| Exam Board | OCR |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2006 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Curve Sketching |
| Type | Reflection or vertical transformation |
| Difficulty | Easy -1.2 This is a straightforward C1 question requiring basic algebraic expansion, identification of roots from factored form, and sketching cubic curves with known intercepts. All steps are routine with no problem-solving or novel insight required—easier than average A-level questions. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.02n Sketch curves: simple equations including polynomials |
| Answer | Marks | Guidance |
|---|---|---|
| \((x-4)(x-3)(x+1) = (x^2 - 7x + 12)(x+1)\) | B1 | |
| \(= x^3 + x^2 - 7x^2 - 7x + 12x + 12\) | M1 | |
| \(= x^3 - 6x^2 + 5x + 12\) | A1 | 3 marks; \(x^2 - 7x + 12\) or \(x^2 - 2x - 3\) or \(x^2 - 3x - 4\) seen; Attempt to multiply a quadratic by a linear factor or attempt to list an 8 term expansion of all 3 brackets; \(x^3 - 6x^2 + 5x + 12\) (AG) obtained (no wrong working seen) |
| Answer | Marks | Guidance |
|---|---|---|
| Graph with +ve cubic with 3 roots (not 3 line segments) | B1 | |
| \((0, 12)\) labelled or indicated on \(y\)-axis | B1 | |
| \((-1, 0)\), \((3, 0)\), \((4, 0)\) labelled or indicated on \(x\)-axis | B1 | 3 marks total |
| Reflect their (ii) in either \(x\)- or \(y\)-axis | M1 | 2 marks |
| Reflect their (ii) in \(x\)-axis | A1 |
## (i)
$(x-4)(x-3)(x+1) = (x^2 - 7x + 12)(x+1)$ | B1 |
$= x^3 + x^2 - 7x^2 - 7x + 12x + 12$ | M1 |
$= x^3 - 6x^2 + 5x + 12$ | A1 | 3 marks; $x^2 - 7x + 12$ or $x^2 - 2x - 3$ or $x^2 - 3x - 4$ seen; Attempt to multiply a quadratic by a linear factor or attempt to list an 8 term expansion of all 3 brackets; $x^3 - 6x^2 + 5x + 12$ (AG) obtained (no wrong working seen)
## (ii)-(iii)
Graph with +ve cubic with 3 roots (not 3 line segments) | B1 |
$(0, 12)$ labelled or indicated on $y$-axis | B1 |
$(-1, 0)$, $(3, 0)$, $(4, 0)$ labelled or indicated on $x$-axis | B1 | 3 marks total
Reflect their (ii) in either $x$- or $y$-axis | M1 | 2 marks
Reflect their (ii) in $x$-axis | A1 |
---\begin{enumerate}[label=(\roman*)]
\item By expanding the brackets, show that
$$(x - 4)(x - 3)(x + 1) = x^3 - 6x^2 + 5x + 12.$$ [3]
\item Sketch the curve
$$y = x^3 - 6x^2 + 5x + 12,$$
giving the coordinates of the points where the curve meets the axes. Label the curve $C_1$. [3]
\item On the same diagram as in part (ii), sketch the curve
$$y = -x^3 + 6x^2 - 5x - 12.$$
Label this curve $C_2$. [2]
\end{enumerate}
\hfill \mbox{\textit{OCR C1 2006 Q4 [8]}}