| Exam Board | OCR MEI |
|---|---|
| Module | Paper 1 (Paper 1) |
| Year | 2024 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Small angle approximation |
| Type | Approximate area under curve |
| Difficulty | Challenging +1.2 This question requires small angle approximations (sin θ ≈ θ, cos θ ≈ 1 - θ²/2) applied to composite functions, then integration of the resulting expression including √x and polynomial terms. While it involves multiple steps and careful algebraic manipulation, the techniques are standard A-level further maths content with clear scaffolding in part (a). The integration in part (b) is routine once the approximation is established. |
| Spec | 1.05e Small angle approximations: sin x ~ x, cos x ~ 1-x^2/2, tan x ~ x1.05l Double angle formulae: and compound angle formulae1.08d Evaluate definite integrals: between limits |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Using \(\sin x \approx x\) and \(\cos x \approx 1 - \frac{1}{2}x^2\): \(\sqrt{\sin 4x} + 2\cos 2x \approx \sqrt{4x} + 2\left(1 - \frac{1}{2}(2x)^2\right)\) | M1 | Uses both given small angle approximations. Must see \(\sqrt{4x}\) and \(\frac{1}{2}(2x)^2\) or \(\frac{1}{2} \times 4x^2\); condone missing brackets. Also allow clear use of \(\cos 2x = 1 - 2\sin^2 x\) and \(\sin x \approx x\) |
| So \(y \approx 2\sqrt{x} + 2 - 4x^2\) | A1 | AG — convincing argument to reach given answer |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\int_0^{0.1} \left(\sqrt{\sin 4x} + 2\cos 2x\right) dx \approx \int_0^{0.1} \left(2x^{\frac{1}{2}} + 2 - 4x^2\right) dx\) | M1 | Attempts to integrate the expression in powers of \(x\). Must be seen |
| \(= \left[\dfrac{2x^{\frac{3}{2}}}{\frac{3}{2}} + 2x - \dfrac{4}{3}x^3\right]_0^{0.1}\) | A1 A1 | At least 2 correct terms (no FT from (a) as answer given); Fully correct indefinite integral. Need not be simplified |
| \(= \left(\dfrac{4}{3}(0.1)^{\frac{3}{2}} + 0.2 - \dfrac{0.004}{3}\right) - 0 = 0.24083\ldots\) | A1 | Correct value from correct indefinite integral. Allow 0.24 or better if method is clear. Do not allow 0.24059… obtained by integrating original function by calculator |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Use trapezium rule to find an area | M1 | Also allow for a single trapezium |
| Correct \(\frac{h}{2}\) for the number of strips used | A1 | e.g. \(\frac{0.1}{2}, \frac{0.05}{2}, \frac{0.025}{2}, \frac{0.01}{2}\) used for 1, 2, 4, 10 strips |
| At least 3 correct ordinates used | A1 | Must be ordinates from the approximating function, e.g. 2, 2.4372… and 2.5924… seen |
| Area is approximately 0.24 | A1 | Accept awrt 0.24 |
## Question 8:
### Part (a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Using $\sin x \approx x$ and $\cos x \approx 1 - \frac{1}{2}x^2$: $\sqrt{\sin 4x} + 2\cos 2x \approx \sqrt{4x} + 2\left(1 - \frac{1}{2}(2x)^2\right)$ | M1 | Uses both given small angle approximations. Must see $\sqrt{4x}$ and $\frac{1}{2}(2x)^2$ or $\frac{1}{2} \times 4x^2$; condone missing brackets. Also allow clear use of $\cos 2x = 1 - 2\sin^2 x$ and $\sin x \approx x$ |
| So $y \approx 2\sqrt{x} + 2 - 4x^2$ | A1 | AG — convincing argument to reach given answer |
### Part (b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\int_0^{0.1} \left(\sqrt{\sin 4x} + 2\cos 2x\right) dx \approx \int_0^{0.1} \left(2x^{\frac{1}{2}} + 2 - 4x^2\right) dx$ | M1 | Attempts to integrate the expression in powers of $x$. Must be seen |
| $= \left[\dfrac{2x^{\frac{3}{2}}}{\frac{3}{2}} + 2x - \dfrac{4}{3}x^3\right]_0^{0.1}$ | A1 A1 | At least 2 correct terms (no FT from (a) as answer given); Fully correct indefinite integral. Need not be simplified |
| $= \left(\dfrac{4}{3}(0.1)^{\frac{3}{2}} + 0.2 - \dfrac{0.004}{3}\right) - 0 = 0.24083\ldots$ | A1 | Correct value from correct indefinite integral. Allow 0.24 or better if method is clear. Do not allow 0.24059… obtained by integrating original function by calculator |
**Alternative (Trapezium Rule):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| Use trapezium rule to find an area | M1 | Also allow for a single trapezium |
| Correct $\frac{h}{2}$ for the number of strips used | A1 | e.g. $\frac{0.1}{2}, \frac{0.05}{2}, \frac{0.025}{2}, \frac{0.01}{2}$ used for 1, 2, 4, 10 strips |
| At least 3 correct ordinates used | A1 | Must be ordinates from the approximating function, e.g. 2, 2.4372… and 2.5924… seen |
| Area is approximately 0.24 | A1 | Accept awrt 0.24 |
---8 The equation of a curve is $\mathrm { y } = \sqrt { \sin 4 \mathrm { x } } + 2 \cos 2 \mathrm { x }$, where $x$ is in radians.
\begin{enumerate}[label=(\alph*)]
\item Show that, for small values of $x , y \approx 2 \sqrt { x } + 2 - 4 x ^ { 2 }$.
The diagram shows the region bounded by the curve $\mathrm { y } = \sqrt { \sin 4 \mathrm { x } } + 2 \cos 2 \mathrm { x }$, the axes and the line $x = 0.1$.\\
\includegraphics[max width=\textwidth, alt={}, center]{1d0ca3d5-6529-435f-a0b8-50ea4859adde-07_499_881_589_223}
\item In this question you must show detailed reasoning.
Use the approximation in part (a) to estimate the area of this region.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Paper 1 2024 Q8 [6]}}