Question 5 - A-Level Maths

OCR Further Discrete AS 2018 June — Question 5 16 marks

Exam Board	OCR
Module	Further Discrete AS (Further Discrete AS)
Year	2018
Session	June
Marks	16
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Linear Programming
Type	Graphical optimization with objective line
Difficulty	Standard +0.3 This is a standard linear programming question with straightforward constraints and systematic enumeration. Parts (i)-(iii) involve basic arithmetic checks, part (iv) requires simple algebraic expressions, and parts (v)-(vi) involve forming and minimizing an objective function. While multi-part and requiring careful bookkeeping, it follows a predictable structure with no novel insights needed—slightly easier than average for Further Maths Decision content.
Spec	7.06a LP formulation: variables, constraints, objective function 7.06b Slack variables: converting inequalities to equations 7.06c Working with constraints: algebra and ad hoc methods 7.06d Graphical solution: feasible region, two variables 7.06e Sensitivity analysis: effect of changing coefficients

5 Greetings cards are sold in luxury, standard and economy packs.
The table shows the cost of each pack and number of cards of each kind in the pack.

Pack	Cost (£)	Handmade cards	Cards with flowers	Cards with animals	Other cards	Total number of cards
Luxury	6.50	10	5	5	0	20
Standard	5.00	5	10	5	10	30
Economy	4.00	0	10	10	20	40

Alice needs 25 cards, of which at least 8 must be handmade cards, at least 8 must be cards with flowers and at least 4 must be cards with animals.

Explain why Alice will need to buy at least two packs of cards. Alice does not want to spend more than \(\pounds 12\) on the cards.
(a) List the combinations of packs that satisfy all Alice's requirements.
(b) Which of these is the cheapest? Ben offers to buy any cards that Alice buys but does not need. He will pay 12 pence for each handmade card and 5 pence for any other card. Alice does not want her net expenditure (the amount she spends minus the amount that Ben pays her) on the cards to be more than \(\pounds 12\).
Show that Alice could now buy two luxury packs. Alice decides to buy exactly 2 packs, of which \(x\) are luxury packs, \(y\) are standard packs and the rest are economy packs.
Give an expression, in terms of \(x\) and \(y\) only, for the number of cards of each type that Alice buys. Alice wants to minimise her net expenditure.
Find, and simplify, an expression for Alice's minimum net expenditure in pence, in terms of \(x\) and \(y\). You may assume that Alice buys enough cards to satisfy her own requirements.
Find Alice's minimum net expenditure.

Show mark scheme Show mark scheme source

Question 5:

Part (i)

Answer	Marks	Guidance
Answer	Marks	Guidance
One luxury pack would not give enough cards with flowers	B1	Explaining why one luxury pack is not enough; One luxury pack only has 20 cards (Alice needs 25 cards)
One standard or economy would not give enough handmade cards	B1 [2]	Explaining why one standard pack or one economy pack is not enough; One standard pack only has 5 handmade cards and an economy pack has none (Alice needs 8); No pack has 8 handmade cards and 8 cards with flowers

Part (ii)(a)

Answer	Marks	Guidance
Answer	Marks	Guidance
1 luxury and 1 standard	M1	Any one of these
1 luxury and 1 economy	A1	All three and no others (or any extras have been rejected)
2 standard	[2]	In any form, e.g. LS or SL

Part (ii)(b)

Answer	Marks	Guidance
Answer	Marks	Guidance
2 standard	B1 [1]

Part (iii)

Answer	Marks	Guidance
Answer	Marks	Guidance
Two luxury packs satisfies all the requirements but costs £13. However Ben will buy back 15 cards. Ben can buy back up to 12 handmade cards \((12\times12\text{p}) + (3\times5\text{p}) = £1.59\)	M1	Description or calculation showing which surplus cards Ben buys back; e.g. £13 and a multiple of 12p or a multiple of 5p or a combination of both (shown)
Giving net cost £11.41; Or a valid calculation, with Ben buying 15 cards, with explanation of how many of each type he bought	A1 [2]	e.g. calc leading to one of 11.48, 11.55, 11.62, 11.69, 11.76 (in any form); e.g. calc leading to one of 11.56, 11.68, 11.80, 11.92 (in any form)

Part (iv)

Answer	Marks	Guidance
Answer	Marks	Guidance
Handmade: \(10x + 5y\)	B1	Correct expression for handmade
Flowers: \(5x+10y+10(2-x-y) = 20-5x\)	M1	Using \(2-x-y\) appropriately
Animals: \(5x+5y+10(2-x-y) = 20-5x-5y\)	A1	Correct expression for one of flowers, animals, other; Or flowers \(= 5x + 10(2-x)\)
Other: \(10y+20(2-x-y) = 40-20x-10y\)	A1 [4]	All of these three correct, simplified

Part (v)

Answer	Marks	Guidance
Answer	Marks	Guidance
\(650x + 500y + 400(2-x-y)\)	M1	\(250x + 100y + 800\); Accept \(650x + 500y\)
\(-12(10x+5y-8) - 5(10x+25y+40(2-x-y)-17)\)	M1	\(30x + 15y - 219\); Accept \(-12(\ldots) - 5(\ldots)\)
\(= 581 + 280x + 115y\)	A1 [3]	cao

