| Exam Board | OCR |
|---|---|
| Module | Further Discrete AS (Further Discrete AS) |
| Session | Specimen |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear Programming |
| Type | Graphical optimization with objective line |
| Difficulty | Standard +0.8 This is a multi-part linear programming question requiring constraint formulation, graphical representation, and optimization across different scenarios. While the individual techniques are standard for Further Maths Decision, the question demands careful tracking of multiple constraints, systematic enumeration of integer solutions, and adaptation when conditions change (parts iii-iv). The complexity and length elevate it above average difficulty. |
| Spec | 7.06a LP formulation: variables, constraints, objective function7.06b Slack variables: converting inequalities to equations7.06d Graphical solution: feasible region, two variables |
| Type | Cost (£) | Milk chocolate | Plain chocolate | White chocolate | Nutty chocolate |
| Assorted | 2.00 | 5 | 5 | 5 | 5 |
| No Nuts | 1.00 | 5 | 8 | 7 | 0 |
| Speciality | 2.50 | 5 | 4 | 9 | 2 |
| Answer | Marks | Guidance |
|---|---|---|
| 8 | (i) | 5 milk chocolate in each type, 4(cid:117)5(cid:32)20 |
| [1] | 2.2a | Using numbers of milk chocolate |
| Answer | Marks | Guidance |
|---|---|---|
| 8 | (ii) | 2 so that there are at least 12 nutty chocolate |
| truffles | B1 | |
| [1] | 2.2a | 2 (Assorted boxes) and using numbers |
| Answer | Marks | Guidance |
|---|---|---|
| 8 | (iii) | Type Cost |
| Answer | Marks |
|---|---|
| 3(cid:117) Assorted and 1(cid:117) No Nuts | M1 |
| Answer | Marks |
|---|---|
| [3] | 1.1 |
| Answer | Marks |
|---|---|
| 1.1 | n |
| Answer | Marks |
|---|---|
| cao | Need not show costs, and if |
| Answer | Marks | Guidance |
|---|---|---|
| Type | Cost | |
| Assorted | No Nuts | Speciality |
| 8 | (iv) | (a) |
| Answer | Marks |
|---|---|
| x(cid:116)0 and y(cid:116)0 (and integer-valued) | M1 |
| Answer | Marks |
|---|---|
| [2] | 3.1b |
| 1.2 | One of 8x(cid:14)4y(cid:116)10 and 7x(cid:14)9y(cid:116)16, |
| Answer | Marks |
|---|---|
| Both of these correct, in any form | Allow strict inequalities, but not |
| Answer | Marks | Guidance |
|---|---|---|
| 8 | (iv) | (b) |
| Answer | Marks |
|---|---|
| 4 | M1 |
| Answer | Marks |
|---|---|
| [2] | 3.3 |
| Answer | Marks |
|---|---|
| c | Plotting nlines and identifying feasible |
| Answer | Marks |
|---|---|
| mscaled axes | For reference: |
| Answer | Marks | Guidance |
|---|---|---|
| 8 | (iv) | (c) |
| Answer | Marks |
|---|---|
| Cheapest solution costs Narendra £5.50 | M1 |
| Answer | Marks |
|---|---|
| [3] | 3.4 |
| Answer | Marks |
|---|---|
| 3.4 | Calculating costs for any integer- |
| Answer | Marks |
|---|---|
| £5.50 | May go straight to solution |
| Answer | Marks |
|---|---|
| 8(iii) | e |
| Answer | Marks |
|---|---|
| 8(iv)(a) | n |
| Answer | Marks |
|---|---|
| e | n |
Question 8:
8 | (i) | 5 milk chocolate in each type, 4(cid:117)5(cid:32)20 | B1
[1] | 2.2a | Using numbers of milk chocolate
truffles
8 | (ii) | 2 so that there are at least 12 nutty chocolate
truffles | B1
[1] | 2.2a | 2 (Assorted boxes) and using numbers
of nutty chocolate truffles
8 | (iii) | Type Cost
Assorted No Nuts Speciality £
4 0 0 8.00
3 1 0 7.00
3 0 1 8.50
2 1 1 7.50
2 0 2 9.00
p
S
3(cid:117) Assorted and 1(cid:117) No Nuts | M1
A1
e
B1
[3] | 1.1
1.1
i
c
1.1 | n
Identifying at least three of these five
caeses for the number of boxes of the
different types bought
m
All five cases, and no others, unless
explicitly rejected as being infeasible
cao | Need not show costs, and if
given do not penalise calculation
errors
May be marked in list
Type | Cost
Assorted | No Nuts | Speciality | £
8 | (iv) | (a) | P: 8x(cid:14)4y(cid:116)10oe
W: 7x(cid:14)9y(cid:116)16oe
x(cid:116)0 and y(cid:116)0 (and integer-valued) | M1
