| Exam Board | OCR MEI |
|---|---|
| Module | AS Paper 1 (AS Paper 1) |
| Year | 2022 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Inequalities |
| Type | Combined linear and quadratic inequalities |
| Difficulty | Standard +0.3 This is a straightforward multi-part question involving basic linear modeling and solving simultaneous linear inequalities. Parts (a)-(c) require simple substitution and interpretation. Part (d) involves setting up and solving two linear inequalities (supply ≥ demand), which is a standard AS-level technique requiring no novel insight—slightly easier than average due to the scaffolded structure and routine algebraic manipulation. |
| Spec | 1.02c Simultaneous equations: two variables by elimination and substitution1.02z Models in context: use functions in modelling |
| Answer | Marks | Guidance |
|---|---|---|
| When \(x = 0\), number of cakes is 190 | B1 | Allow www |
| Answer | Marks | Guidance |
|---|---|---|
| \((1.20, 50)\) gives \(50 = 1.2m + c\) | M1 | Setting up simultaneous equations for \(m\) and \(c\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(m = 75,\ c = -40\) | A1 | Allow for values given or \(y = 75x - 40\) seen |
| Answer | Marks | Guidance |
|---|---|---|
| \(m = \frac{140-50}{2.4-1.2}\) | M1 | Using data to calculate \(m\) |
| \(m = 75,\ c = -40\) | A1 | Allow for values given or \(y = 75x - 40\) seen |
| Answer | Marks | Guidance |
|---|---|---|
| [When \(x\) is small,] \(y\) is negative and number of cakes donated cannot be negative | E1 | Negative \(y\)-values and \(y\) cannot be negative must both be stated or implied |
| Answer | Marks | Guidance |
|---|---|---|
| Upper bound for demand to exceed supply | M1 | Attempt to find one of the bounds for \(x\) using \(y = 75x - 40\) |
| Answer | Marks | Guidance |
|---|---|---|
| So \(x < \frac{46}{29}\) | A1 | Accept \(x < 1.586\) or \(x < 1.59\) or \(x < 1.58\) or \(x \leq 1.56\); Allow use of \(<\) or \(\leq\); Note prices £1.58 and £1.57 do not lead to integer values, fully correct answer is \(x \leq 1.56\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(x > \frac{40}{75}\) | A1 | Accept \(0.533 < x\) or \(0.53 < x\) or \(0.54 < x\) or \(0.56 \leq x\); Allow use of \(<\) or \(\leq\); Note prices £0.54 and £0.55 do not lead to integer values, fully correct answer is \(0.56 \leq x\) |
# Question 8:
## Part (a):
When $x = 0$, number of cakes is 190 | **B1** | Allow www
## Part (b):
$(1.20, 50)$ gives $50 = 1.2m + c$ | **M1** | Setting up simultaneous equations for $m$ and $c$
$(2.40, 140)$ gives $140 = 2.4m + c$
$m = 75,\ c = -40$ | **A1** | Allow for values given or $y = 75x - 40$ seen
**Alternative method:**
$m = \frac{140-50}{2.4-1.2}$ | **M1** | Using data to calculate $m$
$m = 75,\ c = -40$ | **A1** | Allow for values given or $y = 75x - 40$ seen
## Part (c):
[When $x$ is small,] $y$ is negative and number of cakes donated cannot be negative | **E1** | Negative $y$-values and $y$ cannot be negative must both be stated or implied
## Part (d):
Upper bound for demand to exceed supply | **M1** | Attempt to find one of the bounds for $x$ using $y = 75x - 40$
Lower bound for $y$ positive
$190 - 70x = 75x - 40$
So $x < \frac{46}{29}$ | **A1** | Accept $x < 1.586$ or $x < 1.59$ or $x < 1.58$ or $x \leq 1.56$; Allow use of $<$ or $\leq$; Note prices £1.58 and £1.57 do not lead to integer values, fully correct answer is $x \leq 1.56$
$y = 75x - 40 > 0$
$x > \frac{40}{75}$ | **A1** | Accept $0.533 < x$ or $0.53 < x$ or $0.54 < x$ or $0.56 \leq x$; Allow use of $<$ or $\leq$; Note prices £0.54 and £0.55 do not lead to integer values, fully correct answer is $0.56 \leq x$
---8 A team of volunteers donates cakes for sale at a charity stall. The number of cakes that can be sold depends on the price. A model for this is $\mathrm { y } = 190 - 70 \mathrm { x }$, where $y$ cakes can be sold when the price of a cake is $\pounds$ x.
\begin{enumerate}[label=(\alph*)]
\item Find how many cakes could be given away for free according to this model.
The number of volunteers who are willing to donate cakes goes up as the price goes up. If the cakes sell for $\pounds 1.20$ they will donate 50 cakes, but if they sell for $\pounds 2.40$ they will donate 140 cakes. They use the linear model $\mathrm { y } = \mathrm { mx } + \mathrm { c }$ to relate the number of cakes donated, $y$, to the price of a cake, $\pounds x$.
\item Find the values of the constants $m$ and $c$ for which this linear model fits the two data points.
\item Explain why the model is not suitable for very low prices.
\item The team would like to sell all the cakes that they donate.
Find the set of possible prices that the cakes could have to achieve this.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI AS Paper 1 2022 Q8 [7]}}