| Exam Board | SPS |
|---|---|
| Module | SPS SM Pure (SPS SM Pure) |
| Year | 2024 |
| Session | June |
| Marks | 10 |
| Topic | Stationary points and optimisation |
| Type | Find range where function increasing/decreasing |
| Difficulty | Standard +0.3 This is a straightforward A-level calculus question requiring differentiation of a polynomial with a fractional power, solving dy/dx > 0, finding a tangent equation, and computing an area using integration. All techniques are standard with no novel problem-solving required, though the multi-part nature and exact arithmetic with surds adds minor complexity above the most routine questions. |
| Spec | 1.07i Differentiate x^n: for rational n and sums1.07n Stationary points: find maxima, minima using derivatives1.08e Area between curve and x-axis: using definite integrals |
17.
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\caption{Figure 3}
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\section*{In this question you must show detailed reasoning. Solutions relying entirely on calculator technology are not acceptable.}
Figure 3 shows a sketch of part of the curve $C$ with equation
$$y = \frac { 2 } { 3 } x ^ { 2 } - 9 \sqrt { x } + 13 \quad x \geq 0$$
\begin{enumerate}[label=(\alph*)]
\item Find, using calculus, the range of values of $x$ for which $y$ is increasing.
The point $P$ lies on $C$ and has coordinates (9, 40).\\
The line $l$ is the tangent to $C$ at the point $P$.\\
The finite region $R$, shown shaded in Figure 3, is bounded by the curve $C$, the line $l$, the $x$-axis and the $y$-axis.
\item Find, using calculus, the exact area of $R$.\\
(6)
ADDITIONAL SHEET
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\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM Pure 2024 Q17 [10]}}