Question 6 - A-Level Maths

OCR MEI D1 2005 January — Question 6 16 marks

Exam Board	OCR MEI
Module	D1 (Decision Mathematics 1)
Year	2005
Session	January
Marks	16
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Linear Programming
Type	Graphical optimization with objective line
Difficulty	Moderate -0.5 This is a standard linear programming question requiring routine graphical methods and vertex identification. While it has multiple parts, each follows textbook procedures: formulating constraints, sketching the feasible region, and optimizing different objective functions. The context is straightforward and the mathematical techniques are core D1 content with no novel problem-solving required.
Spec	7.06a LP formulation: variables, constraints, objective function 7.06b Slack variables: converting inequalities to equations 7.06d Graphical solution: feasible region, two variables 7.06e Sensitivity analysis: effect of changing coefficients

6 A recipe for jam states that the weight of sugar used must be between the weight of fruit used and four thirds of the weight of fruit used. Georgia has 10 kg of fruit available and 11 kg of sugar.

Define two variables and formulate inequalities in those variables to model this information.
Draw a graph to represent your inequalities.
Find the vertices of your feasible region and identify the points which would represent the best mix of ingredients under each of the following circumstances.
(A) There is to be as much jam as possible, given that the weight of jam produced is the sum of the weights of the fruit and the sugar.
(B) There is to be as much jam as possible, given that it is to have the lowest possible proportion of sugar.
(C) There is to be as much jam as possible, given that it is to have the highest possible proportion of sugar.
(D) Fruit costs \(\pounds 1\) per kg, sugar costs 50 p per kg and the objective is to produce as much jam as possible within a budget of \(\pounds 15\).

Show mark scheme Show mark scheme source

Question 6:

Part (i)

Answer	Marks	Guidance
Answer	Mark	Guidance
Let \(f\) be the number of kg of fruit used. Let \(s\) be the number of kg of sugar used.	B1
\(s \geq f\)	B1
\(s \leq \frac{4f}{3}\)	B1
\(f \leq 10\)	B1
\(s \leq 11\)	B1

Part (ii)

Answer	Marks	Guidance
Answer	Mark	Guidance
Labels and scales on axes	B1
Line \(s = f\) drawn	B1
Line \(s = \frac{4f}{3}\) drawn	B1	\(\checkmark\) for \(\frac{3}{4}\) gradient
Lines \(f = 10\) and \(s = 11\) drawn	B1
Correct shading of feasible region	B1

Part (iii)

Answer	Marks	Guidance
Answer	Mark	Guidance
Vertices: \((0,0)\), \((8.25, 11)\), \((10, 11)\), \((10, 10)\)	B1\(\checkmark\)

Part (iii)(a)

Answer	Marks	Guidance
Answer	Mark	Guidance
\(21\) kg at \((10, 11)\)	B1\(\checkmark\)

Part (iii)(b)

Answer	Marks	Guidance
Answer	Mark	Guidance
\(20\) kg at \(50\%\) concentration at \((10, 10)\)	B1\(\checkmark\)

Part (iii)(c)

Answer	Marks	Guidance
Answer	Mark	Guidance
\(19.25\) kg at \(\frac{4}{7}\) concentration at \((8.25, 11)\)	B1\(\checkmark\)

Part (iii)(d)

Answer	Marks	Guidance
Answer	Mark	Guidance
\(20.5\) kg at \((9.5, 11)\)	M1 A1	cao

# Question 6:

## Part (i)

| Answer | Mark | Guidance |
|--------|------|----------|
| Let $f$ be the number of kg of fruit used. Let $s$ be the number of kg of sugar used. | B1 | |
| $s \geq f$ | B1 | |
| $s \leq \frac{4f}{3}$ | B1 | |
| $f \leq 10$ | B1 | |
| $s \leq 11$ | B1 | |

## Part (ii)

| Answer | Mark | Guidance |
|--------|------|----------|
| Labels and scales on axes | B1 | |
| Line $s = f$ drawn | B1 | |
| Line $s = \frac{4f}{3}$ drawn | B1 | $\checkmark$ for $\frac{3}{4}$ gradient |
| Lines $f = 10$ and $s = 11$ drawn | B1 | |
| Correct shading of feasible region | B1 | |

## Part (iii)

| Answer | Mark | Guidance |
|--------|------|----------|
| Vertices: $(0,0)$, $(8.25, 11)$, $(10, 11)$, $(10, 10)$ | B1$\checkmark$ | |

## Part (iii)(a)

| Answer | Mark | Guidance |
|--------|------|----------|
| $21$ kg at $(10, 11)$ | B1$\checkmark$ | |

## Part (iii)(b)

| Answer | Mark | Guidance |
|--------|------|----------|
| $20$ kg at $50\%$ concentration at $(10, 10)$ | B1$\checkmark$ | |

## Part (iii)(c)

| Answer | Mark | Guidance |
|--------|------|----------|
| $19.25$ kg at $\frac{4}{7}$ concentration at $(8.25, 11)$ | B1$\checkmark$ | |

## Part (iii)(d)

| Answer | Mark | Guidance |
|--------|------|----------|
| $20.5$ kg at $(9.5, 11)$ | M1 A1 | cao |

Show LaTeX source

6 A recipe for jam states that the weight of sugar used must be between the weight of fruit used and four thirds of the weight of fruit used. Georgia has 10 kg of fruit available and 11 kg of sugar.
\begin{enumerate}[label=(\roman*)]
\item Define two variables and formulate inequalities in those variables to model this information.
\item Draw a graph to represent your inequalities.
\item Find the vertices of your feasible region and identify the points which would represent the best mix of ingredients under each of the following circumstances.\\
(A) There is to be as much jam as possible, given that the weight of jam produced is the sum of the weights of the fruit and the sugar.\\
(B) There is to be as much jam as possible, given that it is to have the lowest possible proportion of sugar.\\
(C) There is to be as much jam as possible, given that it is to have the highest possible proportion of sugar.\\
(D) Fruit costs $\pounds 1$ per kg, sugar costs 50 p per kg and the objective is to produce as much jam as possible within a budget of $\pounds 15$.
\end{enumerate}

\hfill \mbox{\textit{OCR MEI D1 2005 Q6 [16]}}

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Q1 8 Q2 8 Q3 8 Q4 16 Q5 16 Q6 16