| Exam Board | Edexcel |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Session | Specimen |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Tangents, normals and gradients |
| Type | Normal meets curve/axis — further geometry |
| Difficulty | Moderate -0.8 This is a straightforward C1 differentiation question requiring standard techniques: finding dy/dx, evaluating at given points, finding normal equations, and calculating distance. All steps are routine with clear guidance ('show that') and no problem-solving insight needed beyond applying learned procedures. |
| Spec | 1.07i Differentiate x^n: for rational n and sums1.07m Tangents and normals: gradient and equations |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{dy}{dx} = 3x^2 - 5 - 2x^{-2}\) | M1 A2(1.0) | |
| At both \(A\) and \(B\), \(\frac{dy}{dx} = 3 \times 1 - 5 - \frac{2}{1} (= -4)\) | M1 A1 | (5 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| Gradient of normal \(= \frac{1}{4}\) | M1 A1ft | |
| \(y - (-2) = \frac{1}{4}(x - 1)\) and \(4y = x - 9\) | M1 A1 | (4 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| Normal at \(A\) meets y-axis where \(x = 0\): \(y = -\frac{9}{4}\) | B1 | |
| Similarly for normal at \(B\): \(4y = x + 9\) and \(y = \frac{9}{4}\) | M1 A1 | |
| Length of \(PQ = \frac{9}{4} + \frac{9}{4} = \frac{9}{2}\) | A1 | (4 marks) |
## Part (a)
$\frac{dy}{dx} = 3x^2 - 5 - 2x^{-2}$ | M1 A2(1.0) |
At both $A$ and $B$, $\frac{dy}{dx} = 3 \times 1 - 5 - \frac{2}{1} (= -4)$ | M1 A1 | (5 marks)
## Part (b)
Gradient of normal $= \frac{1}{4}$ | M1 A1ft |
$y - (-2) = \frac{1}{4}(x - 1)$ and $4y = x - 9$ | M1 A1 | (4 marks)
## Part (c)
Normal at $A$ meets y-axis where $x = 0$: $y = -\frac{9}{4}$ | B1 |
Similarly for normal at $B$: $4y = x + 9$ and $y = \frac{9}{4}$ | M1 A1 |
Length of $PQ = \frac{9}{4} + \frac{9}{4} = \frac{9}{2}$ | A1 | (4 marks)
**Total: 13 marks**The curve $C$ has equation $y = x^3 - 5x + \frac{2}{x}$, $x \neq 0$.
The points $A$ and $B$ both lie on $C$ and have coordinates $(1, -2)$ and $(-1, 2)$ respectively.
\begin{enumerate}[label=(\alph*)]
\item Show that the gradient of $C$ at $A$ is equal to the gradient of $C$ at $B$. [5]
\item Show that an equation for the normal to $C$ at $A$ is $4y = x - 9$. [4]
\end{enumerate}
The normal to $C$ at $A$ meets the $y$-axis at the point $P$. The normal to $C$ at $B$ meets the $y$-axis at the point $Q$.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item Find the length of $PQ$. [4]
\end{enumerate}
\hfill \mbox{\textit{Edexcel C1 Q10 [13]}}