| Exam Board | Edexcel |
|---|---|
| Module | Paper 2 (Paper 2) |
| Year | 2024 |
| Session | June |
| Marks | 3 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Small angle approximation |
| Type | Simplify single fraction to numerical value |
| Difficulty | Standard +0.3 This question requires applying small angle approximations (tan θ ≈ θ, cos θ ≈ 1 - θ²/2) to simplify a fraction, which is a standard technique. The main challenge is correctly handling the coefficients (2θ and 3θ) and simplifying the algebra, but this is routine practice for students who have learned the approximations. It's slightly above average difficulty due to the algebraic manipulation required, but remains a straightforward application question. |
| 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 |
| One of \(\theta \tan 2\theta = \theta \times 2\theta\) or \(1 - \cos 3\theta = 1 - \left(1 - \frac{(3\theta)^2}{2}\right)\) or equivalents | B1 | May be seen in numerator/denominator separately or within the fraction; do not condone missing brackets e.g. \(1 - \frac{3\theta^2}{2}\) unless recovered |
| \(\frac{\theta \tan 2\theta}{1 - \cos 3\theta} = \frac{\theta \times 2\theta}{1 - \left(1 - \frac{(3\theta)^2}{2}\right)}\) | M1 | Must attempt both correct small angle approximations; must have \(\tan 2\theta = 2\theta\) and \(\cos 3\theta = 1 - \frac{(3\theta)^2}{2}\); condone poor bracketing |
| \(= \frac{4}{9}\) or exact equivalent | A1 | Do not allow rounded decimals e.g. 0.444; allow recurring decimals clearly indicated e.g. \(0.\dot{4}\); do not allow \(\frac{2}{4.5}\) |
# Question 5:
| Answer/Working | Mark | Guidance |
|---|---|---|
| One of $\theta \tan 2\theta = \theta \times 2\theta$ or $1 - \cos 3\theta = 1 - \left(1 - \frac{(3\theta)^2}{2}\right)$ or equivalents | B1 | May be seen in numerator/denominator separately or within the fraction; do not condone missing brackets e.g. $1 - \frac{3\theta^2}{2}$ unless recovered |
| $\frac{\theta \tan 2\theta}{1 - \cos 3\theta} = \frac{\theta \times 2\theta}{1 - \left(1 - \frac{(3\theta)^2}{2}\right)}$ | M1 | Must attempt both correct small angle approximations; must have $\tan 2\theta = 2\theta$ and $\cos 3\theta = 1 - \frac{(3\theta)^2}{2}$; condone poor bracketing |
| $= \frac{4}{9}$ or exact equivalent | A1 | Do not allow rounded decimals e.g. 0.444; allow recurring decimals clearly indicated e.g. $0.\dot{4}$; do not allow $\frac{2}{4.5}$ |
---\begin{enumerate}
\item Given that $\theta$ is small and in radians, use the small angle approximations to find an approximate numerical value of
\end{enumerate}
$$\frac { \theta \tan 2 \theta } { 1 - \cos 3 \theta }$$
\hfill \mbox{\textit{Edexcel Paper 2 2024 Q5 [3]}}