| Exam Board | Edexcel |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2012 |
| Session | January |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Factor & Remainder Theorem |
| Type | Two unknowns with show-that step |
| Difficulty | Moderate -0.3 This is a straightforward application of the Remainder Theorem requiring students to substitute values, set up two simultaneous equations, and solve them. The 'show that' in part (a) guides students through the process, making it slightly easier than average. While it involves algebraic manipulation across multiple steps, it follows a standard textbook pattern with no novel insight required. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Marks | Guidance |
| \(f(-2) = -8 + 4a - 2b + 3 = 7\) | M1 | M1: attempts \(f(\pm 2) = 7\) or long division putting remainder equal to 7 |
| so \(2a - b = 6\) | A1 (2) | A1: correct equation with remainder = 7 and no wrong working |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Marks | Guidance |
| \(f(1) = 1 + a + b + 3 = 4\) | M1, A1 | M1: attempts \(f(\pm 1) = 4\) or long division putting remainder equal to 4. A1: correct equation with remainder = 4 and powers calculated correctly |
| Solve two linear equations to give \(a = 2\) and \(b = -2\) | M1, A1 (4) | M1: solving simultaneous equations (may be implied by correct answers). A1 cao: explicit values of \(a\) and \(b\) needed |
# Question 5:
## Part (a)
| Working | Marks | Guidance |
|---------|-------|----------|
| $f(-2) = -8 + 4a - 2b + 3 = 7$ | M1 | M1: attempts $f(\pm 2) = 7$ or long division putting remainder equal to 7 |
| so $2a - b = 6$ | A1 (2) | A1: correct equation with remainder = 7 and no wrong working |
## Part (b)
| Working | Marks | Guidance |
|---------|-------|----------|
| $f(1) = 1 + a + b + 3 = 4$ | M1, A1 | M1: attempts $f(\pm 1) = 4$ or long division putting remainder equal to 4. A1: correct equation with remainder = 4 and powers calculated correctly |
| Solve two linear equations to give $a = 2$ and $b = -2$ | M1, A1 (4) | M1: solving simultaneous equations (may be implied by correct answers). A1 cao: explicit values of $a$ and $b$ needed |\begin{enumerate}
\item $\mathrm { f } ( x ) = x ^ { 3 } + a x ^ { 2 } + b x + 3$, where $a$ and $b$ are constants.
\end{enumerate}
Given that when $\mathrm { f } ( x )$ is divided by $( x + 2 )$ the remainder is 7 ,\\
(a) show that $2 a - b = 6$
Given also that when $\mathrm { f } ( x )$ is divided by $( x - 1 )$ the remainder is 4 ,\\
(b) find the value of $a$ and the value of $b$.\\
\hfill \mbox{\textit{Edexcel C2 2012 Q5 [6]}}