| Exam Board | Edexcel |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2023 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Simultaneous equations |
| Type | Linear simultaneous equations |
| Difficulty | Easy -1.3 This is a straightforward application of perimeter and area formulas for rectangles, leading to two simple simultaneous equations (one linear, one quadratic). The solution requires basic algebraic manipulation and solving a quadratic equation—standard GCSE/early A-level techniques with no conceptual challenges or novel problem-solving required. |
| Spec | 1.02c Simultaneous equations: two variables by elimination and substitution |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(2x + 2y = 350\) | B1 | Accept any correct equivalent e.g. \(2(x+y)=350\), \(x+y=175\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(xy = 7350\) | B1 | Accept any correct equivalent e.g. \(y = \frac{7350}{x}\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(x(175-x) = 7350\) or \((175-y)y = 7350\) | M1 | Substitutes to produce a 2 or 3 term quadratic in one variable. Terms do not need collecting. |
| \(x^2 - 175x + 7350 = 0 \Rightarrow (x-70)(x-105) = 0 \Rightarrow x = ...\) | dM1 | Depends on first M1. Solves 3TQ by any suitable method |
| \(x = 70\) or \(105\) | A1 | Correct simplified roots. Not concerned which is \(x\) and \(y\) for this mark |
| \((x > y \Rightarrow)\ x = 105,\ y = 70\) | A1 | Both \(x\) and \(y\) correct and correctly assigned, all previous marks scored |
# Question 2:
## Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $2x + 2y = 350$ | **B1** | Accept any correct equivalent e.g. $2(x+y)=350$, $x+y=175$ |
## Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $xy = 7350$ | **B1** | Accept any correct equivalent e.g. $y = \frac{7350}{x}$ |
## Part (c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $x(175-x) = 7350$ or $(175-y)y = 7350$ | **M1** | Substitutes to produce a 2 or 3 term quadratic in one variable. Terms do not need collecting. |
| $x^2 - 175x + 7350 = 0 \Rightarrow (x-70)(x-105) = 0 \Rightarrow x = ...$ | **dM1** | Depends on first M1. Solves 3TQ by any suitable method |
| $x = 70$ or $105$ | **A1** | Correct simplified roots. Not concerned which is $x$ and $y$ for this mark |
| $(x > y \Rightarrow)\ x = 105,\ y = 70$ | **A1** | Both $x$ and $y$ correct and correctly assigned, all previous marks scored |
**Special Case:** Trial and Improvement or correct answers with no quadratic formed: M1M1A0A0. Values wrong way round or both sets offered: M1M0A0A0.
---\begin{enumerate}
\item In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.
\end{enumerate}
A rectangular sports pitch has length $x$ metres and width $y$ metres, where $x > y$ Given that the perimeter of the pitch is 350 m ,\\
(a) write down an equation linking $x$ and $y$
Given also that the area of the pitch is $7350 \mathrm {~m} ^ { 2 }$\\
(b) write down a second equation linking $x$ and $y$\\
(c) hence find the value of $x$ and the value of $y$
\hfill \mbox{\textit{Edexcel P1 2023 Q2 [6]}}