| Exam Board | OCR MEI |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2006 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Factor & Remainder Theorem |
| Type | Verify factor then sketch or analyse curve |
| Difficulty | Moderate -0.8 This is a straightforward multi-part question testing standard Factor Theorem techniques. Part (i) is direct substitution, (ii) is routine polynomial division, (iii) follows mechanically from previous parts, (iv) is a standard cubic sketch, and (v) reduces to solving the already-factorised equation. All steps are textbook procedures with no problem-solving insight required, making it easier than average but not trivial due to the multiple techniques involved. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.02n Sketch curves: simple equations including polynomials |
| Answer | Marks | Guidance |
|---|---|---|
| \(f(-2)\) used \(-8 + 36 - 40 + 12 = 0\) | M1 mark | or M1 for division by \((x + 2)\) attempted as far as \(x^2 + \text{...}\) then A1 for \(x^2 + 7x + 6\) with no remainder |
| 2 marks |
| Answer | Marks | Guidance |
|---|---|---|
| divn attempted as far as \(x^2 + 3x\) | M1 mark | or inspection with \(b = 3\) or \(c = 2\) found; B2 for correct answer |
| \(x^2 + 3x + 2\) or \((x + 2)(x + 1)\) | A1 mark | |
| 2 marks |
| Answer | Marks | Guidance |
|---|---|---|
| \((x + 2)(x + 6)(x + 1)\) | 2 marks | allow seen earlier; M1 for \((x + 2)(x + 1)\) with \(2\) turning pts; no 3rd tp |
| 2 marks |
| Answer | Marks | Guidance |
|---|---|---|
| sketch of cubic the right way up through \(12\) marked on \(y\) axis | G1 mark | curve must extend to \(x > 0\) |
| intercepts \(-6, -2, -1\) on \(x\) axis | G1 mark | condone no graph for \(x < -6\) |
| 3 marks |
| Answer | Marks | Guidance |
|---|---|---|
| \([x](x^2 + 9x + 20)\) | M1 mark | or other partial factorisation |
| \([x](x + 4)(x + 5)\) | M1 mark | |
| \(x = 0, -4, -5\) | A1 mark | or B1 for each root found e.g. using factor theorem |
| 3 marks |
**Part (i)**
$f(-2)$ used $-8 + 36 - 40 + 12 = 0$ | M1 mark | or M1 for division by $(x + 2)$ attempted as far as $x^2 + \text{...}$ then A1 for $x^2 + 7x + 6$ with no remainder
| | 2 marks |
**Part (ii)**
divn attempted as far as $x^2 + 3x$ | M1 mark | or inspection with $b = 3$ or $c = 2$ found; B2 for correct answer
$x^2 + 3x + 2$ or $(x + 2)(x + 1)$ | A1 mark |
| | 2 marks |
**Part (iii)**
$(x + 2)(x + 6)(x + 1)$ | 2 marks | allow seen earlier; M1 for $(x + 2)(x + 1)$ with $2$ turning pts; no 3rd tp
| | 2 marks |
**Part (iv)**
sketch of cubic the right way up through $12$ marked on $y$ axis | G1 mark | curve must extend to $x > 0$
intercepts $-6, -2, -1$ on $x$ axis | G1 mark | condone no graph for $x < -6$
| | 3 marks |
**Part (v)**
$[x](x^2 + 9x + 20)$ | M1 mark | or other partial factorisation
$[x](x + 4)(x + 5)$ | M1 mark |
$x = 0, -4, -5$ | A1 mark | or B1 for each root found e.g. using factor theorem
| | 3 marks |
---You are given that $\text{f}(x) = x^3 + 9x^2 + 20x + 12$.
\begin{enumerate}[label=(\roman*)]
\item Show that $x = -2$ is a root of $\text{f}(x) = 0$. [2]
\item Divide $\text{f}(x)$ by $x + 6$. [2]
\item Express $\text{f}(x)$ in fully factorised form. [2]
\item Sketch the graph of $y = \text{f}(x)$. [3]
\item Solve the equation $\text{f}(x) = 12$. [3]
\end{enumerate}
\hfill \mbox{\textit{OCR MEI C1 2006 Q12 [12]}}