| Exam Board | Edexcel |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2024 |
| Session | January |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Function Transformations |
| Type | Transformation of specific function type |
| Difficulty | Standard +0.8 This question combines routine sketching of a reciprocal function (part a) with a more challenging algebraic problem (part b) requiring students to use the discriminant condition for a line not intersecting a curve. Part (b) requires setting up a quadratic equation, applying discriminant < 0, and solving a quadratic inequality—going beyond standard textbook exercises. |
| Spec | 1.02n Sketch curves: simple equations including polynomials1.02o Sketch reciprocal curves: y=a/x and y=a/x^21.02q Use intersection points: of graphs to solve equations |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Shape in quadrant one (monotonically decreasing, not crossing axes) | M1 | Must not cross either axis; tolerant of functions not reaching \(x\)-axis or extending left to \(y\)-axis |
| Fully correct shape and position | A1 | Look for curve in quadrants 1, 3, 4 with vertical asymptote in Q1 and Q4; left branch not dipping below intercept |
| \(C\) cuts \(y\)-axis at \(-\frac{4}{k}\) | B1 | Must be on negative \(y\)-axis as \(-\frac{4}{k}\) or \(\left(0,-\frac{4}{k}\right)\) but not \(\left(-\frac{4}{k},0\right)\) |
| \(C\) has vertical asymptote at \(x = k\) | B1 | Must be to right of \(y\)-axis, marked \(x=k\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\frac{4}{x-k} = 9-x \Rightarrow x^2-(9+k)x+9k+4=0\) | M1, A1 | M1: equate curve and line, attempt quadratic; A1: simplified collected quadratic |
| Uses \(b^2-4ac < 0 \Rightarrow (9+k)^2 - 4\times1\times(9k+4)<0\) | dM1 | Attempts discriminant \(< 0\) with both \(b\) and \(c\) in \(k\) |
\(k^2-18k+65<0 \Rightarrow (k-13)(k-5)<0 \Rightarrow 5| ddM1, A1 |
ddM1: solves quadratic in \(k\), chooses inside region; A1: CSO \(5 |
|
# Question 7:
## Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Shape in quadrant one (monotonically decreasing, not crossing axes) | M1 | Must not cross either axis; tolerant of functions not reaching $x$-axis or extending left to $y$-axis |
| Fully correct shape and position | A1 | Look for curve in quadrants 1, 3, 4 with vertical asymptote in Q1 and Q4; left branch not dipping below intercept |
| $C$ cuts $y$-axis at $-\frac{4}{k}$ | B1 | Must be on negative $y$-axis as $-\frac{4}{k}$ or $\left(0,-\frac{4}{k}\right)$ but not $\left(-\frac{4}{k},0\right)$ |
| $C$ has vertical asymptote at $x = k$ | B1 | Must be to right of $y$-axis, marked $x=k$ |
## Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\frac{4}{x-k} = 9-x \Rightarrow x^2-(9+k)x+9k+4=0$ | M1, A1 | M1: equate curve and line, attempt quadratic; A1: simplified collected quadratic |
| Uses $b^2-4ac < 0 \Rightarrow (9+k)^2 - 4\times1\times(9k+4)<0$ | dM1 | Attempts discriminant $< 0$ with both $b$ and $c$ in $k$ |
| $k^2-18k+65<0 \Rightarrow (k-13)(k-5)<0 \Rightarrow 5<k<13$ | ddM1, A1 | ddM1: solves quadratic in $k$, chooses inside region; A1: CSO $5<k<13$; accept $k>5$ and $k<13$, $(5,13)$, $k\in(5,13)$; NOT $k>5$ or $k<13$ |\begin{enumerate}
\item (a) Sketch the graph of the curve $C$ with equation
\end{enumerate}
$$y = \frac { 4 } { x - k }$$
where $k$ is a positive constant.\\
Show on your sketch
\begin{itemize}
\item the coordinates of any points where $C$ cuts the coordinate axes
\item the equation of the vertical asymptote to $C$
\end{itemize}
Given that the straight line with equation $y = 9 - x$ does not cross or touch $C$\\
(b) find the range of values of $k$.
\hfill \mbox{\textit{Edexcel P1 2024 Q7 [9]}}