| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2011 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Addition & Double Angle Formulae |
| Type | Prove identity then solve equation and evaluate integral |
| Difficulty | Standard +0.8 This question requires proving a non-trivial trigonometric identity using double angle formulae (requiring multiple applications and algebraic manipulation), then applying it to solve an equation and evaluate a definite integral. The proof demands systematic use of cos 2θ and cos 4θ expansions, the equation-solving requires careful algebraic manipulation of the quartic form, and the integration application is non-routine. This is more challenging than standard textbook exercises but accessible to well-prepared P3 students. |
| Spec | 1.05l Double angle formulae: and compound angle formulae1.05o Trigonometric equations: solve in given intervals1.05p Proof involving trig: functions and identities1.08d Evaluate definite integrals: between limits |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Express \(\cos 4\theta\) as \(2\cos^2 2\theta - 1\) or \(\cos^2 2\theta - \sin^2 2\theta\) or \(1 - 2\sin^2 2\theta\) | B1 | |
| Express \(\cos 4\theta\) in terms of \(\cos\theta\) | M1 | |
| Obtain \(8\cos^4\theta - 8\cos^2\theta + 1\) | A1 | |
| Use \(\cos 2\theta = 2\cos^2\theta - 1\) to obtain given answer \(8\cos^4\theta - 3\) | A1 | AG |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| State or imply \(\cos^4\theta = \frac{1}{2}\) | B1 | |
| Obtain \(0.572\) | B1 | |
| Obtain \(-0.572\) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Integrate and obtain form \(k_1\theta + k_2\sin 4\theta + k_3\sin 2\theta\) | M1 | |
| Obtain \(\frac{3}{8}\theta + \frac{1}{32}\sin 4\theta + \frac{1}{4}\sin 2\theta\) | A1 | |
| Obtain \(\frac{3}{32}\pi + \frac{1}{4}\) following completely correct work | A1 |
# Question 9:
## Part (i)
| Answer/Working | Mark | Guidance |
|---|---|---|
| Express $\cos 4\theta$ as $2\cos^2 2\theta - 1$ or $\cos^2 2\theta - \sin^2 2\theta$ or $1 - 2\sin^2 2\theta$ | B1 | |
| Express $\cos 4\theta$ in terms of $\cos\theta$ | M1 | |
| Obtain $8\cos^4\theta - 8\cos^2\theta + 1$ | A1 | |
| Use $\cos 2\theta = 2\cos^2\theta - 1$ to obtain given answer $8\cos^4\theta - 3$ | A1 | AG |
**Total: [4]**
## Part (ii)(a)
| Answer/Working | Mark | Guidance |
|---|---|---|
| State or imply $\cos^4\theta = \frac{1}{2}$ | B1 | |
| Obtain $0.572$ | B1 | |
| Obtain $-0.572$ | B1 | |
**Total: [3]**
## Part (ii)(b)
| Answer/Working | Mark | Guidance |
|---|---|---|
| Integrate and obtain form $k_1\theta + k_2\sin 4\theta + k_3\sin 2\theta$ | M1 | |
| Obtain $\frac{3}{8}\theta + \frac{1}{32}\sin 4\theta + \frac{1}{4}\sin 2\theta$ | A1 | |
| Obtain $\frac{3}{32}\pi + \frac{1}{4}$ following completely correct work | A1 | |
**Total: [3]**
---9 (i) Prove the identity $\cos 4 \theta + 4 \cos 2 \theta \equiv 8 \cos ^ { 4 } \theta - 3$.\\
(ii) Hence
\begin{enumerate}[label=(\alph*)]
\item solve the equation $\cos 4 \theta + 4 \cos 2 \theta = 1$ for $- \frac { 1 } { 2 } \pi \leqslant \theta \leqslant \frac { 1 } { 2 } \pi$,
\item find the exact value of $\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \cos ^ { 4 } \theta \mathrm {~d} \theta$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2011 Q9 [10]}}