| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2020 |
| Session | March |
| 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) requires applying triple angle formulas and algebraic manipulation to prove an identity—a standard technique for P3 level. Part (b) is a straightforward application using the proven identity to solve cot 2x = 2, requiring only basic inverse trig. This is a typical multi-part identity question with moderate algebraic complexity but no novel insight required. |
| Spec | 1.08i Integration by parts |
| Answer | Marks |
|---|---|
| Express LHS correctly as a single fraction | B1 |
| Use \(\cos(A\pm B)\) formula to simplify the numerator | M1 |
| Use \(\sin 2A\) formula to simplify the denominator | M1 |
| Obtain the given result | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Obtain an equation in \(\tan 2x\) and use correct method to solve for \(x\) | M1 | |
| Obtain answer e.g. \(0.232\) | A1 | |
| Obtain second answer e.g. \(1.80\) | A1 | Ignore answers outside the given interval |
## Question 5(a):
| Express LHS correctly as a single fraction | B1 | |
| Use $\cos(A\pm B)$ formula to simplify the numerator | M1 | |
| Use $\sin 2A$ formula to simplify the denominator | M1 | |
| Obtain the given result | A1 | |
## Question 5(b):
| Obtain an equation in $\tan 2x$ and use correct method to solve for $x$ | M1 | |
| Obtain answer e.g. $0.232$ | A1 | |
| Obtain second answer e.g. $1.80$ | A1 | Ignore answers outside the given interval |
---5
\begin{enumerate}[label=(\alph*)]
\item Show that $\frac { \cos 3 x } { \sin x } + \frac { \sin 3 x } { \cos x } = 2 \cot 2 x$.
\item Hence solve the equation $\frac { \cos 3 x } { \sin x } + \frac { \sin 3 x } { \cos x } = 4$, for $0 < x < \pi$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2020 Q5 [7]}}