| Exam Board | Edexcel |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Tangents, normals and gradients |
| Type | Tangent meets curve/axis — further geometry |
| Difficulty | Moderate -0.8 This is a straightforward C1 differentiation question requiring basic polynomial differentiation, solving a quadratic equation, finding a tangent equation, and calculating distance between two points. All techniques are routine with clear signposting and no problem-solving insight required. The multi-part structure guides students through standard procedures, making it easier than the average A-level question. |
| Spec | 1.07i Differentiate x^n: for rational n and sums1.07m Tangents and normals: gradient and equations |
| Answer | Marks |
|---|---|
| \(\frac{dy}{dx} = 3x^2 - 10x + 5\) | M1 A1 (2) |
| Answer | Marks |
|---|---|
| \((3x-1)(x-3) = 0\) → \(x = \frac{1}{3}\) | M1 A1 (2) |
| Answer | Marks |
|---|---|
| When \(x = 3\): \(y = 27 - 45 + 15 + 2 = -1\) | B1 |
| \(y + 1 = 2(x-3)\) → \(y = 2x - 7\) | M1 A1 (3) |
| Answer | Marks |
|---|---|
| \(R\): \(x = 0\) \(y = -7\); \(S\): \(y = 0\) \(x = 3.5\) (Both for M1) | M1 A1 ft |
| \(RS = \sqrt{(72) + \left(\frac{7}{2}\right)^2} = \frac{7}{2}\sqrt{5}\) (or equivalent) | M1 A1 (4) |
## Part (a)
$\frac{dy}{dx} = 3x^2 - 10x + 5$ | M1 A1 (2)
## Part (b)
$3x^2 - 10x + 5 = 2$ → $3x^2 - 10x + 3 = 0$
$(3x-1)(x-3) = 0$ → $x = \frac{1}{3}$ | M1 A1 (2)
## Part (c)
When $x = 3$: $y = 27 - 45 + 15 + 2 = -1$ | B1
$y + 1 = 2(x-3)$ → $y = 2x - 7$ | M1 A1 (3)
## Part (d)
$R$: $x = 0$ $y = -7$; $S$: $y = 0$ $x = 3.5$ (Both for M1) | M1 A1 ft
$RS = \sqrt{(72) + \left(\frac{7}{2}\right)^2} = \frac{7}{2}\sqrt{5}$ (or equivalent) | M1 A1 (4)
**Total: 11 marks**
---A curve $C$ has equation $y = x^3 - 5x^2 + 5x + 2$.
\begin{enumerate}[label=(\alph*)]
\item Find $\frac{dy}{dx}$ in terms of $x$. [2]
\end{enumerate}
The points $P$ and $Q$ lie on $C$. The gradient of $C$ at both $P$ and $Q$ is 2. The $x$-coordinate of $P$ is 3.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Find the $x$-coordinate of $Q$. [2]
\item Find an equation for the tangent to $C$ at $P$, giving your answer in the form $y = mx + c$, where $m$ and $c$ are constants. [3]
\end{enumerate}
This tangent intersects the coordinate axes at the points $R$ and $S$.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{3}
\item Find the length of $RS$, giving your answer as a surd. [4]
\end{enumerate}
\hfill \mbox{\textit{Edexcel C1 Q9 [11]}}