| Exam Board | SPS |
|---|---|
| Module | SPS SM (SPS SM) |
| Year | 2022 |
| Session | January |
| Marks | 10 |
| Topic | Areas by integration |
| Type | Combined region areas |
| Difficulty | Standard +0.8 This question requires multiple integrated steps: finding the derivative to get the gradient at P, determining the normal line equation, setting up the area calculation as the difference between the area under the normal line and the area under the curve (both from x=2 to x=4), and performing integration including a rational function term. While each individual step is standard A-level technique, the combination of calculus, coordinate geometry, and careful region identification makes this moderately challenging, requiring sustained multi-step reasoning and algebraic manipulation to reach the given answer. |
| Spec | 1.03a Straight lines: equation forms y=mx+c, ax+by+c=01.07l Derivative of ln(x): and related functions1.07m Tangents and normals: gradient and equations1.08b Integrate x^n: where n != -1 and sums1.08e Area between curve and x-axis: using definite integrals |
10.
\begin{figure}[h]
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\includegraphics[alt={},max width=\textwidth]{4eb48b49-816b-4a08-9f7f-c20313c4d1c9-22_659_970_141_614}
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\caption{Figure 4}
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Figure 4 shows a sketch of part of the curve $C$ with equation
$$y = \frac { 32 } { x ^ { 2 } } + 3 x - 8 , \quad x > 0$$
The point $P ( 4,6 )$ lies on $C$.\\
The line $l$ is the normal to $C$ at the point $P$.\\
The region $R$, shown shaded in Figure 4, is bounded by the line $l$, the curve $C$, the line with equation $x = 2$ and the $x$-axis.
Show that the area of $R$ is 46\\
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\hfill \mbox{\textit{SPS SPS SM 2022 Q10 [10]}}