| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2024 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Reciprocal Trig & Identities |
| Type | Double angle with reciprocal functions |
| Difficulty | Standard +0.3 This is a multi-part question testing standard A-level techniques: (a) is a routine identity proof using double angle formulas, (b) requires substituting the proven identity and solving a quadratic in sin θ, and (c) uses double angle formulas to simplify before integrating. All parts follow predictable patterns with no novel insight required, making it slightly easier than average. |
| Spec | 1.05h Reciprocal trig functions: sec, cosec, cot definitions and graphs1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05o Trigonometric equations: solve in given intervals1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Express left-hand side in terms of \(\sin\theta\) and \(\cos\theta\) using \(\cosec 2\theta = \dfrac{1}{\sin 2\theta}\) | M1 | |
| Obtain \(\dfrac{1}{\cos\theta}\) and confirm \(\sec\theta\) | A1 | Answer given – necessary detail needed. |
| Total: 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Attempt to obtain quadratic equation in \(\sec\theta\) or \(\cos\theta\) only | *M1 | |
| Obtain \(\sec^2\theta - 1 + \frac{7}{2}\sec\theta = 8\) involving one trigonometric ratio | A1 | Or equivalent, may be unsimplified, but reduce to \(2\sec^2\theta + 7\sec\theta - 18 = 0\) or \(18\cos^2\theta - 7\cos\theta - 2 = 0\) |
| Attempt to solve 3-term quadratic equation for \(\sec\theta\), using a correct method, to find at least one value of \(\theta\) | *DM1 | Or equivalent using \(\cos\theta\) |
| Obtain any two of the four correct solutions \(\pm 0.952,\ \pm 1.76\) | A1 | Or greater accuracy. |
| Obtain remaining two correct solutions | A1 | Or greater accuracy; and no others between \(-\pi\) and \(\pi\) |
| Total: 5 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Identify integrand as \(2\sec^2\dfrac{1}{2}x\) | B1 | |
| Integrate \(k\sec^2\dfrac{1}{2}x\) to obtain \(2k\tan\dfrac{1}{2}x\) | M1 | |
| Obtain correct \(4\tan\dfrac{1}{2}x\) | A1 | Condone omission of \(\ldots + c\) |
| Total: 3 |
## Question 7(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Express left-hand side in terms of $\sin\theta$ and $\cos\theta$ using $\cosec 2\theta = \dfrac{1}{\sin 2\theta}$ | M1 | |
| Obtain $\dfrac{1}{\cos\theta}$ and confirm $\sec\theta$ | A1 | Answer given – necessary detail needed. |
| **Total: 2** | | |
---
## Question 7(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Attempt to obtain quadratic equation in $\sec\theta$ or $\cos\theta$ only | *M1 | |
| Obtain $\sec^2\theta - 1 + \frac{7}{2}\sec\theta = 8$ involving one trigonometric ratio | A1 | Or equivalent, may be unsimplified, but reduce to $2\sec^2\theta + 7\sec\theta - 18 = 0$ or $18\cos^2\theta - 7\cos\theta - 2 = 0$ |
| Attempt to solve 3-term quadratic equation for $\sec\theta$, using a correct method, to find at least one value of $\theta$ | *DM1 | Or equivalent using $\cos\theta$ |
| Obtain any two of the four correct solutions $\pm 0.952,\ \pm 1.76$ | A1 | Or greater accuracy. |
| Obtain remaining two correct solutions | A1 | Or greater accuracy; and no others between $-\pi$ and $\pi$ |
| **Total: 5** | | |
---
## Question 7(c):
| Answer | Mark | Guidance |
|--------|------|----------|
| Identify integrand as $2\sec^2\dfrac{1}{2}x$ | B1 | |
| Integrate $k\sec^2\dfrac{1}{2}x$ to obtain $2k\tan\dfrac{1}{2}x$ | M1 | |
| Obtain correct $4\tan\dfrac{1}{2}x$ | A1 | Condone omission of $\ldots + c$ |
| **Total: 3** | | |7
\begin{enumerate}[label=(\alph*)]
\item Prove that $2 \sin \theta \operatorname { cosec } 2 \theta \equiv \sec \theta$.
\item Solve the equation $\tan ^ { 2 } \theta + 7 \sin \theta \operatorname { cosec } 2 \theta = 8$ for $- \pi < \theta < \pi$.\\
\includegraphics[max width=\textwidth, alt={}, center]{76df3465-9617-4f2b-a8b7-f474b2817504-12_2725_37_136_2010}
\item Find $\int 8 \sin ^ { 2 } \frac { 1 } { 2 } x \operatorname { cosec } ^ { 2 } x \mathrm {~d} x$.\\
If you use the following page to complete the answer to any question, the question number must be clearly shown.\\
\includegraphics[max width=\textwidth, alt={}, center]{76df3465-9617-4f2b-a8b7-f474b2817504-14_2715_35_143_2012}
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2024 Q7 [10]}}