| Exam Board | Edexcel |
|---|---|
| Module | D1 (Decision Mathematics 1) |
| Year | 2024 |
| Session | January |
| Marks | 17 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear Programming |
| Type | Three-variable constraint reduction |
| Difficulty | Standard +0.3 This is a standard D1 linear programming question requiring formulation, graphical representation, and vertex method application. While it involves three variables and multiple constraints, the techniques are routine for this module. The constraint reduction in part (b) requires some algebraic insight but follows a predictable pattern. The graphical work and vertex method are textbook procedures, making this slightly easier than average overall. |
| Spec | 7.06a LP formulation: variables, constraints, objective function7.06d Graphical solution: feasible region, two variables |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \((P=)\ 160x + 75y + 125z\) and maximise | B1 | |
| \(x + y + z \leqslant 100\) | B1 | |
| \(\frac{1}{4}(x+y+z) \leqslant x \Rightarrow 3x - y - z \geqslant 0\) | M1 | |
| \(3z \leqslant 5y\) | M1 | |
| \(2x + 1.5y + 0.75z \leqslant 138\ (\Rightarrow 8x+6y+3z \leqslant 552)\ (120x+90y+45z \leqslant 8280)\) | M1 | |
| \(3x-y-z \geqslant 0\); \(3z \leqslant 5y\); \(8x+6y+3z \leqslant 552\); \(x \geqslant 0,\ y \geqslant 0,\ z \geqslant 0\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Substitute \(x+y+z=100\) into \(P=160x+75y+125z\) and simplify | M1 | |
| \(P = 5(7x-10y) + 12500\) so maximising a (positive) multiple of \(7x-10y\) is equivalent to minimising the negative of this expression, that is, \(-(7x-10y) = -7x+10y\) | A1* |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Line 1 drawn correctly | B1 | |
| Line 2 drawn correctly | B1 | |
| Line 3 drawn correctly | B1 | |
| Region \(R\) correctly identified | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Vertices of \(R\) are \(\left(25, \frac{225}{8}\right),\ \left(25, \frac{127}{3}\right),\ (36, 24)\) | M1 A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Vertices including \(\left(25, \frac{225}{8}\right)\), \(\left(25, \frac{127}{3}\right)\), \((36, 24)\) | dM1 | Attempt to find exact coordinates of their \(R\) (must be finding at least two pairs of coordinates) — dependent on at least two B marks in (c) |
| \(\frac{-7x+10y}{4}\): values \(\frac{425}{4}\) (106.25), \(\frac{745}{3}\) (248.3), \(-12\) | dM1 | Point testing (objective line method is M0) — testing at least two of their vertices of their \(R\) in either \(-7x+10y\) or \(160x+75y+125z\) — dependent on previous M mark |
| \(\frac{160x+75y+125z}{4}\): values \(\frac{47875}{4}\) (11,968.75), \(\frac{33775}{3}\) (11,258.3), \(12560\) | ||
| 36 acres for crop A, 24 for crop B and 40 acres for crop C | A1 | All three values of \(P\) found (either exact or decimal equivalent for either objective function) and stating the correct allocation of the three crops. Allocation must be in context for crop A, B, C not \(x=, y=, z=\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Maximum expected profit is (£) 12 560 | A1 | Maximum expected profit stated correctly (units not required) |
| (5) | ||
| 17 marks total |
| Answer | Marks | Guidance |
|---|---|---|
| Mark | Criteria | Guidance |
| a1B1 | CAO — expression \((160x + 75y + 125z)\) together with 'max' or 'maximise' not 'maximum' | |
| a2B1 | CAO \((x + y + z \leqslant 100)\) | |
| a1M1 | \(\frac{1}{4}(x+y+z) \mathrel{\square} x\) where \(\square\) is any inequality or equals — brackets must be present or implied by later working (accept correct equivalent unsimplified forms) | |
| a2M1 | \(3z \mathrel{\square} 5y\) where \(\square\) is any inequality or equals. Also allow \(5z \leqslant 3y\) for this mark (accept correct equivalent unsimplified forms) | |
| a3M1 | \(2x + 1.5y + 0.75z \mathrel{\square} 138\) (oe) where \(\square\) is any inequality or equals. Time may be converted to minutes (must be all 4 values) (accept correct equivalent unsimplified forms) | |
| a1A1 | All three correct constraints \((3x \geqslant y+z,\ 3z \leqslant 5y,\ 8x+6y+3z \leqslant 552)\) correct — must have integer coefficients with only one term in each variable. Condone omission of trivial constraints \(x \geqslant 0,\ y \geqslant 0,\ z \geqslant 0\) (Accept e.g. \(24x+18y+9z \leqslant 1656\) oe if working in minutes) | |
