| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2023 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Quadratic trigonometric equations |
| Type | Show then solve: tan/sin/cos identity manipulation |
| Difficulty | Moderate -0.3 This is a standard trigonometric equation requiring routine manipulation using tan θ = sin θ/cos θ and sin²θ + cos²θ = 1 to convert to quadratic form, followed by solving a straightforward quadratic equation. The algebraic steps are mechanical and well-practiced at A-level, making it slightly easier than average but not trivial due to the multi-step nature. |
| 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 |
| \(5\cos^2\theta - \sin^2\theta + \cos\theta = 0\) | M1 | Multiply by \(\cos\theta\) and replace \(\tan\theta\) by \(\frac{\sin\theta}{\cos\theta}\) |
| \(5\cos^2\theta - (1-\cos^2\theta) + \cos\theta = 0\) | M1 | |
| \(6\cos^2\theta + \cos\theta - 1 = 0\) | A1 | Missing '\(= 0\)' can be condoned if '\(= 0\)' appears earlier |
| 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \((3\cos\theta - 1)(2\cos\theta + 1) = 0\) | M1 | Must have 3 term quadratic, expect \(\cos\theta = \frac{1}{3}, -\frac{1}{2}\). Factors (OE) must be shown |
| \(\theta = \{1.23\}\); \(\{2.09\) or \(\frac{2\pi}{3}\}\); \(\{5.05\) and \(4.19\) (allow \(\frac{4\pi}{3})\}\) | A1 A1 FT | For A1 FT is for both \(2\pi - 1\)st solutions |
| 4 |
## Question 3(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $5\cos^2\theta - \sin^2\theta + \cos\theta = 0$ | M1 | Multiply by $\cos\theta$ and replace $\tan\theta$ by $\frac{\sin\theta}{\cos\theta}$ |
| $5\cos^2\theta - (1-\cos^2\theta) + \cos\theta = 0$ | M1 | |
| $6\cos^2\theta + \cos\theta - 1 = 0$ | A1 | Missing '$= 0$' can be condoned if '$= 0$' appears earlier |
| | **3** | |
## Question 3(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $(3\cos\theta - 1)(2\cos\theta + 1) = 0$ | M1 | Must have 3 term quadratic, expect $\cos\theta = \frac{1}{3}, -\frac{1}{2}$. Factors (OE) must be shown |
| $\theta = \{1.23\}$; $\{2.09$ or $\frac{2\pi}{3}\}$; $\{5.05$ and $4.19$ (allow $\frac{4\pi}{3})\}$ | A1 A1 FT | For A1 FT is for both $2\pi - 1$st solutions |
| | **4** | |3
\begin{enumerate}[label=(\alph*)]
\item Show that the equation
$$5 \cos \theta - \sin \theta \tan \theta + 1 = 0$$
may be expressed in the form $a \cos ^ { 2 } \theta + b \cos \theta + c = 0$, where $a$, $b$ and $c$ are constants to be found.
\item Hence solve the equation $5 \cos \theta - \sin \theta \tan \theta + 1 = 0$ for $0 < \theta < 2 \pi$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2023 Q3 [7]}}