| Exam Board | Edexcel |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Tangents, normals and gradients |
| Type | Tangent parallel to given line |
| Difficulty | Standard +0.3 This is a straightforward C1 integration and differentiation question requiring standard techniques: integrate a polynomial, use a point to find the constant, verify a point lies on the curve, find a tangent equation, and solve for where the derivative equals a given value. All steps are routine with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.07m Tangents and normals: gradient and equations1.08b Integrate x^n: where n != -1 and sums |
| Answer | Marks | Guidance |
|---|---|---|
| Integrate: \(y = x^3 - 10x^2 + 29x\ (+C)\) | M1 M1 | |
| \(6 = 8 - 40 + 58 + C \Rightarrow C = -20\ (y = x^3 - 10x^2 + 29x - 20)\) | M1 A1 | (4 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| Subs. \(x = 4\): \(64 - 160 + 116 - 20 = 0\) | M1 A1 | (2 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| At \(x = 2\), \(\frac{\mathrm{d}y}{\mathrm{d}x} = 12 - 40 + 29 = 1\) | B1 | |
| Tangent: \(y - 6 = x - 2\) \((y = x + 4)\) | M1 A1 | (3 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{\mathrm{d}y}{\mathrm{d}x} = 1\) | M1 | |
| \(3x^2 - 20x + 28 = 0\) | M1 | |
| \((3x - 14)(x - 2) = 0\) | M1 A1 | |
| \(x = \frac{14}{3}\) | A1 | (5 marks) |
## Question 9:
### Part (a)
Integrate: $y = x^3 - 10x^2 + 29x\ (+C)$ | M1 M1 |
$6 = 8 - 40 + 58 + C \Rightarrow C = -20\ (y = x^3 - 10x^2 + 29x - 20)$ | M1 A1 | (4 marks)
### Part (b)
Subs. $x = 4$: $64 - 160 + 116 - 20 = 0$ | M1 A1 | (2 marks)
### Part (c)
At $x = 2$, $\frac{\mathrm{d}y}{\mathrm{d}x} = 12 - 40 + 29 = 1$ | B1 |
Tangent: $y - 6 = x - 2$ $(y = x + 4)$ | M1 A1 | (3 marks)
### Part (d)
$\frac{\mathrm{d}y}{\mathrm{d}x} = 1$ | M1 |
$3x^2 - 20x + 28 = 0$ | M1 |
$(3x - 14)(x - 2) = 0$ | M1 A1 |
$x = \frac{14}{3}$ | A1 | (5 marks)9. The curve $C$ has equation $y = \mathrm { f } ( x )$. Given that $\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { 2 } - 20 x + 29$ and that $C$ passes through the point $P ( 2,6 )$,
\begin{enumerate}[label=(\alph*)]
\item find $y$ in terms of $x$.
\item Verify that $C$ passes through the point $( 4,0 )$.
\item Find an equation of the tangent to $C$ at $P$.
The tangent to $C$ at the point $Q$ is parallel to the tangent at $P$.
\item Calculate the exact $x$-coordinate of $Q$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C1 Q9 [14]}}