| Exam Board | Edexcel |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2015 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Curve Sketching |
| Type | Factorise then sketch |
| Difficulty | Moderate -0.8 This is a straightforward C1 question requiring basic factorisation (taking out common factor x, then difference of two squares), plotting x-intercepts, and using the distance formula. All techniques are routine with no problem-solving insight needed, making it easier than average but not trivial due to the multi-part structure. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.02n Sketch curves: simple equations including polynomials1.10f Distance between points: using position vectors |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(9x - 4x^3 = x(9-4x^2)\) or \(-x(4x^2-9)\) | B1 | Takes out common factor of \(x\) or \(-x\) correctly |
| \(9 - 4x^2 = (3+2x)(3-2x)\) or \(4x^2-9=(2x-3)(2x+3)\) | M1 | \(9-4x^2 = (\pm3\pm2x)(\pm3\pm2x)\) or \(4x^2-9=(\pm2x\pm3)(\pm2x\pm3)\) |
| \(9x - 4x^3 = x(3+2x)(3-2x)\) | A1 | Cao but allow equivalents e.g. \(x(-3-2x)(-3+2x)\) or \(-x(2x+3)(2x-3)\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Cubic shape with one max and one minimum | M1 | Correct \(\cap\cup\) or \(\cup\cap\) cubic shape |
| Any line or curve passing through (not touching) the origin | B1 | |
| Correct shape in all four quadrants passing through \((-1.5, 0)\) and \((1.5, 0)\) | A1 | Allow \((0,-1.5)\) and \((0,1.5)\) or just \(-1.5\) and \(1.5\) if positioned correctly. Must be on diagram. Allow \(\sqrt{\frac{9}{4}}\) for 1.5 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(A=(-2, 14)\), \(B=(1, 5)\) | B1 B1 | B1: \(y=14\) or \(y=5\); B1: \(y=14\) and \(y=5\). Must be seen or used in (c). |
| \((AB=)\sqrt{(-2-1)^2+(14-5)^2}\ (=\sqrt{90})\) | M1 | Correct use of Pythagoras including square root. Must be correct expression for their \(A\) and \(B\) if correct formula not quoted. |
| \(AB = 3\sqrt{10}\) | A1 | cao |
## Question 8:
### Part (a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $9x - 4x^3 = x(9-4x^2)$ or $-x(4x^2-9)$ | B1 | Takes out common factor of $x$ or $-x$ correctly |
| $9 - 4x^2 = (3+2x)(3-2x)$ or $4x^2-9=(2x-3)(2x+3)$ | M1 | $9-4x^2 = (\pm3\pm2x)(\pm3\pm2x)$ or $4x^2-9=(\pm2x\pm3)(\pm2x\pm3)$ |
| $9x - 4x^3 = x(3+2x)(3-2x)$ | A1 | Cao but allow equivalents e.g. $x(-3-2x)(-3+2x)$ or $-x(2x+3)(2x-3)$ |
> Note: $9x(1-\frac{2}{3}x)(1+\frac{2}{3}x)$ scores full marks.
### Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Cubic shape with one max and one minimum | M1 | Correct $\cap\cup$ or $\cup\cap$ cubic shape |
| Any line or curve passing through (not touching) the origin | B1 | |
| Correct shape in all four quadrants passing through $(-1.5, 0)$ and $(1.5, 0)$ | A1 | Allow $(0,-1.5)$ and $(0,1.5)$ or just $-1.5$ and $1.5$ if positioned correctly. Must be on diagram. Allow $\sqrt{\frac{9}{4}}$ for 1.5 |
### Part (c):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $A=(-2, 14)$, $B=(1, 5)$ | B1 B1 | B1: $y=14$ or $y=5$; B1: $y=14$ and $y=5$. Must be seen or used in (c). |
| $(AB=)\sqrt{(-2-1)^2+(14-5)^2}\ (=\sqrt{90})$ | M1 | Correct use of Pythagoras including square root. Must be correct expression for their $A$ and $B$ if correct formula not quoted. |
| $AB = 3\sqrt{10}$ | A1 | cao |
> Special case: Use of $4x^3-9x$ gives $(-2,-14)$ and $(1,-5)$: allow max B0B0M1A1.
---\begin{enumerate}
\item (a) Factorise completely $9 x - 4 x ^ { 3 }$\\
(b) Sketch the curve $C$ with equation
\end{enumerate}
$$y = 9 x - 4 x ^ { 3 }$$
Show on your sketch the coordinates at which the curve meets the $x$-axis.
The points $A$ and $B$ lie on $C$ and have $x$ coordinates of - 2 and 1 respectively.\\
(c) Show that the length of $A B$ is $k \sqrt { } 10$ where $k$ is a constant to be found.
\hfill \mbox{\textit{Edexcel C1 2015 Q8 [10]}}