| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2022 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Reciprocal Trig & Identities |
| Type | Double angle with reciprocal functions |
| Difficulty | Standard +0.8 Part (a) requires systematic application of double angle formulas (cos 4θ = 2cos²2θ - 1, then cos 2θ = 2cos²θ - 1) with careful algebraic manipulation to reach the quartic form. Part (b) involves substituting into the proven identity to get 8cos⁴θ = 7, then solving a straightforward equation. This is moderately challenging due to the multi-step proof requiring strategic formula choices, but follows standard A-level techniques without requiring novel insight. |
| Spec | 1.05l Double angle formulae: and compound angle formulae1.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Express \(\cos 4\theta\) in terms of \(\cos 2\theta\) and/or \(\sin 2\theta\) | B1 | |
| Express \(\cos 2\theta\) in terms of \(\cos\theta\) and/or \(\sin\theta\) | B1 | Anywhere |
| Expand to obtain a correct expression in terms of \(\cos\theta\) | B1 | e.g. \(2(2\cos^2\theta - 1)^2 - 1 + 4(2\cos^2\theta - 1) + 3\) |
| Reduce correctly to \(\cos 4\theta + 4\cos 2\theta + 3 \equiv 8\cos^4\theta\) | B1 | AG |
| Total | 4 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Use the identity and carry out method to calculate a root | M1 | \(8\cos^4\theta - 3 = 4\) |
| Obtain answer, e.g. \(14.7°\) | A1 | |
| Obtain second answer, e.g. \(165.3°\), and no other in the given interval | A1 FT | Ignore answers outside the given interval; treat answers in radians as a misread |
| Total | 3 |
## Question 6(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Express $\cos 4\theta$ in terms of $\cos 2\theta$ and/or $\sin 2\theta$ | B1 | |
| Express $\cos 2\theta$ in terms of $\cos\theta$ and/or $\sin\theta$ | B1 | Anywhere |
| Expand to obtain a correct expression in terms of $\cos\theta$ | B1 | e.g. $2(2\cos^2\theta - 1)^2 - 1 + 4(2\cos^2\theta - 1) + 3$ |
| Reduce correctly to $\cos 4\theta + 4\cos 2\theta + 3 \equiv 8\cos^4\theta$ | B1 | AG |
| **Total** | **4** | |
## Question 6(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Use the identity and carry out method to calculate a root | M1 | $8\cos^4\theta - 3 = 4$ |
| Obtain answer, e.g. $14.7°$ | A1 | |
| Obtain second answer, e.g. $165.3°$, and no other in the given interval | A1 FT | Ignore answers outside the given interval; treat answers in radians as a misread |
| **Total** | **3** | |6
\begin{enumerate}[label=(\alph*)]
\item Prove the identity $\cos 4 \theta + 4 \cos 2 \theta + 3 \equiv 8 \cos ^ { 4 } \theta$.
\item Hence solve the equation $\cos 4 \theta + 4 \cos 2 \theta = 4$ for $0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2022 Q6 [7]}}