| Exam Board | SPS |
|---|---|
| Module | SPS FM (SPS FM) |
| Year | 2020 |
| Session | June |
| Marks | 9 |
| Topic | Product & Quotient Rules |
| Type | Find stationary points and nature |
| Difficulty | Standard +0.3 This is a straightforward application of the product rule to find a stationary point, followed by a standard integration. Part (a) requires differentiating xe^(-2x) using the product rule, setting equal to zero, and solving a simple equation. Part (b) involves integration by parts, which is routine for Further Maths students. The question is slightly easier than average because it follows a standard template with clear steps and no conceptual surprises. |
| Spec | 1.06a Exponential function: a^x and e^x graphs and properties1.07q Product and quotient rules: differentiation1.08e Area between curve and x-axis: using definite integrals1.08i Integration by parts |
6.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{0e7cab3d-c1e6-4420-93b4-eca5af704432-06_758_1227_280_443}
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\caption{Figure 3}
\end{center}
\end{figure}
In this question you must show all stages of your working.
\section*{Solutions relying on calculator technology are not acceptable.}
Figure 3 shows a sketch of part of the curve with equation
$$y = x e ^ { - 2 x }$$
The point $P ( a , b )$ is the turning point of the curve.
\begin{enumerate}[label=(\alph*)]
\item Find the value of $a$ and the exact value of $b$
The finite region $R$, shown shaded in Figure 3, is bounded by the curve, the line with equation $y = b$ and the $y$-axis.
\item Find the exact area of $R$.
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM 2020 Q6 [9]}}