| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2021 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Reciprocal Trig & Identities |
| Type | Double angle with reciprocal functions |
| Difficulty | Standard +0.3 Part (a) is a guided algebraic proof using the Pythagorean identity and double angle formula with clear scaffolding. Part (b) applies this result to solve a straightforward equation requiring basic rearrangement and inverse trig. This is slightly easier than average as the structure is heavily guided and uses 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 |
| Expand the square and equate to 1 | B1 | |
| Use correct double angle formula | M1 | Need to see \(\frac{4}{2}\) or \(\sin 2\theta = 2\sin\theta\cos\theta\) stated |
| Obtain \(\cos^4\theta + \sin^4\theta = 1 - \frac{1}{2}\sin^2 2\theta\) | A1 | Obtain the given result correctly |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Use the identity and carry out a method for finding a root | M1 | \(\left(1 - \frac{1}{2}\sin^2 2\theta = \frac{5}{9}\right)\) |
| Obtain answer \(35.3°\) | A1 | Must be correct if overspecified: 35.264... |
| Obtain a second answer, e.g. \(54.7°\) | A1 FT | [e.g \(90° - \textit{their } 35.3°\)] Do not FT if mixing degrees and radians |
| Obtain the remaining answers, e.g. \(144.7°\) and \(125.3°\) and no others in the given interval | A1 FT | [e.g. \(180° - ..\) and \(180° - ..\)] Ignore answers outside the given interval. Treat answers in radians as a misread. \((0.615, 0.955, 2.19, 2.53)\) Do not FT if mixing degrees and radians |
## Question 8(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Expand the square and equate to 1 | B1 | |
| Use correct double angle formula | M1 | Need to see $\frac{4}{2}$ or $\sin 2\theta = 2\sin\theta\cos\theta$ stated |
| Obtain $\cos^4\theta + \sin^4\theta = 1 - \frac{1}{2}\sin^2 2\theta$ | A1 | Obtain the **given result** correctly |
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## Question 8(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Use the identity and carry out a method for finding a root | M1 | $\left(1 - \frac{1}{2}\sin^2 2\theta = \frac{5}{9}\right)$ |
| Obtain answer $35.3°$ | A1 | Must be correct if overspecified: 35.264... |
| Obtain a second answer, e.g. $54.7°$ | A1 FT | [e.g $90° - \textit{their } 35.3°$] Do not FT if mixing degrees and radians |
| Obtain the remaining answers, e.g. $144.7°$ and $125.3°$ and no others in the given interval | A1 FT | [e.g. $180° - ..$ and $180° - ..$] Ignore answers outside the given interval. Treat answers in radians as a misread. $(0.615, 0.955, 2.19, 2.53)$ Do not FT if mixing degrees and radians |
---8
\begin{enumerate}[label=(\alph*)]
\item By first expanding $\left( \cos ^ { 2 } \theta + \sin ^ { 2 } \theta \right) ^ { 2 }$, show that
$$\cos ^ { 4 } \theta + \sin ^ { 4 } \theta \equiv 1 - \frac { 1 } { 2 } \sin ^ { 2 } 2 \theta .$$
\item Hence solve the equation
$$\cos ^ { 4 } \theta + \sin ^ { 4 } \theta = \frac { 5 } { 9 } ,$$
for $0 ^ { \circ } < \theta < 180 ^ { \circ }$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2021 Q8 [7]}}