| Exam Board | OCR |
|---|---|
| Module | H240/01 (Pure Mathematics) |
| Year | 2020 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Topic | Modulus function |
| Type | Find range of k for number of roots |
| Difficulty | Standard +0.3 This is a standard modulus function question requiring graph sketching and finding intersection conditions. Part (a) is routine coordinate finding. Part (b)(i) requires understanding that two distinct intersections occur when the line intersects both branches of the V-shape, leading to inequalities on the gradient. Part (b)(ii) involves solving two linear equations with modulus. While it requires careful case analysis and algebraic manipulation, this is a well-practiced technique at A-level with no novel insight needed, making it slightly easier than average. |
| Spec | 1.02l Modulus function: notation, relations, equations and inequalities1.02q Use intersection points: of graphs to solve equations1.02s Modulus graphs: sketch graph of |ax+b|1.02t Solve modulus equations: graphically with modulus function |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \((1.5,\ 0)\) | B1 | Allow \(x = 1.5\); unless contradicted with non-zero \(y\)-coord |
| \((0,\ 3)\) | B1 | Allow \(y = 3\); unless contradicted with non-zero \(x\)-coord |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(a < 2\) | B1 | Allow for answer of form \(k < a < 2\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(0 = 1.5a + 2\) | M1 | Attempt to find value of \(a\) at their \(x\) intersection |
| \(a = -\frac{4}{3}\) | A1 | Obtain \(-\frac{4}{3}\) (condone any inequality sign, equals sign or no sign) |
| \(-\frac{4}{3} < a < 2\) | A1 | Correct final inequality; formal set notation not required |
| [4] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(2x - 3 = ax + 2\), \(x = \frac{5}{2-a}\) | B1 | Correct point of intersection – allow any exact equiv; OR M1 square both sides and attempt to solve |
| \(3 - 2x = ax + 2\), \((2+a)x = 1\) | M1 | Attempt to solve linear equation with \(2x\) and \(ax\) of different signs |
| \(x = \frac{1}{2+a}\) | A1 | Correct point of intersection – allow any exact equiv; Max 2 out of 3 if additional roots |
| [3] |
# Question 9:
## Part (a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $(1.5,\ 0)$ | B1 | Allow $x = 1.5$; unless contradicted with non-zero $y$-coord |
| $(0,\ 3)$ | B1 | Allow $y = 3$; unless contradicted with non-zero $x$-coord |
**[2 marks]**
## Part (b)(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $a < 2$ | B1 | Allow for answer of form $k < a < 2$ |
# Mark Scheme Extraction - H240/01 November 2020
---
## Question [preceding Q1] - Part (i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $0 = 1.5a + 2$ | M1 | Attempt to find value of $a$ at their $x$ intersection |
| $a = -\frac{4}{3}$ | A1 | Obtain $-\frac{4}{3}$ (condone any inequality sign, equals sign or no sign) |
| $-\frac{4}{3} < a < 2$ | A1 | Correct final inequality; formal set notation not required |
| | **[4]** | |
## Question [preceding Q1] - Part (ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $2x - 3 = ax + 2$, $x = \frac{5}{2-a}$ | B1 | Correct point of intersection – allow any exact equiv; OR M1 square both sides and attempt to solve |
| $3 - 2x = ax + 2$, $(2+a)x = 1$ | M1 | Attempt to solve linear equation with $2x$ and $ax$ of different signs |
| $x = \frac{1}{2+a}$ | A1 | Correct point of intersection – allow any exact equiv; Max 2 out of 3 if additional roots |
| | **[3]** | |
---9\\
\begin{tikzpicture}[>=latex, thick]
% Draw horizontal x-axis
\draw[->] (-3, 0) -- (4, 0) node[below right] {$x$};
% Draw vertical y-axis
\draw[->] (0, -1) -- (0, 4.5) node[above left] {$y$};
% Origin label
\node[below left] at (0, 0) {$O$};
% Draw the V-shaped graph
\draw (-0.85, 3.7) -- (1, 0) -- (2.85, 3.7);
\end{tikzpicture}
The diagram shows the graph of $y = | 2 x - 3 |$.
\begin{enumerate}[label=(\alph*)]
\item State the coordinates of the points of intersection with the axes.
\item Given that the graphs of $y = | 2 x - 3 |$ and $y = a x + 2$ have two distinct points of intersection, determine
\begin{enumerate}[label=(\roman*)]
\item the set of possible values of $a$,
\item the $x$-coordinates of the points of intersection of these graphs, giving your answers in terms of $a$.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{OCR H240/01 2020 Q9 [9]}}