| b1M1 | Substitute \(x+y+z=100\) into their linear objective function and simplify to a single term in \(x\) and a single term in \(y\) only | |
| b1A1* | Explaining why maximising the correct objective \(35x - 50y\) (+12500) is equivalent to minimising \(-7x + 10y\) | |
| c1B1 | Any two lines correctly drawn: \((5x+3y=252)\) should pass within half a small square of \((18, 54)\) and \((42, 14)\); \((3x+8y=300)\) should pass within half a small square of \((20, 30)\) and \((100, 0)\); \((x+y=100)\) should pass within half a small square of \((0, 100)\), \((50, 50)\) and \((100, 0)\); \((x=25\) must be drawn through the middle of the small square from \((24, 0)\) to \((26, 0))\) | |
| c2B1 | Any three lines correctly drawn | |
| c3B1 | All four lines correctly drawn (penalise any poorly drawn lines with the loss of this mark) | |
| c4B1 | Correct \(R\) labelled — dependent on all three previous B marks | |
| d1M1 | Attempt to find the exact coordinates (must be finding at least two pairs of coordinates) of their \(R\) — dependent on at least two B marks in (c) | |
| d1A1 | CAO — all three exact coordinates of the correct \(R\) (accept 28.125 and \(42.\dot{3}\) but not 42.3 or 42.33) | |
| d2M1 | Point testing (objective line method is M0) — testing at least two of their vertices of their \(R\) in either \(-7x+10y\) or \(160x+75y+125z\) — dependent on previous M mark | |
| d2A1 | All three values of \(P\) found (either exact or decimal equivalent for either objective function) and stating the correct allocation of the three crops. Allocation must be in context for crop A, B, C not \(x=,\ y=,\ z=\) | |
| d3A1 | Maximum expected profit stated correctly (units not required) |
## Question 7(a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $(P=)\ 160x + 75y + 125z$ and maximise | B1 | |
| $x + y + z \leqslant 100$ | B1 | |
| $\frac{1}{4}(x+y+z) \leqslant x \Rightarrow 3x - y - z \geqslant 0$ | M1 | |
| $3z \leqslant 5y$ | M1 | |
| $2x + 1.5y + 0.75z \leqslant 138\ (\Rightarrow 8x+6y+3z \leqslant 552)\ (120x+90y+45z \leqslant 8280)$ | M1 | |
| $3x-y-z \geqslant 0$; $3z \leqslant 5y$; $8x+6y+3z \leqslant 552$; $x \geqslant 0,\ y \geqslant 0,\ z \geqslant 0$ | A1 | |
---
## Question 7(b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Substitute $x+y+z=100$ into $P=160x+75y+125z$ and simplify | M1 | |
| $P = 5(7x-10y) + 12500$ so maximising a (positive) multiple of $7x-10y$ is equivalent to minimising the negative of this expression, that is, $-(7x-10y) = -7x+10y$ | A1* | |
---
## Question 7(c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Line 1 drawn correctly | B1 | |
| Line 2 drawn correctly | B1 | |
| Line 3 drawn correctly | B1 | |
| Region $R$ correctly identified | B1 | |
---
## Question 7(d)(i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Vertices of $R$ are $\left(25, \frac{225}{8}\right),\ \left(25, \frac{127}{3}\right),\ (36, 24)$ | M1 A1 | |
# Question 7:
## Part (ii)
| Answer | Mark | Guidance |
|--------|------|----------|
| Vertices including $\left(25, \frac{225}{8}\right)$, $\left(25, \frac{127}{3}\right)$, $(36, 24)$ | dM1 | Attempt to find exact coordinates of their $R$ (must be finding at least two pairs of coordinates) — dependent on at least two B marks in (c) |
| $\frac{-7x+10y}{4}$: values $\frac{425}{4}$ (106.25), $\frac{745}{3}$ (248.3), $-12$ | dM1 | Point testing (objective line method is M0) — testing at least two of their vertices of their $R$ in either $-7x+10y$ or $160x+75y+125z$ — dependent on previous M mark |
| $\frac{160x+75y+125z}{4}$: values $\frac{47875}{4}$ (11,968.75), $\frac{33775}{3}$ (11,258.3), $12560$ | | |
| 36 acres for crop A, 24 for crop B and 40 acres for crop C | A1 | All three values of $P$ found (either exact or decimal equivalent for either objective function) and stating the correct allocation of the three crops. Allocation must be in context for crop A, B, C not $x=, y=, z=$ |
## Part (iii)
| Answer | Mark | Guidance |
|--------|------|----------|
| Maximum expected profit is (£) 12 560 | A1 | Maximum expected profit stated correctly (units not required) |
| | **(5)** | |
| | **17 marks total** | |
---
## Notes for Question 7
| Mark | Criteria | Guidance |
|------|----------|----------|
| a1B1 | CAO — expression $(160x + 75y + 125z)$ together with 'max' or 'maximise' not 'maximum' | |
| a2B1 | CAO $(x + y + z \leqslant 100)$ | |
