| Exam Board | Edexcel |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2019 |
| Session | January |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Curve Sketching |
| Type | Polynomial intersection with algebra |
| Difficulty | Moderate -0.3 This is a straightforward P1 question involving standard curve sketching of factored polynomials (finding intercepts), equating two functions and simplifying algebraically, then solving a quadratic using the quadratic formula. All steps are routine techniques with no novel insight required, making it slightly easier than average for A-level. |
| Spec | 1.02f Solve quadratic equations: including in a function of unknown1.02n Sketch curves: simple equations including polynomials1.02q Use intersection points: of graphs to solve equations |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\cap\) shaped quadratic | B1 | General shape; do not be concerned with parts which appear linear |
| Intercepts at \(O\) and \(3\) | B1 | Quadratic crosses \(x\)-axis at \(O\) and \(3\); accept mark of 3 on \(x\)-axis; origin need not be labelled |
| Negative cubic | B1 | |
| Intercepts at \(O\), \(2\) and \(5\) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Sets \(x(x-2)(5-x) = x(3-x)\) | M1 | |
| \(3x - x^2 = -x^3 + 7x^2 - 10x \Rightarrow \pm(x^3 - 8x^2 + 13x)(=0)\) OR \(\pm x\{(x-2)(5-x)-(3-x)\}(=0)\) | dM1 | |
| Proceeds to \(x(x^2 - 8x + 13) = 0\) * | A1* | Reaches given answer with no errors |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Solves \(x^2 - 8x + 13 = 0 \Rightarrow x = 4 \pm \sqrt{3}\) | M1 A1 | |
| Substitutes \(x = "4 - \sqrt{3}"\) into \(y = x(3-x)\) | M1 | |
| \(y = (4-\sqrt{3})(-1+\sqrt{3}) = -4 + 4\sqrt{3} + \sqrt{3} - 3 = \ldots\) | M1 | |
| \(y = -7 + 5\sqrt{3}\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Negative cubic with maximum and minimum | B1 | General shape only; condone parts appearing linear |
| Cubic crossing \(x\)-axis at \(O\), \(2\) and \(5\) | B1 | Accept 2 and 5 marked on \(x\)-axis; origin need not be labelled |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Sets equations equal to each other | M1 | |
| Multiplies out and collects terms on one side (unsimplified); \(x\) must be factor of each term | dM1 | Condone errors in multiplying out; condone invisible brackets; condone absence of "=0" |
| Proceeds to \(x(x^2 - 8x + 13) = 0\) with no errors including brackets | A1* | Must see \((x-2)(5-x)\) multiplied out, terms collected, and factor of \(x\) taken out |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Solves \(x^2 - 8x + 13 = 0\) by completing the square or formula | M1 | Solutions need not be exact for this mark |
| \((x=)\ 4 \pm \sqrt{3}\) | A1 | Must be of form \(\frac{a \pm \sqrt{b}}{c}\) or simplified further |
| Substitutes \(x = 4 - \sqrt{3}\) (lower value) into either equation to find \(y\) | M1 | Can be awarded with rounded decimal solutions; check \(y\) on calculator if \(x\) value incorrect |
| Uses rules of surds to form exact, simplified \(y\)-coordinate; shows working with surds before simplifying; evidence of form \(d + f\sqrt{g}\) | M1 | Must show surd working e.g. from \(y=(a-\sqrt{b})(c+\sqrt{b}) = ac \mathbf{+a\sqrt{b}-c\sqrt{b}-b}\); if no working shown M0 A0 follow |
| \((4-\sqrt{3},\ -7+5\sqrt{3})\) or exact equivalent | A1 | Must be the only coordinate stated as final answer |
