| Exam Board | OCR |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Session | Specimen |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Factor & Remainder Theorem |
| Type | Polynomial with equal remainders |
| Difficulty | Standard +0.3 This is a structured multi-part question on polynomial remainders and factorization using the Factor/Remainder Theorem. Part (i) requires equating f(2) = f(-2) to find b, part (ii) uses f(1) = 0 to find a, part (iii) involves polynomial division, and part (iv) requires analyzing the discriminant of a quadratic. All techniques are standard C2 content with clear scaffolding, making it slightly easier than average despite being worth 11 marks total. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.02k Simplify rational expressions: factorising, cancelling, algebraic division |
| Answer | Marks | Guidance |
|---|---|---|
| \(8 + 4a + 2b - 6 = -8 + 4a - 2b - 6\) | M1 | For equating \(f(2)\) and \(f(-2)\) |
| A1 | For correct equation | |
| Hence \(4b = -16 \Rightarrow b = -4\) | A1 | 3 |
| Answer | Marks | Guidance |
|---|---|---|
| \(f(1) = 1 + a + b = 0\) | M1 | For equating \(f(1)\) to 0 (not \(f(-1)\)) |
| A1 | 2 | For correct value |
| Hence \(a = 9\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(f(x) = (x-1)(x^2 + 10x^2 + 10x + 6)\) | M1 | For quadratic factor with \(x^2\) and/or \(x\) terms \(\pm 6\) OK |
| A1 | For trinomial with both these terms correct | |
| A1 | 3 | For completely correct factorisation |
| Answer | Marks | Guidance |
|---|---|---|
| The discriminant of the quadratic is 76 | M1 | For evaluating the discriminant |
| M1 | For using positive discriminant to deduce that there are 2 roots from the quadratic factor | |
| A1 | 3 | For completely correct explanation of 3 roots |
| Total: 11 |
## Part (i)
$8 + 4a + 2b - 6 = -8 + 4a - 2b - 6$ | M1 | For equating $f(2)$ and $f(-2)$
| A1 | | For correct equation
Hence $4b = -16 \Rightarrow b = -4$ | A1 | 3 | For showing given answer correctly
## Part (ii)
$f(1) = 1 + a + b = 0$ | M1 | For equating $f(1)$ to 0 (not $f(-1)$)
| A1 | 2 | For correct value
Hence $a = 9$ | |
## Part (iii)
$f(x) = (x-1)(x^2 + 10x^2 + 10x + 6)$ | M1 | For quadratic factor with $x^2$ and/or $x$ terms $\pm 6$ OK
| A1 | | For trinomial with both these terms correct
| A1 | 3 | For completely correct factorisation
## Part (iv)
The discriminant of the quadratic is 76 | M1 | For evaluating the discriminant
| M1 | For using positive discriminant to deduce that there are 2 roots from the quadratic factor
| A1 | 3 | For completely correct explanation of 3 roots
| **Total: 11** | |The cubic polynomial $x^3 + ax^2 + bx - 6$ is denoted by f$(x)$.
\begin{enumerate}[label=(\roman*)]
\item The remainder when f$(x)$ is divided by $(x - 2)$ is equal to the remainder when f$(x)$ is divided by $(x + 2)$. Show that $b = -4$. [3]
\item Given also that $(x - 1)$ is a factor of f$(x)$, find the value of $a$. [2]
\item With these values of $a$ and $b$, express f$(x)$ as a product of a linear factor and a quadratic factor. [3]
\item Hence determine the number of real roots of the equation f$(x) = 0$, explaining your reasoning. [3]
\end{enumerate}
\hfill \mbox{\textit{OCR C2 Q9 [11]}}