| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2024 |
| Session | March |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Standard trigonometric equations |
| Type | Prove identity then solve |
| Difficulty | Moderate -0.3 Part (a) is a straightforward algebraic manipulation of a trigonometric identity requiring expansion of brackets and use of sin²θ + cos²θ = 1. Part (b) uses the result to solve a cubic equation in tan θ, which factors easily to give tan θ = 0 or tan²θ = 2/5. This is a standard two-part question slightly easier than average due to the routine algebraic techniques and straightforward solving process. |
| Spec | 1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05p Proof involving trig: functions and identities |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Expand bracket to obtain 3 terms and use correct identity | M1 | \(\theta\) may be missing or another symbol used. |
| Use identity \(\frac{\sin\theta}{\cos\theta} = \tan\theta\) | M1 | Does not require any further explanation. \(\theta\) may be missing or another symbol used. |
| Conclude with \(2\tan\theta\) | A1 | WWW, AG |
| Total: 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Attempt solution of \(5\tan^3\theta = 2\tan\theta\) to obtain at least one value of \(\tan\theta\) | M1 | SOI. Can be awarded if \(\tan\theta\) is cancelled and ignored. |
| Obtain at least two of \(0, \pm 32.3\) | A1 | Or greater accuracy. SC B1 if no method shown. |
| Obtain all three values | A1 | Or greater accuracy; and no others in \(-90°< \theta < 90°\) range. Other units SC B1 only for all 3 angles. SC B1 if no method shown. |
| Total: 3 |
## Question 4(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Expand bracket to obtain 3 terms and use correct identity | M1 | $\theta$ may be missing or another symbol used. |
| Use identity $\frac{\sin\theta}{\cos\theta} = \tan\theta$ | M1 | Does not require any further explanation. $\theta$ may be missing or another symbol used. |
| Conclude with $2\tan\theta$ | A1 | WWW, AG |
| **Total: 3** | | |
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## Question 4(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Attempt solution of $5\tan^3\theta = 2\tan\theta$ to obtain at least one value of $\tan\theta$ | M1 | SOI. Can be awarded if $\tan\theta$ is cancelled and ignored. |
| Obtain at least two of $0, \pm 32.3$ | A1 | Or greater accuracy. **SC B1** if no method shown. |
| Obtain all three values | A1 | Or greater accuracy; and no others in $-90°< \theta < 90°$ range. Other units **SC B1** only for all 3 angles. **SC B1** if no method shown. |
| **Total: 3** | | |
---4
\begin{enumerate}[label=(\alph*)]
\item Prove that $\frac { ( \sin \theta + \cos \theta ) ^ { 2 } - 1 } { \cos ^ { 2 } \theta } \equiv 2 \tan \theta$.\\
\includegraphics[max width=\textwidth, alt={}]{b5eb378d-a9cb-40e0-9203-374b58f1dcf9-05_63_1569_333_328} ..............................................................................................................................................\\
\includegraphics[max width=\textwidth, alt={}, center]{b5eb378d-a9cb-40e0-9203-374b58f1dcf9-05_65_1570_511_324}\\
\includegraphics[max width=\textwidth, alt={}, center]{b5eb378d-a9cb-40e0-9203-374b58f1dcf9-05_62_1570_603_324}\\
\includegraphics[max width=\textwidth, alt={}, center]{b5eb378d-a9cb-40e0-9203-374b58f1dcf9-05_72_1570_685_324}\\
\includegraphics[max width=\textwidth, alt={}]{b5eb378d-a9cb-40e0-9203-374b58f1dcf9-05_72_1570_776_324} ...................................................................................................................................... .\\
\includegraphics[max width=\textwidth, alt={}]{b5eb378d-a9cb-40e0-9203-374b58f1dcf9-05_76_1572_952_322} ........................................................................................................................................\\
\includegraphics[max width=\textwidth, alt={}, center]{b5eb378d-a9cb-40e0-9203-374b58f1dcf9-05_72_1570_1137_324}\\
\includegraphics[max width=\textwidth, alt={}, center]{b5eb378d-a9cb-40e0-9203-374b58f1dcf9-05_74_1572_1226_322}\\
\includegraphics[max width=\textwidth, alt={}, center]{b5eb378d-a9cb-40e0-9203-374b58f1dcf9-05_77_1575_1315_319}
\item Hence solve the equation $\frac { ( \sin \theta + \cos \theta ) ^ { 2 } - 1 } { \cos ^ { 2 } \theta } = 5 \tan ^ { 3 } \theta$ for $- 90 ^ { \circ } < \theta < 90 ^ { \circ }$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2024 Q4 [6]}}