| Exam Board | Edexcel |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2012 |
| Session | January |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Curve Sketching |
| Type | Sketch then find derivative/gradient/tangent |
| Difficulty | Moderate -0.8 This is a straightforward C1 question requiring routine differentiation using the product rule (or expansion), basic curve sketching of a cubic with clear roots, and gradient evaluation at specific points. Part (d) involves recognizing a horizontal translation. All techniques are standard with no problem-solving insight required, making it easier than average. |
| Spec | 1.02n Sketch curves: simple equations including polynomials1.07i Differentiate x^n: for rational n and sums1.07m Tangents and normals: gradient and equations1.07q Product and quotient rules: differentiation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \([y = x^3 + 2x^2]\) so \(\frac{dy}{dx} = 3x^2 + 4x\) | M1A1 | M1 for attempt to multiply out then differentiate \(x^n \to x^{n-1}\); do not award for \(2x(x+2)\) or \(2x(1+2)\) etc; A1 both terms correct (if \(+c\) or extra term included score A0) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Correct shape \(\smallsmile\) | B1 | Must have 2 clear turning points |
| Touching \(x\)-axis at origin | B1 | Graph touching at origin (not crossing or ending) |
| Through and not touching or stopping at \(-2\) on \(x\)-axis | B1 | Passing through (not stopping/touching at) \(-2\); SC: B0B0B1 for \(y=x^3\) or cubic with straight line between \((-2,0)\) and \((0,0)\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| At \(x=-2\): \(\frac{dy}{dx} = 3(-2)^2 + 4(-2) = 4\) | M1 | For attempt at \(y'(0)\) or \(y'(-2)\); follow through their 0 or \(-2\) and their \(y'(x)\); or correct statement of zero gradient for identified point touching \(x\)-axis |
| At \(x=0\): \(\frac{dy}{dx} = 0\) (Both values correct) | A1 | For both correct answers |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Horizontal translation (touches \(x\)-axis still) | M1 | For horizontal translation of their (b); should still touch \(x\)-axis |
| \(k-2\) and \(k\) marked on positive \(x\)-axis | B1 | For \(k\) and \(k-2\) on positive \(x\)-axis; curve must pass through \(k-2\) and touch at \(k\) |
| \(k^2(2-k)\) (o.e.) marked on negative \(y\)-axis | B1 | For correct intercept on negative \(y\)-axis in terms of \(k\); allow \((0, 2k^2-k^3)\) |
## Question 8:
**(a)**
| Answer/Working | Marks | Guidance |
|---|---|---|
| $[y = x^3 + 2x^2]$ so $\frac{dy}{dx} = 3x^2 + 4x$ | M1A1 | M1 for attempt to multiply out then differentiate $x^n \to x^{n-1}$; do not award for $2x(x+2)$ or $2x(1+2)$ etc; A1 both terms correct (if $+c$ or extra term included score A0) |
**(b)**
| Answer/Working | Marks | Guidance |
|---|---|---|
| Correct shape $\smallsmile$ | B1 | Must have 2 clear turning points |
| Touching $x$-axis at origin | B1 | Graph touching at origin (not crossing or ending) |
| Through and not touching or stopping at $-2$ on $x$-axis | B1 | Passing through (not stopping/touching at) $-2$; SC: B0B0B1 for $y=x^3$ or cubic with straight line between $(-2,0)$ and $(0,0)$ |
**(c)**
| Answer/Working | Marks | Guidance |
|---|---|---|
| At $x=-2$: $\frac{dy}{dx} = 3(-2)^2 + 4(-2) = 4$ | M1 | For attempt at $y'(0)$ or $y'(-2)$; follow through their 0 or $-2$ and their $y'(x)$; or correct statement of zero gradient for identified point touching $x$-axis |
| At $x=0$: $\frac{dy}{dx} = 0$ (Both values correct) | A1 | For both correct answers |
**(d)**
| Answer/Working | Marks | Guidance |
|---|---|---|
| Horizontal translation (touches $x$-axis still) | M1 | For horizontal translation of their (b); should still touch $x$-axis |
| $k-2$ and $k$ marked on positive $x$-axis | B1 | For $k$ and $k-2$ on positive $x$-axis; curve must pass through $k-2$ and touch at $k$ |
| $k^2(2-k)$ (o.e.) marked on negative $y$-axis | B1 | For correct intercept on negative $y$-axis in terms of $k$; allow $(0, 2k^2-k^3)$ |
---8. The curve $C _ { 1 }$ has equation
$$y = x ^ { 2 } ( x + 2 )$$
\begin{enumerate}[label=(\alph*)]
\item Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$
\item Sketch $C _ { 1 }$, showing the coordinates of the points where $C _ { 1 }$ meets the $x$-axis.
\item Find the gradient of $C _ { 1 }$ at each point where $C _ { 1 }$ meets the $x$-axis.
The curve $C _ { 2 }$ has equation
$$y = ( x - k ) ^ { 2 } ( x - k + 2 )$$
where $k$ is a constant and $k > 2$
\item Sketch $C _ { 2 }$, showing the coordinates of the points where $C _ { 2 }$ meets the $x$ and $y$ axes.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C1 2012 Q8 [10]}}