| Exam Board | OCR |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2010 |
| Session | January |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Factor & Remainder Theorem |
| Type | Single polynomial, two remainder/factor conditions |
| Difficulty | Moderate -0.3 This is a standard C2 factor/remainder theorem question requiring students to set up two simultaneous equations (f(-3)=0 and f(2)=35) and solve for a and b, then perform polynomial division. While it involves multiple steps, the techniques are routine and commonly practiced at this level, 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) | \(f(-3) = 0 \Rightarrow -54 + 9a - 3b + 15 = 0\) | M1 |
| \(3a - b = 13\) | A1 | Obtain \(3a - b = 13\), or unsimplified equiv |
| \(f(2) = 35 \Rightarrow 16 + 4a + 2b + 15 = 35\) | M1 | Attempt f(2) and equate to 35, or equiv method |
| \(2a + b = 2\) | A1 | Obtain \(2a + b = 2\), or unsimplified equiv |
| Hence \(a = 3, b = -4\) | M1, A1 | Attempt to solve simultaneous eqns; Obtain \(a = 3, b = -4\) |
| (ii) | \(f(x) = (x+3)(2x^2 - 3x + 5)\) | M1, A1 |
| ie quotient is \((2x^2 - 3x + 5)\) | A1 | Obtain \(2x^2 - 3x + 5\) (state or imply as quotient) |
(i) | $f(-3) = 0 \Rightarrow -54 + 9a - 3b + 15 = 0$ | M1 | Attempt f(-3) and equate to 0, or equiv method |
| $3a - b = 13$ | A1 | Obtain $3a - b = 13$, or unsimplified equiv |
| $f(2) = 35 \Rightarrow 16 + 4a + 2b + 15 = 35$ | M1 | Attempt f(2) and equate to 35, or equiv method |
| $2a + b = 2$ | A1 | Obtain $2a + b = 2$, or unsimplified equiv |
| Hence $a = 3, b = -4$ | M1, A1 | Attempt to solve simultaneous eqns; Obtain $a = 3, b = -4$ |
(ii) | $f(x) = (x+3)(2x^2 - 3x + 5)$ | M1, A1 | Attempt complete division by $(x+3)$, or equiv; Obtain $2x^2 - 3x + c$ or $2x^2 + bx + 5$, from correct f(x) |
| ie quotient is $(2x^2 - 3x + 5)$ | A1 | Obtain $2x^2 - 3x + 5$ (state or imply as quotient) |6 The cubic polynomial $\mathrm { f } ( x )$ is given by
$$\mathrm { f } ( x ) = 2 x ^ { 3 } + a x ^ { 2 } + b x + 15$$
where $a$ and $b$ are constants. It is given that ( $x + 3$ ) is a factor of $\mathrm { f } ( x )$ and that, when $\mathrm { f } ( x )$ is divided by ( $x - 2$ ), the remainder is 35 .\\
(i) Find the values of $a$ and $b$.\\
(ii) Using these values of $a$ and $b$, divide $\mathrm { f } ( x )$ by ( $x + 3$ ).
\hfill \mbox{\textit{OCR C2 2010 Q6 [9]}}