| Exam Board | SPS |
|---|---|
| Module | SPS SM Pure (SPS SM Pure) |
| Year | 2021 |
| Session | September |
| Marks | 9 |
| Topic | Stationary points and optimisation |
| Type | Find stationary points coordinates |
| Difficulty | Moderate -0.3 This is a straightforward multi-part question combining standard integration for area between curves, basic coordinate geometry, and routine differentiation from first principles. Part (i)(A) requires solving x^4 = 8x (simple algebra), part (i)(B) is a standard definite integration exercise, and part (ii) is a textbook application of the first principles definition. All techniques are routine A-level procedures with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.07g Differentiation from first principles: for small positive integer powers of x1.08e Area between curve and x-axis: using definite integrals1.08f Area between two curves: using integration |
4.
\begin{enumerate}[label=(\roman*)]
\item \begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{cee51b6b-40d2-4abb-acf7-47c73a919bf9-10_656_776_210_721}
\captionsetup{labelformat=empty}
\caption{Fig. 12}
\end{center}
\end{figure}
Fig. 12 shows part of the curve $y = x ^ { 4 }$ and the line $y = 8 x$, which intersect at the origin and the point P .\\
(A) Find the coordinates of P , and show that the area of triangle OPQ is 16 square units.\\
(B) Find the area of the region bounded by the line and the curve.
\item If $f ( x ) = x ^ { 3 }$, find $f ^ { \prime } ( x )$ from first principles.
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM Pure 2021 Q4 [9]}}