| Exam Board | SPS |
|---|---|
| Module | SPS SM Pure (SPS SM Pure) |
| Year | 2024 |
| Session | September |
| Marks | 8 |
| Topic | Standard Integrals and Reverse Chain Rule |
| Type | Area under curve using integration |
| Difficulty | Standard +0.8 This question requires finding a stationary point by differentiation (including negative fractional powers), then computing a definite integral with multiple terms including x^(-3/2). While the techniques are standard A-level, the combination of steps, algebraic manipulation of fractional powers, and requirement for exact answers without calculator reliance makes this moderately challenging—above average but not exceptionally difficult. |
| Spec | 1.07i Differentiate x^n: for rational n and sums1.07n Stationary points: find maxima, minima using derivatives1.08b Integrate x^n: where n != -1 and sums1.08d Evaluate definite integrals: between limits1.08e Area between curve and x-axis: using definite integrals |
13.
In this question you must show detailed reasoning.\\
Solutions relying entirely on calculator technology are not acceptable.
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\includegraphics[alt={},max width=\textwidth]{8b616e3c-db87-430e-91c1-63a24e2f9593-28_633_725_475_676}
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\caption{Figure 2}
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Figure 2 shows a sketch of the curve with equation
$$y = \frac { 1 } { 2 } x ^ { 2 } + \frac { 1458 } { \sqrt { x ^ { 3 } } } - 74 \quad x > 0$$
The point $P$ is the only stationary point on the curve.\\
The line $l$ passes through the point $P$ and is parallel to the $x$-axis.\\
The region $R$, shown shaded in Figure 2, is bounded by the curve, the line $l$ and the line with equation $x = 4$
Use algebraic integration to find the exact area of $R$.\\
(8)
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\hfill \mbox{\textit{SPS SPS SM Pure 2024 Q13 [8]}}