| Exam Board | Edexcel |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2013 |
| Session | January |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Curve Sketching |
| Type | Sketch rational function from transformation |
| Difficulty | Moderate -0.8 This is a straightforward C1 transformation question requiring a vertical translation of a given reciprocal curve, sketching a linear function, and solving a simple quadratic equation. All techniques are routine with no problem-solving insight needed, making it easier than average but not trivial due to the multi-part nature and need for accuracy in multiple representations. |
| Spec | 1.02o Sketch reciprocal curves: y=a/x and y=a/x^21.02q Use intersection points: of graphs to solve equations1.02w Graph transformations: simple transformations of f(x) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(y = \frac{2}{x}\) translated up or down | M1 | Curve implies \(y\)-axis as asymptote, does not change shape significantly; horizontal asymptote implied |
| \(y = \frac{2}{x} - 5\) in correct position | A1 | Crosses positive \(x\)-axis; both sections move by almost same amount |
| Intersection with \(x\)-axis at \(\left(\frac{2}{5}, 0\right)\) only | B1 | Independent mark; accept \(2/5\) or \(0.4\) shown on \(x\)-axis |
| \(y = 4x+2\): straight line, positive gradient, positive \(y\)-intercept | B1 | Must be attempt at straight line with positive gradient and positive \(y\)-intercept |
| Intersection with \(x\)-axis at \(\left(-\frac{1}{2}, 0\right)\) and \(y\)-axis at \((0, 2)\) | B1 | Accept \(x=-1/2\) or \(-0.5\) and \(y=2\) on axes |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Asymptote \(x=0\) (or \(y\)-axis) stated correctly | B1 | Independent of (a) |
| Asymptote \(y=-5\) stated correctly | B1 | These two lines only; not to fit their graph; lose second B mark for extra asymptotes |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\frac{2}{x} - 5 = 4x+2\) leading to \(4x^2 + 7x - 2 = 0\) | M1 | Either correct equation set up |
| Attempt to solve 3-term quadratic | dM1 | Factorising, formula, completing the square, or implied by correct answers; depends on previous mark |
| \(x = -2,\ \frac{1}{4}\) | A1 | Need both correct \(x\) values |
| When \(x=-2,\ y=-6\); when \(x=\frac{1}{4},\ y=3\) | M1A1 | M1: at least one attempt to find second variable; A1: both correct second variable answers |
## Question 6:
### Part (a)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $y = \frac{2}{x}$ translated up or down | M1 | Curve implies $y$-axis as asymptote, does not change shape significantly; horizontal asymptote implied |
| $y = \frac{2}{x} - 5$ in correct position | A1 | Crosses positive $x$-axis; both sections move by almost same amount |
| Intersection with $x$-axis at $\left(\frac{2}{5}, 0\right)$ only | B1 | Independent mark; accept $2/5$ or $0.4$ shown on $x$-axis |
| $y = 4x+2$: straight line, positive gradient, positive $y$-intercept | B1 | Must be attempt at straight line with positive gradient and positive $y$-intercept |
| Intersection with $x$-axis at $\left(-\frac{1}{2}, 0\right)$ and $y$-axis at $(0, 2)$ | B1 | Accept $x=-1/2$ or $-0.5$ and $y=2$ on axes |
### Part (b)
| Answer/Working | Mark | Guidance |
|---|---|---|
| Asymptote $x=0$ (or $y$-axis) stated correctly | B1 | Independent of (a) |
| Asymptote $y=-5$ stated correctly | B1 | These two lines only; not to fit their graph; lose second B mark for extra asymptotes |
### Part (c)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\frac{2}{x} - 5 = 4x+2$ leading to $4x^2 + 7x - 2 = 0$ | M1 | Either correct equation set up |
| Attempt to solve 3-term quadratic | dM1 | Factorising, formula, completing the square, or implied by correct answers; depends on previous mark |
| $x = -2,\ \frac{1}{4}$ | A1 | Need both correct $x$ values |
| When $x=-2,\ y=-6$; when $x=\frac{1}{4},\ y=3$ | M1A1 | M1: at least one attempt to find second variable; A1: both correct second variable answers |
---6.
\begin{tikzpicture}[>=latex]
% Draw horizontal x-axis
\draw[->] (-4.5, 0) -- (4.5, 0) node[below] {$x$};
% Draw vertical y-axis
\draw[->] (0, -4.5) -- (0, 4.5) node[left] {$y$};
% Origin label
\node[below left] at (0, 0) {$O$};
% Plot the hyperbola branches: y = 2/x
% Top-right branch (Quadrant 1)
\draw[domain=0.28:4.2, smooth, samples=100] plot (\x, {1.2/\x});
% Bottom-left branch (Quadrant 3)
\draw[domain=-4.2:-0.28, smooth, samples=100] plot (\x, {1.2/\x});
\end{tikzpicture}
Figure 1 shows a sketch of the curve with equation $y = \frac { 2 } { x } , x \neq 0$
The curve $C$ has equation $y = \frac { 2 } { x } - 5 , x \neq 0$, and the line $l$ has equation $y = 4 x + 2$
\begin{enumerate}[label=(\alph*)]
\item Sketch and clearly label the graphs of $C$ and $l$ on a single diagram.
On your diagram, show clearly the coordinates of the points where $C$ and $l$ cross the coordinate axes.
\item Write down the equations of the asymptotes of the curve $C$.
\item Find the coordinates of the points of intersection of $y = \frac { 2 } { x } - 5$ and $y = 4 x + 2$
\end{enumerate}
\hfill \mbox{\textit{Edexcel C1 2013 Q6 [12]}}