| Exam Board | Edexcel |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Curve Sketching |
| Type | Sketch transformations from algebraic function |
| Difficulty | Moderate -0.8 This is a routine C1 curve sketching question requiring straightforward algebraic expansion to verify factorization, plotting a cubic using given roots and y-intercept, and applying standard transformations (horizontal shift and reflection). All techniques are mechanical with no problem-solving or novel insight required, making it easier than average but not trivial due to the multi-part nature and transformation work. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.02n Sketch curves: simple equations including polynomials |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(\text{LHS} = (x+1)(x^2 - 7x + 12)\) | M1 | |
| \(= x^3 - 7x^2 + 12x + x^2 - 7x + 12\) | M1 | |
| \(= x^3 - 6x^2 + 5x + 12 = \text{RHS}\) | A1 | |
| (b) [Curve sketch showing: local maximum at \((-1, 0)\), local minimum at \((3, 0)\), x-intercepts at \((-1, 0)\), \((3, 0)\), \((4, 0)\), y-intercept at \((0, 12)\)] | B3 | |
| (c) (i) [Sketch showing roots at \(x = -4, 0, 1\)] | B2 | |
| (ii) [Sketch showing roots at \(x = -4, -3, 1\)] | B2 | (10) |
**(a)** $\text{LHS} = (x+1)(x^2 - 7x + 12)$ | M1 |
$= x^3 - 7x^2 + 12x + x^2 - 7x + 12$ | M1 |
$= x^3 - 6x^2 + 5x + 12 = \text{RHS}$ | A1 |
**(b)** [Curve sketch showing: local maximum at $(-1, 0)$, local minimum at $(3, 0)$, x-intercepts at $(-1, 0)$, $(3, 0)$, $(4, 0)$, y-intercept at $(0, 12)$] | B3 |
**(c)** **(i)** [Sketch showing roots at $x = -4, 0, 1$] | B2 |
**(ii)** [Sketch showing roots at $x = -4, -3, 1$] | B2 | (10)$\text{f}(x) = x^3 - 6x^2 + 5x + 12$.
\begin{enumerate}[label=(\alph*)]
\item Show that
$$(x + 1)(x - 3)(x - 4) \equiv x^3 - 6x^2 + 5x + 12.$$ [3]
\item Sketch the curve $y = \text{f}(x)$, showing the coordinates of any points of intersection with the coordinate axes. [3]
\item Showing the coordinates of any points of intersection with the coordinate axes, sketch on separate diagrams the curves
\begin{enumerate}[label=(\roman*)]
\item $y = \text{f}(x + 3)$,
\item $y = \text{f}(-x)$. [4]
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{Edexcel C1 Q8 [10]}}