| Exam Board | Edexcel |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2019 |
| Session | October |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Trigonometric equations in context |
| Type | Solve shifted trig equation |
| Difficulty | Standard +0.3 Part (i) is a routine trig equation requiring basic angle manipulation and calculator work. Part (ii)(a) uses the arithmetic sequence property (common difference) to derive an identity—straightforward algebra. Part (ii)(b) requires solving a quadratic in sin α and selecting the correct quadrant solution. All techniques are standard P2 material with no novel insight required, making this slightly easier than average. |
| Spec | 1.04h Arithmetic sequences: nth term and sum formulae1.05j 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 |
| \(3\sin(2\theta-10°)=1 \Rightarrow (2\theta-10°)=\arcsin\left(\frac{1}{3}\right)\) | M1 | Proceeding to \(x=\arcsin\left(\frac{1}{3}\right)\); implied by awrt 19.5° or 160.5°. Allow awrt 0.340 rad. |
| \(\theta = \frac{19.47+10}{2},\ \frac{160.53+10}{2}\) | dM1 | Correct order of operations leading to one answer for \(\theta\). Cannot score by adding 10 to angle in radians. |
| \(\theta =\) awrt \(14.7°, 85.3°\) | A1, A1 | First A1: one of the values. Second A1: both values and no others in range. |
| (4) | Note: Solutions based entirely on graphical or numerical methods score 0 marks. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Writes \(\frac{1}{\tan\alpha}-\sin\alpha = 2\sin\alpha - \frac{1}{\tan\alpha}\) | M1 | Uses terms of AP to set up correct equation. Condone mixed variables and poor notation. |
| \(\frac{2}{\tan\alpha}=3\sin\alpha \Rightarrow \frac{2\cos\alpha}{\sin\alpha}=3\sin\alpha \Rightarrow 2\cos\alpha=3\sin^2\alpha\) | dM1 A1* | dM1: uses \(\tan\alpha=\frac{\sin\alpha}{\cos\alpha}\) within equation. A1*: proceeds to given answer with no errors. Equation must start involving \(\tan\alpha\). |
| (3) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(2\cos\alpha = 3\sin^2\alpha \Rightarrow 2\cos\alpha = 3(1-\cos^2\alpha)\) | M1 | Attempts to use \(\sin^2\alpha+\cos^2\alpha=1\) |
| \(3\cos^2\alpha + 2\cos\alpha - 3 = 0\) | A1 | "=0" may be implied by later work; terms must be collected on one side. |
| Attempts to solve \(3\cos^2\alpha+2\cos\alpha-3=0 \Rightarrow \cos\alpha=\frac{-2\pm\sqrt{40}}{6}\) | dM1 A1 | Use formula/completing the square. Award for \((\cos\alpha=)\ \frac{-1\pm\sqrt{10}}{3}\) or awrt 0.72 or awrt \(-1.4\). Do not award for attempted factorisation unless quadratic factorises. |
| \(\alpha = 5.517\) radians | A1 | awrt 5.517 radians and no others in the given range. |
| (5) |
## Question 9:
### Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $3\sin(2\theta-10°)=1 \Rightarrow (2\theta-10°)=\arcsin\left(\frac{1}{3}\right)$ | M1 | Proceeding to $x=\arcsin\left(\frac{1}{3}\right)$; implied by awrt 19.5° or 160.5°. Allow awrt 0.340 rad. |
| $\theta = \frac{19.47+10}{2},\ \frac{160.53+10}{2}$ | dM1 | Correct order of operations leading to one answer for $\theta$. Cannot score by adding 10 to angle in radians. |
| $\theta =$ awrt $14.7°, 85.3°$ | A1, A1 | First A1: one of the values. Second A1: both values and no others in range. |
| | **(4)** | **Note: Solutions based entirely on graphical or numerical methods score 0 marks.** |
### Part (ii)(a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Writes $\frac{1}{\tan\alpha}-\sin\alpha = 2\sin\alpha - \frac{1}{\tan\alpha}$ | M1 | Uses terms of AP to set up correct equation. Condone mixed variables and poor notation. |
| $\frac{2}{\tan\alpha}=3\sin\alpha \Rightarrow \frac{2\cos\alpha}{\sin\alpha}=3\sin\alpha \Rightarrow 2\cos\alpha=3\sin^2\alpha$ | dM1 A1* | dM1: uses $\tan\alpha=\frac{\sin\alpha}{\cos\alpha}$ within equation. A1*: proceeds to given answer with no errors. Equation must start involving $\tan\alpha$. |
| | **(3)** | |
### Part (ii)(b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $2\cos\alpha = 3\sin^2\alpha \Rightarrow 2\cos\alpha = 3(1-\cos^2\alpha)$ | M1 | Attempts to use $\sin^2\alpha+\cos^2\alpha=1$ |
| $3\cos^2\alpha + 2\cos\alpha - 3 = 0$ | A1 | "=0" may be implied by later work; terms must be collected on one side. |
| Attempts to solve $3\cos^2\alpha+2\cos\alpha-3=0 \Rightarrow \cos\alpha=\frac{-2\pm\sqrt{40}}{6}$ | dM1 A1 | Use formula/completing the square. Award for $(\cos\alpha=)\ \frac{-1\pm\sqrt{10}}{3}$ or awrt 0.72 or awrt $-1.4$. Do not award for attempted factorisation unless quadratic factorises. |
| $\alpha = 5.517$ radians | A1 | awrt 5.517 radians and no others in the given range. |
| | **(5)** | |
---9. Solutions based entirely on graphical or numerical methods are not acceptable in this question.\\
(i) Solve, for $0 \leqslant \theta < 180 ^ { \circ }$, the equation
$$3 \sin \left( 2 \theta - 10 ^ { \circ } \right) = 1$$
giving your answers to one decimal place.\\
(ii) The first three terms of an arithmetic sequence are
$$\sin \alpha , \frac { 1 } { \tan \alpha } \text { and } 2 \sin \alpha$$
where $\alpha$ is a constant.
\begin{enumerate}[label=(\alph*)]
\item Show that $2 \cos \alpha = 3 \sin ^ { 2 } \alpha$
Given that $\pi < \alpha < 2 \pi$,
\item find, showing all working, the value of $\alpha$ to 3 decimal places.
\begin{center}
\end{center}
\end{enumerate}
\hfill \mbox{\textit{Edexcel P2 2019 Q9 [12]}}