| Exam Board | Edexcel |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Curve Sketching |
| Type | Polynomial with line intersection |
| Difficulty | Moderate -0.8 This is a straightforward C1 curve sketching question requiring basic polynomial expansion to find intercepts, sketching a cubic and linear function, and reading intersection information from the graph. All techniques are routine with no problem-solving insight needed beyond standard textbook methods. |
| Spec | 1.02m Graphs of functions: difference between plotting and sketching1.02n Sketch curves: simple equations including polynomials1.02q Use intersection points: of graphs to solve equations |
| Answer | Marks | Guidance |
|---|---|---|
| (a) Graph showing cubic curve \(y = (x - 1)^2(x - 5)\) passing through \((1, 0)\), \((5, 0)\), with local maximum between them and point \((0, 5)\). Also showing line \(y = 8 - 2x\) with point \((4, 0)\) and \((0, 8)\). | B3 | |
| B2 | ||
| (b) the graphs intersect at exactly one point \(\therefore\) one solution | B1 | |
| (c) \(n = 4\) | B1 | (7 marks) |
**(a)** Graph showing cubic curve $y = (x - 1)^2(x - 5)$ passing through $(1, 0)$, $(5, 0)$, with local maximum between them and point $(0, 5)$. Also showing line $y = 8 - 2x$ with point $(4, 0)$ and $(0, 8)$. | B3 |
| B2 |
**(b)** the graphs intersect at exactly one point $\therefore$ one solution | B1 |
**(c)** $n = 4$ | B1 | (7 marks)\begin{enumerate}[label=(\alph*)]
\item Sketch on the same diagram the graphs of $y = (x - 1)^2(x - 5)$ and $y = 8 - 2x$.
Label on your diagram the coordinates of any points where each graph meets the coordinate axes. [5]
\item Explain how your diagram shows that there is only one solution, $\alpha$, to the equation
$$(x - 1)^2(x - 5) = 8 - 2x.$$ [1]
\item State the integer, $n$, such that
$$n < \alpha < n + 1.$$ [1]
\end{enumerate}
\hfill \mbox{\textit{Edexcel C1 Q5 [7]}}