## Question 11(a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\cap$ shaped quadratic | B1 | General shape; do not be concerned with parts which appear linear |
| Intercepts at $O$ and $3$ | B1 | Quadratic crosses $x$-axis at $O$ and $3$; accept mark of 3 on $x$-axis; origin need not be labelled |
| Negative cubic | B1 | |
| Intercepts at $O$, $2$ and $5$ | B1 | |
---
## Question 11(b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Sets $x(x-2)(5-x) = x(3-x)$ | M1 | |
| $3x - x^2 = -x^3 + 7x^2 - 10x \Rightarrow \pm(x^3 - 8x^2 + 13x)(=0)$ OR $\pm x\{(x-2)(5-x)-(3-x)\}(=0)$ | dM1 | |
| Proceeds to $x(x^2 - 8x + 13) = 0$ * | A1* | Reaches given answer with no errors |
---
## Question 11(c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Solves $x^2 - 8x + 13 = 0 \Rightarrow x = 4 \pm \sqrt{3}$ | M1 A1 | |
| Substitutes $x = "4 - \sqrt{3}"$ into $y = x(3-x)$ | M1 | |
| $y = (4-\sqrt{3})(-1+\sqrt{3}) = -4 + 4\sqrt{3} + \sqrt{3} - 3 = \ldots$ | M1 | |
| $y = -7 + 5\sqrt{3}$ | A1 | |
# Mark Scheme Extraction
## Question (ii) - Cubic Graph:
| Answer/Working | Mark | Guidance |
|---|---|---|
| Negative cubic with maximum and minimum | B1 | General shape only; condone parts appearing linear |
| Cubic crossing $x$-axis at $O$, $2$ and $5$ | B1 | Accept 2 and 5 marked on $x$-axis; origin need not be labelled |
## Question (b) - Setting Equations Equal:
| Answer/Working | Mark | Guidance |
|---|---|---|
| Sets equations equal to each other | M1 | |
| Multiplies out and collects terms on one side (unsimplified); $x$ must be factor of each term | dM1 | Condone errors in multiplying out; condone invisible brackets; condone absence of "=0" |
| Proceeds to $x(x^2 - 8x + 13) = 0$ with no errors including brackets | A1* | Must see $(x-2)(5-x)$ multiplied out, terms collected, and factor of $x$ taken out |
## Question (c) - Solving Quadratic:
| Answer/Working | Mark | Guidance |
|---|---|---|
| Solves $x^2 - 8x + 13 = 0$ by completing the square or formula | M1 | Solutions need not be exact for this mark |
| $(x=)\ 4 \pm \sqrt{3}$ | A1 | Must be of form $\frac{a \pm \sqrt{b}}{c}$ or simplified further |
| Substitutes $x = 4 - \sqrt{3}$ (lower value) into either equation to find $y$ | M1 | Can be awarded with rounded decimal solutions; check $y$ on calculator if $x$ value incorrect |
| Uses rules of surds to form exact, simplified $y$-coordinate; shows working with surds before simplifying; evidence of form $d + f\sqrt{g}$ | M1 | Must show surd working e.g. from $y=(a-\sqrt{b})(c+\sqrt{b}) = ac \mathbf{+a\sqrt{b}-c\sqrt{b}-b}$; if no working shown M0 A0 follow |
| $(4-\sqrt{3},\ -7+5\sqrt{3})$ or exact equivalent | A1 | Must be the only coordinate stated as final answer |
---11. (a) On Diagram 1 sketch the graphs of
\begin{enumerate}[label=(\roman*)]
\item $y = x ( 3 - x )$
\item $y = x ( x - 2 ) ( 5 - x )$\\
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
$$y = x ( 3 - x ) \text { and } y = x ( x - 2 ) ( 5 - x )$$
are given by the solutions to the equation $x \left( x ^ { 2 } - 8 x + 13 \right) = 0$
The point $P$ lies on both curves. Given that $P$ lies in the first quadrant,\\
(c) find, using algebra and showing your working, the exact coordinates of $P$.\\
\begin{center}
\includegraphics[max width=\textwidth, alt={}]{c8f8d35d-c2dd-4a1f-a4bb-a4fa06413d12-23_824_1211_296_370}
\end{center}
\section*{Diagram 1}
\end{enumerate}
\hfill \mbox{\textit{Edexcel P1 2019 Q11 [12]}}