| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2024 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Addition & Double Angle Formulae |
| Type | Prove identity then solve equation |
| Difficulty | Standard +0.8 This question requires fluent manipulation of both double angle and addition formulae, algebraic simplification to prove a non-trivial identity, then solving a resulting equation. The identity proof involves multiple formula applications and careful algebra, while part (b) requires solving a rational equation in tan θ with domain restrictions. More demanding than routine formula application but less challenging than proof-heavy or multi-concept integration problems. |
| 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 | Marks | Guidance |
| State \(3\tan 2\theta = \dfrac{6\tan\theta}{1-\tan^2\theta}\) | B1 | Allow \(3\tan 2\theta = 3\left(\dfrac{2\tan\theta}{1-\tan^2\theta}\right)\) |
| State \(\tan(\theta + 45°) = \dfrac{\tan\theta + 1}{1 - \tan\theta}\) | B1 | |
| Attempt to express left-hand side in terms of \(\tan\theta\) as a single fraction | M1 | Condone sign errors in identities. Allow if \(\tan 45°\) is not evaluated. Allow if 2 separate terms with a common denominator |
| Confirm \(\dfrac{\tan^2\theta + 8\tan\theta + 1}{1 - \tan^2\theta}\) | A1 | Answer given – necessary detail needed |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Solve quadratic equation \(5\tan^2\theta + 8\tan\theta - 3 = 0\) to obtain at least one value of \(\theta\) | M1 | Must be using a correct method |
| Obtain \(17.4\) | A1 | Or greater accuracy |
| Obtain \(117.6\) | A1 | Or greater accuracy; and no others between \(0°\) and \(180°\) |
## Question 4(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| State $3\tan 2\theta = \dfrac{6\tan\theta}{1-\tan^2\theta}$ | B1 | Allow $3\tan 2\theta = 3\left(\dfrac{2\tan\theta}{1-\tan^2\theta}\right)$ |
| State $\tan(\theta + 45°) = \dfrac{\tan\theta + 1}{1 - \tan\theta}$ | B1 | |
| Attempt to express left-hand side in terms of $\tan\theta$ as a single fraction | M1 | Condone sign errors in identities. Allow if $\tan 45°$ is not evaluated. Allow if 2 separate terms with a common denominator |
| Confirm $\dfrac{\tan^2\theta + 8\tan\theta + 1}{1 - \tan^2\theta}$ | A1 | Answer given – necessary detail needed |
## Question 4(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Solve quadratic equation $5\tan^2\theta + 8\tan\theta - 3 = 0$ to obtain at least one value of $\theta$ | M1 | Must be using a correct method |
| Obtain $17.4$ | A1 | Or greater accuracy |
| Obtain $117.6$ | A1 | Or greater accuracy; and no others between $0°$ and $180°$ |
---4
\begin{enumerate}[label=(\alph*)]
\item Show that $3 \tan 2 \theta + \tan \left( \theta + 45 ^ { \circ } \right) \equiv \frac { \tan ^ { 2 } \theta + 8 \tan \theta + 1 } { 1 - \tan ^ { 2 } \theta }$.
\item Hence solve the equation $3 \tan 2 \theta + \tan \left( \theta + 45 ^ { \circ } \right) = 4$ for $0 ^ { \circ } < \theta < 180 ^ { \circ }$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2024 Q4 [7]}}