| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2007 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Factor & Remainder Theorem |
| Type | One unknown constant: find it then solve |
| Difficulty | Moderate -0.8 This is a straightforward application of the factor theorem requiring substitution to find 'a', then polynomial division and solving a quadratic. All steps are routine procedures with no problem-solving insight needed, making it easier than average but not trivial due to the algebraic manipulation involved. |
| Spec | 1.02f Solve quadratic equations: including in a function of unknown1.02j Manipulate polynomials: expanding, factorising, division, factor theorem |
| Answer | Marks | Guidance |
|---|---|---|
| Substitute \(x = -2\) and equate to zero | M1 | |
| Obtain answer \(a = 3\) | A1 | [2] |
| Answer | Marks | Guidance |
|---|---|---|
| At any stage state that \(x = -2\) is a solution | B1 | |
| EITHER: Attempt division by \(x + 2\) and reach a partial quotient of \(3x^2 + kx\) | M1 | |
| Obtain quadratic factor \(3x^2 + 2x - 1\) | A1 | |
| Obtain solutions \(x = -1\) and \(x = \frac{1}{3}\) | A1 | |
| OR: Obtain solution \(x = -1\) by trial or inspection | B1 | |
| Obtain solution \(x = \frac{1}{3}\) similarly | B2 | [4] |
**(i)**
Substitute $x = -2$ and equate to zero | M1 |
Obtain answer $a = 3$ | A1 | [2]
**(ii)**
At any stage state that $x = -2$ is a solution | B1 |
EITHER: Attempt division by $x + 2$ and reach a partial quotient of $3x^2 + kx$ | M1 |
Obtain quadratic factor $3x^2 + 2x - 1$ | A1 |
Obtain solutions $x = -1$ and $x = \frac{1}{3}$ | A1 |
OR: Obtain solution $x = -1$ by trial or inspection | B1 |
Obtain solution $x = \frac{1}{3}$ similarly | B2 | [4]5 The polynomial $3 x ^ { 3 } + 8 x ^ { 2 } + a x - 2$, where $a$ is a constant, is denoted by $\mathrm { p } ( x )$. It is given that $( x + 2 )$ is a factor of $\mathrm { p } ( x )$.\\
(i) Find the value of $a$.\\
(ii) When $a$ has this value, solve the equation $\mathrm { p } ( x ) = 0$.
\hfill \mbox{\textit{CAIE P2 2007 Q5 [6]}}