| Exam Board | Edexcel |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2024 |
| Session | January |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Reciprocal Trig & Identities |
| Type | Multiple angle equations |
| Difficulty | Standard +0.3 This is a straightforward multi-part question testing standard techniques. Part (i) requires converting sin x tan x to sin²x/cos x and solving a quadratic in cos x—routine manipulation. Part (ii) involves reading amplitude from a graph, finding where a transformed sine curve equals 2 (its midline), and locating a maximum—all standard textbook exercises requiring recall of transformations rather than problem-solving insight. |
| Spec | 1.05f Trigonometric function graphs: symmetries and periodicities1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| States or uses \(\tan x = \dfrac{\sin x}{\cos x}\) | B1 | May be seen as \(\sin x\tan x \Rightarrow \tan^2 x\cos x\) |
| \(\sin x\tan x=5 \Rightarrow \sin^2 x=5\cos x \Rightarrow 1-\cos^2 x=5\cos x\) | M1A1 | Uses \(\pm\sin^2 x\pm\cos^2 x=\pm1\) to set up quadratic in \(\cos x\) only |
| \(\cos^2 x+5\cos x-1=0 \Rightarrow \cos x=\dfrac{-5\pm\sqrt{29}}{2} \Rightarrow x=\) awrt \(78.9°, 281.1°\) | M1dM1A1 | Must be in degrees; no other values in range |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(A=5\) | B1 | Check by the question |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(2\theta-\dfrac{3\pi}{8}=\dfrac{3\pi}{2} \Rightarrow \theta=\ldots\) | M1 | Must use correct order of operations; mixture of radians/degrees without recovery: M0 |
| \(\theta=\dfrac{15\pi}{16}\) | A1 | Must be exact |
| \(y\) coordinate \(Q=-3\) (or \(2-\)"\(A\)") | B1ft | May be seen on diagram |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Sets \(0=\text{"5"}\sin\!\left(2\theta-\dfrac{3\pi}{8}\right)+2 \Rightarrow \sin\!\left(2\theta-\dfrac{3\pi}{8}\right)=\pm\dfrac{2}{\text{"5"}}\) | M1 | May be implied; solutions with no working score 0 |
| \(\sin\!\left(2\theta-\dfrac{3\pi}{8}\right)=\pm\dfrac{2}{5} \Rightarrow 2\theta-\dfrac{3\pi}{8}=\arcsin\!\left(\pm\dfrac{2}{5}\right)=\ldots\) | dM1 | Allow use of \(X\) for \(2\theta\) or \(2\theta-\dfrac{3\pi}{8}\) |
| One of \(\theta=0.38, 2.4, 3.5, 5.5, 6.7, 8.6, 9.8\ldots\) | A1 | |
| \(\theta=\) awrt \(5.51\) | A1 |
| Answer | Marks |
|---|---|
| dM1 | Proceeds from \(\sin\left(2\theta - \frac{3\pi}{8}\right) = \pm\frac{2}{5}\) to \(2\theta - \frac{3\pi}{8} = \arcsin\left(\pm\frac{2}{5}\right) = \ldots\) which is one of the values below: |
| Answer | Marks | Guidance |
|---|---|---|
| A1 | Any one of the values in the table provided M1dM1 has been scored. Do not withhold this mark if other incorrect angles are seen. | |
| \(\theta\) | Radians (awrt): \(0.38, 2.4, 3.5, 5.5, 6.7, 8.6, 9.8, 11.8, 12.9\) | Degrees (awrt): \(22, 136, 202, 316, 382, 496, 562, 676, 742\) |
| A1 | \(\theta = \) awrt \(5.51\) only (must be in radians). Can only be scored from correct working and all previous marks are scored. |
# Question 9:
## Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| States or uses $\tan x = \dfrac{\sin x}{\cos x}$ | B1 | May be seen as $\sin x\tan x \Rightarrow \tan^2 x\cos x$ |
| $\sin x\tan x=5 \Rightarrow \sin^2 x=5\cos x \Rightarrow 1-\cos^2 x=5\cos x$ | M1A1 | Uses $\pm\sin^2 x\pm\cos^2 x=\pm1$ to set up quadratic in $\cos x$ only |
