| Exam Board | Edexcel |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2010 |
| Session | January |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Curve Sketching |
| Type | Factorise then sketch |
| Difficulty | Moderate -0.8 This is a straightforward C1 question testing basic factorisation (common factor then difference of squares), routine curve sketching with clear intercepts, finding a line through two points, and distance formula application. All parts are standard textbook exercises requiring only direct application of techniques with no problem-solving insight needed. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.02n Sketch curves: simple equations including polynomials1.03a Straight lines: equation forms y=mx+c, ax+by+c=01.10f Distance between points: using position vectors |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(x(x^2 - 4)\) | B1 | Factor \(x\) seen in correct factorised form |
| \(= x(x-2)(x+2)\) | M1 A1 | Attempt to factorise quadratic. Accept \((x-0)\) or \((x+0)\) instead of \(x\) at any stage |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Shape \(\bigcap\!\!\!\bigcup\) (2 turning points required) | B1 | 2nd and 3rd B marks not dependent on 1st B mark, but are dependent on a sketch having been attempted |
| Through (or touching) origin | B1 | |
| Crossing \(x\)-axis at \((-2, 0)\) and \((2, 0)\) | B1 | Not a turning point. Allow \(-2\) and \(2\) on \(x\)-axis. Also allow \((0,-2)\) and \((0,2)\) if marked on \(x\)-axis |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Either \(y = 3\) (at \(x = -1\)) or \(y = 15\) (at \(x = 3\)) | B1 | Allow if seen elsewhere |
| Gradient \(= \frac{\text{"15"}-\text{"3"}}{3-(-1)} (= 3)\) | M1 | Attempt correct gradient formula with their \(y\) values |
| \(y - \text{"15"} = m(x-3)\) or \(y - \text{"3"} = m(x-(-1))\) with any value of \(m\) | M1 | Equation of line through \((3, \text{"15"})\) or \((-1, \text{"3"})\) in any form |
| \(y - 15 = 3(x-3)\) or correct equation in any form | A1 | 1st A: correct equation in any form |
| \(y = 3x + 6\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(AB = \sqrt{(\text{"15"}-\text{"3"})^2 + (3-(-1))^2}\) | M1 | With their non-zero \(y\) values. Square root is required |
| \(= \sqrt{160} = 4\sqrt{10}\) | A1 | Ignore \(\pm\) if seen. \(\sqrt{16}\sqrt{10}\) need not be seen |
## Question 9:
**(a)**
| Answer/Working | Marks | Guidance |
|---|---|---|
| $x(x^2 - 4)$ | B1 | Factor $x$ seen in correct factorised form |
| $= x(x-2)(x+2)$ | M1 A1 | Attempt to factorise quadratic. Accept $(x-0)$ or $(x+0)$ instead of $x$ at any stage |
**(b)**
| Answer/Working | Marks | Guidance |
|---|---|---|
| Shape $\bigcap\!\!\!\bigcup$ (2 turning points required) | B1 | 2nd and 3rd B marks not dependent on 1st B mark, but are dependent on a sketch having been attempted |
| Through (or touching) origin | B1 | |
| Crossing $x$-axis at $(-2, 0)$ and $(2, 0)$ | B1 | Not a turning point. Allow $-2$ and $2$ on $x$-axis. Also allow $(0,-2)$ and $(0,2)$ if marked on $x$-axis |
**(c)**
| Answer/Working | Marks | Guidance |
|---|---|---|
| Either $y = 3$ (at $x = -1$) or $y = 15$ (at $x = 3$) | B1 | Allow if seen elsewhere |
| Gradient $= \frac{\text{"15"}-\text{"3"}}{3-(-1)} (= 3)$ | M1 | Attempt correct gradient formula with their $y$ values |
| $y - \text{"15"} = m(x-3)$ or $y - \text{"3"} = m(x-(-1))$ with any value of $m$ | M1 | Equation of line through $(3, \text{"15"})$ or $(-1, \text{"3"})$ in any form |
| $y - 15 = 3(x-3)$ or correct equation in any form | A1 | 1st A: correct equation in any form |
| $y = 3x + 6$ | A1 | |
**(d)**
| Answer/Working | Marks | Guidance |
|---|---|---|
| $AB = \sqrt{(\text{"15"}-\text{"3"})^2 + (3-(-1))^2}$ | M1 | With their non-zero $y$ values. Square root is required |
| $= \sqrt{160} = 4\sqrt{10}$ | A1 | Ignore $\pm$ if seen. $\sqrt{16}\sqrt{10}$ need not be seen |\begin{enumerate}
\item (a) Factorise completely $x ^ { 3 } - 4 x$\\
(b) Sketch the curve $C$ with equation
\end{enumerate}
$$y = x ^ { 3 } - 4 x ,$$
showing the coordinates of the points at which the curve meets the $x$-axis.
The point $A$ with $x$-coordinate - 1 and the point $B$ with $x$-coordinate 3 lie on the curve $C$.\\
(c) Find an equation of the line which passes through $A$ and $B$, giving your answer in the form $y = m x + c$, where $m$ and $c$ are constants.\\
(d) Show that the length of $A B$ is $k \sqrt { } 10$, where $k$ is a constant to be found.\\
\hfill \mbox{\textit{Edexcel C1 2010 Q9 [13]}}