| Exam Board | Edexcel |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Session | Specimen |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Tangents, normals and gradients |
| Type | Tangent parallel to given line |
| Difficulty | Easy -1.2 This is a straightforward C1 differentiation question requiring basic polynomial differentiation, point verification by substitution, and finding where gradients are equal. All techniques are routine with no problem-solving insight needed, making it easier than average but not trivial due to the multi-part structure. |
| Spec | 1.07i Differentiate x^n: for rational n and sums1.07m Tangents and normals: gradient and equations |
| Answer | Marks | Guidance |
|---|---|---|
| \(f(x) = x^3 - 4x^2 + 6x + C\) | M1 A1 | |
| \(5 = 27 - 36 + 18 + C\) and \(C = -4\) | M1 A1 | (4 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| \(x = 2: y = 8 - 16 + 12 - 4 = 0\) | M1 A1 | (2 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| \(f'(3) = 27 - 24 + 6 = 9\), Parallel therefore equal gradient | B1, M1 | |
| \(3x^2 - 8x + 6 = 9\) and \(3x^2 - 8x - 3 = 0\) | M1 | |
| \((3x + 1)(x - 3) = 0\) and \(Q: x = -\frac{1}{3}\) | M1 A1 | (5 marks) |
## Part (a)
$f(x) = x^3 - 4x^2 + 6x + C$ | M1 A1 |
$5 = 27 - 36 + 18 + C$ and $C = -4$ | M1 A1 | (4 marks)
## Part (b)
$x = 2: y = 8 - 16 + 12 - 4 = 0$ | M1 A1 | (2 marks)
## Part (c)
$f'(3) = 27 - 24 + 6 = 9$, Parallel therefore equal gradient | B1, M1 |
$3x^2 - 8x + 6 = 9$ and $3x^2 - 8x - 3 = 0$ | M1 |
$(3x + 1)(x - 3) = 0$ and $Q: x = -\frac{1}{3}$ | M1 A1 | (5 marks)
**Total: 11 marks**
---The curve $C$ has equation $y = \text{f}(x)$ and the point $P(3, 5)$ lies on $C$.
Given that
$$\text{f}(x) = 3x^2 - 8x + 6,$$
\begin{enumerate}[label=(\alph*)]
\item find $\text{f}'(x)$. [4]
\item Verify that the point $(2, 0)$ lies on $C$. [2]
\end{enumerate}
The point $Q$ also lies on $C$, and the tangent to $C$ at $Q$ is parallel to the tangent to $C$ at $P$.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item Find the $x$-coordinate of $Q$. [5]
\end{enumerate}
\hfill \mbox{\textit{Edexcel C1 Q9 [11]}}