| Exam Board | Edexcel |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2021 |
| Session | October |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Inequalities |
| Type | Write inequalities from graph |
| Difficulty | Moderate -0.3 Part (i) is a routine rational inequality requiring simple algebraic manipulation. Part (ii) requires finding the intersection point, determining the line equation, and writing two inequalities to define a region—standard P1 content but involves multiple coordinated steps. Overall slightly easier than average due to straightforward techniques. |
| Spec | 1.02g Inequalities: linear and quadratic in single variable1.02i Represent inequalities: graphically on coordinate plane |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\frac{3}{x} > 4 \Rightarrow 3x > 4x^2 \Rightarrow x(4x-3) < 0 \Rightarrow 0, \frac{3}{4}\) | B1 | For the two critical values \(0\) and \(\frac{3}{4}\) |
| \(0 < x < \frac{3}{4}\) | M1 | Chooses the inside region for their critical values |
| A1 | \(0 < x < \frac{3}{4}\). Award for exact equivalents such as \(x > 0\) and \(x < \frac{3}{4}\) or \(\left\{x : x > 0 \cap x < \frac{3}{4}\right\}\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(y - 0 = 3(x + 5)\) | B1 | Correct equation for line \(l\). May be implied by \(y > 3x + 15\) or \(y = 3x + k\) with \(k = 15\) |
| E.g. \(y < 2x^2 - 50\), \(y > 3x +\) "15" | M1 | Two of: \(y < 2x^2 - 50\), \(y > 3x +\) "15", \(x < a\) where \(-5 \lesssim a \lesssim 6.5\). Follow through on their straight line provided gradient is 3 with numerical "15" |
| E.g. \(y < 2x^2 - 50\), \(y > 3x + 15\), \(x < -5\) | A1 | Fully defines region. Also allow set notation with \(\cap\) between inequalities. Do not allow inequalities in terms of \(R\) |
## Question 3(i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\frac{3}{x} > 4 \Rightarrow 3x > 4x^2 \Rightarrow x(4x-3) < 0 \Rightarrow 0, \frac{3}{4}$ | B1 | For the two critical values $0$ and $\frac{3}{4}$ |
| $0 < x < \frac{3}{4}$ | M1 | Chooses the inside region for their critical values |
| | A1 | $0 < x < \frac{3}{4}$. Award for exact equivalents such as $x > 0$ and $x < \frac{3}{4}$ or $\left\{x : x > 0 \cap x < \frac{3}{4}\right\}$ |
---
## Question 3(ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $y - 0 = 3(x + 5)$ | B1 | Correct equation for line $l$. May be implied by $y > 3x + 15$ or $y = 3x + k$ with $k = 15$ |
| E.g. $y < 2x^2 - 50$, $y > 3x +$ "15" | M1 | Two of: $y < 2x^2 - 50$, $y > 3x +$ "15", $x < a$ where $-5 \lesssim a \lesssim 6.5$. Follow through on their straight line provided gradient is 3 with numerical "15" |
| E.g. $y < 2x^2 - 50$, $y > 3x + 15$, $x < -5$ | A1 | Fully defines region. Also allow set notation with $\cap$ between inequalities. Do not allow inequalities in terms of $R$ |
---3. (i) Solve\\
(ii)
$$\frac { 3 } { x } > 4$$
Figure 1 shows a sketch of the curve $C$ and the straight line $l$.\\
The infinite region $R$, shown shaded in Figure 1, lies in quadrants 2 and 3 and is bounded by $C$ and $l$ only.
Given that
\begin{itemize}
\item $\quad l$ has a gradient of 3
\item $C$ has equation $y = 2 x ^ { 2 } - 50$
\item $\quad C$ and $l$ intersect on the negative $x$-axis\\
use inequalities to define the region $R$.
\end{itemize}
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{f1e1d4f5-dd27-4839-a6f3-f6906666302c-06_643_652_575_648}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\end{figure}
Figure 1 shows a sketch of the curve $C$ and the straight line $l$.\\
The infinite region $R$, shown shaded in Figure 1, lies in quadrants 2 and 3 and is bounded by $C$ and $l$ only.\\
Given that
\begin{itemize}
\item $l$ has a gradient of 3
\item $C$ has equation $y = 2 x ^ { 2 } - 50$
\item $C$ and $l$ intersect on the negative $x$-axis\\
use inequalities to define the region $R$.\\
\end{itemize}
\begin{center}
\end{center}
\begin{itemize}
\item
\end{itemize}
\hfill \mbox{\textit{Edexcel P1 2021 Q3 [6]}}