| Exam Board | OCR |
|---|---|
| Module | H240/01 (Pure Mathematics) |
| Year | 2022 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Topic | Areas Between Curves |
| Type | Area with Inequality Constraints |
| Difficulty | Moderate -0.8 This is a straightforward multi-part question requiring standard techniques: solving simultaneous quadratic equations by equating and factorising, sketching a linear function, and shading an inequality region. All components are routine A-level procedures with no problem-solving insight required, making it easier than average. |
| Spec | 1.02c Simultaneous equations: two variables by elimination and substitution1.02i Represent inequalities: graphically on coordinate plane |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| DR: \(2x^2 - 8x + 6 = 0\), \(x^2 - 4x + 3 = 0\) | M1 | Equate, and rearrange to three term quadratic. Attempt to gather like terms, but not necessarily on same side. Condone no '\(= 0\)' |
| \((x-1)(x-3) = 0\) | M1 | Attempt to solve quadratic. If factorising then expansion should give \(x^2\) and one other term correct. Quadratic formula should be correct – allow one slip when substituting. Completing the square needs to go as far as \(x - p = \pm\sqrt{q}\) |
| \(x = 1,\ x = 3\) | A1 | Obtain both correct \(x\) values. Or one correct \((x,y)\) coordinate following a correct factorisation |
| \((1, 0)\) and \((3, 4)\) | A1 | Obtain both correct pairs of coordinates. Allow as e.g. \(x=1, y=0\) as long as pairings are clear |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Attempt graph of \(y = 2x - 2\), with positive gradient and negative intercept | M1 | No need for line to actually intersect with negative \(y\)-axis as long as it goes beneath positive \(x\)-axis |
| Graph of \(y = 2x - 2\) passing through both points of intersection of the two quadratic graphs | A1 | Must pass through both points |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Correct region labelled with R, or otherwise clearly identified | B1FT | FT any straight line that splits the overlap area into two finite regions, with the lower region identified. Allow for straight line with negative gradient as well, but not \(x = k\) |
# Question 3:
## Part (a)
| Answer | Marks | Guidance |
|--------|-------|----------|
| DR: $2x^2 - 8x + 6 = 0$, $x^2 - 4x + 3 = 0$ | **M1** | Equate, and rearrange to three term quadratic. Attempt to gather like terms, but not necessarily on same side. Condone no '$= 0$' |
| $(x-1)(x-3) = 0$ | **M1** | Attempt to solve quadratic. If factorising then expansion should give $x^2$ and one other term correct. Quadratic formula should be correct – allow one slip when substituting. Completing the square needs to go as far as $x - p = \pm\sqrt{q}$ |
| $x = 1,\ x = 3$ | **A1** | Obtain both correct $x$ values. Or one correct $(x,y)$ coordinate following a correct factorisation |
| $(1, 0)$ and $(3, 4)$ | **A1** | Obtain both correct pairs of coordinates. Allow as e.g. $x=1, y=0$ as long as pairings are clear |
## Part (b)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Attempt graph of $y = 2x - 2$, with positive gradient and negative intercept | **M1** | No need for line to actually intersect with negative $y$-axis as long as it goes beneath positive $x$-axis |
| Graph of $y = 2x - 2$ passing through both points of intersection of the two quadratic graphs | **A1** | Must pass through both points |
## Part (c)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Correct region labelled with R, or otherwise clearly identified | **B1FT** | FT any straight line that splits the overlap area into two finite regions, with the lower region identified. Allow for straight line with negative gradient as well, but not $x = k$ |
---3
\begin{enumerate}[label=(\alph*)]
\item In this question you must show detailed reasoning.\\
Find the coordinates of the points of intersection of the curves with equations $y = x ^ { 2 } - 2 x + 1$ and $y = - x ^ { 2 } + 6 x - 5$.
\item The diagram shows the curves $y = x ^ { 2 } - 2 x + 1$ and $y = - x ^ { 2 } + 6 x - 5$.
This diagram is repeated in the Printed Answer Booklet.\\
\includegraphics[max width=\textwidth, alt={}, center]{38b515c2-4764-4b51-a1f5-9b48d46610f0-5_377_542_603_322}
On the diagram in the Printed Answer Booklet, draw the line $y = 2 x - 2$.
\item Show on your diagram in the Printed Answer Booklet the region of the $x - y$ plane within which all three of the following inequalities are satisfied.\\
$y \geqslant x ^ { 2 } - 2 x + 1 \quad y \leqslant - x ^ { 2 } + 6 x - 5 \quad y \leqslant 2 x - 2$\\
You should indicate the region for which all the inequalities hold by labelling the region $R$.[1]
\end{enumerate}
\hfill \mbox{\textit{OCR H240/01 2022 Q3 [7]}}