| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2013 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Polynomial Division & Manipulation |
| Type | Finding Constants from Remainder Conditions |
| Difficulty | Moderate -0.3 This is a straightforward application of the Remainder Theorem to find two unknowns, followed by polynomial division and solving a quadratic. Part (i) requires setting up two simultaneous equations from p(3)=14 and p(-2)=24, which is routine. Part (ii) involves recognizing that x²+2x-8 factors as (x-2)(x+4), performing polynomial division, and solving the resulting equation. While multi-step, each component is standard A-level technique with no novel insight required, making it slightly easier than average. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.02k Simplify rational expressions: factorising, cancelling, algebraic division |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Substitute \(x = 3\) and equate to 14 \((9a + 3b + 35 = 14)\) | M1 | |
| Substitute \(x = -2\) and equate to 24 \((4a - 2b = 24)\) | M1 | |
| Obtain a correct equation in any form | A1 | |
| Solve a relevant pair of equations for \(a\) or for \(b\) | M1 | |
| Obtain \(a = 1\) and \(b = -10\) | A1 | [5] |
| (ii) Attempt division by \(x^2 + 2x - 8\) and reach a partial quotient of \(x - k\) | M1 | |
| Obtain quotient \(x - 1\) with no errors seen (can be done by observation) | A1 | |
| Correct solution method for quadratic e.g. factorisation | M1 | |
| All solutions \(x = 1, x = 2\) and \(x = -4\) given and no others CWO | A1 | [4] |
**(i)** Substitute $x = 3$ and equate to 14 $(9a + 3b + 35 = 14)$ | M1 |
Substitute $x = -2$ and equate to 24 $(4a - 2b = 24)$ | M1 |
Obtain a correct equation in any form | A1 |
Solve a relevant pair of equations for $a$ or for $b$ | M1 |
Obtain $a = 1$ and $b = -10$ | A1 | [5]
**(ii)** Attempt division by $x^2 + 2x - 8$ and reach a partial quotient of $x - k$ | M1 |
Obtain quotient $x - 1$ with no errors seen (can be done by observation) | A1 |
Correct solution method for quadratic e.g. factorisation | M1 |
All solutions $x = 1, x = 2$ and $x = -4$ given and no others CWO | A1 | [4]
---4 (i) The polynomial $x ^ { 3 } + a x ^ { 2 } + b x + 8$, where $a$ and $b$ are constants, is denoted by $\mathrm { p } ( x )$. It is given that when $\mathrm { p } ( x )$ is divided by $( x - 3 )$ the remainder is 14 , and that when $\mathrm { p } ( x )$ is divided by $( x + 2 )$ the remainder is 24 . Find the values of $a$ and $b$.\\
(ii) When $a$ and $b$ have these values, find the quotient when $\mathrm { p } ( x )$ is divided by $x ^ { 2 } + 2 x - 8$ and hence solve the equation $\mathrm { p } ( x ) = 0$.
\hfill \mbox{\textit{CAIE P2 2013 Q4 [9]}}