A1
[2] | 3.1b
1.2 | One of 8x(cid:14)4y(cid:116)10 and 7x(cid:14)9y(cid:116)16,
in any form
Both of these correct, in any form | Allow strict inequalities, but not
an equality or with inequality
reversed
8 | (iv) | (b) | y
4
x
4 | M1
A1
e
[2] | 3.3
1.1
i
c | Plotting nlines and identifying feasible
region
e
All lines and feasible region correct on
mscaled axes | For reference:
(0, 2.5)
(0.59, 1.32) or better
(2.28 to 2.29, 0)
8 | (iv) | (c) | p
S
(0, 3) + 2 small milk + 2 small nutty (cid:159)£9.50
(1, 2) + 2 small milk + 2 small nutty (cid:159)£8.00
(2, 1) + 2 small milk + 3 small nutty (cid:159)£7.00
(3, 0) + 2 small milk + 3 small nutty (cid:159)£5.50
(1, 1) + 3 small milk + 3 small nutty (cid:159)£6.50
Cheapest solution costs Narendra £5.50 | M1
M1
E1
[3] | 3.4
3.2a
3.4 | Calculating costs for any integer-
valued feasible solution
Including sufficient small packs to deal
with ‘milk’ and ‘nutty’ requirements
£5.50 | May go straight to solution
(from logical reasoning or
calculation)
E0 if exterior points used as if
feasible
E0 if correct answer given with
insufficient evidence to justify it
PMT
Y534 Mark Scheme June 20XX
Assessment Objectives (AO) Grid
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M = Modelling
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Y534 Mark Scheme June 20XX
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Y534 Mark Scheme June 20XX
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AS Level Further Mathematics A
Unit Y534 Discrete Mathematics
Printed Answer Booklet
Date – Morning/Afternoon
Time allowed: 1 hour 15 minutes
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number number
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INFORMATION
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© OCR 2017 Y534 Turn over
603/1329/8 B10039/1.2
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© OCR 2017 Y5348 A sweet shop sells three different types of boxes of chocolate truffles. The cost of each type of box and the number of truffles of each variety in each type of box are given in the table below.
\begin{center}
\begin{tabular}{|l|l|l|l|l|l|}
\hline
Type & Cost (£) & Milk chocolate & Plain chocolate & White chocolate & Nutty chocolate \\
\hline
Assorted & 2.00 & 5 & 5 & 5 & 5 \\
\hline
No Nuts & 1.00 & 5 & 8 & 7 & 0 \\
\hline
Speciality & 2.50 & 5 & 4 & 9 & 2 \\
\hline
\end{tabular}
\end{center}
Narendra wants to buy some boxes of truffles so that in total he has at least 20 milk chocolate, 10 plain chocolate, 16 white chocolate and 12 nutty chocolate truffles.
\begin{enumerate}[label=(\roman*)]
\item Explain why Narendra needs to buy at least four boxes of truffles.
\item Narendra decides that he will buy exactly four boxes. Determine the minimum number of Assorted boxes that Narendra must buy.
\item For your answer in part (ii),
\begin{itemize}
\item list all the feasible solutions and
\item find the cheapest solution.
\end{itemize}
Narendra finds that the sweet shop has sold out of Assorted boxes, but he then spots that it also sells small boxes of milk chocolate truffles and small boxes of nutty chocolate truffles. Each small box contains 4 truffles (all of one variety) and costs $\pounds 0.50$.
He decides to buy $x$ boxes of No Nuts and $y$ boxes of Speciality, where $x + y < 4$, so that he has at least 10 plain chocolate and 16 white chocolate truffles. He will then buy as many small boxes as he needs to give a total of at least 20 milk chocolate and 12 nutty chocolate truffles.
\item (a) Set up constraints on the values of $x$ and $y$.\\
(b) Represent the feasible region graphically.\\
(c) Hence determine the cheapest cost for Narendra.
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OCR is part of the
\end{enumerate}
\hfill \mbox{\textit{OCR Further Discrete AS Q8 [12]}}