| Exam Board | OCR |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2011 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Factor & Remainder Theorem |
| Type | Find factor then solve |
| Difficulty | Moderate -0.8 This is a straightforward application of the factor theorem requiring testing small integer values (likely ±1, ±2, ±5, ±10), followed by polynomial division and solving a quadratic. It's a standard C2 textbook exercise with no novel insight required, making it easier than average but not trivial since it requires multiple routine steps. |
| Spec | 1.02f Solve quadratic equations: including in a function of unknown1.02j Manipulate polynomials: expanding, factorising, division, factor theorem |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(f(1) = 1\), \(f(-1) = 21\), \(f(2) = 0\), hence \((x - 2)\) is a factor | M1 | Attempt use of factor theorem at least once. Just substituting at least one value for \(x\) is enough for M1, showing either the working or the result, or both. Just stating \(f(a) = k\) is enough – don't need to see term by term evaluation. If result is inconsistent with the \(f(a)\) being attempted, then we do need to see evidence of method used. |
| Answer | Marks | Guidance |
|---|---|---|
| Obtain factor of \((x - 2)\) | A1 | 2 |
| Answer | Marks | Guidance |
|---|---|---|
| (ii) \(f(x) = (x - 2)(x^2 + 3x - 5)\) with \(x = \frac{-3 \pm \sqrt{29}}{2}\) or \(x = 2\) | M1 | Attempt complete division by a linear factor, or equivalent ie inspection or coefficient matching. Need linear factor of form \((x \pm a), a \neq 0\). Allow if factor different to their answer to (i), ino answer to (i). Must be complete attempt at all three terms. If long division then need to be subtracting lower line; if coefficient matching then expansion must give at least one correct coefficient for the two middle terms in the cubic. |
| Obtain \(x^2 + 3x + c\) or \(x^2 + bx - 5\) | A1 | Obtain \(x^2\) and one other correct term. Just having two correct terms does not imply M1 – need to look at method used for third term. If coefficient matching allow for stating values eg \(a = 1\) etc. If quadratic factor given with minimal working in (ii), there may be more evidence of method shown in (i). |
| Obtain \(x^2 + 3x - 5\) | A1 | Could appear as quotient in long division, or as part of product if inspection. If coefficient matching, must now be explicitly stated rather than just \(a = 1, b = 3, c = -5\). |
| Attempt to solve quadratic equation | M1 | Using quadratic formula or completing the square – see extra guidance sheet. Quadratic must come from division attempt, even if this was not good enough for first M1. |
| Obtain \(\frac{1}{2}(-3 \pm \sqrt{29})\) | A1 | Or \(\frac{5}{1/2}\)/... from completing the square. Ignore terminology and ignore if subsequently given as factors, as long as seen fully simplified as roots. |
| B1 6 | State 2 as root, at any point | Must be stated in this part, not just in part (i). Ignore terminology. |
**(i)** $f(1) = 1$, $f(-1) = 21$, $f(2) = 0$, hence $(x - 2)$ is a factor | M1 | Attempt use of factor theorem at least once. Just substituting at least one value for $x$ is enough for M1, showing either the working or the result, or both. Just stating $f(a) = k$ is enough – don't need to see term by term evaluation. If result is inconsistent with the $f(a)$ being attempted, then we do need to see evidence of method used.
No conclusion required.
M0 A0 for division attempts, even if considering remainder.
**Obtain factor of $(x - 2)$** | A1 | 2 | Allow A1 for sight of $(x - 2)$, even if $x = 2$ also present. No words required, but penalise if used incorrectly as A0 if explicitly labelled as 'root'. A0 if $(x - 2)$ not seen in this part, even if subsequently used in (ii).
SR B1 for $(x - 2)$ stated with no justification, and no incorrect terminology.
**(ii)** $f(x) = (x - 2)(x^2 + 3x - 5)$ with $x = \frac{-3 \pm \sqrt{29}}{2}$ or $x = 2$ | M1 | Attempt complete division by a linear factor, or equivalent ie inspection or coefficient matching. Need linear factor of form $(x \pm a), a \neq 0$. Allow if factor different to their answer to (i), ino answer to (i). Must be complete attempt at all three terms. If long division then need to be subtracting lower line; if coefficient matching then expansion must give at least one correct coefficient for the two middle terms in the cubic.
**Obtain $x^2 + 3x + c$ or $x^2 + bx - 5$** | A1 | Obtain $x^2$ and one other correct term. Just having two correct terms does not imply M1 – need to look at method used for third term. If coefficient matching allow for stating values eg $a = 1$ etc. If quadratic factor given with minimal working in (ii), there may be more evidence of method shown in (i).
**Obtain $x^2 + 3x - 5$** | A1 | Could appear as quotient in long division, or as part of product if inspection. If coefficient matching, must now be explicitly stated rather than just $a = 1, b = 3, c = -5$.
**Attempt to solve quadratic equation** | M1 | Using quadratic formula or completing the square – see extra guidance sheet. Quadratic must come from division attempt, even if this was not good enough for first M1.
**Obtain $\frac{1}{2}(-3 \pm \sqrt{29})$** | A1 | Or $\frac{5}{1/2}$/... from completing the square. Ignore terminology and ignore if subsequently given as factors, as long as seen fully simplified as roots.
**B1 6** | **State 2 as root, at any point** | Must be stated in this part, not just in part (i). Ignore terminology.
---6 The cubic polynomial $\mathrm { f } ( x )$ is defined by $\mathrm { f } ( x ) = x ^ { 3 } + x ^ { 2 } - 11 x + 10$.\\
(i) Use the factor theorem to find a factor of $\mathrm { f } ( x )$.\\
(ii) Hence solve the equation $\mathrm { f } ( x ) = 0$, giving each root in an exact form.
\hfill \mbox{\textit{OCR C2 2011 Q6 [8]}}