| Exam Board | OCR MEI |
|---|---|
| Module | AS Paper 2 (AS Paper 2) |
| Year | 2024 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Areas by integration |
| Type | Area involving fractional powers |
| Difficulty | Moderate -0.8 This is a straightforward area under curve question requiring basic integration. Part (a) is simple verification by substitution, and part (b) involves direct integration of a given function between two provided limits with no problem-solving or insight needed—easier than average A-level questions. |
| Spec | 1.08a Fundamental theorem of calculus: integration as reverse of differentiation1.08e Area between curve and x-axis: using definite integrals |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(5\sqrt{4} - 4 - 6 = 10 - 10 = 0\) | B1 | 1.1 - Command word 'verify' so attempts at solving score zero. Need to see substitution of given value(s) and minimal processing to obtain 0 on RHS |
| \(5\sqrt{9} - 9 - 6 = 15 - 15 = 0\) | B1 | 1.1 - Need to see substitution of given value(s) and minimal processing to obtain 0 on RHS |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\int_{(4)}^{(9)} \left(5x^{\frac{1}{2}} - x - 6\right) dx\) | M1 | 1.1 - Attempting the integration (ignore limits) and obtaining at least two terms of the correct form in \(\alpha x^{\frac{3}{2}} + \beta x^2 + \gamma x\) where \(\alpha, \beta, \gamma \in \mathbb{R}\) |
| \(\frac{10}{3}x^{\frac{3}{2}} - \frac{x^2}{2} - 6x\) o.e. | A1 | 1.1 - Fully correct integrated expression - coefficients need not be simplified but must be correct |
| \(\left[\frac{10}{3}9^{\frac{3}{2}} - \frac{9^2}{2} - 6(9)\right] - \left[\frac{10}{3}4^{\frac{3}{2}} - \frac{4^2}{2} - 6(4)\right]\) | M1 | 1.1 - Must show substitution of correct limits into integrand of correct form and subtract the right way. NB \(-4.5 - \left(-\frac{16}{3}\right)\) |
| \(-\frac{9}{2} - \left(-\frac{16}{3}\right) = \frac{5}{6}\) | A1 | 1.1 - At least one line of working required before seeing the final answer. Accept equivalents or a decimal – awrt 0.833 |
## Question 11:
### Part (a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $5\sqrt{4} - 4 - 6 = 10 - 10 = 0$ | B1 | 1.1 - Command word 'verify' so attempts at solving score zero. Need to see substitution of given value(s) and minimal processing to obtain 0 on RHS |
| $5\sqrt{9} - 9 - 6 = 15 - 15 = 0$ | B1 | 1.1 - Need to see substitution of given value(s) and minimal processing to obtain 0 on RHS |
**Total: [2]**
### Part (b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\int_{(4)}^{(9)} \left(5x^{\frac{1}{2}} - x - 6\right) dx$ | M1 | 1.1 - Attempting the integration (ignore limits) and obtaining **at least two terms of the correct form** in $\alpha x^{\frac{3}{2}} + \beta x^2 + \gamma x$ where $\alpha, \beta, \gamma \in \mathbb{R}$ |
| $\frac{10}{3}x^{\frac{3}{2}} - \frac{x^2}{2} - 6x$ o.e. | A1 | 1.1 - Fully correct integrated expression - coefficients need not be simplified but must be correct |
| $\left[\frac{10}{3}9^{\frac{3}{2}} - \frac{9^2}{2} - 6(9)\right] - \left[\frac{10}{3}4^{\frac{3}{2}} - \frac{4^2}{2} - 6(4)\right]$ | M1 | 1.1 - Must show substitution of correct limits into integrand of correct form and subtract the right way. NB $-4.5 - \left(-\frac{16}{3}\right)$ |
| $-\frac{9}{2} - \left(-\frac{16}{3}\right) = \frac{5}{6}$ | A1 | 1.1 - **At least one line of working required before seeing the final answer.** Accept equivalents or a decimal – awrt 0.833 |
**Total: [4]**
---\begin{enumerate}[label=(\alph*)]
\item Verify that the curve cuts the $x$-axis at $x = 4$ and at $x = 9$.
The curve does not cut or touch the $x$-axis at any other points.
\item Determine the exact area bounded by the curve and the $x$-axis.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI AS Paper 2 2024 Q11 [6]}}