| Exam Board | OCR MEI |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2009 |
| Session | January |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Polynomial Division & Manipulation |
| Type | Division then Solve Polynomial Equation |
| Difficulty | Moderate -0.3 This is a structured multi-part question with clear guidance at each step. Part (ii) is straightforward algebraic manipulation (multiplying by x), part (iii) involves routine polynomial division given that x=1 is a root, followed by checking discriminant or using quadratic formula. All techniques are standard C1 content with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.02q Use intersection points: of graphs to solve equations1.07n Stationary points: find maxima, minima using derivatives |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Any correct \(y\) value calculated from quadratic seen or implied by plots | B1 | for \(x \neq 0\) or 1; may be for neg \(x\) or eg min at \((2.5, -1.25)\) |
| \((0,5)(1,1)(2,-1)(3,-1)(4,1)\) and \((5,5)\) plotted | P2 | tol 1mm; P1 for 4 correct [including \((2.5,-1.25)\) if plotted]; plots may be implied by curve within 1mm of correct position |
| Good quality smooth parabola within 1mm of their points | C1 | allow for correct points only; accept graph on graph paper, not insert |
| Total: 4 marks |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(x^2 - 5x + 5 = \frac{1}{x}\) | M1 | |
| \(x^3 - 5x^2 + 5x = 1\) and completion to given answer | M1 | |
| Total: 2 marks |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Division of \(x^3 - 5x^2 + 5x - 1\) by \(x - 1\) as far as \(x^3 - x^2\) used in working | M1 | or inspection e.g. \((x-1)(x^2\ldots+1)\) or equating coefficients with two correct coefficients found |
| \(x^2 - 4x + 1\) obtained | A1 | |
| Use of \(b^2 - 4ac\) or formula with quadratic factor | M1 | or \((x-2)^2 = 3\); may be implied by correct roots or \(\sqrt{12}\) obtained |
| \(\sqrt{12}\) obtained and comment re shows other roots (real and) irrational | A2 | [A1 for \(\sqrt{12}\) and A1 for comment]; NB A2 available only for correct quadratic factor used; if wrong factor used, allow A1 ft for obtaining two irrational roots or for their discriminant and comment re irrational [no ft if their discriminant is negative] |
| or \(2 \pm \sqrt{3}\) or \(\frac{4 \pm \sqrt{12}}{2}\) obtained isw | ||
| Total: 5 marks |
# Question 13:
## Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Any correct $y$ value calculated from quadratic seen or implied by plots | B1 | for $x \neq 0$ or 1; may be for neg $x$ or eg min at $(2.5, -1.25)$ |
| $(0,5)(1,1)(2,-1)(3,-1)(4,1)$ and $(5,5)$ plotted | P2 | tol 1mm; P1 for 4 correct [including $(2.5,-1.25)$ if plotted]; plots may be implied by curve within 1mm of correct position |
| Good quality smooth parabola within 1mm of their points | C1 | allow for correct points only; accept graph on graph paper, not insert |
| **Total: 4 marks** | | |
## Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $x^2 - 5x + 5 = \frac{1}{x}$ | M1 | |
| $x^3 - 5x^2 + 5x = 1$ and completion to given answer | M1 | |
| **Total: 2 marks** | | |
## Part (iii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Division of $x^3 - 5x^2 + 5x - 1$ by $x - 1$ as far as $x^3 - x^2$ used in working | M1 | or inspection e.g. $(x-1)(x^2\ldots+1)$ or equating coefficients with two correct coefficients found |
| $x^2 - 4x + 1$ obtained | A1 | |
| Use of $b^2 - 4ac$ or formula with quadratic factor | M1 | or $(x-2)^2 = 3$; may be implied by correct roots or $\sqrt{12}$ obtained |
| $\sqrt{12}$ obtained and comment re shows other roots (real and) irrational | A2 | [A1 for $\sqrt{12}$ and A1 for comment]; NB A2 available only for correct quadratic factor used; if wrong factor used, allow A1 ft for obtaining two irrational roots or for their discriminant and comment re irrational [no ft if their discriminant is negative] |
| or $2 \pm \sqrt{3}$ or $\frac{4 \pm \sqrt{12}}{2}$ obtained isw | | |
| **Total: 5 marks** | | |13 Answer part (i) of this question on the insert provided.
The insert shows the graph of $y = \frac { 1 } { x }$.\\
(i) On the insert, on the same axes, plot the graph of $y = x ^ { 2 } - 5 x + 5$ for $0 \leqslant x \leqslant 5$.\\
(ii) Show algebraically that the $x$-coordinates of the points of intersection of the curves $y = \frac { 1 } { x }$ and $y = x ^ { 2 } - 5 x + 5$ satisfy the equation $x ^ { 3 } - 5 x ^ { 2 } + 5 x - 1 = 0$.\\
(iii) Given that $x = 1$ at one of the points of intersection of the curves, factorise $x ^ { 3 } - 5 x ^ { 2 } + 5 x - 1$ into a linear and a quadratic factor.
Show that only one of the three roots of $x ^ { 3 } - 5 x ^ { 2 } + 5 x - 1 = 0$ is rational.
\hfill \mbox{\textit{OCR MEI C1 2009 Q13 [11]}}