| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2010 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Factor & Remainder Theorem |
| Type | Single polynomial, two remainder/factor conditions |
| Difficulty | Moderate -0.8 This is a straightforward application of the Factor and Remainder Theorems requiring students to set up two simultaneous equations (p(1)=0 and p(2)=10), solve for constants a and b, then factor or use the factor theorem to solve the cubic. It's more routine than average A-level questions since it follows a standard template with clear signposting and involves only algebraic manipulation without requiring geometric insight or novel problem-solving approaches. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Substitute \(x = 1\), equate to zero and obtain a correct equation in any form | B1 | |
| Substitute \(x = 2\) and equate to 10 | M1 | |
| Obtain a correct equation in any form | A1 | |
| Solve a relevant pair of equations for \(a\) or for \(b\) | M1 | |
| Obtain \(a = -17\) and \(b = 12\) | A1 | [5] |
| (ii) At any stage, state that \(x = 1\) is a solution | B1 | |
| EITHER: Attempt division by \(x - 1\) and reach a partial quotient of \(3x^2 + 5x\) | M1 | |
| Obtain quotient \(3x^2 + 5x - 12\) | A1 | |
| Obtain solutions \(x = -3\) and \(x = \frac{4}{3}\) | A1 | |
| OR: Obtain solution \(x = -3\) by trial and error or inspection | B1 | |
| Obtain solution \(x = \frac{4}{3}\) | B2 | |
| [If an attempt at the quadratic factor is made by inspection, the M1 is earned if it reaches an unknown factor of \(3x^2 + 5x + \lambda\) and an equation in \(\lambda\)] | [4] |
**(i)** Substitute $x = 1$, equate to zero and obtain a correct equation in any form | B1 |
Substitute $x = 2$ and equate to 10 | M1 |
Obtain a correct equation in any form | A1 |
Solve a relevant pair of equations for $a$ or for $b$ | M1 |
Obtain $a = -17$ and $b = 12$ | A1 | [5]
**(ii)** At any stage, state that $x = 1$ is a solution | B1 |
EITHER: Attempt division by $x - 1$ and reach a partial quotient of $3x^2 + 5x$ | M1 |
Obtain quotient $3x^2 + 5x - 12$ | A1 |
Obtain solutions $x = -3$ and $x = \frac{4}{3}$ | A1 |
OR: Obtain solution $x = -3$ by trial and error or inspection | B1 |
Obtain solution $x = \frac{4}{3}$ | B2 |
[If an attempt at the quadratic factor is made by inspection, the M1 is earned if it reaches an unknown factor of $3x^2 + 5x + \lambda$ and an equation in $\lambda$] | [4]7 The polynomial $3 x ^ { 3 } + 2 x ^ { 2 } + a x + b$, where $a$ and $b$ are constants, is denoted by $\mathrm { p } ( x )$. It is given that $( x - 1 )$ is a factor of $\mathrm { p } ( x )$, and that when $\mathrm { p } ( x )$ is divided by $( x - 2 )$ the remainder is 10 .\\
(i) Find the values of $a$ and $b$.\\
(ii) When $a$ and $b$ have these values, solve the equation $\mathrm { p } ( x ) = 0$.
\hfill \mbox{\textit{CAIE P2 2010 Q7 [9]}}