| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2023 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Reciprocal Trig & Identities |
| Type | Double angle with reciprocal functions |
| Difficulty | Standard +0.3 This is a structured multi-part question requiring standard manipulations of reciprocal and double angle identities, followed by routine equation solving and integration. Part (a) involves algebraic manipulation using sin 2θ = 2sin θ cos θ and simplifying with cosec θ = 1/sin θ. Part (b) substitutes the proven identity to solve a quadratic in cos θ. Part (c) integrates the simplified form directly. While it requires multiple techniques, each step follows standard procedures without requiring novel insight, making it slightly easier than average. |
| Spec | 1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05k Further identities: sec^2=1+tan^2 and cosec^2=1+cot^21.05l Double angle formulae: and compound angle formulae1.05o Trigonometric equations: solve in given intervals1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use \(\cosec\theta = \dfrac{1}{\sin\theta}\) | B1 | |
| Express in terms of \(\sin\theta\) and \(\cos\theta\) only | M1 | Dependent on B1 |
| Obtain given result \(4+6\cos\theta-4\cos^2\theta\) with sufficient detail | A1 | AG |
| Total: 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Attempt use of formula to solve 3-term quadratic equation as far as \(\cos\theta=k_1\) | M1 | where \(-1 < k_1 < 1\) |
| Solve \(4\cos^2\theta - 6\cos\theta - 7 = 0\) to obtain at least \(\cos\theta = -0.770...\) | A1 | or exact equivalent \(\cos\theta = \dfrac{6-\sqrt{148}}{8}\) |
| Obtain \(-2.45\) | A1 | or greater accuracy; and no others between \(-\pi\) and \(0\) |
| Total: 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Express \(\cos^2\theta\) term in the form \(k_2+k_3\cos 2\theta\) | M1 | where \(k_2 k_3 \neq 0\) |
| Obtain integrand \(6\cos\theta + 2 - 2\cos 2\theta\) | A1 | Following the 3-term expression in \(\cos\theta\) from part (a) |
| Integrate to obtain correct \(6\sin\theta + 2\theta - \sin 2\theta\) | A1 | Condone absence of \(+c\), but all in terms of \(\theta\) |
| Total: 3 |
## Question 6(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use $\cosec\theta = \dfrac{1}{\sin\theta}$ | B1 | |
| Express in terms of $\sin\theta$ and $\cos\theta$ only | M1 | Dependent on B1 |
| Obtain given result $4+6\cos\theta-4\cos^2\theta$ with sufficient detail | A1 | AG |
| **Total: 3** | | |
## Question 6(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Attempt use of formula to solve 3-term quadratic equation as far as $\cos\theta=k_1$ | M1 | where $-1 < k_1 < 1$ |
| Solve $4\cos^2\theta - 6\cos\theta - 7 = 0$ to obtain at least $\cos\theta = -0.770...$ | A1 | or exact equivalent $\cos\theta = \dfrac{6-\sqrt{148}}{8}$ |
| Obtain $-2.45$ | A1 | or greater accuracy; and no others between $-\pi$ and $0$ |
| **Total: 3** | | |
## Question 6(c):
| Answer | Mark | Guidance |
|--------|------|----------|
| Express $\cos^2\theta$ term in the form $k_2+k_3\cos 2\theta$ | M1 | where $k_2 k_3 \neq 0$ |
| Obtain integrand $6\cos\theta + 2 - 2\cos 2\theta$ | A1 | Following the 3-term expression in $\cos\theta$ from part (a) |
| Integrate to obtain correct $6\sin\theta + 2\theta - \sin 2\theta$ | A1 | Condone absence of $+c$, but all in terms of $\theta$ |
| **Total: 3** | | |6
\begin{enumerate}[label=(\alph*)]
\item Show that $\operatorname { cosec } \theta \left( 3 \sin 2 \theta + 4 \sin ^ { 3 } \theta \right) \equiv 4 + 6 \cos \theta - 4 \cos ^ { 2 } \theta$.
\item Solve the equation
$$\operatorname { cosec } \theta \left( 3 \sin 2 \theta + 4 \sin ^ { 3 } \theta \right) + 3 = 0$$
for $- \pi < \theta < 0$.
\item Find $\int \operatorname { cosec } \theta \left( 3 \sin 2 \theta + 4 \sin ^ { 3 } \theta \right) \mathrm { d } \theta$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2023 Q6 [9]}}