| Exam Board | OCR |
|---|---|
| Module | Further Discrete AS (Further Discrete AS) |
| Year | 2024 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear Programming |
| Type | Graphical optimization with objective line |
| Difficulty | Moderate -0.3 This is a straightforward linear programming question requiring standard setup of constraints and graphical solution. The context is simple (buying tokens with a budget), constraints are given explicitly, and the graphical method is routine for Further Maths students. Slightly easier than average due to small numbers and clear structure, though the 'already has' tokens add minor complexity to constraint formulation. |
| Spec | 7.06a LP formulation: variables, constraints, objective function7.06b Slack variables: converting inequalities to equations7.06d Graphical solution: feasible region, two variables |
| Answer | Marks | Guidance |
|---|---|---|
| 6 | (a) | Maximise 1.5x + 4y |
| Answer | Marks |
|---|---|
| x – 2 and y – 1 are non-negative (integers) | B1 |
| Answer | Marks |
|---|---|
| [4] | 1.1 |
| Answer | Marks |
|---|---|
| 1.1 | P = 1.5x + 4y (allow + constant) |
| Answer | Marks |
|---|---|
| (b) | Check integer-valued points near their vertices |
| Answer | Marks |
|---|---|
| Buy 5 silver tokens 0 gold tokens | B1 |
| Answer | Marks |
|---|---|
| [5] | 1.1 |
| Answer | Marks |
|---|---|
| 3.2a | Mark grids first, tolerance + 0.5 grid square |
Question 6:
6 | (a) | Maximise 1.5x + 4y
Subject to 5 ≤ x + y ≤ 8
2(x – 2) + 6(y – 1) ≤ 10
x – 2 and y – 1 are non-negative (integers) | B1
B1
B1
B1
[4] | 1.1
1.1
3.1b
1.1 | P = 1.5x + 4y (allow + constant)
Or as two separate inequalities
Inequality for amount spent, 2x + 6y ≤ 20 o.e.
Allow x, y 0
Ignore extras
BOD if < used for ≤ and/or > used for
(b) | Check integer-valued points near their vertices
(3, 2) (4,2) (5, 0) (7, 1) (8, 0)
12.5 14 7.5 14.5 12
Max is £14.50 when x = 7, y = 1
7 – 2 = 5 and 1 – 1 = 0
Buy 5 silver tokens 0 gold tokens | B1
B1
B1
M1
A1
[5] | 1.1
1.1
3.4
3.4
3.2a | Mark grids first, tolerance + 0.5 grid square
2x + 6y < 20 drawn correctly
5 < x + y and x + y < 8 both correct, BOD scales if not labelled
Shading to give a FR (between parallels and below other line)
Allow
At least one (of theirs) correct
3
Or profit line y = 0.25P – 0.375x, gradient - written or drawn
8
May be implied from correct answer or from x = 7, y = 1 or (7, 1)
or from 14.50
Solution in context, 5 silver
May imply 0 gold tokens if not mentioned6 Beth wants to buy some tokens for use in a game.\\
Each token is either a silver token or a gold token.\\
Silver tokens and gold tokens have different points values in the game.\\
Silver tokens have a value of 1.5 points each.\\
Gold tokens have a value of 4 points each.\\
Beth already has 2 silver tokens and 1 gold token.\\
She also has $\pounds 10$ that can be spent on buying more tokens.\\
Silver tokens can be bought for $\pounds 2$ each.\\
Gold tokens can be bought for $\pounds 6$ each.\\
After buying some tokens, Beth has $x$ silver tokens and $y$ gold tokens.\\
She now has a total of at least 5 tokens and no more than 8 tokens.
\begin{enumerate}[label=(\alph*)]
\item Set up an LP formulation in $x$ and $y$ for the problem of maximising the points value of tokens that she finishes with.
\item Use a graphical method to determine how many tokens of each type Beth should buy to maximise the points value of her tokens.
\end{enumerate}
\hfill \mbox{\textit{OCR Further Discrete AS 2024 Q6 [9]}}