| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2012 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Polynomial Division & Manipulation |
| Type | Division then Solve Polynomial Equation |
| Difficulty | Standard +0.3 This is a straightforward polynomial division followed by solving a quadratic equation. The division by a quadratic is routine (though slightly more involved than linear division), and the 'hence' part simply requires factoring or using the quadratic formula. It's slightly above average difficulty due to the quartic polynomial and quadratic divisor, but requires no novel insight—just careful algebraic manipulation. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.02k Simplify rational expressions: factorising, cancelling, algebraic division |
| Answer | Marks |
|---|---|
| (i) Attempt division by \(x^2 - 3x + 2\) or equivalent, and reach a partial quotient of \(x^2 + kx\) | M1 |
| Obtain partial quotient \(x^2 - x\) | A1 |
| Obtain \(x^2 - x - 2\) with no errors seen | A1 |
| [3] | |
| (ii) Correct solution method for either quadratic e.g. factorisation | M1 |
| One correct solution from solving quadratic or inspection | B1 |
| All solutions \(x = 2, y = 1\) and \(x = -1\) given and no others | A1 |
| [3] |
**(i)** Attempt division by $x^2 - 3x + 2$ or equivalent, and reach a partial quotient of $x^2 + kx$ | M1 |
Obtain partial quotient $x^2 - x$ | A1 |
Obtain $x^2 - x - 2$ with no errors seen | A1 |
| | [3] |
**(ii)** Correct solution method for either quadratic e.g. factorisation | M1 |
One correct solution from solving quadratic or inspection | B1 |
All solutions $x = 2, y = 1$ and $x = -1$ given and no others | A1 |
| | [3] |3 The polynomial $x ^ { 4 } - 4 x ^ { 3 } + 3 x ^ { 2 } + 4 x - 4$ is denoted by $\mathrm { p } ( x )$.\\
(i) Find the quotient when $\mathrm { p } ( x )$ is divided by $x ^ { 2 } - 3 x + 2$.\\
(ii) Hence solve the equation $\mathrm { p } ( x ) = 0$.
\hfill \mbox{\textit{CAIE P2 2012 Q3 [6]}}