| Exam Board | SPS |
|---|---|
| Module | SPS SM (SPS SM) |
| Year | 2022 |
| Session | January |
| Marks | 13 |
| Topic | Curve Sketching |
| Type | Area between curve and line |
| Difficulty | Standard +0.3 This is a straightforward multi-part question covering standard A-level techniques: evaluating a polynomial, factorising using the factor theorem, sketching a cubic with a repeated root, and finding area using integration. All parts follow routine procedures with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.02n Sketch curves: simple equations including polynomials1.08b Integrate x^n: where n != -1 and sums1.08e Area between curve and x-axis: using definite integrals |
9.
$$\mathrm { f } ( x ) = x ^ { 3 } - 9 x ^ { 2 } + 24 x - 16$$
\begin{enumerate}[label=(\alph*)]
\item Evaluate $\mathrm { f } ( 1 )$ and hence state a linear factor of $\mathrm { f } ( x )$.
\item Show that $\mathrm { f } ( x )$ can be expressed in the form
$$\mathrm { f } ( x ) = ( x + p ) ( x + q ) ^ { 2 } ,$$
where $p$ and $q$ are integers to be found.
\item Sketch the curve $y = \mathrm { f } ( x )$ in the space provided.
\item Using integration, find the area of the region enclosed by the curve $y = \mathrm { f } ( x )$ and the $x$-axis.\\[0pt]
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\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM 2022 Q9 [13]}}