| Exam Board | SPS |
|---|---|
| Module | SPS FM (SPS FM) |
| Year | 2022 |
| Session | November |
| Marks | 11 |
| Topic | Curve Sketching |
| Type | Find tangent to polynomial curve |
| Difficulty | Moderate -0.8 This is a straightforward multi-part question testing basic differentiation and curve sketching. Part (a) requires factorising a cubic to find x-intercepts, part (b) needs finding dy/dx at the origin, and part (c) involves solving a cubic equation after substituting the tangent line equation. All techniques are routine A-level procedures with no novel problem-solving required, making it easier than average. |
| Spec | 1.02f Solve quadratic equations: including in a function of unknown1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.02q Use intersection points: of graphs to solve equations1.07i Differentiate x^n: for rational n and sums1.07m Tangents and normals: gradient and equations |
4.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{d8940254-0663-413e-a802-71519742cfcc-08_721_982_114_347}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\end{figure}
Figure 1 shows the curve $C$ with the equation $y = x ^ { 3 } + 3 x ^ { 2 } - 4 x$ and the straight line $l$.
The curve $C$ crosses the $x$-axis at the origin, $O$, and at the points $A$ and $B$.
\begin{enumerate}[label=(\alph*)]
\item Find the coordinates of $A$ and $B$.
The line $l$ is the tangent to $C$ at $O$.
\item Find an equation for $l$.
\item Find the coordinates of the point where $l$ intersects $C$ again.\\[0pt]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM 2022 Q4 [11]}}