| Exam Board | Edexcel |
|---|---|
| Module | C12 (Core Mathematics 1 & 2) |
| Year | 2018 |
| Session | June |
| 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 substitution of x-values and solving simultaneous equations. Part (a) is a 'show that' which guides students to one equation, and part (b) involves standard simultaneous equation solving. While it requires multiple steps, the techniques are routine and commonly practiced in C1/C2 with no novel insight needed. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Attempts \(f(\pm\frac{1}{2})\) and puts equal to 1, or long division with remainder \(= 1\) | M1 | |
| \(f(\frac{1}{2}) = a(\frac{1}{8}) + 2(\frac{1}{4}) + b(\frac{1}{2}) - 3 = 1\) so \(a + 4b = 28^*\) | A1 [2] | cao; note answer is printed so working must be correct; accept \(1a + 4b = 28\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Attempts \(f(\pm 1)\) and puts equal to \(-17\), or long division with remainder \(= -17\) | M1 | |
| \(\Rightarrow -3 - b - a + 2 = -17\) so \(a + b = 16\) | A1 | Correct unsimplified equation; powers of \(-1\) must be processed correctly |
| Solve simultaneous equations to give values for \(a\) and \(b\) | dM1 | Dependent on previous M; solves their equation with \(a+4b=28\) |
| \(a = 12\) and \(b = 4\) | A1 [4] | Correct values |
# Question 2:
## Part (a)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Attempts $f(\pm\frac{1}{2})$ and puts equal to 1, or long division with remainder $= 1$ | M1 | |
| $f(\frac{1}{2}) = a(\frac{1}{8}) + 2(\frac{1}{4}) + b(\frac{1}{2}) - 3 = 1$ so $a + 4b = 28^*$ | A1 [2] | cao; note answer is printed so working must be correct; accept $1a + 4b = 28$ |
## Part (b)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Attempts $f(\pm 1)$ and puts equal to $-17$, or long division with remainder $= -17$ | M1 | |
| $\Rightarrow -3 - b - a + 2 = -17$ so $a + b = 16$ | A1 | Correct unsimplified equation; powers of $-1$ must be processed correctly |
| Solve simultaneous equations to give values for $a$ and $b$ | dM1 | Dependent on previous M; solves their equation with $a+4b=28$ |
| $a = 12$ and $b = 4$ | A1 [4] | Correct values |
---2.
$$f ( x ) = a x ^ { 3 } + 2 x ^ { 2 } + b x - 3$$
where $a$ and $b$ are constants.\\
When $\mathrm { f } ( x )$ is divided by ( $2 x - 1$ ) the remainder is 1
\begin{enumerate}[label=(\alph*)]
\item Show that
$$a + 4 b = 28$$
When $\mathrm { f } ( x )$ is divided by $( x + 1 )$ the remainder is - 17
\item Find the value of $a$ and the value of $b$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C12 2018 Q2 [6]}}