| Exam Board | Edexcel |
|---|---|
| Module | Paper 2 (Paper 2) |
| Session | Specimen |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Small angle approximation |
| Type | Identify error in approximation usage |
| Difficulty | Moderate -0.8 This question tests understanding of small angle approximations with a common error (forgetting to convert degrees to radians). Part (a) is routine algebraic manipulation, while part (b) requires identifying a procedural mistake and correcting it—straightforward for students who understand the context of small angle approximations. The error is obvious once students recognize that θ must be in radians, making this easier than average. |
| Spec | 1.05e Small angle approximations: sin x ~ x, cos x ~ 1-x^2/2, tan x ~ x |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Attempts to substitute \(\cos\theta \approx 1 - \frac{1}{2}\theta^2\) into either \(1 + 4\cos\theta\) or \(3\cos^2\theta\) | M1 | AO1.1b |
| \(1 + 4\cos\theta + 3\cos^2\theta \approx 1 + 4\left(1-\frac{1}{2}\theta^2\right) + 3\left(1-\frac{1}{2}\theta^2\right)^2\) | ||
| \(= 1 + 4\left(1-\frac{1}{2}\theta^2\right) + 3\left(1 - \theta^2 + \frac{1}{4}\theta^4\right)\) | M1 | AO1.1b; Substitutes and attempts to apply \(\left(1-\frac{1}{2}\theta^2\right)^2\). Note: not required to write/refer to \(\theta^4\) term |
| \(= 1 + 4 - 2\theta^2 + 3 - 3\theta^2 + \frac{3}{4}\theta^4\) | ||
| \(= 8 - 5\theta^2\) | A1* | AO2.1; Correct proof, no errors in working. Note: not required to write/refer to \(\theta^4\) term |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| E.g. Adele is working in degrees and not radians; Adele should substitute \(\theta = \frac{5\pi}{180}\) and not \(\theta = 5\) into the approximation | B1 | AO2.3; See scheme |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(8 - 5\left(\frac{5\pi}{180}\right)^2 \approx 7.962\), so \(\theta = 5°\) gives a good approximation | B1 | AO2.4; Substitutes \(\theta = \frac{5\pi}{180}\) or \(\frac{\pi}{36}\) into \(8-5\theta^2\) to give answer \(\approx 7.962\) and an appropriate conclusion |
## Question 2:
### Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Attempts to substitute $\cos\theta \approx 1 - \frac{1}{2}\theta^2$ into either $1 + 4\cos\theta$ or $3\cos^2\theta$ | M1 | AO1.1b |
| $1 + 4\cos\theta + 3\cos^2\theta \approx 1 + 4\left(1-\frac{1}{2}\theta^2\right) + 3\left(1-\frac{1}{2}\theta^2\right)^2$ | | |
| $= 1 + 4\left(1-\frac{1}{2}\theta^2\right) + 3\left(1 - \theta^2 + \frac{1}{4}\theta^4\right)$ | M1 | AO1.1b; Substitutes and attempts to apply $\left(1-\frac{1}{2}\theta^2\right)^2$. Note: not required to write/refer to $\theta^4$ term |
| $= 1 + 4 - 2\theta^2 + 3 - 3\theta^2 + \frac{3}{4}\theta^4$ | | |
| $= 8 - 5\theta^2$ | A1* | AO2.1; Correct proof, no errors in working. Note: not required to write/refer to $\theta^4$ term |
### Part (b)(i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| E.g. Adele is working in degrees and not radians; Adele should substitute $\theta = \frac{5\pi}{180}$ and not $\theta = 5$ into the approximation | B1 | AO2.3; See scheme |
### Part (b)(ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $8 - 5\left(\frac{5\pi}{180}\right)^2 \approx 7.962$, so $\theta = 5°$ gives a good approximation | B1 | AO2.4; Substitutes $\theta = \frac{5\pi}{180}$ or $\frac{\pi}{36}$ into $8-5\theta^2$ to give answer $\approx 7.962$ **and** an appropriate conclusion |
---\begin{enumerate}
\item (a) Given that $\theta$ is small, use the small angle approximation for $\cos \theta$ to show that
\end{enumerate}
$$1 + 4 \cos \theta + 3 \cos ^ { 2 } \theta \approx 8 - 5 \theta ^ { 2 }$$
Adele uses $\theta = 5 ^ { \circ }$ to test the approximation in part (a).\\
Adele's working is shown below.
Using my calculator, $1 + 4 \cos \left( 5 ^ { \circ } \right) + 3 \cos ^ { 2 } \left( 5 ^ { \circ } \right) = 7.962$, to 3 decimal places.\\
Using the approximation $8 - 5 \theta ^ { 2 }$ gives $8 - 5 ( 5 ) ^ { 2 } = - 117$\\
Therefore, $1 + 4 \cos \theta + 3 \cos ^ { 2 } \theta \approx 8 - 5 \theta ^ { 2 }$ is not true for $\theta = 5 ^ { \circ }$\\
(b) (i) Identify the mistake made by Adele in her working.\\
(ii) Show that $8 - 5 \theta ^ { 2 }$ can be used to give a good approximation to $1 + 4 \cos \theta + 3 \cos ^ { 2 } \theta$ for an angle of size $5 ^ { \circ }$\\
(2)
\hfill \mbox{\textit{Edexcel Paper 2 Q2 [5]}}