| Exam Board | Edexcel |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2007 |
| Session | January |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Implicit equations and differentiation |
| Type | Find stationary points |
| Difficulty | Standard +0.3 This is a straightforward implicit differentiation question requiring standard technique to find dy/dx, then solving a simple trigonometric equation (cos x = 0) within given bounds. The steps are routine for C4 level with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.07s Parametric and implicit differentiation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\sin x + \cos y = 0.5\) | (equation *) | |
| \(\cos x - \sin y \frac{dy}{dx} = 0\) | M1 | Differentiates implicitly to include \(\pm \sin y \frac{dy}{dx}\). Ignore \(\left(\frac{dy}{dx} =\right)\) |
| \(\frac{dy}{dx} = \frac{\cos x}{\sin y}\) | A1 cso | |
| [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\frac{dy}{dx} = 0 \Rightarrow \frac{\cos x}{\sin y} = 0 \Rightarrow \cos x = 0\) | M1\(\sqrt{}\) | Candidate realises they need to solve 'their numerator' \(= 0\), or candidate sets \(\frac{dy}{dx} = 0\) in (eqn #) and attempts to solve the resulting equation |
| giving \(x = -\frac{\pi}{2}\) or \(x = \frac{\pi}{2}\) | A1 | both \(x = -\frac{\pi}{2}, \frac{\pi}{2}\) or \(x = \pm 90°\) or awrt \(x = \pm 1.57\) required here |
| When \(x = -\frac{\pi}{2}\): \(\sin\left(-\frac{\pi}{2}\right) + \cos y = 0.5\) | M1 | Substitutes either their \(x = \frac{\pi}{2}\) or \(x = -\frac{\pi}{2}\) into eqn * |
| When \(x = \frac{\pi}{2}\): \(\sin\left(\frac{\pi}{2}\right) + \cos y = 0.5\) | ||
| \(\Rightarrow \cos y = 1.5 \Rightarrow y\) has no solutions | A1 | Only one of \(y = \frac{2\pi}{3}\) or \(-\frac{2\pi}{3}\) or \(120°\) or \(-120°\) or awrt \(-2.09\) or awrt \(2.09\) |
| \(\Rightarrow \cos y = -0.5 \Rightarrow y = \frac{2\pi}{3}\) or \(-\frac{2\pi}{3}\) | ||
| 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 | Only exact coordinates of \(\left(\frac{\pi}{2}, \frac{2\pi}{3}\right)\) and \(\left(\frac{\pi}{2}, -\frac{2\pi}{3}\right)\). Do not award if candidate states other coordinates inside the required range. |
| [5] |
## Question 5:
### Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\sin x + \cos y = 0.5$ | | (equation *) |
| $\cos x - \sin y \frac{dy}{dx} = 0$ | M1 | Differentiates implicitly to include $\pm \sin y \frac{dy}{dx}$. Ignore $\left(\frac{dy}{dx} =\right)$ |
| $\frac{dy}{dx} = \frac{\cos x}{\sin y}$ | A1 cso | |
| | **[2]** | |
### Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\frac{dy}{dx} = 0 \Rightarrow \frac{\cos x}{\sin y} = 0 \Rightarrow \cos x = 0$ | M1$\sqrt{}$ | Candidate realises they need to solve 'their numerator' $= 0$, or candidate sets $\frac{dy}{dx} = 0$ in (eqn #) and attempts to solve the resulting equation |
| giving $x = -\frac{\pi}{2}$ or $x = \frac{\pi}{2}$ | A1 | both $x = -\frac{\pi}{2}, \frac{\pi}{2}$ or $x = \pm 90°$ or awrt $x = \pm 1.57$ required here |
| When $x = -\frac{\pi}{2}$: $\sin\left(-\frac{\pi}{2}\right) + \cos y = 0.5$ | M1 | Substitutes either their $x = \frac{\pi}{2}$ or $x = -\frac{\pi}{2}$ into eqn * |
| When $x = \frac{\pi}{2}$: $\sin\left(\frac{\pi}{2}\right) + \cos y = 0.5$ | | |
| $\Rightarrow \cos y = 1.5 \Rightarrow y$ has no solutions | A1 | Only one of $y = \frac{2\pi}{3}$ or $-\frac{2\pi}{3}$ or $120°$ or $-120°$ or awrt $-2.09$ or awrt $2.09$ |
| $\Rightarrow \cos y = -0.5 \Rightarrow y = \frac{2\pi}{3}$ or $-\frac{2\pi}{3}$ | | |
| 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 | Only exact coordinates of $\left(\frac{\pi}{2}, \frac{2\pi}{3}\right)$ and $\left(\frac{\pi}{2}, -\frac{2\pi}{3}\right)$. Do not award if candidate states other coordinates inside the required range. |
| | **[5]** | |
---5. A set of curves is given by the equation $\sin x + \cos y = 0.5$.
\begin{enumerate}[label=(\alph*)]
\item Use implicit differentiation to find an expression for $\frac { \mathrm { d } y } { \mathrm {~d} x }$.
For $- \pi < x < \pi$ and $- \pi < y < \pi$,
\item find the coordinates of the points where $\frac { \mathrm { d } y } { \mathrm {~d} x } = 0$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C4 2007 Q5 [7]}}