| Exam Board | Edexcel |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Implicit equations and differentiation |
| Type | Find tangent equation at point |
| Difficulty | Standard +0.3 This is a standard implicit differentiation question from C4. Part (a) requires applying the chain rule to differentiate sin 2x and tan y implicitly, then rearranging—a routine technique. Part (b) involves substituting a point to find the gradient and using point-slope form, which is straightforward verification. While it requires careful algebraic manipulation, it follows a well-practiced procedure with no novel insight needed, making it slightly easier than average. |
| Spec | 1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.07m Tangents and normals: gradient and equations1.07s Parametric and implicit differentiation |
| Answer | Marks |
|---|---|
| \(4\cos 2x - \sec^2 y \frac{dy}{dx} = 0\) | M1 A2 |
| \(\frac{dy}{dx} = 4\cos 2x \cos^2 y\) | M1 A1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(\text{grad} = 4 \times \frac{1}{2} \times \frac{1}{4} = \frac{1}{2}\) | B1 | |
| \(\therefore y - \frac{\pi}{3} = \frac{1}{2}\left(x-\frac{\pi}{6}\right)\) | M1 | |
| \(y - \frac{\pi}{3} = \frac{1}{2}x - \frac{\pi}{12}\) | M1 | |
| \(y = \frac{1}{2}x + \frac{\pi}{4}\) | A1 | (8 marks) |
## (a)
$4\cos 2x - \sec^2 y \frac{dy}{dx} = 0$ | M1 A2 |
$\frac{dy}{dx} = 4\cos 2x \cos^2 y$ | M1 A1 |
## (b)
$\text{grad} = 4 \times \frac{1}{2} \times \frac{1}{4} = \frac{1}{2}$ | B1 |
$\therefore y - \frac{\pi}{3} = \frac{1}{2}\left(x-\frac{\pi}{6}\right)$ | M1 |
$y - \frac{\pi}{3} = \frac{1}{2}x - \frac{\pi}{12}$ | M1 |
$y = \frac{1}{2}x + \frac{\pi}{4}$ | A1 | **(8 marks)**
---A curve has the equation
$$2 \sin 2x - \tan y = 0.$$
\begin{enumerate}[label=(\alph*)]
\item Find an expression for $\frac{dy}{dx}$ in its simplest form in terms of $x$ and $y$. [5]
\item Show that the tangent to the curve at the point $\left(\frac{\pi}{6}, \frac{\pi}{3}\right)$ has the equation
$$y = \frac{1}{2}x + \frac{\pi}{4}.$$ [3]
\end{enumerate}
\hfill \mbox{\textit{Edexcel C4 Q3 [8]}}