| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2021 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Addition & Double Angle Formulae |
| Type | Derive triple angle then solve equation |
| Difficulty | Standard +0.8 This question requires deriving tan(4θ) using double angle formulae twice, then algebraic manipulation to reach a quartic in tan θ, followed by solving a quadratic-in-tan²θ. The derivation demands careful symbolic manipulation across multiple steps, and the solving phase requires handling both positive and negative roots with appropriate range restrictions. More demanding than standard formula application but less than proof-heavy AEA questions. |
| Spec | 1.05l Double angle formulae: and compound angle formulae1.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Use double angle formula to express \(\tan 4\theta\) in terms of \(\tan 2\theta\) | M1 | |
| Use double angle formula to express result in terms of \(\tan\theta\) | M1 | |
| Obtain a correct equation in \(\tan\theta\) in any form | A1 | |
| Obtain the given answer | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Solve for \(\tan\theta\) and obtain a value of \(\theta\) | M1 | |
| Obtain answer, e.g. \(53.5°\) | A1 | |
| Obtain second answer, e.g. \(126.5°\) and no other in the interval | A1 | Ignore answers outside the given interval. Treat answers in radians as a misread. |
## Question 5(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Use double angle formula to express $\tan 4\theta$ in terms of $\tan 2\theta$ | M1 | |
| Use double angle formula to express result in terms of $\tan\theta$ | M1 | |
| Obtain a correct equation in $\tan\theta$ in any form | A1 | |
| Obtain the given answer | A1 | |
## Question 5(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Solve for $\tan\theta$ and obtain a value of $\theta$ | M1 | |
| Obtain answer, e.g. $53.5°$ | A1 | |
| Obtain second answer, e.g. $126.5°$ and no other in the interval | A1 | Ignore answers outside the given interval. Treat answers in radians as a misread. |5
\begin{enumerate}[label=(\alph*)]
\item By first expanding $\tan ( 2 \theta + 2 \theta )$, show that the equation $\tan 4 \theta = \frac { 1 } { 2 } \tan \theta$ may be expressed as $\tan ^ { 4 } \theta + 2 \tan ^ { 2 } \theta - 7 = 0$.
\item Hence solve the equation $\tan 4 \theta = \frac { 1 } { 2 } \tan \theta$, for $0 ^ { \circ } < \theta < 180 ^ { \circ }$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2021 Q5 [7]}}