| Exam Board | OCR MEI |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Factor & Remainder Theorem |
| Type | Find constants from coefficient conditions |
| Difficulty | Moderate -0.3 This is a straightforward two-equation system requiring expansion to find one coefficient condition and direct application of the remainder theorem. Both techniques are standard C1 content with no conceptual difficulty, though it requires careful algebraic manipulation across multiple steps. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.04a Binomial expansion: (a+b)^n for positive integer n |
| Answer | Marks | Guidance |
|---|---|---|
| \(5 + 2k\) soi | M1 | Allow M1 for expansion with \(5x^3 + 2kx^3\) and no other \(x^3\) terms, or M1 for \((29-5)/2\) soi |
| \(k = 12\) | A1 | |
| Attempt at \(f(3)\) | M1 | Must substitute 3 for \(x\) in cubic not product, or long division as far as obtaining \(x^2 + x\) in quotient |
| \(27 + 36 + m = 59\) o.e. | A1 | Or from division \(m - (-63) = 59\) o.e., or for \(27 + 3k + m = 59\) or ft their \(k\) |
| \(m = -4\) cao | A1 |
## Question 8:
$5 + 2k$ soi | **M1** | Allow M1 for expansion with $5x^3 + 2kx^3$ and no other $x^3$ terms, or M1 for $(29-5)/2$ soi
$k = 12$ | **A1** |
Attempt at $f(3)$ | **M1** | Must substitute 3 for $x$ in cubic not product, or long division as far as obtaining $x^2 + x$ in quotient
$27 + 36 + m = 59$ o.e. | **A1** | Or from division $m - (-63) = 59$ o.e., or for $27 + 3k + m = 59$ or ft their $k$
$m = -4$ cao | **A1** |
---8 You are given that
\begin{itemize}
\item the coefficient of $x ^ { 3 }$ in the expansion of $\left( 5 + 2 x ^ { 2 } \right) \left( x ^ { 3 } + k x + m \right)$ is 29 ,
\item when $x ^ { 3 } + k x + m$ is divided by ( $x - 3$ ), the remainder is 59 .
\end{itemize}
Find the values of $k$ and $m$.
\hfill \mbox{\textit{OCR MEI C1 Q8 [5]}}