| Exam Board | SPS |
|---|---|
| Module | SPS SM Mechanics (SPS SM Mechanics) |
| Year | 2022 |
| Session | February |
| Marks | 9 |
| Topic | Areas by integration |
| Type | Area between curve and line |
| Difficulty | Standard +0.3 This is a standard calculus question requiring differentiation to find a tangent line, verification of an intersection, and integration to find an area between curves. All techniques are routine A-level methods with straightforward algebra. The multi-part structure and 9 total marks indicate moderate length, but no step requires novel insight or particularly challenging manipulation. |
| Spec | 1.07a Derivative as gradient: of tangent to curve1.07m Tangents and normals: gradient and equations1.08e Area between curve and x-axis: using definite integrals |
In this question you should show all stages of your working.
Solutions relying entirely on calculator technology are not acceptable.
\includegraphics{figure_2}
Figure 2
Figure 2 shows a sketch of part of the curve $C$ with equation
$$y = x^3 - 10x^2 + 27x - 23$$
The point $P(5, -13)$ lies on $C$
The line $l$ is the tangent to $C$ at $P$
\begin{enumerate}[label=(\alph*)]
\item Use differentiation to find the equation of $l$, giving your answer in the form $y = mx + c$ where $m$ and $c$ are integers to be found.
[4]
\item Hence verify that $l$ meets $C$ again on the $y$-axis.
[1]
\end{enumerate}
The finite region $R$, shown shaded in Figure 2, is bounded by the curve $C$ and the line $l$.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item Use algebraic integration to find the exact area of $R$.
[4]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM Mechanics 2022 Q5 [9]}}