| Exam Board | OCR |
|---|---|
| Module | H240/01 (Pure Mathematics) |
| Year | 2020 |
| Session | November |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Topic | Small angle approximation |
| Type | Simplify expression to polynomial form |
| Difficulty | Moderate -0.3 This is a straightforward application of standard small angle approximations (cos θ ≈ 1 - θ²/2, tan θ ≈ θ, sin θ ≈ θ) with basic algebraic expansion. Part (a) is routine substitution and simplification, while part (b) requires only direct substitution of the result into a linear equation. Slightly easier than average due to minimal problem-solving required. |
| Spec | 1.05e Small angle approximations: sin x ~ x, cos x ~ 1-x^2/2, tan x ~ x |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(2\left(1-\frac{1}{2}\theta^2\right)+(1-\theta)^2\) | B1 | Correct statement; first term could possibly already be expanded |
| \(2-\theta^2+1-2\theta+\theta^2\) | M1 | Attempt to expand and simplify given expression |
| \(=3-2\theta\) A.G. | A1 | Obtain given answer; Max of M1A1 if neither \((1-\theta)^2\) nor \(1-2\tan\theta+\tan^2\theta\) seen |
| [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(3-2\theta=28\theta\) | M1 | Use \(28\sin\theta\approx28\theta\) and attempt to solve |
| \(\theta=0.1\) | A1 | Obtain 0.1 oe; BOD if \(0.1°\); ISW once 0.1 seen |
| [2] |
# Question 1:
## Part (a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $2\left(1-\frac{1}{2}\theta^2\right)+(1-\theta)^2$ | B1 | Correct statement; first term could possibly already be expanded |
| $2-\theta^2+1-2\theta+\theta^2$ | M1 | Attempt to expand and simplify given expression |
| $=3-2\theta$ **A.G.** | A1 | Obtain given answer; Max of M1A1 if neither $(1-\theta)^2$ nor $1-2\tan\theta+\tan^2\theta$ seen |
| **[3]** | | |
## Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $3-2\theta=28\theta$ | M1 | Use $28\sin\theta\approx28\theta$ and attempt to solve |
| $\theta=0.1$ | A1 | Obtain 0.1 oe; BOD if $0.1°$; ISW once 0.1 seen |
| **[2]** | | |
---1
\begin{enumerate}[label=(\alph*)]
\item For a small angle $\theta$, where $\theta$ is in radians, show that $2 \cos \theta + ( 1 - \tan \theta ) ^ { 2 } \approx 3 - 2 \theta$.
\item Hence determine an approximate solution to $2 \cos \theta + ( 1 - \tan \theta ) ^ { 2 } = 28 \sin \theta$.
\end{enumerate}
\hfill \mbox{\textit{OCR H240/01 2020 Q1 [5]}}