| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2021 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Factor & Remainder Theorem |
| Type | One factor, one non-zero remainder |
| Difficulty | Moderate -0.8 This is a straightforward application of the factor and remainder theorems requiring students to set up two simultaneous equations (p(-2)=0 and p(-3)=-55), solve for a and b, then factorise. It's routine algebraic manipulation with no conceptual challenges beyond standard A-level techniques, making it easier than average but not trivial. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Substitute \(x = -2\) and equate to zero | \*M1 | |
| Substitute \(x = -3\) and equate to \(-55\) | \*M1 | |
| Obtain \(-8a - 2b - 10 = 0\) and \(-27a - 3b - 10 = -55\) | A1 | OE |
| Solve a pair of relevant simultaneous linear equations to find \(a\) or \(b\) | DM1 | Dependent at least one M1 |
| Obtain \(a = 4\) and \(b = -21\) | A1 | |
| Total | 5 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Divide by \(x + 2\) at least as far as the \(x\) term | M1 | Or using identity or inspection |
| Obtain \(4x^2 - 8x - 5\) | A1 | |
| Conclude \((x + 2)(2x + 1)(2x - 5)\) | A1 | Condone inclusion of \(= 0\) |
| Total | 3 |
**Question 1(a):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| Substitute $x = -2$ and equate to zero | \*M1 | |
| Substitute $x = -3$ and equate to $-55$ | \*M1 | |
| Obtain $-8a - 2b - 10 = 0$ and $-27a - 3b - 10 = -55$ | A1 | OE |
| Solve a pair of relevant simultaneous linear equations to find $a$ or $b$ | DM1 | Dependent at least one M1 |
| Obtain $a = 4$ and $b = -21$ | A1 | |
| **Total** | **5** | |
---
**Question 1(b):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| Divide by $x + 2$ at least as far as the $x$ term | M1 | Or using identity or inspection |
| Obtain $4x^2 - 8x - 5$ | A1 | |
| Conclude $(x + 2)(2x + 1)(2x - 5)$ | A1 | Condone inclusion of $= 0$ |
| **Total** | **3** | |
---1 The polynomial $\mathrm { p } ( x )$ is defined by
$$\mathrm { p } ( x ) = a x ^ { 3 } + b x - 10$$
where $a$ and $b$ are constants. It is given that $( x + 2 )$ is a factor of $\mathrm { p } ( x )$ and that the remainder is - 55 when $\mathrm { p } ( x )$ is divided by $( x + 3 )$.
\begin{enumerate}[label=(\alph*)]
\item Find the values of $a$ and $b$.
\item Hence factorise $\mathrm { p } ( x )$ completely.
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2021 Q1 [8]}}