| Exam Board | OCR |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2006 |
| Session | January |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Polynomial Division & Manipulation |
| Type | Finding Constants from Remainder Conditions |
| Difficulty | Moderate -0.3 Part (i) is a straightforward polynomial long division exercise requiring standard technique. Part (ii) requires setting the remainder to zero and solving simultaneous equations, which is a routine application once division is understood. This is slightly easier than average as it's a standard two-part question testing a core C4 skill with no conceptual surprises. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Quotient \(= 3x + \ldots\) | B1 | For correct leading term in quotient |
| For evidence of correct division process | M1 | Or for cubic \(\equiv (x^2-2x+5)(gx+h)(+\ldots)\) |
| \(3x+4\) | A1 | For correct quotient |
| \(-6x-13\) | A1 | 4 For correct remainder ISW |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(a = 7\) | B1\(\sqrt{}\) | Follow through if rem in (i) is \(Px+Q\), then B1\(\sqrt{}\) for \(a=1-P\) |
| \(b = 20\) | B1\(\sqrt{}\) | 2 and B1\(\sqrt{}\) for \(b=7-Q\) |
| [SR: If B0+B0, award B1\(\sqrt{}\) for \(a=1+P\) AND \(b=7+Q\); also SR B1 for \(a=20, b=7\)] |
# Question 3:
## Part (i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Quotient $= 3x + \ldots$ | B1 | For correct leading term in quotient |
| For evidence of correct division process | M1 | Or for cubic $\equiv (x^2-2x+5)(gx+h)(+\ldots)$ |
| $3x+4$ | A1 | For correct quotient |
| $-6x-13$ | A1 | **4** For correct remainder ISW |
## Part (ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $a = 7$ | B1$\sqrt{}$ | Follow through if rem in (i) is $Px+Q$, then B1$\sqrt{}$ for $a=1-P$ |
| $b = 20$ | B1$\sqrt{}$ | **2** and B1$\sqrt{}$ for $b=7-Q$ |
| [SR: If B0+B0, award B1$\sqrt{}$ for $a=1+P$ AND $b=7+Q$; also SR B1 for $a=20, b=7$] | | |3 (i) Find the quotient and the remainder when $3 x ^ { 3 } - 2 x ^ { 2 } + x + 7$ is divided by $x ^ { 2 } - 2 x + 5$.\\
(ii) Hence, or otherwise, determine the values of the constants $a$ and $b$ such that, when $3 x ^ { 3 } - 2 x ^ { 2 } + a x + b$ is divided by $x ^ { 2 } - 2 x + 5$, there is no remainder.
\hfill \mbox{\textit{OCR C4 2006 Q3 [6]}}