| Exam Board | Edexcel |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2019 |
| Session | January |
| Marks | 3 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Curve Sketching |
| Type | Area between curve and line |
| Difficulty | Moderate -0.8 This question requires finding intersection points and writing inequalities to define a region, but involves only straightforward algebraic manipulation (solving 2y=x with y=2x-x²/8, finding x-intercept by setting y=0) and interpreting a given diagram. It's a standard P1 exercise testing basic coordinate geometry and inequality notation with no problem-solving insight required, making it easier than average. |
| Spec | 1.02i Represent inequalities: graphically on coordinate plane |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Either \(2y \leq x\) or \(y \geq 2x - \frac{1}{8}x^2\) | B1 | Either inequality sufficient; may be written in equivalent correct form. NB: Inequalities cannot be in terms of R |
| \(2x - \frac{1}{8}x^2 = 0 \Rightarrow x = 16 \Rightarrow x < \ldots\) or \(x \leq \ldots\) | M1 | Attempts to find upper bound for \(x\). Solves quadratic intersection with \(x\)-axis, uses value to write \(x <\ldots\) or \(x \leq \ldots\). Do not need \(x=0\); ignore lower bound e.g. \(0 < x < \ldots\) |
| \(x < 16\), \(2y \leq x\) and \(y \geq 2x - \frac{1}{8}x^2\) | A1 | Allow \(A \leq x < 16\) where \(A \leq 12\). Note: \(y \leq \frac{x}{2}\) for \(2y \leq x\), or \(2x - \frac{1}{8}x^2 \leq y \leq \frac{x}{2}\) accepted |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Either \(2y < x\) or \(y > 2x - \frac{1}{8}x^2\) | B1 | |
| \(2x - \frac{1}{8}x^2 = 0 \Rightarrow x = 16 \Rightarrow x < \ldots\) or \(x \leq \ldots\) | M1 | |
| \(x \leq 16\), \(2y < x\) and \(y > 2x - \frac{1}{8}x^2\) | A1 |
## Question 4:
| Answer | Mark | Guidance |
|--------|------|----------|
| Either $2y \leq x$ or $y \geq 2x - \frac{1}{8}x^2$ | B1 | Either inequality sufficient; may be written in equivalent correct form. **NB: Inequalities cannot be in terms of R** |
| $2x - \frac{1}{8}x^2 = 0 \Rightarrow x = 16 \Rightarrow x < \ldots$ or $x \leq \ldots$ | M1 | Attempts to find upper bound for $x$. Solves quadratic intersection with $x$-axis, uses value to write $x <\ldots$ or $x \leq \ldots$. Do not need $x=0$; ignore lower bound e.g. $0 < x < \ldots$ |
| $x < 16$, $2y \leq x$ and $y \geq 2x - \frac{1}{8}x^2$ | A1 | Allow $A \leq x < 16$ where $A \leq 12$. Note: $y \leq \frac{x}{2}$ for $2y \leq x$, or $2x - \frac{1}{8}x^2 \leq y \leq \frac{x}{2}$ accepted |
**Alt1** (dashed = $\leq$ or $\geq$, solid = $<$ or $>$):
| Answer | Mark | Guidance |
|--------|------|----------|
| Either $2y < x$ or $y > 2x - \frac{1}{8}x^2$ | B1 | |
| $2x - \frac{1}{8}x^2 = 0 \Rightarrow x = 16 \Rightarrow x < \ldots$ or $x \leq \ldots$ | M1 | |
| $x \leq 16$, $2y < x$ and $y > 2x - \frac{1}{8}x^2$ | A1 | |
---4.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{c8f8d35d-c2dd-4a1f-a4bb-a4fa06413d12-08_857_857_251_548}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\end{figure}
Figure 1 shows a line $l _ { 1 }$ with equation $2 y = x$ and a curve $C$ with equation $y = 2 x - \frac { 1 } { 8 } x ^ { 2 }$ The region $R$, shown unshaded in Figure 1, is bounded by the line $l _ { 1 }$, the curve $C$ and a line $l _ { 2 }$
Given that $l _ { 2 }$ is parallel to the $y$-axis and passes through the intercept of $C$ with the positive $x$-axis, identify the inequalities that define $R$.\\
\hfill \mbox{\textit{Edexcel P1 2019 Q4 [3]}}