| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2021 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Reciprocal Trig & Identities |
| Type | Convert equation to quadratic form |
| Difficulty | Standard +0.3 This is a standard A-level question requiring routine application of double angle formulas and reciprocal trig identities to convert to a quadratic in tan θ, then solve. The algebraic manipulation is straightforward with clear steps, making it slightly easier than average but not trivial due to the multi-step process and need for careful algebraic handling. |
| Spec | 1.05l Double angle formulae: and compound angle formulae1.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use correct trig formulae and express equation in terms of \(\tan\theta\) | M1 | |
| Obtain a correct equation in \(\tan\theta\) in any form | A1 | e.g. \(\frac{1-\tan^2\theta}{2\tan\theta} + \frac{1}{\tan\theta} = 2\) |
| Reduce to \(\tan^2\theta + 4\tan\theta - 3 = 0\), or 3-term equivalent | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Solve a 3-term quadratic for \(\tan\theta\) and calculate \(\theta\) | M1 | \(\left(\tan\theta = -2 \pm \sqrt{7}\right)\) |
| Obtain answer, e.g. \(0.573\) | A1 | Must be 3 d.p. |
| Obtain second answer, e.g. \(1.783\) and no other | A1 | Ignore answers outside the given interval. Treat answers in degrees as a misread. \((32.9°, 102.1°)\) |
## Question 5(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use correct trig formulae and express equation in terms of $\tan\theta$ | M1 | |
| Obtain a correct equation in $\tan\theta$ in any form | A1 | e.g. $\frac{1-\tan^2\theta}{2\tan\theta} + \frac{1}{\tan\theta} = 2$ |
| Reduce to $\tan^2\theta + 4\tan\theta - 3 = 0$, or 3-term equivalent | A1 | |
## Question 5(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Solve a 3-term quadratic for $\tan\theta$ and calculate $\theta$ | M1 | $\left(\tan\theta = -2 \pm \sqrt{7}\right)$ |
| Obtain answer, e.g. $0.573$ | A1 | Must be 3 d.p. |
| Obtain second answer, e.g. $1.783$ and no other | A1 | Ignore answers outside the given interval. Treat answers in degrees as a misread. $(32.9°, 102.1°)$ |5
\begin{enumerate}[label=(\alph*)]
\item Show that the equation
$$\cot 2 \theta + \cot \theta = 2$$
can be expressed as a quadratic equation in $\tan \theta$.
\item Hence solve the equation $\cot 2 \theta + \cot \theta = 2$, for $0 < \theta < \pi$, giving your answers correct to 3 decimal places.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2021 Q5 [6]}}