| Exam Board | AQA |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2010 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Implicit equations and differentiation |
| Type | Find dy/dx at a point |
| Difficulty | Standard +0.3 Part (a) is straightforward substitution and solving a simple cubic equation. Part (b) requires implicit differentiation with a product rule and trigonometric derivative, then substitution—standard C4 technique but with multiple steps. The question is slightly above average difficulty due to the implicit differentiation mechanics, but follows a predictable pattern with no novel insight required. |
| Spec | 1.07s Parametric and implicit differentiation |
| Answer | Marks | Guidance |
|---|---|---|
| \(x^3 + \cos\pi = 7 \Rightarrow x^3 - 1 = 7\) | M1 | |
| \(x = 2\) | A1 | 2 marks |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{d}{dx}(x^3 y) = 3x^2 y + x^3\frac{dy}{dx}\) | M1 | 2 terms added, one with \(\frac{dy}{dx}\) |
| \(\frac{d}{dx}(\cos\pi y) = -\pi\sin(\pi y)\frac{dy}{dx}\) | B1 | |
| At \((2,1)\): \(3 \times 4 + 8\frac{dy}{dx} - \pi\sin\pi\frac{dy}{dx} = 0\) | M1 | Substitute candidate's \(x\) from (a) and \(y = 1\) with 0 on RHS and both derivatives attempted and no extra derivatives |
| \(\frac{dy}{dx} = -\frac{3}{2}\) | A1 | 5 marks |
**6(a)**
$x^3 + \cos\pi = 7 \Rightarrow x^3 - 1 = 7$ | M1
$x = 2$ | A1 | 2 marks | CSO
**6(b)**
$\frac{d}{dx}(x^3 y) = 3x^2 y + x^3\frac{dy}{dx}$ | M1 | 2 terms added, one with $\frac{dy}{dx}$
$\frac{d}{dx}(\cos\pi y) = -\pi\sin(\pi y)\frac{dy}{dx}$ | B1
At $(2,1)$: $3 \times 4 + 8\frac{dy}{dx} - \pi\sin\pi\frac{dy}{dx} = 0$ | M1 | Substitute candidate's $x$ from (a) and $y = 1$ with 0 on RHS and both derivatives attempted and no extra derivatives
$\frac{dy}{dx} = -\frac{3}{2}$ | A1 | 5 marks | CSO; OE
---A curve has equation $x^3 y + \cos(\pi y) = 7$.
\begin{enumerate}[label=(\alph*)]
\item Find the exact value of the $x$-coordinate at the point on the curve where $y = 1$. [2 marks]
\item Find the gradient of the curve at the point where $y = 1$. [5 marks]
\end{enumerate}
\hfill \mbox{\textit{AQA C4 2010 Q6 [7]}}