| Exam Board | OCR MEI |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Curve Sketching |
| Type | Horizontal translation of cubic with root finding |
| Difficulty | Moderate -0.8 This is a straightforward C1 question testing basic understanding of roots, factorisation, and simple transformations. All parts follow standard procedures: writing factors from roots, expanding to find coefficients, sketching a cubic, and applying translations. The coefficient of x³ is given, making part (i) routine. The transformations in (iii) and (iv) are elementary applications of translation rules with no conceptual challenges. |
| Spec | 1.02j 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 |
|---|---|---|
| \((2x+1)(x+2)(x-5)\) | M1 | Or \((x+\frac{1}{2})(x+2)(x-5)\); need not be written as product; throughout, ignore \(=0\); for all Ms condone missing brackets if used correctly |
| Correct expansion of two linear factors of their product of three linear factors | M1 | — |
| Expansion of their linear and quadratic factors | M1 | Dep on first M1; ft one error in previous expansion; condone one error in this expansion; or for direct expansion of all three factors, allow M2 for \(2x^3 - 10x^2 + 4x^2 + x^2 - 20x - 5x + 2x - 10\) [or half all these], or M1 if one or two errors |
| \([y =]\ 2x^3 - 5x^2 - 23x - 10\) or \(a = -5,\ b = -23\) and \(c = -10\) | A1 | Condone poor notation when 'doubling' to reach expression with \(2x^3\); for attempt at setting up three simultaneous equations in \(a, b, c\): M1 for at least two equations e.g. \(250 + 25a + 5b + c = 0\), \(-16 + 4a - 2b + c = 0\), \(-\frac{1}{4} + \frac{1}{4}a - \frac{1}{2}b + c = 0\); then M2 for correctly eliminating any two variables or M1 for correctly eliminating one variable; then A1 for values |
| Answer | Marks | Guidance |
|---|---|---|
| Graph of cubic correct way up | B1 | Must not be ruled; no curving back; condone slight 'flicking out' at ends; allow min on \(y\)-axis or in 3rd/4th quadrants; condone some 'doubling' or 'feathering' |
| Crossing \(x\)-axis at \(-2,\ -\frac{1}{2}\) and \(5\) | B1 | B0 if stops at \(x\)-axis; mark intent for intersections with both axes; allow if no graph but marked on \(x\)-axis |
| Crossing \(y\)-axis at \(-10\) or ft their cubic in (i) | B1 | Or \(x=0, y=-10\) or ft; allow if no graph but B0 for graph nowhere near their indicated \(-10\) |
| Answer | Marks | Guidance |
|---|---|---|
| \((0, -18)\); accept \(-18\) or ft their constant \(-8\) | 1 mark | Or ft their intersection on \(y\)-axis \(-8\) |
| Answer | Marks | Guidance |
|---|---|---|
| Roots at \(2.5,\ 1,\ 8\) | M1 | Or attempt to substitute \((x-3)\) in \((2x+1)(x+2)(x-5)\) or in \((x+\frac{1}{2})(x+2)(x-5)\) or in their unfactorised form of \(f(x)\); attempt need not be simplified |
| \((2x-5)(x-1)(x-8)\) | A1 | Accept \(2(x-2.5)\) instead of \((2x-5)\); M0 for use of \((x+3)\) or roots \(-3.5, -5, 2\) but then allow SC1 for \((2x+7)(x+5)(x-2)\) |
| \((0, -40)\); accept \(-40\) | B2 | M1 for \(-5 \times -1 \times -8\) or ft or for \(f(-3)\) attempted or \(g(0)\) attempted or for their answer ft from their factorised form; e.g. M1 for \((0, -70)\) or \(-70\) after \((2x+7)(x+5)(x-2)\); after M0, allow SC1 for \(f(3) = -70\) |
