _ { 2 }$ meet at the origin, and at the points $P$ and $Q$\\
Given that the $x$ coordinate of the point $P$ is negative,\\
(c) using algebra and showing all stages of your working, find the coordinates of $P$\\
(3)\\
\hfill \mbox{\textit{SPS SPS SM Pure 2024 Q6 [7]}}