| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2024 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Quadratic trigonometric equations |
| Type | Show then solve: compound angle substitution |
| Difficulty | Standard +0.3 This is a standard trigonometric equation requiring conversion to quadratic form using tan θ = sin θ/cos θ and sin²θ + cos²θ = 1, followed by routine application to a double angle case. The algebraic manipulation is straightforward and the technique is commonly practiced in P1, making it slightly easier than average. |
| Spec | 1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Use identity \(\tan\theta = \frac{\sin\theta}{\cos\theta}\) | M1 | |
| Use identity \(\cos^2\theta = 1 - \sin^2\theta\) | M1 | |
| \(\pm(5\sin^2\theta + 7\sin\theta - 6 = 0)\) | A1 | |
| 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Attempt solution of their 3 term equation and correct process to find at least 1 value of \(\sin x\) or \(\sin 2x\) or \(\sin\theta\) | M1 | Expect \((5s-3)(s+2) = 0\), \(s = 3/5\) |
| \(x = 18.4\) | A1 | Or greater accuracy. B1 SC if no solution to the quadratic. |
| \(x = 71.6\) or \((90 - \text{their } 18.4)\) or greater accuracy; and no other solutions for \(0° < x < 180°\) | A1FT | WWW. B1 SC FT if no solution to the quadratic. B1 SC both correct in radians, 0.322, 1.25. |
| 3 |
## Question 4(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Use identity $\tan\theta = \frac{\sin\theta}{\cos\theta}$ | M1 | |
| Use identity $\cos^2\theta = 1 - \sin^2\theta$ | M1 | |
| $\pm(5\sin^2\theta + 7\sin\theta - 6 = 0)$ | A1 | |
| | **3** | |
## Question 4(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Attempt solution of their 3 term equation and correct process to find at least 1 value of $\sin x$ or $\sin 2x$ or $\sin\theta$ | M1 | Expect $(5s-3)(s+2) = 0$, $s = 3/5$ |
| $x = 18.4$ | A1 | Or greater accuracy. **B1 SC** if no solution to the quadratic. |
| $x = 71.6$ or $(90 - \text{their } 18.4)$ or greater accuracy; and no other solutions for $0° < x < 180°$ | A1FT | WWW. **B1 SC** FT if no solution to the quadratic. **B1 SC** both correct in radians, 0.322, 1.25. |
| | **3** | |4
\begin{enumerate}[label=(\alph*)]
\item Show that the equation $\cos \theta ( 7 \tan \theta - 5 \cos \theta ) = 1$ can be written in the form $a \sin ^ { 2 } \theta + b \sin \theta + c = 0$, where $a , b$ and $c$ are integers to be found.
\item Hence solve the equation $\cos 2 x ( 7 \tan 2 x - 5 \cos 2 x ) = 1$ for $0 ^ { \circ } < x < 180 ^ { \circ }$.\\
\includegraphics[max width=\textwidth, alt={}, center]{d6976a4b-aecf-43f1-a3f2-bcad37d03585-06_2718_35_141_2012}\\
\includegraphics[max width=\textwidth, alt={}, center]{d6976a4b-aecf-43f1-a3f2-bcad37d03585-07_2714_33_144_22}
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2024 Q4 [6]}}