| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2016 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Polynomial Division & Manipulation |
| Type | Finding Constants from Remainder Conditions |
| Difficulty | Moderate -0.3 Part (i) is a straightforward polynomial long division requiring standard algebraic manipulation. Part (ii) requires understanding that for a factor, the remainder must be zero, leading to a simple system of two linear equations in two unknowns. This is slightly easier than average as it's a routine application of polynomial division with no conceptual surprises or extended reasoning. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.02k Simplify rational expressions: factorising, cancelling, algebraic division |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Carry out division, or equivalent, at least as far as quotient \(2x+k\) | M1 | |
| Obtain quotient \(2x-3\) | A1 | |
| Obtain remainder \(-25x+18\) | A1 | [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Subtract remainder of form \(ax+b\) \((ab\neq 0)\) from \(2x^3-7x^2-9x+3\) or multiply their quotient by \(x^2-2x+5\) | M1 | |
| Obtain \(p=16\) and \(q=-15\) | A1 | [2] |
## Question 2:
### Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Carry out division, or equivalent, at least as far as quotient $2x+k$ | M1 | |
| Obtain quotient $2x-3$ | A1 | |
| Obtain remainder $-25x+18$ | A1 | [3] |
### Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Subtract remainder of form $ax+b$ $(ab\neq 0)$ from $2x^3-7x^2-9x+3$ or multiply their quotient by $x^2-2x+5$ | M1 | |
| Obtain $p=16$ and $q=-15$ | A1 | [2] |
---2 (i) Find the quotient and remainder when $2 x ^ { 3 } - 7 x ^ { 2 } - 9 x + 3$ is divided by $x ^ { 2 } - 2 x + 5$.\\
(ii) Hence find the values of the constants $p$ and $q$ such that $x ^ { 2 } - 2 x + 5$ is a factor of $2 x ^ { 3 } - 7 x ^ { 2 } + p x + q$.
\hfill \mbox{\textit{CAIE P2 2016 Q2 [5]}}