| Exam Board | Edexcel |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2019 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Inequalities |
| Type | Perimeter or area constraint inequality |
| Difficulty | Moderate -0.8 This is a straightforward P1 inequality question requiring basic perimeter and area calculations from a diagram, forming a linear inequality (part a) and a quadratic inequality (part b), then combining constraints. The algebraic manipulation is routine and the problem-solving is guided by the question structure with no novel insight required. |
| Spec | 1.02g Inequalities: linear and quadratic in single variable1.02h Express solutions: using 'and', 'or', set and interval notation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Perimeter \(= 2\times 5x + 2\times(6x-2)\) | M1 | Attempt at perimeter. Scored for sight of \(5x+2x-1+2x+6x-2+\) additional term(s) involving \(x\). Individual lengths may not be seen, so imply for sight of total \(ax+b\) where \(a>15\) |
| \(2\times5x + 2\times(6x-2) > 29 \Rightarrow 22x > 33\) | dM1 | Sets \(P>29\) and attempts to solve proceeding to \(ax>c\) |
| \(x > \frac{33}{22} \Rightarrow x > 1.5\) | A1* | cso with at least one correct intermediate (simplified) line \(22x>33\) or \(x>\frac{33}{22}\) before \(x>1.5\) seen |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Area \(= 2x(2x-1) + 3x(6x-2)\) | M1 | Attempt at area of garden. Look for sum of two areas \(2x(2x-1)+\ldots x(6x-2)\) condoning slips |
| \(A < 72 \Rightarrow 22x^2 - 8x - 72 < 0\) | A1 | Correct and simplified equality or inequality, condoning \(\leftrightarrow\). E.g. \(22x^2-8x-72<0\) oe |
| \(11x^2 - 4x - 36 \Rightarrow x = -\frac{18}{11}, 2\) | M1 | Valid attempt to find critical values of their 3TQ. Allow factorisation, formula, completing the square or calculator |
| Chooses inside region | ddM1 | Dependent on both M's. Choosing inside region for their critical values. Condone replacing negative root with 0, 0.5 or 1.5 |
| \(-\frac{18}{11} < x < 2\) | A1 | Allow \(0 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(1.5 < x < 2\) | B1 | Accept \((1.5, 2)\), \(x>1.5\) and \(x<2\), \(x>1.5 \cap x<2\). Do not allow \(x>1.5\) or \(x<2\); \(x>1.5,\ x<2\) |
## Question 3:
### Part (a)
| Answer/Working | Marks | Guidance |
|---|---|---|
| Perimeter $= 2\times 5x + 2\times(6x-2)$ | M1 | Attempt at perimeter. Scored for sight of $5x+2x-1+2x+6x-2+$ additional term(s) involving $x$. Individual lengths may not be seen, so imply for sight of total $ax+b$ where $a>15$ |
| $2\times5x + 2\times(6x-2) > 29 \Rightarrow 22x > 33$ | dM1 | Sets $P>29$ and attempts to solve proceeding to $ax>c$ |
| $x > \frac{33}{22} \Rightarrow x > 1.5$ | A1* | cso with at least one correct intermediate (simplified) line $22x>33$ or $x>\frac{33}{22}$ before $x>1.5$ seen |
**(3 marks)**
### Part (b)
| Answer/Working | Marks | Guidance |
|---|---|---|
| Area $= 2x(2x-1) + 3x(6x-2)$ | M1 | Attempt at area of garden. Look for sum of two areas $2x(2x-1)+\ldots x(6x-2)$ condoning slips |
| $A < 72 \Rightarrow 22x^2 - 8x - 72 < 0$ | A1 | Correct and simplified equality or inequality, condoning $\leftrightarrow$. E.g. $22x^2-8x-72<0$ oe |
| $11x^2 - 4x - 36 \Rightarrow x = -\frac{18}{11}, 2$ | M1 | Valid attempt to find critical values of their 3TQ. Allow factorisation, formula, completing the square or calculator |
| Chooses inside region | ddM1 | Dependent on both M's. Choosing inside region for their critical values. Condone replacing negative root with 0, 0.5 or 1.5 |
| $-\frac{18}{11} < x < 2$ | A1 | Allow $0<x<2$ or $0.5<x<2$ due to context |
**(5 marks)**
### Part (c)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $1.5 < x < 2$ | B1 | Accept $(1.5, 2)$, $x>1.5$ and $x<2$, $x>1.5 \cap x<2$. Do not allow $x>1.5$ or $x<2$; $x>1.5,\ x<2$ |
**(1 mark)**
---3.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{5eee32af-9b0e-428c-bbc6-1feef44e0e1e-06_881_974_255_495}
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\caption{Figure 1}
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\end{figure}
Figure 1 shows the plan of a garden. The marked angles are right angles.\\
The six edges are straight lines.\\
The lengths shown in the diagram are given in metres.
Given that the perimeter of the garden is greater than 29 m ,
\begin{enumerate}[label=(\alph*)]
\item show that $x > 1.5 \mathrm {~m}$
Given also that the area of the garden is less than $72 \mathrm {~m} ^ { 2 }$,
\item form and solve a quadratic inequality in $x$.
\item Hence state the range of possible values of $x$.\\
\href{http://www.dynamicpapers.com}{www.dynamicpapers.com}
\begin{center}
\end{center}
\end{enumerate}
\hfill \mbox{\textit{Edexcel P1 2019 Q3 [9]}}