| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2012 |
| Session | November |
| Marks | 8 |
| 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 from two conditions, followed by polynomial long division. Both parts require standard techniques with no novel insight—slightly easier than average due to the mechanical nature of the algebra involved. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.02k Simplify rational expressions: factorising, cancelling, algebraic division |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Substitute \(x = -1\), equate to zero and obtain a correct equation in any form | B1 | |
| Substitute \(x = 3\) and equate to 12 | M1 | |
| Obtain a correct equation in any form | A1 | |
| Solve a relevant pair of equations for \(a\) or for \(b\) | M1 | |
| Obtain \(a = -4\) and \(b = 6\) | A1 | [5] |
| (ii) Attempt division by \(x^2 - 2\) and reach a partial quotient of \(2x - k\) | M1 | |
| Obtain quotient \(2x - 4\) | A1 | |
| Obtain remainder –2 | A1 | [3] |
(i) Substitute $x = -1$, equate to zero and obtain a correct equation in any form | B1 |
Substitute $x = 3$ and equate to 12 | M1 |
Obtain a correct equation in any form | A1 |
Solve a relevant pair of equations for $a$ or for $b$ | M1 |
Obtain $a = -4$ and $b = 6$ | A1 | [5]
(ii) Attempt division by $x^2 - 2$ and reach a partial quotient of $2x - k$ | M1 |
Obtain quotient $2x - 4$ | A1 |
Obtain remainder –2 | A1 | [3]7 The polynomial $2 x ^ { 3 } - 4 x ^ { 2 } + a x + b$, where $a$ and $b$ are constants, is denoted by $\mathrm { p } ( x )$. It is given that when $\mathrm { p } ( x )$ is divided by $( x + 1 )$ the remainder is 4 , and that when $\mathrm { p } ( x )$ is divided by $( x - 3 )$ the remainder is 12 .\\
(i) Find the values of $a$ and $b$.\\
(ii) When $a$ and $b$ have these values, find the quotient and remainder when $\mathrm { p } ( x )$ is divided by ( $x ^ { 2 } - 2$ ).
\hfill \mbox{\textit{CAIE P2 2012 Q7 [8]}}