| Exam Board | Edexcel |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2008 |
| Session | January |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Curve Sketching |
| Type | Expand from factored form |
| Difficulty | Moderate -0.8 This is a straightforward C1 question requiring routine algebraic expansion, basic curve sketching from factored form (finding intercepts), and solving a simple derivative equation. All techniques are standard with no problem-solving insight needed, making it easier than average but not trivial due to the multi-part structure. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.02n Sketch curves: simple equations including polynomials1.07a Derivative as gradient: of tangent to curve |
| Answer | Marks |
|---|---|
| (a) Shape \(\bigcup\) (drawn anywhere) | B1 |
| Minimum at (1, 0) (perhaps labelled 1 on \(x\)-axis) | B1 |
| \((-3, 0)\) (or \(-3\) shown on –ve \(y\)-axis) | B1 |
| \((0, 3)\) (or 3 shown on +ve \(y\)-axis) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| (b) \(y = (x+3)(x^2 - 2x + 1)\) | M1 | Marks can be awarded if this is seen in part (a) |
| \(= x^3 + x^2 - 5x + 3\) | A1also |
| Answer | Marks |
|---|---|
| (c) \(\frac{dy}{dx} = 3x^2 + 2x - 5\) | M1, A1 |
| \(3x^2 + 2x - 5 = 3\) or \(3x^2 + 2x - 8 = 0\) | M1 |
| \((3x - 4)(x+2) = 0\) | M1 |
| \(x = ...\) | M1 |
| \(x = \frac{4}{3}\) (or exact equiv.) | A1, A1 |
| \(x = -2\) | A1, A1 |
(a) Shape $\bigcup$ (drawn anywhere) | B1 |
Minimum at (1, 0) (perhaps labelled 1 on $x$-axis) | B1 |
$(-3, 0)$ (or $-3$ shown on –ve $y$-axis) | B1 |
$(0, 3)$ (or 3 shown on +ve $y$-axis) | B1 |
**Note:** The max. can be anywhere.
**Total for (a): 4 marks**
(b) $y = (x+3)(x^2 - 2x + 1)$ | M1 | Marks can be awarded if this is seen in part (a)
$= x^3 + x^2 - 5x + 3$ | A1also |
**Total for (b): 2 marks**
(c) $\frac{dy}{dx} = 3x^2 + 2x - 5$ | M1, A1 |
$3x^2 + 2x - 5 = 3$ or $3x^2 + 2x - 8 = 0$ | M1 |
$(3x - 4)(x+2) = 0$ | M1 |
$x = ...$ | M1 |
$x = \frac{4}{3}$ (or exact equiv.) | A1, A1 |
$x = -2$ | A1, A1 |
**Total for (c): 6 marks**
**Total: 12 marks**
---\begin{enumerate}
\item The curve $C$ has equation
\end{enumerate}
$$y = ( x + 3 ) ( x - 1 ) ^ { 2 }$$
(a) Sketch $C$ showing clearly the coordinates of the points where the curve meets the coordinate axes.\\
(b) Show that the equation of $C$ can be written in the form
$$y = x ^ { 3 } + x ^ { 2 } - 5 x + k ,$$
where $k$ is a positive integer, and state the value of $k$.
There are two points on $C$ where the gradient of the tangent to $C$ is equal to 3 .\\
(c) Find the $x$-coordinates of these two points.
\hfill \mbox{\textit{Edexcel C1 2008 Q10 [12]}}