| Exam Board | Edexcel |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Implicit equations and differentiation |
| Type | Find normal equation at point |
| Difficulty | Standard +0.3 This is a standard C4 implicit differentiation question requiring finding dy/dx, calculating the gradient at a point, finding the normal equation, and solving simultaneous equations. While it involves multiple steps (implicit differentiation, gradient of normal, substitution back into curve equation), each step follows routine procedures taught in C4 with no novel insight required. Slightly easier than average due to straightforward algebraic manipulation. |
| Spec | 1.07s Parametric and implicit differentiation |
| Answer | Marks | Guidance |
|---|---|---|
| \((a)\) \(6x - 2 + y + x\frac{dy}{dx} + 2y\frac{dy}{dx} = 0\) | M1 A2 | |
| \((-1, 3) \Rightarrow -6 - 2 + 3 - \frac{dy}{dx} + 6\frac{dy}{dx} = 0, \frac{dy}{dx} = 1\) | M1 A1 | |
| grad of normal \(= -1\) | M1 | |
| \(\therefore y - 3 = -(x + 1)\) | A1 | |
| \(y = 2 - x\) | A1 | |
| \((b)\) sub. \(\Rightarrow 3x^2 - 2x + x(2 - x) + (2 - x)^2 - 11 = 0\) | M1 | |
| \(3x^2 - 4x - 7 = 0\) | A1 | |
| \((3x - 7)(x + 1) = 0\) | M1 | |
| \(x = -1\) (at \(P\)) or \(\frac{7}{3} \therefore (\frac{7}{3}, -\frac{1}{3})\) | A1 | (11 marks) |
$(a)$ $6x - 2 + y + x\frac{dy}{dx} + 2y\frac{dy}{dx} = 0$ | M1 A2 |
$(-1, 3) \Rightarrow -6 - 2 + 3 - \frac{dy}{dx} + 6\frac{dy}{dx} = 0, \frac{dy}{dx} = 1$ | M1 A1 |
grad of normal $= -1$ | M1 |
$\therefore y - 3 = -(x + 1)$ | A1 |
$y = 2 - x$ | A1 |
$(b)$ sub. $\Rightarrow 3x^2 - 2x + x(2 - x) + (2 - x)^2 - 11 = 0$ | M1 |
$3x^2 - 4x - 7 = 0$ | A1 |
$(3x - 7)(x + 1) = 0$ | M1 |
$x = -1$ (at $P$) or $\frac{7}{3} \therefore (\frac{7}{3}, -\frac{1}{3})$ | A1 | (11 marks)3. A curve has the equation
$$3 x ^ { 2 } - 2 x + x y + y ^ { 2 } - 11 = 0$$
The point $P$ on the curve has coordinates $( - 1,3 )$.
\begin{enumerate}[label=(\alph*)]
\item Show that the normal to the curve at $P$ has the equation $y = 2 - x$.
\item Find the coordinates of the point where the normal to the curve at $P$ meets the curve again.\\
3. continued
\end{enumerate}
\hfill \mbox{\textit{Edexcel C4 Q3 [11]}}