| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2019 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Addition & Double Angle Formulae |
| Type | Solve equation with sec/cosec/cot |
| Difficulty | Moderate -0.3 Part (a) requires converting sec and cosec to sin/cos, then using the double angle formula sin(2α)=2sinαcosα - a straightforward manipulation. Part (b) uses the standard addition formula for sin(A+B)+sin(A-B) which simplifies directly to 2sinAcosB. Both parts are routine applications of standard formulae with minimal problem-solving, though they require careful algebraic manipulation and are slightly above pure recall. |
| Spec | 1.05h Reciprocal trig functions: sec, cosec, cot definitions and graphs1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05l Double angle formulae: and compound angle formulae1.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Express equation as \(\dfrac{1}{\cos\alpha \sin\alpha} = 7\) | B1 | OE; May be implied by subsequent work |
| Attempt use of identity for \(\sin 2\alpha\) or attempt to obtain a quadratic equation in terms of any one of: \(\sin^2\alpha,\ \cos^2\alpha,\ \cot^2\alpha\) or \(\tan^2\alpha\) | M1 | From equation of form \(\sin 2\alpha = k\) where \(0 < k < 1\) or from use of correct identities |
| Obtain \(\sin 2\alpha = \frac{2}{7}\) or a correct 3 term quadratic equation, equated to zero in any one of: \(\sin^2\alpha,\ \cos^2\alpha,\ \cot^2\alpha\) or \(\tan^2\alpha\) | A1 | |
| Attempt correct process to find at least one correct value of \(\alpha\) | M1 | |
| Obtain \(8.3\) and \(81.7\) and no others between \(0\) and \(90\) | A1 | |
| 5 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Simplify left-hand side to obtain \(2\sin\beta\cos 20°\) | B1 | |
| Attempt to form equation where \(\tan\beta\) is only variable, \(\tan\beta \neq 3\) | M1 | |
| Obtain \(\tan\beta = \dfrac{3}{\cos 20°}\) | A1 | OE |
| Obtain \(\beta = 72.6\) and no others between \(0\) and \(90\) | A1 | |
| 5 |
## Question 6(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Express equation as $\dfrac{1}{\cos\alpha \sin\alpha} = 7$ | **B1** | OE; May be implied by subsequent work |
| Attempt use of identity for $\sin 2\alpha$ or attempt to obtain a quadratic equation in terms of any one of: $\sin^2\alpha,\ \cos^2\alpha,\ \cot^2\alpha$ or $\tan^2\alpha$ | **M1** | From equation of form $\sin 2\alpha = k$ where $0 < k < 1$ or from use of correct identities |
| Obtain $\sin 2\alpha = \frac{2}{7}$ or a correct 3 term quadratic equation, equated to zero in any one of: $\sin^2\alpha,\ \cos^2\alpha,\ \cot^2\alpha$ or $\tan^2\alpha$ | **A1** | |
| Attempt correct process to find at least one correct value of $\alpha$ | **M1** | |
| Obtain $8.3$ and $81.7$ and no others between $0$ and $90$ | **A1** | |
| | **5** | |
---
## Question 6(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Simplify left-hand side to obtain $2\sin\beta\cos 20°$ | **B1** | |
| Attempt to form equation where $\tan\beta$ is only variable, $\tan\beta \neq 3$ | **M1** | |
| Obtain $\tan\beta = \dfrac{3}{\cos 20°}$ | **A1** | OE |
| Obtain $\beta = 72.6$ and no others between $0$ and $90$ | **A1** | |
| | **5** | |
---6
\begin{enumerate}[label=(\alph*)]
\item Showing all necessary working, solve the equation
$$\sec \alpha \operatorname { cosec } \alpha = 7$$
for $0 ^ { \circ } < \alpha < 90 ^ { \circ }$.
\item Showing all necessary working, solve the equation
$$\sin \left( \beta + 20 ^ { \circ } \right) + \sin \left( \beta - 20 ^ { \circ } \right) = 6 \cos \beta$$
for $0 ^ { \circ } < \beta < 90 ^ { \circ }$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2019 Q6 [9]}}