| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2010 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Factor & Remainder Theorem |
| Type | Polynomial with equal remainders |
| Difficulty | Moderate -0.8 This is a straightforward application of the Remainder Theorem requiring students to set p(-1) = p(2) to find a constant, then verify a factor and perform polynomial division. The question involves only routine algebraic manipulation with no problem-solving insight needed, making it easier than average but not trivial since it requires understanding the Remainder Theorem concept and careful arithmetic across multiple steps. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem |
| Answer | Marks | Guidance |
|---|---|---|
| Substitute \(x = -1\) OR \(x = 2\) correctly | M1 | |
| Equate remainders to obtain correct equation \(5 - a = 26 + 2a\) or equivalent | A1 | |
| Obtain \(a = -7\) | A1 | [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Attempt division by \(x - 1\) and reach a partial quotient of \(x^2 + kx\) | M1 | |
| Obtain quotient \(x^2 + 5x - 2\) | A1 | |
| EITHER Show remainder is zero OR substitute \(x = 1\) to obtain zero | B1 | [3] |
**(i)**
| Substitute $x = -1$ OR $x = 2$ correctly | M1 |
| Equate remainders to obtain correct equation $5 - a = 26 + 2a$ or equivalent | A1 |
| Obtain $a = -7$ | A1 | [3] |
**(ii)**
| Attempt division by $x - 1$ and reach a partial quotient of $x^2 + kx$ | M1 |
| Obtain quotient $x^2 + 5x - 2$ | A1 |
| EITHER Show remainder is zero OR substitute $x = 1$ to obtain zero | B1 | [3] |3 The polynomial $x ^ { 3 } + 4 x ^ { 2 } + a x + 2$, where $a$ is a constant, is denoted by $\mathrm { p } ( x )$. It is given that the remainder when $\mathrm { p } ( x )$ is divided by ( $x + 1$ ) is equal to the remainder when $\mathrm { p } ( x )$ is divided by ( $x - 2$ ).\\
(i) Find the value of $a$.\\
(ii) When $a$ has this value, show that $( x - 1 )$ is a factor of $\mathrm { p } ( x )$ and find the quotient when $\mathrm { p } ( x )$ is divided by $( x - 1 )$.
\hfill \mbox{\textit{CAIE P2 2010 Q3 [6]}}