| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2022 |
| Session | March |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Addition & Double Angle Formulae |
| Type | Prove identity then find exact value and solve/integrate |
| Difficulty | Standard +0.3 Part (a) is a straightforward identity proof using standard double angle formulae (sin 2θ = 2sin θ cos θ, cos 2θ = cos²θ - sin²θ) and cot θ = cos θ/sin θ. Part (b) applies the proven identity with direct substitution. Part (c) requires rearranging to use the identity, then solving a standard trigonometric equation. All steps use routine A-level techniques with no novel insight required, making this slightly easier than average. |
| Spec | 1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05l Double angle formulae: and compound angle formulae1.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use at least two of \(\sin 2\theta = 2\sin\theta\cos\theta\), \(\cos 2\theta = \cos^2\theta - \sin^2\theta\), \(\cot\theta = \frac{\cos\theta}{\sin\theta}\) | B1 | OE |
| Express LHS in terms of \(\sin\theta\) and \(\cos\theta\) only and attempt valid simplification | M1 | |
| Obtain \(\cos^2\theta + \sin^2\theta\) or equivalent and hence 1 | A1 | AG – necessary detail needed |
| Total | 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Substitute \(\theta = \frac{1}{12}\pi\) and show or imply \(\sin\frac{1}{6}\pi\cot\frac{1}{12}\pi = 1 + \cos\frac{1}{6}\pi\) | M1 | |
| Obtain \(1 + \frac{1}{2}\sqrt{3}\) or exact equivalent | A1 | |
| Total | 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use the identity from part (a) to obtain \(-2\cos 2\theta = 0\) or equivalent | M1 | Or alternative starting again, using valid simplification and reaching single trigonometric function |
| Obtain \(\theta = \frac{1}{4}\pi\) | A1 | |
| Total | 2 |
## Question 4(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use at least two of $\sin 2\theta = 2\sin\theta\cos\theta$, $\cos 2\theta = \cos^2\theta - \sin^2\theta$, $\cot\theta = \frac{\cos\theta}{\sin\theta}$ | B1 | OE |
| Express LHS in terms of $\sin\theta$ and $\cos\theta$ only and attempt valid simplification | M1 | |
| Obtain $\cos^2\theta + \sin^2\theta$ or equivalent and hence 1 | A1 | AG – necessary detail needed |
| **Total** | **3** | |
## Question 4(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Substitute $\theta = \frac{1}{12}\pi$ and show or imply $\sin\frac{1}{6}\pi\cot\frac{1}{12}\pi = 1 + \cos\frac{1}{6}\pi$ | M1 | |
| Obtain $1 + \frac{1}{2}\sqrt{3}$ or exact equivalent | A1 | |
| **Total** | **2** | |
## Question 4(c):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use the identity from part (a) to obtain $-2\cos 2\theta = 0$ or equivalent | M1 | Or alternative starting again, using valid simplification and reaching single trigonometric function |
| Obtain $\theta = \frac{1}{4}\pi$ | A1 | |
| **Total** | **2** | |4
\begin{enumerate}[label=(\alph*)]
\item Show that $\sin 2 \theta \cot \theta - \cos 2 \theta \equiv 1$.
\item Hence find the exact value of $\sin \frac { 1 } { 6 } \pi \cot \frac { 1 } { 12 } \pi$.
\item Find the smallest positive value of $\theta$ (in radians) satisfying the equation
$$\sin 2 \theta \cot \theta - 3 \cos 2 \theta = 1 .$$
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2022 Q4 [7]}}