| Exam Board | OCR MEI |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Curve Sketching |
| Type | Horizontal translation of cubic with root finding |
| Difficulty | Moderate -0.3 This is a multi-part question covering standard C1 techniques: expanding brackets, sketching a cubic from factored form, applying a horizontal translation, and solving a cubic equation with one given root. While it requires multiple steps, each component is routine and follows textbook procedures with no novel problem-solving required. The translation and root-finding are straightforward applications of standard methods, making it slightly easier than average. |
| 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 polynomials1.02w Graph transformations: simple transformations of f(x) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| expansion of one pair of brackets | M1 | eg \([(x+1)](x^2 - 6x + 8)\); need not be simplified |
| correct 6 term expansion | M1 | eg \(x^3 - 6x^2 + 8x + x^2 - 6x + 8\); or M2 for correct 8 term expansion: \(x^3 - 4x^2 + x^2 - 2x^2 + 8x - 4x - 2x + 8\), M1 if one error; allow equivalent marks working backwards to factorisation, by long division or factor theorem etc; or M1 for all three roots checked by factor theorem and M1 for comparing coeffts of \(x^3\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| cubic the correct way up | G1 | with two tps and extending beyond the axes at 'ends' |
| \(x\)-axis: \(-1, 2, 4\) shown | G1 | |
| \(y\)-axis \(8\) shown | G1 | ignore a second graph which is a translation of the correct graph |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \([y=](x-2)(x-5)(x-7)\) isw or \((x-3)^3 - 5(x-3)^2 + 2(x-3) + 8\) isw or \(x^3 - 14x^2 + 59x - 70\) | 2 | M1 if one slip or for \([y=]f(x-3)\) or for roots identified at \(2, 5, 7\) or for translation \(3\) to the left allow M1 for complete attempt: \((x+4)(x+1)(x-1)\) isw or \((x+3)^3 - 5(x+3)^2 + 2(x+3) + 8\) isw |
| \((0, -70)\) or \(y = -70\) | 1 | allow 1 for \((0,-4)\) or \(y=-4\) after \(f(x+3)\) used |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(27 - 45 + 6 + 8 = -4\) or \(27 - 45 + 6 + 12 = 0\) | B1 | or correct long division of \(x^3 - 5x^2 + 2x + 12\) by \((x-3)\) with no remainder or of \(x^3 - 5x^2 + 2x + 8\) with rem \(-4\) |
| long division of \(f(x)\) or their \(f(x)+4\) by \((x-3)\) attempted as far as \(x^3 - 3x^2\) in working | M1 | or inspection with two terms correct eg \((x-3)(x^2 \ldots - 4)\) |
| \(x^2 - 2x - 4\) obtained | A1 | |
| \([x=]\dfrac{2 \pm \sqrt{(-2)^2 - 4\times(-4)}}{2}\) or \((x-1)^2 = 5\) | M1 | dep on previous M1 earned; for attempt at formula or comp square on their other 'factor' |
| \(\dfrac{2 \pm \sqrt{20}}{2}\) o.e. isw or \(1 \pm \sqrt{5}\) | A1 |
## Question 3(iA):
| Answer/Working | Mark | Guidance |
|---|---|---|
| expansion of one pair of brackets | M1 | eg $[(x+1)](x^2 - 6x + 8)$; need not be simplified |
| correct 6 term expansion | M1 | eg $x^3 - 6x^2 + 8x + x^2 - 6x + 8$; or M2 for correct 8 term expansion: $x^3 - 4x^2 + x^2 - 2x^2 + 8x - 4x - 2x + 8$, M1 if one error; allow equivalent marks working backwards to factorisation, by long division or factor theorem etc; or M1 for all three roots checked by factor theorem and M1 for comparing coeffts of $x^3$ |
## Question 3(iB):
| Answer/Working | Mark | Guidance |
|---|---|---|
| cubic the correct way up | G1 | with two tps and extending beyond the axes at 'ends' |
| $x$-axis: $-1, 2, 4$ shown | G1 | |
| $y$-axis $8$ shown | G1 | ignore a second graph which is a translation of the correct graph |
## Question 3(iC):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $[y=](x-2)(x-5)(x-7)$ isw or $(x-3)^3 - 5(x-3)^2 + 2(x-3) + 8$ isw or $x^3 - 14x^2 + 59x - 70$ | 2 | M1 if one slip or for $[y=]f(x-3)$ or for roots identified at $2, 5, 7$ or for translation $3$ to the left allow M1 for complete attempt: $(x+4)(x+1)(x-1)$ isw or $(x+3)^3 - 5(x+3)^2 + 2(x+3) + 8$ isw |
| $(0, -70)$ or $y = -70$ | 1 | allow 1 for $(0,-4)$ or $y=-4$ after $f(x+3)$ used |
## Question 3(ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $27 - 45 + 6 + 8 = -4$ or $27 - 45 + 6 + 12 = 0$ | B1 | or correct long division of $x^3 - 5x^2 + 2x + 12$ by $(x-3)$ with no remainder or of $x^3 - 5x^2 + 2x + 8$ with rem $-4$ |
| long division of $f(x)$ or their $f(x)+4$ by $(x-3)$ attempted as far as $x^3 - 3x^2$ in working | M1 | or inspection with two terms correct eg $(x-3)(x^2 \ldots - 4)$ |
| $x^2 - 2x - 4$ obtained | A1 | |
| $[x=]\dfrac{2 \pm \sqrt{(-2)^2 - 4\times(-4)}}{2}$ or $(x-1)^2 = 5$ | M1 | dep on previous M1 earned; for attempt at formula or comp square on their other 'factor' |
| $\dfrac{2 \pm \sqrt{20}}{2}$ o.e. isw or $1 \pm \sqrt{5}$ | A1 | |
---3
\begin{enumerate}[label=(\roman*)]
\item You are given that $\mathrm { f } ( x ) = ( x + 1 ) ( x - 2 ) ( x - 4 )$.\\
(A) Show that $\mathrm { f } ( x ) = x ^ { 3 } - 5 x ^ { 2 } + 2 x + 8$.\\
(B) Sketch the graph of $y = \mathrm { f } ( x )$.\\
(C) The graph of $y = \mathrm { f } ( x )$ is translated by $\binom { 3 } { 0 }$.
State an equation for the resulting graph. You need not simplify your answer.\\
Find the coordinates of the point at which the resulting graph crosses the $y$-axis.
\item Show that 3 is a root of $x ^ { 3 } - 5 x ^ { 2 } + 2 x + 8 = - 4$. Hence solve this equation completely, giving the other roots in surd form.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI C1 Q3 [13]}}