| Exam Board | SPS |
|---|---|
| Module | SPS SM Pure (SPS SM Pure) |
| Year | 2023 |
| Session | September |
| Marks | 7 |
| Topic | Areas by integration |
| Type | Combined region areas |
| Difficulty | Standard +0.3 This is a straightforward area under curves question requiring factorization to find the third root, integration of a polynomial, and splitting the integral at x-intercepts to handle absolute values. All steps are standard A-level techniques with no novel insight required, making it slightly easier than average. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.08b Integrate x^n: where n != -1 and sums1.08e Area between curve and x-axis: using definite integrals |
5.
The curve with equation $y = x ^ { 3 } - 7 x + 6$ is sketched below.\\
\includegraphics[max width=\textwidth, alt={}, center]{a1b449df-1096-4b3a-8306-fca410a7e530-10_428_627_342_810}
The curve intersects the $x$-axis at the points $A ( - 3,0 ) , B ( 1,0 )$ and $C$.
\begin{enumerate}[label=(\alph*)]
\item Find the coordinates of $C$.\\[0pt]
[1 mark]
\item Find $\int \left( x ^ { 3 } - 7 x + 6 \right) \mathrm { d } x$\\[0pt]
[2 marks]
\item Find the total area of the shaded regions enclosed by the curve and the $x$-axis.\\[0pt]
[4 marks]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM Pure 2023 Q5 [7]}}