| a1M1 | $\frac{1}{4}(x+y+z) \mathrel{\square} x$ where $\square$ is any inequality or equals — brackets must be present or implied by later working (accept correct equivalent unsimplified forms) | |
| a2M1 | $3z \mathrel{\square} 5y$ where $\square$ is any inequality or equals. Also allow $5z \leqslant 3y$ for this mark (accept correct equivalent unsimplified forms) | |
| a3M1 | $2x + 1.5y + 0.75z \mathrel{\square} 138$ (oe) where $\square$ is any inequality or equals. Time may be converted to minutes (must be all 4 values) (accept correct equivalent unsimplified forms) | |
| a1A1 | All three correct constraints $(3x \geqslant y+z,\ 3z \leqslant 5y,\ 8x+6y+3z \leqslant 552)$ correct — must have integer coefficients with only one term in each variable. Condone omission of trivial constraints $x \geqslant 0,\ y \geqslant 0,\ z \geqslant 0$ (Accept e.g. $24x+18y+9z \leqslant 1656$ oe if working in minutes) | |
| b1M1 | Substitute $x+y+z=100$ into their linear objective function and simplify to a single term in $x$ and a single term in $y$ only | |
| b1A1* | Explaining why maximising the correct objective $35x - 50y$ (+12500) is equivalent to minimising $-7x + 10y$ | |
| c1B1 | Any two lines correctly drawn: $(5x+3y=252)$ should pass within half a small square of $(18, 54)$ and $(42, 14)$; $(3x+8y=300)$ should pass within half a small square of $(20, 30)$ and $(100, 0)$; $(x+y=100)$ should pass within half a small square of $(0, 100)$, $(50, 50)$ and $(100, 0)$; $(x=25$ must be drawn through the middle of the small square from $(24, 0)$ to $(26, 0))$ | |
| c2B1 | Any three lines correctly drawn | |
| c3B1 | All four lines correctly drawn (penalise any poorly drawn lines with the loss of this mark) | |
| c4B1 | Correct $R$ labelled — dependent on all three previous B marks | |
| d1M1 | Attempt to find the exact coordinates (must be finding at least two pairs of coordinates) of their $R$ — dependent on at least two B marks in (c) | |
| d1A1 | CAO — all three exact coordinates of the correct $R$ (accept 28.125 and $42.\dot{3}$ but not 42.3 or 42.33) | |
| d2M1 | Point testing (objective line method is M0) — testing at least two of their vertices of their $R$ in either $-7x+10y$ or $160x+75y+125z$ — dependent on previous M mark | |
| d2A1 | All three values of $P$ found (either exact or decimal equivalent for either objective function) and stating the correct allocation of the three crops. Allocation must be in context for crop A, B, C not $x=,\ y=,\ z=$ | |
| d3A1 | Maximum expected profit stated correctly (units not required) | |7. A farmer has 100 acres of land available that can be used for planting three crops: A, B and C .
It takes 2 hours to plant each acre of crop A, 1.5 hours to plant each acre of crop B and 45 minutes to plant each acre of crop C . The farmer has 138 hours available for planting.
At least one quarter of the total crops planted must be crop A.\\
For every three acres of crop B planted, at most five acres of crop C will be planted.\\
The farmer expects a profit of $\pounds 160$ for each acre of crop A planted, $\pounds 75$ for each acre of crop B planted and $\pounds 125$ for each acre of crop C planted.
The farmer wishes to maximise the profit from planting these three crops.\\
Let $x , y$ and $z$ represent the number of acres of land used for planting crop A, crop B, and crop C respectively.
\begin{enumerate}[label=(\alph*)]
\item Formulate this information as a linear programming problem. State the objective, and list the constraints as simplified inequalities with integer coefficients.
The farmer decides that all 100 acres of available land will be used for planting the three crops.
\item Explain why the maximum total profit is achieved when $- 7 x + 10 y$ is minimised.
The farmer's decision to use all 100 acres reduces the constraints of the problem to the following:
$$\begin{aligned}
x & \geqslant 25 \\
3 x + 8 y & \geqslant 300 \\
x + y & \leqslant 100 \\
5 x + 3 y & \leqslant 252 \\
y & \geqslant 0
\end{aligned}$$
\item Represent these constraints on Diagram 1 in the answer book. Hence determine, and label, the feasible region, $R$.
\item \begin{enumerate}[label=(\roman*)]
\item Determine the exact coordinates of each of the vertices of $R$.
\item Apply the vertex method to determine how the 100 acres should be used for planting the three crops.
\item Hence find the corresponding maximum expected profit.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{Edexcel D1 2024 Q7 [17]}}