| Exam Board | WJEC |
|---|---|
| Module | Unit 3 (Unit 3) |
| Year | 2024 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Small angle approximation |
| Type | Reciprocal expansion to polynomial form |
| Difficulty | Standard +0.3 This question tests small angle approximations and binomial expansion of (1+x)^(-1), both standard C3/C4 techniques. Part (a) is straightforward substitution of cos θ ≈ 1 - θ²/2 and sin θ ≈ θ. Part (b) requires recognizing the form for binomial expansion and expanding to θ² terms—routine application with no novel insight required, though the algebraic manipulation earns it a slightly above-average rating. |
| Spec | 1.05e Small angle approximations: sin x ~ x, cos x ~ 1-x^2/2, tan x ~ x |
\begin{enumerate}[label=(\alph*)]
\item Given that $\theta$ is small, show that $2\cos\theta + \sin\theta - 1 \approx 1 + \theta - \theta^2$. [2]
\item Hence, when $\theta$ is small, show that
$$\frac{1}{2\cos\theta + \sin\theta - 1} \approx 1 + a\theta + b\theta^2,$$
where $a$, $b$ are constants to be found. [4]
\end{enumerate}
\hfill \mbox{\textit{WJEC Unit 3 2024 Q12 [6]}}