| Exam Board | Edexcel |
|---|---|
| Module | PMT Mocks (PMT Mocks) |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Areas by integration |
| Type | Exact area with surds |
| Difficulty | Standard +0.3 This is a straightforward integration question requiring simplification of the integrand into powers of x, followed by standard integration and evaluation at limits. The algebraic manipulation (dividing by √x) and surd arithmetic are routine A-level skills, making this slightly easier than average despite the 'show that' format. |
| Spec | 1.02b Surds: manipulation and rationalising denominators1.07i Differentiate x^n: for rational n and sums1.08e Area between curve and x-axis: using definite integrals |
| Answer | Marks | Guidance |
|---|---|---|
| M1 | Correct attempt to write \(\frac{12x - x^2}{\sqrt{x}}\) as a sum of terms with indices. Look for the terms with the correct index. Example: \(\frac{12x}{x^{\frac{1}{2}}} - \frac{x^2}{x^{\frac{1}{2}}} = ax^{\frac{1}{2}} - bx^{\frac{3}{2}}\) | |
| M1 | Integrates \(x^n \to x^{n+1}\) at least 1 correct index. Example: at least 1 of \(ax^{\frac{1}{2}} \to x^{\frac{3}{2}}\), \(bx^{\frac{3}{2}} \to x^{\frac{5}{2}}\) | |
| A1 | \(8x^{\frac{3}{2}} - \frac{2}{5}x^{\frac{5}{2}}\) (+c) | Allow unsimplified e.g. \(\int 12x^{\frac{1}{2}} - x^{\frac{3}{2}} \, dx = 12 \times \frac{2}{3}x^{\frac{3}{2}} - \frac{2}{5}x^{\frac{5}{2}}\) |
| M1 | Substitutes the limits 8 and 4 to their \(8x^{\frac{3}{2}} - \frac{2}{5}x^{\frac{5}{2}}\) and subtracts either way round. There is no requirement to evaluate but 8 and 4 must be substituted either way round with evidence of subtraction condoning omission of brackets. Example: \(8x^{\frac{3}{2}} - \frac{2}{5}x^{\frac{5}{2}} = 8 \times 8^{\frac{3}{2}} - \frac{2}{5} \times 8^{\frac{3}{2}} - 8 \times 4^{\frac{3}{2}} - \frac{2}{5} \times 4^{\frac{3}{2}}\) | |
| A1 | Correct working shown leading to \(\frac{128}{5}(3\sqrt{2} - 2)\). |
(5 marks)
**M1** | Correct attempt to write $\frac{12x - x^2}{\sqrt{x}}$ as a sum of terms with indices. Look for the terms with the correct index. Example: $\frac{12x}{x^{\frac{1}{2}}} - \frac{x^2}{x^{\frac{1}{2}}} = ax^{\frac{1}{2}} - bx^{\frac{3}{2}}$
**M1** | Integrates $x^n \to x^{n+1}$ at least 1 correct index. Example: at least 1 of $ax^{\frac{1}{2}} \to x^{\frac{3}{2}}$, $bx^{\frac{3}{2}} \to x^{\frac{5}{2}}$
**A1** | $8x^{\frac{3}{2}} - \frac{2}{5}x^{\frac{5}{2}}$ (+c) | Allow unsimplified e.g. $\int 12x^{\frac{1}{2}} - x^{\frac{3}{2}} \, dx = 12 \times \frac{2}{3}x^{\frac{3}{2}} - \frac{2}{5}x^{\frac{5}{2}}$
**M1** | Substitutes the limits 8 and 4 to their $8x^{\frac{3}{2}} - \frac{2}{5}x^{\frac{5}{2}}$ and subtracts either way round. There is no requirement to evaluate but 8 and 4 must be substituted either way round with evidence of subtraction condoning omission of brackets. Example: $8x^{\frac{3}{2}} - \frac{2}{5}x^{\frac{5}{2}} = 8 \times 8^{\frac{3}{2}} - \frac{2}{5} \times 8^{\frac{3}{2}} - 8 \times 4^{\frac{3}{2}} - \frac{2}{5} \times 4^{\frac{3}{2}}$
**A1** | Correct working shown leading to $\frac{128}{5}(3\sqrt{2} - 2)$.
---8.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{d37eaba2-0a25-4abf-b2c8-1e08673229fb-14_1090_1205_274_456}
\captionsetup{labelformat=empty}
\caption{Figure 2\\
Figure 2 shows a sketch of part of the curve with equation
$$y = \frac { 12 x - x ^ { 2 } } { \sqrt { x } } , \quad x > 0$$
The region $R$, shows shaded in figure 2, is bounded by the curve, the line with equation $x = 4$, the $x$-axis and the line with equation $x = 8$.\\
Show that the area of the shaded region $R$ is $\frac { 128 } { 5 } ( 3 \sqrt { 2 } - 2 )$.}
\end{center}
\end{figure}
(5)\\
\hfill \mbox{\textit{Edexcel PMT Mocks Q8 [5]}}