| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2015 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Factor & Remainder Theorem |
| Type | Two polynomials, shared factor or separate conditions |
| Difficulty | Standard +0.3 Part (i) is straightforward application of factor theorem and remainder theorem to set up two simultaneous equations in a and b. Part (ii) requires finding the maximum of a cubic function using differentiation, which is standard A-level calculus. The question involves multiple steps but uses routine techniques with no novel insight required. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.07n Stationary points: find maxima, minima using derivatives |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Substitute \(x = -2\) in \(f(x)\) and equate to zero to obtain \(-8 + 4a + b = 0\) or equiv | B1 | |
| Substitute \(x = -1\) in \(g(x)\) and equate to \(-18\) | M1 | |
| Obtain \(-1 + b - a = -18\) or equivalent | A1 | |
| Solve a pair of linear equations for \(a\) or \(b\) | DM1 | |
| Obtain \(a = 5, b = -12\) | A1 | [5] |
| (ii) Simplify \(g(x) - f(x)\) to obtain form \(kx^2 + c\) where \(k < 0\) | M1 | |
| Obtain \(-17x^2 + 7\) and state 7, following their value of \(c\) | A1∨ | [2] |
(i) Substitute $x = -2$ in $f(x)$ and equate to zero to obtain $-8 + 4a + b = 0$ or equiv | B1 |
Substitute $x = -1$ in $g(x)$ and equate to $-18$ | M1 |
Obtain $-1 + b - a = -18$ or equivalent | A1 |
Solve a pair of linear equations for $a$ or $b$ | DM1 |
Obtain $a = 5, b = -12$ | A1 | [5]
(ii) Simplify $g(x) - f(x)$ to obtain form $kx^2 + c$ where $k < 0$ | M1 |
Obtain $-17x^2 + 7$ and state 7, following their value of $c$ | A1∨ | [2]
---4 The polynomials $\mathrm { f } ( x )$ and $\mathrm { g } ( x )$ are defined by
$$\mathrm { f } ( x ) = x ^ { 3 } + a x ^ { 2 } + b \quad \text { and } \quad \mathrm { g } ( x ) = x ^ { 3 } + b x ^ { 2 } - a$$
where $a$ and $b$ are constants. It is given that ( $x + 2$ ) is a factor of $\mathrm { f } ( x )$. It is also given that, when $\mathrm { g } ( x )$ is divided by $( x + 1 )$, the remainder is - 18 .\\
(i) Find the values of $a$ and $b$.\\
(ii) When $a$ and $b$ have these values, find the greatest possible value of $\mathrm { g } ( x ) - \mathrm { f } ( x )$ as $x$ varies.
\hfill \mbox{\textit{CAIE P2 2015 Q4 [7]}}