| Exam Board | OCR |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2006 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Factor & Remainder Theorem |
| Type | Single polynomial, two remainder/factor conditions |
| Difficulty | Moderate -0.3 This is a standard C2 Factor/Remainder Theorem question requiring systematic application of f(2)=12 and f(-1)=0 to form simultaneous equations, then polynomial division. It's slightly easier than average because it's a routine textbook exercise with clear signposting and no conceptual surprises, though it does require careful algebraic manipulation across multiple steps. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem |
| Answer | Marks | Guidance |
|---|---|---|
| Hence \(a = 5, b = -7\) | M1, A1, M1, A1, M1, A1 | For equating \(f(2)\) to 12. For correct equation \(4a + 2b = 6\). For equating \(f(-1)\) to 0. For correct equation \(a - b = 12\). For attempt to find \(a\) and \(b\). For both values correct |
| Answer | Marks | Guidance |
|---|---|---|
| Remainder is \(8\) | B1, M1, A1, M1, A1 | For correct lead term of \(2x^2\). For complete division attempt or equiv. For completely correct quotient. For attempt at remainder – either division or \(f(-2)\). For correct remainder |
**(i)** $f(2) = 12 \Rightarrow 4a + 2b = 6$
$f(-1) = 0 \Rightarrow a - b = 12$
Hence $a = 5, b = -7$ | M1, A1, M1, A1, M1, A1 | For equating $f(2)$ to 12. For correct equation $4a + 2b = 6$. For equating $f(-1)$ to 0. For correct equation $a - b = 12$. For attempt to find $a$ and $b$. For both values correct | 6 marks
**(ii)** Quotient is $2x^2 + x - 9$
Remainder is $8$ | B1, M1, A1, M1, A1 | For correct lead term of $2x^2$. For complete division attempt or equiv. For completely correct quotient. For attempt at remainder – either division or $f(-2)$. For correct remainder | 5 marks
**Total: 11 marks**
---8 The cubic polynomial $2 x ^ { 3 } + a x ^ { 2 } + b x - 10$ is denoted by $\mathrm { f } ( x )$. It is given that, when $\mathrm { f } ( x )$ is divided by $( x - 2 )$, the remainder is 12 . It is also given that ( $x + 1$ ) is a factor of $\mathrm { f } ( x )$.\\
(i) Find the values of $a$ and $b$.\\
(ii) Divide $\mathrm { f } ( x )$ by ( $x + 2$ ) to find the quotient and the remainder.
\hfill \mbox{\textit{OCR C2 2006 Q8 [11]}}