| $\cos^2 x+5\cos x-1=0 \Rightarrow \cos x=\dfrac{-5\pm\sqrt{29}}{2} \Rightarrow x=$ awrt $78.9°, 281.1°$ | M1dM1A1 | Must be in degrees; no other values in range |
## Part (ii)(a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $A=5$ | B1 | Check by the question |
## Part (ii)(b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $2\theta-\dfrac{3\pi}{8}=\dfrac{3\pi}{2} \Rightarrow \theta=\ldots$ | M1 | Must use correct order of operations; mixture of radians/degrees without recovery: M0 |
| $\theta=\dfrac{15\pi}{16}$ | A1 | Must be exact |
| $y$ coordinate $Q=-3$ (or $2-$"$A$") | B1ft | May be seen on diagram |
## Part (ii)(c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Sets $0=\text{"5"}\sin\!\left(2\theta-\dfrac{3\pi}{8}\right)+2 \Rightarrow \sin\!\left(2\theta-\dfrac{3\pi}{8}\right)=\pm\dfrac{2}{\text{"5"}}$ | M1 | May be implied; solutions with no working score 0 |
| $\sin\!\left(2\theta-\dfrac{3\pi}{8}\right)=\pm\dfrac{2}{5} \Rightarrow 2\theta-\dfrac{3\pi}{8}=\arcsin\!\left(\pm\dfrac{2}{5}\right)=\ldots$ | dM1 | Allow use of $X$ for $2\theta$ or $2\theta-\dfrac{3\pi}{8}$ |
| One of $\theta=0.38, 2.4, 3.5, 5.5, 6.7, 8.6, 9.8\ldots$ | A1 | |
| $\theta=$ awrt $5.51$ | A1 | |
## Question 9 (Trigonometry):
**dM1** | Proceeds from $\sin\left(2\theta - \frac{3\pi}{8}\right) = \pm\frac{2}{5}$ to $2\theta - \frac{3\pi}{8} = \arcsin\left(\pm\frac{2}{5}\right) = \ldots$ which is one of the values below:
$2\theta - \frac{3\pi}{8} = \arcsin\left(-\frac{2}{5}\right) = -0.41, 3.6, 5.9, 9.8, 12.2$
$2\theta - \frac{3\pi}{8} = \arcsin\left(\frac{2}{5}\right) = 0.41, 2.7, 6.7, 9.0, 13.0, 15.3$
May be implied by $2\theta = \arcsin\left(\pm\frac{2}{5}\right) + 1.17\ldots$ or allow the expression $= \dfrac{\arcsin\left(\pm\frac{2}{5}\right) + \frac{3\pi}{8}}{2}$
The sign slip is only condoned before they take arcsin(…)
**A1** | Any one of the values in the table provided M1dM1 has been scored. Do not withhold this mark if other incorrect angles are seen.
| $\theta$ | Radians (awrt): $0.38, 2.4, 3.5, 5.5, 6.7, 8.6, 9.8, 11.8, 12.9$ | Degrees (awrt): $22, 136, 202, 316, 382, 496, 562, 676, 742$ |
**A1** | $\theta = $ awrt $5.51$ only (must be in radians). **Can only be scored from correct working and all previous marks are scored.**
---\begin{enumerate}
\item In this question you must show detailed reasoning.
\end{enumerate}
Solutions relying entirely on calculator technology are not acceptable.\\
(i) Solve, for $0 \leqslant x < 360 ^ { \circ }$, the equation
$$\sin x \tan x = 5$$
giving your answers to one decimal place.\\
(ii)
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{0e08d931-aa1c-48a8-8b39-47096f981950-26_643_736_721_660}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\end{figure}
Figure 1 shows a sketch of part of the curve with equation
$$y = A \sin \left( 2 \theta - \frac { 3 \pi } { 8 } \right) + 2$$
where $A$ is a constant and $\theta$ is measured in radians.\\
The points $P , Q$ and $R$ lie on the curve and are shown in Figure 1.\\
Given that the $y$ coordinate of $P$ is 7\\
(a) state the value of $A$,\\
(b) find the exact coordinates of $Q$,\\
(c) find the value of $\theta$ at $R$, giving your answer to 3 significant figures.
\begin{center}
\end{center}
\hfill \mbox{\textit{Edexcel P2 2024 Q9 [14]}}