| Exam Board | Edexcel |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2010 |
| Session | June |
| Marks | 15 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Curve Sketching |
| Type | Polynomial intersection with algebra |
| Difficulty | Moderate -0.3 This is a structured multi-part question requiring curve sketching from factored forms (straightforward), algebraic manipulation to find intersection points (routine), and solving a quadratic to find exact coordinates in surd form. While part (c) requires completing the square or the quadratic formula with surds, this is standard C1 technique with clear scaffolding through parts (a) and (b). Slightly easier than average due to the guided structure and routine algebraic skills required. |
| Spec | 1.02b Surds: manipulation and rationalising denominators1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.02n Sketch curves: simple equations including polynomials1.02p Interpret algebraic solutions: graphically1.02q Use intersection points: of graphs to solve equations |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\cap\) shape (anywhere on diagram) | B1 | |
| Passing through or stopping at \((0,0)\) and \((4,0)\) only (needn't be \(\cap\) shape) | B1 | |
| Correct shape (\(-\)ve cubic) with a max and min drawn anywhere | B1 | |
| Minimum or maximum at \((0,0)\) | B1 | |
| Passes through or stops at \((7,0)\) but NOT touching; \((7,0)\) should be to right of \((4,0)\) | B1 | Condone \((0,4)\) or \((0,7)\) marked correctly on \(x\)-axis; points must be marked on sketch not in text |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(x(4-x) = x^2(7-x)\), \((0=)x[7x - x^2 - (4-x)]\) | M1 | For forming a suitable equation |
| \((0=)x[7x - x^2 - (4-x)]\) (o.e.) | B1ft | Common factor of \(x\) taken out legitimately |
| \(0 = x(x^2 - 8x + 4)\) | A1 cso | No incorrect working seen; "= 0" required; cancelling \(x\) scores B0A0 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(x = \frac{8 \pm \sqrt{64-16}}{2}\) or \((x \pm 4)^2 - 4^2 + 4 = 0\) | M1 | Some use of correct formula or attempt to complete the square |
| \((x-4)^2 = 12\) | A1 | Fully correct expression; condone \(+\) instead of \(\pm\) |
| \(= \frac{8 \pm 4\sqrt{3}}{2}\) or \((x-4) = \pm 2\sqrt{3}\) | B1 | For simplifying \(\sqrt{48} = 4\sqrt{3}\) or \(\sqrt{12} = 2\sqrt{3}\) |
| \(x = 4 \pm 2\sqrt{3}\) | A1 | Correct solution of form \(p + q\sqrt{3}\) |
| From sketch \(A\) is \(x = 4 - 2\sqrt{3}\) | M1 | Selecting answer in interval \((0,4)\); if no value in \((0,4)\) score M0 |
| \(y = (4 - 2\sqrt{3})(4 - [4 - 2\sqrt{3}])\) | M1 | Attempt at \(y = \ldots\) using \(x\) in correct equation |
| \(= -12 + 8\sqrt{3}\) | A1 | Correct answer; if 2 answers given, A0 |
## Question 10:
### Part (a):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\cap$ shape (anywhere on diagram) | B1 | |
| Passing through or stopping at $(0,0)$ and $(4,0)$ only (needn't be $\cap$ shape) | B1 | |
| Correct shape ($-$ve cubic) with a max and min drawn anywhere | B1 | |
| Minimum or maximum at $(0,0)$ | B1 | |
| Passes through or stops at $(7,0)$ but NOT touching; $(7,0)$ should be to right of $(4,0)$ | B1 | Condone $(0,4)$ or $(0,7)$ marked correctly on $x$-axis; points must be marked on sketch not in text |
### Part (b):
| Answer | Mark | Guidance |
|--------|------|----------|
| $x(4-x) = x^2(7-x)$, $(0=)x[7x - x^2 - (4-x)]$ | M1 | For forming a suitable equation |
| $(0=)x[7x - x^2 - (4-x)]$ (o.e.) | B1ft | Common factor of $x$ taken out legitimately |
| $0 = x(x^2 - 8x + 4)$ | A1 cso | No incorrect working seen; "= 0" required; cancelling $x$ scores B0A0 |
### Part (c):
| Answer | Mark | Guidance |
|--------|------|----------|
| $x = \frac{8 \pm \sqrt{64-16}}{2}$ or $(x \pm 4)^2 - 4^2 + 4 = 0$ | M1 | Some use of correct formula or attempt to complete the square |
| $(x-4)^2 = 12$ | A1 | Fully correct expression; condone $+$ instead of $\pm$ |
| $= \frac{8 \pm 4\sqrt{3}}{2}$ or $(x-4) = \pm 2\sqrt{3}$ | B1 | For simplifying $\sqrt{48} = 4\sqrt{3}$ or $\sqrt{12} = 2\sqrt{3}$ |
| $x = 4 \pm 2\sqrt{3}$ | A1 | Correct solution of form $p + q\sqrt{3}$ |
| From sketch $A$ is $x = 4 - 2\sqrt{3}$ | M1 | Selecting answer in interval $(0,4)$; if no value in $(0,4)$ score M0 |
| $y = (4 - 2\sqrt{3})(4 - [4 - 2\sqrt{3}])$ | M1 | Attempt at $y = \ldots$ using $x$ in correct equation |
| $= -12 + 8\sqrt{3}$ | A1 | Correct answer; if 2 answers given, A0 |
---10. (a) On the axes below sketch the graphs of
\begin{enumerate}[label=(\roman*)]
\item $y = x ( 4 - x )$
\item $y = x ^ { 2 } ( 7 - x )$\\
showing clearly the coordinates of the points where the curves cross the coordinate axes.\\
(b) Show that the $x$-coordinates of the points of intersection of
$$y = x ( 4 - x ) \text { and } y = x ^ { 2 } ( 7 - x )$$
are given by the solutions to the equation $x \left( x ^ { 2 } - 8 x + 4 \right) = 0$
The point $A$ lies on both of the curves and the $x$ and $y$ coordinates of $A$ are both positive.\\
(c) Find the exact coordinates of $A$, leaving your answer in the form ( $p + q \sqrt { } 3 , r + s \sqrt { } 3$ ), where $p , q , r$ and $s$ are integers.\\
\includegraphics[max width=\textwidth, alt={}, center]{65d61b2c-2e47-402e-b08f-2d46bb00c188-14_1178_1203_1407_379}
\end{enumerate}
\hfill \mbox{\textit{Edexcel C1 2010 Q10 [15]}}