| Exam Board | Edexcel |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Areas by integration |
| Type | Area involving fractional powers |
| Difficulty | Standard +0.3 This is a straightforward C2 integration question requiring differentiation to find a minimum (standard quotient rule or rewriting as x^{-2}), then integration of x^{1/2} and x^{-2} using standard power rule. The arithmetic with surds is routine, and the answer is given to verify. Slightly easier than average due to being a standard two-part question with no novel problem-solving required. |
| Spec | 1.07n 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 |
| Answer | Marks |
|---|---|
| \(\frac{dy}{dx} = \frac{1}{2}x^{-\frac{1}{2}} - 16x^{-3}\) | M1 A2 |
| For minimum, \(\frac{1}{2}x^{-\frac{1}{2}} - 16x^{-3} = 0\) | M1 |
| \(\frac{1}{2}x^{-3}(x^{\frac{5}{2}} - 32) = 0\) | |
| \(x^{\frac{5}{2}} = 32\) | A1 |
| \(x = (\sqrt[5]{32})^2 = 4\) | M1 A1 |
| \(\therefore (4, \frac{5}{2})\) |
| Answer | Marks |
|---|---|
| \(= \int_1^9 \left(\sqrt{x} + \frac{8}{x^2}\right) dx\) | |
| \(= \left[\frac{2}{3}x^{\frac{3}{2}} - 8x^{-1}\right]_1^9\) | M1 A2 |
| \(= (18 - \frac{8}{9}) - (\frac{2}{3} - 8)\) | M1 |
| \(= 24\frac{4}{9}\) | A1 |
**(a)**
$\frac{dy}{dx} = \frac{1}{2}x^{-\frac{1}{2}} - 16x^{-3}$ | M1 A2 |
For minimum, $\frac{1}{2}x^{-\frac{1}{2}} - 16x^{-3} = 0$ | M1 |
$\frac{1}{2}x^{-3}(x^{\frac{5}{2}} - 32) = 0$ | |
$x^{\frac{5}{2}} = 32$ | A1 |
$x = (\sqrt[5]{32})^2 = 4$ | M1 A1 |
$\therefore (4, \frac{5}{2})$ | |
**(b)**
$= \int_1^9 \left(\sqrt{x} + \frac{8}{x^2}\right) dx$ | |
$= \left[\frac{2}{3}x^{\frac{3}{2}} - 8x^{-1}\right]_1^9$ | M1 A2 |
$= (18 - \frac{8}{9}) - (\frac{2}{3} - 8)$ | M1 |
$= 24\frac{4}{9}$ | A1 |8.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{089f5506-94ac-489f-b219-e67fa6ca834f-4_536_883_248_486}
\captionsetup{labelformat=empty}
\caption{Figure 3}
\end{center}
\end{figure}
Figure 3 shows the curve with equation $y = \sqrt { x } + \frac { 8 } { x ^ { 2 } } , x > 0$.
\begin{enumerate}[label=(\alph*)]
\item Find the coordinates of the minimum point of the curve.
\item Show that the area of the shaded region bounded by the curve, the $x$-axis and the lines $x = 1$ and $x = 9$ is $24 \frac { 4 } { 9 }$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C2 Q8 [12]}}