## Question 5(i):
$(2x+1)(x+2)(x-5)$ | M1 | Or $(x+\frac{1}{2})(x+2)(x-5)$; need not be written as product; throughout, ignore $=0$; for all Ms condone missing brackets if used correctly
Correct expansion of two linear factors of their product of three linear factors | M1 | —
Expansion of their linear and quadratic factors | M1 | Dep on first M1; ft one error in previous expansion; condone one error in this expansion; or for direct expansion of all three factors, allow M2 for $2x^3 - 10x^2 + 4x^2 + x^2 - 20x - 5x + 2x - 10$ [or half all these], or M1 if one or two errors
$[y =]\ 2x^3 - 5x^2 - 23x - 10$ or $a = -5,\ b = -23$ and $c = -10$ | A1 | Condone poor notation when 'doubling' to reach expression with $2x^3$; for attempt at setting up three simultaneous equations in $a, b, c$: M1 for at least two equations e.g. $250 + 25a + 5b + c = 0$, $-16 + 4a - 2b + c = 0$, $-\frac{1}{4} + \frac{1}{4}a - \frac{1}{2}b + c = 0$; then M2 for correctly eliminating any two variables or M1 for correctly eliminating one variable; then A1 for values
**[4]**
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## Question 5(ii):
Graph of cubic correct way up | B1 | Must not be ruled; no curving back; condone slight 'flicking out' at ends; allow min on $y$-axis or in 3rd/4th quadrants; condone some 'doubling' or 'feathering'
Crossing $x$-axis at $-2,\ -\frac{1}{2}$ and $5$ | B1 | B0 if stops at $x$-axis; mark intent for intersections with both axes; allow if no graph but marked on $x$-axis
Crossing $y$-axis at $-10$ or ft their cubic in (i) | B1 | Or $x=0, y=-10$ or ft; allow if no graph but B0 for graph nowhere near their indicated $-10$
**[3]**
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## Question 5(iii):
$(0, -18)$; accept $-18$ or ft their constant $-8$ | 1 mark | Or ft their intersection on $y$-axis $-8$
**[1]**
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## Question 5(iv):
Roots at $2.5,\ 1,\ 8$ | M1 | Or attempt to substitute $(x-3)$ in $(2x+1)(x+2)(x-5)$ or in $(x+\frac{1}{2})(x+2)(x-5)$ or in their unfactorised form of $f(x)$; attempt need not be simplified
$(2x-5)(x-1)(x-8)$ | A1 | Accept $2(x-2.5)$ instead of $(2x-5)$; M0 for use of $(x+3)$ or roots $-3.5, -5, 2$ but then allow SC1 for $(2x+7)(x+5)(x-2)$
$(0, -40)$; accept $-40$ | B2 | M1 for $-5 \times -1 \times -8$ or ft or for $f(-3)$ attempted or $g(0)$ attempted or for their answer ft from their factorised form; e.g. M1 for $(0, -70)$ or $-70$ after $(2x+7)(x+5)(x-2)$; after M0, allow SC1 for $f(3) = -70$
**[4]**5 A cubic curve has equation $y = \mathrm { f } ( x )$. The curve crosses the $x$-axis where $x = - , \frac { 1 } { 2 }$ and 5 .\\
(i) Write down three linear factors of $\mathrm { f } ( x )$. Hence find the equation of the curve in the form $y = 2 x ^ { 3 } + a x ^ { 2 } + b x + c$.\\
(ii) Sketch the graph of $y = \mathrm { f } ( x )$.\\
(iii) The curve $y = \mathrm { f } ( x )$ is translated by $\binom { 0 } { - 8 }$. State the coordinates of the point where the translated curve intersects the $y$-axis.\\
(iv) The curve $y = \mathrm { f } ( x )$ is translated by $\binom { 3 } { 0 }$ to give the curve $y = \mathrm { g } ( x )$.
Find an expression in factorised form for $\mathrm { g } ( x )$ and state the coordinates of the point where the curve $y = \mathrm { g } ( x )$ intersects the $y$-axis.
\hfill \mbox{\textit{OCR MEI C1 Q5 [12]}}