| Exam Board | SPS |
|---|---|
| Module | SPS SM Pure (SPS SM Pure) |
| Year | 2021 |
| Session | June |
| Marks | 10 |
| Topic | Factor & Remainder Theorem |
| Type | Integration or area using factorised polynomial |
| Difficulty | Moderate -0.3 This is a straightforward multi-part question combining factor theorem, polynomial factorization, and definite integration. Part (a) is routine substitution, part (b) requires polynomial division and factoring a quadratic (both standard techniques), and part (c) involves integrating a cubic between given roots. While it requires multiple steps across different topics, each individual step is a standard A-level procedure with no novel problem-solving or insight required, making it slightly easier than average. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.08e Area between curve and x-axis: using definite integrals |
$$g(x) = 2x^3 + x^2 - 41x - 70$$
\begin{enumerate}[label=(\alph*)]
\item Use the factor theorem to show that $g(x)$ is divisible by $(x - 5)$. [2]
\item Hence, showing all your working, write $g(x)$ as a product of three linear factors. [4]
\end{enumerate}
The finite region $R$ is bounded by the curve with equation $y = g(x)$ and the $x$-axis, and lies below the $x$-axis.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item Find, using algebraic integration, the exact value of the area of $R$. [4]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM Pure 2021 Q10 [10]}}