| Exam Board | OCR |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2014 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Factor & Remainder Theorem |
| Type | Find remainder(s) then factorise |
| Difficulty | Moderate -0.8 This is a straightforward application of the Factor and Remainder Theorem with routine algebraic manipulation. Part (i) requires simple substitution, part (ii) is verification by substitution, part (iii) is polynomial division (which is mechanical), and part (iv) requires solving a quadratic using the formula. All steps are standard textbook procedures with no problem-solving insight required, making it easier than average but not trivial due to the multi-part nature and need for careful algebraic manipulation. |
| Spec | 1.02f Solve quadratic equations: including in a function of unknown1.02j Manipulate polynomials: expanding, factorising, division, factor theorem |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(f(-2) = 12 - 22(-2) + 9(-2)^2 - (-2)^3 = 12 + 44 + 36 + 8\) | M1 | M0 for using \(x = 2\); allow slips in evaluation as long as intention is clear; at least one of second or fourth terms must be of correct sign |
| \(= 100\) | A1 | Do not ISW if subsequently given as \(-100\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(f(3) = 12 - 66 + 81 - 27 = 0\) | B1 | \(12 - 22(3) + 9(3)^2 - (3)^3 = 0\) is enough; B0 for just stating \(f(3) = 0\); if using division must show 0 on last line or make equiv comment such as 'no remainder' |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Attempt complete division by \((3-x)\) or \((x-3)\) or equiv | M1 | Must be complete method; allow M1 if dividing \(x^3 - 9x^2 + 22x - 12\) by \((3-x)\); long division must subtract lower line (allow one slip); synthetic division must use 3 (not \(-3\)) and add within each column (allow one slip) |
| Obtain \(x^2 - 6x + 4\) or \(-x^2 + 6x - 4\) | A1 | Allow A1 even if division is inconsistent; must be explicit and not implied |
| \(f(x) = (3-x)(x^2 - 6x + 4)\) or \((x-3)(-x^2 + 6x - 4)\) | A1 | Must be written as a product; must come from a method with consistent signs in divisor and dividend |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(x = 3\) | B1 | At any point |
| Attempt to find roots of quadratic quotient | M1 | Can gain M1 if using incorrect quotient from (iii) as long as it is a three term quadratic from division attempt by \((3-x)\) or \((x-3)\) |
| \(x = 3 \pm \sqrt{5}\) | A1 | Must be in simplified surd form; allow A1 if from \(-f(x) = 0\) eg \((x-3)(x^2 - 6x + 4) = 0\) |
## Question 7:
### Part (i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $f(-2) = 12 - 22(-2) + 9(-2)^2 - (-2)^3 = 12 + 44 + 36 + 8$ | M1 | M0 for using $x = 2$; allow slips in evaluation as long as intention is clear; at least one of second or fourth terms must be of correct sign |
| $= 100$ | A1 | Do not ISW if subsequently given as $-100$ |
---
### Part (ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $f(3) = 12 - 66 + 81 - 27 = 0$ | B1 | $12 - 22(3) + 9(3)^2 - (3)^3 = 0$ is enough; B0 for just stating $f(3) = 0$; if using division must show 0 on last line or make equiv comment such as 'no remainder' |
---
### Part (iii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Attempt complete division by $(3-x)$ or $(x-3)$ or equiv | M1 | Must be complete method; allow M1 if dividing $x^3 - 9x^2 + 22x - 12$ by $(3-x)$; long division must subtract lower line (allow one slip); synthetic division must use 3 (not $-3$) and add within each column (allow one slip) |
| Obtain $x^2 - 6x + 4$ or $-x^2 + 6x - 4$ | A1 | Allow A1 even if division is inconsistent; must be explicit and not implied |
| $f(x) = (3-x)(x^2 - 6x + 4)$ or $(x-3)(-x^2 + 6x - 4)$ | A1 | Must be written as a product; must come from a method with consistent signs in divisor and dividend |
---
### Part (iv):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $x = 3$ | B1 | At any point |
| Attempt to find roots of quadratic quotient | M1 | Can gain M1 if using incorrect quotient from (iii) as long as it is a three term quadratic from division attempt by $(3-x)$ or $(x-3)$ |
| $x = 3 \pm \sqrt{5}$ | A1 | Must be in simplified surd form; allow A1 if from $-f(x) = 0$ eg $(x-3)(x^2 - 6x + 4) = 0$ |7 The cubic polynomial $\mathrm { f } ( x )$ is defined by $\mathrm { f } ( x ) = 12 - 22 x + 9 x ^ { 2 } - x ^ { 3 }$.\\
(i) Find the remainder when $\mathrm { f } ( x )$ is divided by $( x + 2 )$.\\
(ii) Show that ( $3 - x$ ) is a factor of $\mathrm { f } ( x )$.\\
(iii) Express $\mathrm { f } ( x )$ as the product of a linear factor and a quadratic factor.\\
(iv) Hence solve the equation $\mathrm { f } ( x ) = 0$, giving each root in simplified surd form where appropriate.
\hfill \mbox{\textit{OCR C2 2014 Q7 [9]}}