| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2023 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Reciprocal Trig & Identities |
| Type | Double angle with reciprocal functions |
| Difficulty | Standard +0.8 This question requires multiple trigonometric identities (reciprocal, Pythagorean, double angle) and algebraic manipulation to reach the given form, then solving a quartic in sin θ that factors as a quadratic. While systematic, it demands careful handling of multiple transformations and consideration of domain restrictions for cot θ, making it moderately challenging but still within standard Further Maths scope. |
| Spec | 1.05j 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 |
| Use correct Pythagoras \(\cot^2\theta = \cosec^2\theta - 1\) or \(\cot^2\theta = 1/\sin^2\theta - 1\) or \(\cot^2\theta = \cos^2\theta/\sin^2\theta\) and then \(\cos^2\theta = 1-\sin^2\theta\), together with double angle formula \(\cos 2\theta = 1-2\sin^2\theta\), to obtain an equation in \(\sin\theta\) or \(\sin\theta\) and \(\cosec^2\theta\) | M1 | If consistent omission of brackets, e.g. \((\sin\theta)^2\) written as \(\sin\theta^2\) then SC B1 in place of M1A1. |
| Obtain a correct equation in \(\sin\theta\) in any form | A1 | e.g. \(1/\sin^2\theta - 1 + 2(1-2\sin^2\theta) = 4\) or \(\frac{1-\sin^2}{\sin^2}+2(1-2\sin^2)=4\). If \(\frac{\cos^2}{\sin^2}+2(1-2\sin^2)=4\) then e.g. \(1-\sin^2+2(1-2\sin^2)\sin^2=4\). (missing \(\sin^2\) on right) allow M1A1A0. |
| Reduce to the given answer of \(4\sin^4\theta + 3\sin^2\theta - 1 = 0\) correctly | A1 | AG Must follow from a horizontal equation (no denominators). If \(s=\sin\theta\) used and defined, allow all marks. If not defined, award M1A1A0. |
| Total | 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Solve the given quadratic to obtain a value for \(\theta\) | M1 | \((4\sin^2\theta-1)(\sin^2\theta+1)=0\) and solve for \(\theta\). Incorrect sign in solution of quadratic seen, e.g. \((4\sin^2\theta-1)(\sin^2\theta-1)=0\) then M0 A0 A0 but if only see \((4\sin^2\theta-1)=0\) and nothing incorrect seen allow 3/3. |
| Obtain answer, e.g. \(\theta = 30°\) | A1 | \(\pi/6\) award A0. |
| Obtain three further answers, e.g. \(\theta = 150°\), \(210°\) and \(330°\) and no others in the interval | A1 | Ignore any answers outside interval. \(5\pi/6\ \ 7\pi/6\ \ 11\pi/6\) award A1. |
| Total | 3 |
## Question 6(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Use correct Pythagoras $\cot^2\theta = \cosec^2\theta - 1$ or $\cot^2\theta = 1/\sin^2\theta - 1$ or $\cot^2\theta = \cos^2\theta/\sin^2\theta$ and then $\cos^2\theta = 1-\sin^2\theta$, together with double angle formula $\cos 2\theta = 1-2\sin^2\theta$, to obtain an equation in $\sin\theta$ or $\sin\theta$ and $\cosec^2\theta$ | M1 | If consistent omission of brackets, e.g. $(\sin\theta)^2$ written as $\sin\theta^2$ then SC B1 in place of M1A1. |
| Obtain a correct equation in $\sin\theta$ in any form | A1 | e.g. $1/\sin^2\theta - 1 + 2(1-2\sin^2\theta) = 4$ or $\frac{1-\sin^2}{\sin^2}+2(1-2\sin^2)=4$. If $\frac{\cos^2}{\sin^2}+2(1-2\sin^2)=4$ then e.g. $1-\sin^2+2(1-2\sin^2)\sin^2=4$. (missing $\sin^2$ on right) allow M1A1A0. |
| Reduce to the given answer of $4\sin^4\theta + 3\sin^2\theta - 1 = 0$ correctly | A1 | AG Must follow from a horizontal equation (no denominators). If $s=\sin\theta$ used and defined, allow all marks. If not defined, award M1A1A0. |
| **Total** | **3** | |
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## Question 6(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Solve the given quadratic to obtain a value for $\theta$ | M1 | $(4\sin^2\theta-1)(\sin^2\theta+1)=0$ and solve for $\theta$. Incorrect sign in solution of quadratic seen, e.g. $(4\sin^2\theta-1)(\sin^2\theta-1)=0$ then M0 A0 A0 but if only see $(4\sin^2\theta-1)=0$ and nothing incorrect seen allow 3/3. |
| Obtain answer, e.g. $\theta = 30°$ | A1 | $\pi/6$ award A0. |
| Obtain three further answers, e.g. $\theta = 150°$, $210°$ and $330°$ and no others in the interval | A1 | Ignore any answers outside interval. $5\pi/6\ \ 7\pi/6\ \ 11\pi/6$ award A1. |
| **Total** | **3** | |
---6
\begin{enumerate}[label=(\alph*)]
\item Show that the equation $\cot ^ { 2 } \theta + 2 \cos 2 \theta = 4$ can be written in the form
$$4 \sin ^ { 4 } \theta + 3 \sin ^ { 2 } \theta - 1 = 0$$
\item Hence solve the equation $\cot ^ { 2 } \theta + 2 \cos 2 \theta = 4$, for $0 ^ { \circ } < \theta < 360 ^ { \circ }$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2023 Q6 [6]}}