| Exam Board | Edexcel |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2019 |
| Session | October |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Curve Sketching |
| Type | Parameter values from curve properties |
| Difficulty | Standard +0.8 This question requires students to sketch a transformed reciprocal function, then use the tangent condition to set up and solve a discriminant equation (b² - 4ac = 0). While the individual techniques are standard P1 content, combining curve sketching with the tangent condition and solving the resulting quadratic discriminant problem requires solid algebraic manipulation and conceptual understanding beyond routine exercises. |
| Spec | 1.02n Sketch curves: simple equations including polynomials1.02o Sketch reciprocal curves: y=a/x and y=a/x^21.07m Tangents and normals: gradient and equations |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| Correct shape | B1 | Look for \(y=\frac{1}{x}\) curve translated in any direction. Tolerant with slips of pen near asymptotes, but curve must not bend back on itself |
| States asymptote as \(y = k\) | B1 | Curve has horizontal asymptote above \(x\)-axis with asymptote stated as \(y=k\) on diagram or in text. Do not accept just \(k\) marked on axis |
| States intercept as \(-\frac{4}{k}\) | B1 | Curve crosses (not just touches) negative \(x\)-axis, with intercept marked or stated as \(-\frac{4}{k}\) |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| \(10-2x = \frac{4}{x}+k \Rightarrow 10x-2x^2=4+kx\) | M1 | Equates and attempts to multiply by \(x\) obtaining terms in \(x^2\), \(x\) and constant(s) |
| \(\Rightarrow 2x^2+(k-10)x+4=0\) | A1 | This may be implied by a correct \(a\), \(b\) and \(c\) |
| Attempts \("b^2-4ac"=0 \Rightarrow (k-10)^2-4\times2\times4=0\) | M1 | Withhold if an inequality is applied |
| \(k = 10\pm4\sqrt{2}\) oe | M1 A1 | Do not accept decimal approximations and do not isw if an inequality is later stated |
# Question 6:
## Part (a)
| Working | Mark | Guidance |
|---------|------|----------|
| Correct shape | B1 | Look for $y=\frac{1}{x}$ curve translated in any direction. Tolerant with slips of pen near asymptotes, but curve must not bend back on itself |
| States asymptote as $y = k$ | B1 | Curve has horizontal asymptote above $x$-axis with asymptote **stated** as $y=k$ on diagram or in text. Do not accept just $k$ marked on axis |
| States intercept as $-\frac{4}{k}$ | B1 | Curve crosses (not just touches) negative $x$-axis, with intercept marked or stated as $-\frac{4}{k}$ |
## Part (b)
| Working | Mark | Guidance |
|---------|------|----------|
| $10-2x = \frac{4}{x}+k \Rightarrow 10x-2x^2=4+kx$ | M1 | Equates and attempts to multiply by $x$ obtaining terms in $x^2$, $x$ and constant(s) |
| $\Rightarrow 2x^2+(k-10)x+4=0$ | A1 | This may be implied by a correct $a$, $b$ and $c$ |
| Attempts $"b^2-4ac"=0 \Rightarrow (k-10)^2-4\times2\times4=0$ | M1 | Withhold if an inequality is applied |
| $k = 10\pm4\sqrt{2}$ oe | M1 A1 | Do not accept decimal approximations and do not isw if an inequality is later stated |6. The curve $C$ has equation $y = \frac { 4 } { x } + k$, where $k$ is a positive constant.
\begin{enumerate}[label=(\alph*)]
\item Sketch a graph of $C$, stating the equation of the horizontal asymptote and the coordinates of the point of intersection with the $x$-axis.
The line with equation $y = 10 - 2 x$ is a tangent to $C$.
\item Find the possible values for $k$.\\
$\_\_\_\_$ -
\end{enumerate}
\hfill \mbox{\textit{Edexcel P1 2019 Q6 [8]}}