| Exam Board | Edexcel |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2011 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Curve Sketching |
| Type | Rational curve intersections |
| Difficulty | Moderate -0.3 This is a straightforward C1 curve sketching question requiring students to sketch a cubic (by finding roots and considering end behavior) and a reciprocal function, then count intersections. While it involves multiple steps, each component uses standard techniques with no novel problem-solving required—slightly easier than average due to the routine nature of the tasks. |
| Spec | 1.02n Sketch curves: simple equations including polynomials1.02o Sketch reciprocal curves: y=a/x and y=a/x^21.02q Use intersection points: of graphs to solve equations |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Correct shape (negative cubic) | B1 | |
| Crossing at \((-2, 0)\) | B1 | |
| Through the origin | B1 | |
| Crossing at \((3, 0)\) | B1 | |
| 2 branches in correct quadrants not crossing axes | B1 | |
| One intersection with cubic on each branch | B1 | |
| (6) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| "2" solutions | B1ft | For a value compatible with their sketch. Only allow 0, 2 or 4 |
| Since only "2" intersections | dB1ft | Dependent on value being compatible with sketch. Comment relating number of solutions to number of intersections |
| (2) |
## Question 10:
### Part (a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Correct shape (negative cubic) | B1 | |
| Crossing at $(-2, 0)$ | B1 | |
| Through the origin | B1 | |
| Crossing at $(3, 0)$ | B1 | |
| 2 branches in correct quadrants not crossing axes | B1 | |
| One intersection with cubic on each branch | B1 | |
| | **(6)** | |
### Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| "2" solutions | B1ft | For a value compatible with their sketch. Only allow 0, 2 or 4 |
| Since only "2" intersections | dB1ft | Dependent on value being compatible with sketch. Comment relating number of solutions to number of intersections |
| | **(2)** | |
---10. (a) On the axes below, sketch the graphs of
\begin{enumerate}[label=(\roman*)]
\item $y = x ( x + 2 ) ( 3 - x )$
\item $y = - \frac { 2 } { x }$\\
showing clearly the coordinates of all the points where the curves cross the coordinate axes.\\
(b) Using your sketch state, giving a reason, the number of real solutions to the equation
$$x ( x + 2 ) ( 3 - x ) + \frac { 2 } { x } = 0$$
\includegraphics[max width=\textwidth, alt={}, center]{95e11fd7-765c-477d-800b-7574bc1af81f-13_994_997_1270_479}
\end{enumerate}
\hfill \mbox{\textit{Edexcel C1 2011 Q10 [8]}}