| Exam Board | AQA |
|---|---|
| Module | D1 (Decision Mathematics 1) |
| Year | 2016 |
| Session | June |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear Programming |
| Type | Constraint derivation verification |
| Difficulty | Easy -1.2 This is a straightforward linear programming question requiring basic constraint formulation from word problems. Part (a) is a simple 'show that' verification involving converting 6 hours to minutes (360) and dividing the inequality 10x + 15y ≤ 360 by 5. The remaining parts involve standard textbook LP techniques (writing inequalities, drawing feasible regions, finding optimal vertices). No novel problem-solving or geometric insight required—purely routine application of D1 methods. |
| Spec | 7.06a LP formulation: variables, constraints, objective function7.06b Slack variables: converting inequalities to equations7.06d Graphical solution: feasible region, two variables7.06e Sensitivity analysis: effect of changing coefficients |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(10x + 15y \leq 360\) (simplifies to) \(2x + 3y \leq 72\) | B1 AG | 1 mark total |
| (b) \(x + y \leq 32\) oe | B1 | \(x + y < 32\) AND \(x > 2y\) oe scores SC1 |
| \(x \geq 2y\) oe | B1 | 2 marks total |
| (c) Each line must be ruled to have the B mark available. For all lines, must be correct to ½ square horizontal and vertical at the indicated vertices. | B1 | \(x = 5\) and \(y = 5\) from axes to (5,30) and (30,5) |
| B1 | \(y = \frac{1}{2}x\) (0,0), (20, 10) | |
| B1 | \(x + y = 32\) (32, 0), (0, 32) | |
| B1 | \(2x + 3y = 72\) (0,24), (36,0) | |
| B1 | FR, all 5 lines above correct and region labelled (ignore shading) | |
| Objective line: any line with gradient \(= -\frac{3}{4}\) | M1 | Accept either \(-\frac{3}{4}\) (allow -0.7 to -0.8) or its reciprocal (allow -1.3 to -1.4) for the M1 only |
| A correct line (which may not intersect the axes) | A1 | 7 marks total |
| (d)(i) (24,8) or (27,5) or (21,10) | M1 | Allow values in the range(23-25,7-9) or (26-28,4-6) or (20-22,9-11). PI by further working |
| 24 Luxury and 8 Special | A1 CAO | |
| (ii) (Profit =) £520 | B1 | 3 marks total |
**(a)** $10x + 15y \leq 360$ (simplifies to) $2x + 3y \leq 72$ | B1 AG | 1 mark total | Accept $\frac{x}{6} + \frac{y}{4} \leq 6$ (or correct decimal/fractional equiv.)
**(b)** $x + y \leq 32$ oe | B1 | $x + y < 32$ AND $x > 2y$ oe scores SC1
$x \geq 2y$ oe | B1 | 2 marks total
**(c)** Each line must be ruled to have the B mark available. For all lines, must be correct to ½ square horizontal and vertical at the indicated vertices. | B1 | $x = 5$ and $y = 5$ from axes to (5,30) and (30,5)
| B1 | $y = \frac{1}{2}x$ (0,0), (20, 10)
| B1 | $x + y = 32$ (32, 0), (0, 32)
| B1 | $2x + 3y = 72$ (0,24), (36,0)
| B1 | FR, all 5 lines above correct and region labelled (ignore shading)
Objective line: any line with gradient $= -\frac{3}{4}$ | M1 | Accept either $-\frac{3}{4}$ (allow -0.7 to -0.8) or its reciprocal (allow -1.3 to -1.4) for the M1 only
A correct line (which may not intersect the axes) | A1 | 7 marks total
**(d)(i)** (24,8) or (27,5) or (21,10) | M1 | Allow values in the range(23-25,7-9) or (26-28,4-6) or (20-22,9-11). PI by further working
24 Luxury and 8 Special | A1 CAO |
**(ii)** (Profit =) £520 | B1 | 3 marks total | Must include £
**Total: 13 marks**8 Nerys runs a cake shop. In November and December she sells Christmas hampers. She makes up the hampers herself, in two sizes: Luxury and Special.
Each day, Nerys prepares $x$ Luxury hampers and $y$ Special hampers.\\
It takes Nerys 10 minutes to prepare a Luxury hamper and 15 minutes to prepare a Special hamper. She has 6 hours available, each day, for preparing hampers.
From past experience, Nerys knows that each day:
\begin{itemize}
\item she will need to prepare at least 5 hampers of each size
\item she will prepare at most a total of 32 hampers
\item she will prepare at least twice as many Luxury hampers as Special hampers.
\end{itemize}
Each Luxury hamper that Nerys prepares makes her a profit of $\pounds 15$; each Special hamper makes a profit of $\pounds 20$. Nerys wishes to maximise her daily profit, $\pounds P$.
\begin{enumerate}[label=(\alph*)]
\item Show that $x$ and $y$ must satisfy the inequality $2 x + 3 y \leqslant 72$.
\item In addition to $x \geqslant 5$ and $y \geqslant 5$, write down two more inequalities that model the constraints above.
\item On the grid opposite draw a suitable diagram to enable this problem to be solved graphically. Indicate a feasible region and the direction of an objective line.
\item \begin{enumerate}[label=(\roman*)]
\item Use your diagram to find the number of each type of hamper that Nerys should prepare each day to achieve a maximum profit.
\item Calculate this profit.
\begin{center}
\includegraphics[max width=\textwidth, alt={}]{fb95068f-f76d-492a-b385-bce17b26ae30-27_2490_1719_217_150}
\end{center}
\section*{DO NOT WRITE ON THIS PAGE ANSWER IN THE SPACES PROVIDED}
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA D1 2016 Q8 [13]}}