| Exam Board | OCR MEI |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Curve Sketching |
| Type | Polynomial intersection with algebra |
| Difficulty | Moderate -0.3 This is a straightforward C1 question requiring standard curve sketching (expanding brackets, finding intercepts), simple algebraic manipulation to rearrange an equation, verification of a given root by substitution, and factorization followed by the quadratic formula. All techniques are routine for C1 with no novel problem-solving required, though the multi-part structure and surd answers make it slightly more substantial than the most basic exercises. |
| Spec | 1.02f Solve quadratic equations: including in a function of unknown1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.02n Sketch curves: simple equations including polynomials |
| Answer | Marks |
|---|---|
| Sketch of cubic the correct way up | G1 |
| Curve passing through \((0,0)\) | G1 |
| Curve touching \(x\) axis at \((3,0)\) | G1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(x(x^2 - 6x + 9) = 2\) | M1 | Or \((x^2-3x)(x-3) = 2\) [for one step in expanding brackets] |
| \(x^3 - 6x^2 + 9x = 2\) | M1 | For 2nd step, dep on first M1 |
| Answer | Marks | Guidance |
|---|---|---|
| Subst \(x=2\) in LHS of their equation or in \(x(x-3)^2 = 2\) o.e. working to show consistent | 1 | Or 2 for division of their eqn by \((x-2)\) and showing no remainder |
| Division of their eqn by \((x-2)\) attempted | M1 | Or inspection attempted with \((x^2 + kx + c)\) seen |
| \(x^2 - 4x + 1\) | A1 | |
| Solution of their quadratic by formula or completing square attempted | M1 | Condone ignoring remainder |
| \(x = 2 \pm \sqrt{3}\) or \(\frac{4 \pm \sqrt{12}}{2}\) isw | A2 | A1 for one correct; must be 3 intersections; condone \(x=2\) not marked |
| Locating the roots on intersection of their curve and \(y=2\) | G1 | Mark this when marking sketch graph in (i) |
## Question 4:
### Part (i)
| Sketch of cubic the correct way up | G1 | |
|---|---|---|
| Curve passing through $(0,0)$ | G1 | |
| Curve touching $x$ axis at $(3,0)$ | G1 | |
### Part (ii)
| $x(x^2 - 6x + 9) = 2$ | M1 | Or $(x^2-3x)(x-3) = 2$ [for one step in expanding brackets] |
|---|---|---|
| $x^3 - 6x^2 + 9x = 2$ | M1 | For 2nd step, dep on first M1 |
### Part (iii)
| Subst $x=2$ in LHS of their equation or in $x(x-3)^2 = 2$ o.e. working to show consistent | 1 | Or 2 for division of their eqn by $(x-2)$ and showing no remainder |
|---|---|---|
| Division of their eqn by $(x-2)$ attempted | M1 | Or inspection attempted with $(x^2 + kx + c)$ seen |
| $x^2 - 4x + 1$ | A1 | |
| Solution of their quadratic by formula or completing square attempted | M1 | Condone ignoring remainder |
| $x = 2 \pm \sqrt{3}$ or $\frac{4 \pm \sqrt{12}}{2}$ isw | A2 | A1 for one correct; must be 3 intersections; condone $x=2$ not marked |
| Locating the roots on intersection of their curve and $y=2$ | G1 | Mark this when marking sketch graph in (i) |
---4 (i) Sketch the graph of $y = x ( x - 3 ) ^ { 2 }$.\\
(ii) Show that the equation $x ( x - 3 ) ^ { 2 } = 2$ can be expressed as $x ^ { 3 } - 6 x ^ { 2 } + 9 x - 2 = 0$.\\
(iii) Show that $x = 2$ is one root of this equation and find the other two roots, expressing your answers in surd form.
Show the location of these roots on your sketch graph in part (i).
\hfill \mbox{\textit{OCR MEI C1 Q4 [13]}}