Part (vi)

Answer	Marks	Guidance
Answer	Marks	Guidance
Min when \(x\) is as small as possible and then \(y\) is as small as possible whilst still satisfying Alice's requirements; \(x=0\) and \(y=2\)	M1	Using their objective; Or a fresh start
£8.11	A1 [2]	cao

# Question 5:

## Part (i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| One luxury pack would not give enough cards with flowers | B1 | Explaining why one luxury pack is not enough; One luxury pack only has 20 cards (Alice needs 25 cards) |
| One standard or economy would not give enough handmade cards | B1 [2] | Explaining why one standard pack or one economy pack is not enough; One standard pack only has 5 handmade cards and an economy pack has none (Alice needs 8); No pack has 8 handmade cards and 8 cards with flowers |

## Part (ii)(a)
| Answer | Marks | Guidance |
|--------|-------|----------|
| 1 luxury and 1 standard | M1 | Any one of these |
| 1 luxury and 1 economy | A1 | All three and no others (or any extras have been rejected) |
| 2 standard | [2] | In any form, e.g. LS or SL |

## Part (ii)(b)
| Answer | Marks | Guidance |
|--------|-------|----------|
| 2 standard | B1 [1] | |

## Part (iii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Two luxury packs satisfies all the requirements but costs £13. However Ben will buy back 15 cards. Ben can buy back up to 12 handmade cards $(12\times12\text{p}) + (3\times5\text{p}) = £1.59$ | M1 | Description or calculation showing which surplus cards Ben buys back; e.g. £13 and a multiple of 12p or a multiple of 5p or a combination of both (shown) |
| Giving net cost £11.41; Or a valid calculation, with Ben buying 15 cards, with explanation of how many of each type he bought | A1 [2] | e.g. calc leading to one of 11.48, 11.55, 11.62, 11.69, 11.76 (in any form); e.g. calc leading to one of 11.56, 11.68, 11.80, 11.92 (in any form) |

## Part (iv)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Handmade: $10x + 5y$ | B1 | Correct expression for handmade |
| Flowers: $5x+10y+10(2-x-y) = 20-5x$ | M1 | Using $2-x-y$ appropriately |
| Animals: $5x+5y+10(2-x-y) = 20-5x-5y$ | A1 | Correct expression for one of flowers, animals, other; Or flowers $= 5x + 10(2-x)$ |
| Other: $10y+20(2-x-y) = 40-20x-10y$ | A1 [4] | All of these three correct, simplified |

## Part (v)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $650x + 500y + 400(2-x-y)$ | M1 | $250x + 100y + 800$; Accept $650x + 500y$ |
| $-12(10x+5y-8) - 5(10x+25y+40(2-x-y)-17)$ | M1 | $30x + 15y - 219$; Accept $-12(\ldots) - 5(\ldots)$ |
| $= 581 + 280x + 115y$ | A1 [3] | cao |

## Part (vi)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Min when $x$ is as small as possible and then $y$ is as small as possible whilst still satisfying Alice's requirements; $x=0$ and $y=2$ | M1 | Using their objective; Or a fresh start |
| £8.11 | A1 [2] | cao |

Show LaTeX source

5 Greetings cards are sold in luxury, standard and economy packs.\\
The table shows the cost of each pack and number of cards of each kind in the pack.

\begin{center}
\begin{tabular}{|l|l|l|l|l|l|l|}
\hline
Pack & Cost (£) & Handmade cards & Cards with flowers & Cards with animals & Other cards & Total number of cards \\
\hline
Luxury & 6.50 & 10 & 5 & 5 & 0 & 20 \\
\hline
Standard & 5.00 & 5 & 10 & 5 & 10 & 30 \\
\hline
Economy & 4.00 & 0 & 10 & 10 & 20 & 40 \\
\hline
\end{tabular}
\end{center}

Alice needs 25 cards, of which at least 8 must be handmade cards, at least 8 must be cards with flowers and at least 4 must be cards with animals.
\begin{enumerate}[label=(\roman*)]
\item Explain why Alice will need to buy at least two packs of cards.

Alice does not want to spend more than $\pounds 12$ on the cards.
\item (a) List the combinations of packs that satisfy all Alice's requirements.\\
(b) Which of these is the cheapest?

Ben offers to buy any cards that Alice buys but does not need. He will pay 12 pence for each handmade card and 5 pence for any other card.

Alice does not want her net expenditure (the amount she spends minus the amount that Ben pays her) on the cards to be more than $\pounds 12$.
\item Show that Alice could now buy two luxury packs.

Alice decides to buy exactly 2 packs, of which $x$ are luxury packs, $y$ are standard packs and the rest are economy packs.
\item Give an expression, in terms of $x$ and $y$ only, for the number of cards of each type that Alice buys.

Alice wants to minimise her net expenditure.
\item Find, and simplify, an expression for Alice's minimum net expenditure in pence, in terms of $x$ and $y$. You may assume that Alice buys enough cards to satisfy her own requirements.
\item Find Alice's minimum net expenditure.
\end{enumerate}

\hfill \mbox{\textit{OCR Further Discrete AS 2018 Q5 [16]}}

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