| Exam Board | OCR |
|---|---|
| Module | Further Discrete AS (Further Discrete AS) |
| Year | 2022 |
| Session | June |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear Programming |
| Type | Three-variable constraint reduction |
| Difficulty | Standard +0.8 This is a multi-part linear programming question requiring formulation, graphical solution, and constraint analysis. Part (d) requires identifying when constraints become redundant by finding critical values of k, which demands deeper understanding of constraint interactions beyond routine LP problems. While systematic, this level of constraint analysis is above typical A-level content. |
| Spec | 7.06a LP formulation: variables, constraints, objective function7.06b Slack variables: converting inequalities to equations7.06d Graphical solution: feasible region, two variables7.06e Sensitivity analysis: effect of changing coefficients |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Maximise \(x+y+z\) | B1 | Max \(x+y+z\) |
| Subject to \(6x+7y+8z \leq 360\) | B1 | Jam constraint correct |
| \(4x+3y+2z \leq 180\) | B1 | Custard constraint correct |
| \(x \leq 30\) | B1 | At most 30 batches of type X |
| \(y \geq 2z\) | B1 | At least twice as many batches of type Y as batches of type Z |
| \(x, y, z \geq 0\) | Non-negativity may be implied. Use of strict inequalities (e.g. \(x<30\)) penalised once only |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Set \(z=0\) and graph constraints \(6x+7y \leq 360\), \(4x+3y \leq 180\), \(x \leq 30\) and \(x,y \geq 0\) | Working may be on graph for part (b) | |
| At least two non-trivial boundary lines correct (ft from above) | M1 FT | |
| Feasible region correct: \((0, 51.4)\) to \((18, 36)\) to \((30, 20)\) to \((30, 0)\) | A1 | May assume boundary lines on axes. May see extra lines for part (d) |
| Solving for optimum values of \(x\) and \(y\) from graph (checking vertices or profit line or implied from answer) | M1 FT | |
| 18 batches of type X and 36 batches of type Y | A1 | Correct and in context. Need not state 0 batches of type Z |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| e.g. The baker may not sell them all and then they would be wasted; or there may be customers who want type Z; or the baker may not have enough time | B1 | A valid and relevant practical problem |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(6x+7y=k\) passes through \((30,20)\) when \(k=320\) | M1 | 320 as critical value for \(4x+3y \leq 180\) to become redundant |
| \(6x+7y=k\) passes through \((30,0)\) when \(k=180\) | M1 | 180 as critical value for \(x \leq 30\) to become redundant |
| \(180 \leq k < 320\) | A1 | \(180 \leq k\) and \(k < 320\) |
# Question 5:
## Part (a)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Maximise $x+y+z$ | B1 | Max $x+y+z$ |
| Subject to $6x+7y+8z \leq 360$ | B1 | Jam constraint correct |
| $4x+3y+2z \leq 180$ | B1 | Custard constraint correct |
| $x \leq 30$ | B1 | At most 30 batches of type X |
| $y \geq 2z$ | B1 | At least twice as many batches of type Y as batches of type Z |
| $x, y, z \geq 0$ | | Non-negativity may be implied. Use of strict inequalities (e.g. $x<30$) penalised once only |
## Part (b)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Set $z=0$ and graph constraints $6x+7y \leq 360$, $4x+3y \leq 180$, $x \leq 30$ and $x,y \geq 0$ | | Working may be on graph for part (b) |
| At least two non-trivial boundary lines correct (ft from above) | M1 FT | |
| Feasible region correct: $(0, 51.4)$ to $(18, 36)$ to $(30, 20)$ to $(30, 0)$ | A1 | May assume boundary lines on axes. May see extra lines for part (d) |
| Solving for optimum values of $x$ and $y$ from graph (checking vertices or profit line or implied from answer) | M1 FT | |
| 18 batches of type X and 36 batches of type Y | A1 | Correct and in context. Need not state 0 batches of type Z |
## Part (c)
| Answer | Marks | Guidance |
|--------|-------|----------|
| e.g. The baker may not sell them all and then they would be wasted; or there may be customers who want type Z; or the baker may not have enough time | B1 | A valid and relevant practical problem |
## Part (d)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $6x+7y=k$ passes through $(30,20)$ when $k=320$ | M1 | 320 as critical value for $4x+3y \leq 180$ to become redundant |
| $6x+7y=k$ passes through $(30,0)$ when $k=180$ | M1 | 180 as critical value for $x \leq 30$ to become redundant |
| $180 \leq k < 320$ | A1 | $180 \leq k$ and $k < 320$ |
---5 A baker makes three types of jam-and-custard doughnuts.
\begin{itemize}
\item Each batch of type X uses 6 units of jam and 4 units of custard.
\item Each batch of type Y uses 7 units of jam and 3 units of custard.
\item Each batch of type Z uses 8 units of jam and 2 units of custard.
\end{itemize}
The baker has 360 units of jam and 180 units of custard available.
The baker has plenty of doughnut batter, so this does not restrict the number of batches made.
From past experience the baker knows that they must make at most 30 batches of type X and at least twice as many batches of type Y as batches of type Z .
Let $x =$ number of batches of type X made\\
$y =$ number of batches of type Y made\\
$z =$ number of batches of type Z made.
\begin{enumerate}[label=(\alph*)]
\item Set up an LP formulation for the problem of maximising the total number of batches of doughnuts made.
The baker finds that type Z doughnuts are not popular and decides to make zero batches of type Z .
\item Use a graphical method to find how many batches of each type the baker should make to maximise the total number of batches of doughnuts made.
\item Give a reason why this solution may not be practical.
The baker finds that some of the jam has been used so there are only $k$ units of jam (where $k < 360$ ).\\
There are still 180 units of custard available and the baker still makes zero batches of type Z .
\item Find the values of $k$ if exactly one of the other (non-trivial) constraints is redundant.
Express your answer using inequalities.
\end{enumerate}
\hfill \mbox{\textit{OCR Further Discrete AS 2022 Q5 [13]}}