| Exam Board | Edexcel |
|---|---|
| Module | Paper 2 (Paper 2) |
| Session | Specimen |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Implicit equations and differentiation |
| Type | Find stationary points |
| Difficulty | Standard +0.8 This question requires implicit differentiation to find dy/dx, setting it equal to zero for horizontal tangents, then solving the resulting trigonometric system with domain restrictions. It combines multiple A-level techniques (implicit differentiation, trigonometric equations, domain analysis) and requires careful reasoning about which solutions are valid, making it moderately challenging but within standard Further Maths scope. |
| Spec | 1.05k Further identities: sec^2=1+tan^2 and cosec^2=1+cot^21.07s Parametric and implicit differentiation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Complete process: differentiate, set \(\frac{dy}{dx}=0\), substitute \(x\) into \(\sin x + \cos y = 0.5\) | M1 | 3.1a |
| \(\cos x - \sin y\frac{dy}{dx} = 0\) | B1 | 1.1b — Correct differentiated equation |
| Applies \(\frac{dy}{dx}=0 \Rightarrow \cos x = 0 \Rightarrow x = \ldots\) | M1 | 2.2a |
| At least one of \(x = -\frac{\pi}{2}\) or \(x = \frac{\pi}{2}\) | A1 | 1.1b |
| \(x=\frac{\pi}{2} \Rightarrow \sin\left(\frac{\pi}{2}\right)+\cos y = 0.5 \Rightarrow \cos y = -\frac{1}{2} \Rightarrow y = \frac{2\pi}{3}\) or \(-\frac{2\pi}{3}\) | M1 | 1.1b |
| In specified range: \((x,y)=\left(\frac{\pi}{2},\frac{2\pi}{3}\right)\) and \(\left(\frac{\pi}{2},-\frac{2\pi}{3}\right)\) | A1 | 1.1b |
| \(x=-\frac{\pi}{2} \Rightarrow \sin\left(-\frac{\pi}{2}\right)+\cos y=0.5 \Rightarrow \cos y = 1.5\) has no solutions, so exactly 2 possible points \(P\) | B1 | 2.1 |
## Question 12:
| Answer/Working | Mark | Guidance |
|---|---|---|
| Complete process: differentiate, set $\frac{dy}{dx}=0$, substitute $x$ into $\sin x + \cos y = 0.5$ | M1 | 3.1a |
| $\cos x - \sin y\frac{dy}{dx} = 0$ | B1 | 1.1b — Correct differentiated equation |
| Applies $\frac{dy}{dx}=0 \Rightarrow \cos x = 0 \Rightarrow x = \ldots$ | M1 | 2.2a |
| At least one of $x = -\frac{\pi}{2}$ or $x = \frac{\pi}{2}$ | A1 | 1.1b |
| $x=\frac{\pi}{2} \Rightarrow \sin\left(\frac{\pi}{2}\right)+\cos y = 0.5 \Rightarrow \cos y = -\frac{1}{2} \Rightarrow y = \frac{2\pi}{3}$ or $-\frac{2\pi}{3}$ | M1 | 1.1b |
| In specified range: $(x,y)=\left(\frac{\pi}{2},\frac{2\pi}{3}\right)$ and $\left(\frac{\pi}{2},-\frac{2\pi}{3}\right)$ | A1 | 1.1b |
| $x=-\frac{\pi}{2} \Rightarrow \sin\left(-\frac{\pi}{2}\right)+\cos y=0.5 \Rightarrow \cos y = 1.5$ has no solutions, so exactly 2 possible points $P$ | B1 | 2.1 |
---\begin{enumerate}
\item A curve $C$ is given by the equation
\end{enumerate}
$$\sin x + \cos y = 0.5 \quad - \frac { \pi } { 2 } \leqslant x < \frac { 3 \pi } { 2 } , - \pi < y < \pi$$
A point $P$ lies on $C$.\\
The tangent to $C$ at the point $P$ is parallel to the $x$-axis.\\
Find the exact coordinates of all possible points $P$, justifying your answer.\\
(Solutions based entirely on graphical or numerical methods are not acceptable.)
\hfill \mbox{\textit{Edexcel Paper 2 Q12 [7]}}