AQA D1 2012 January — Question 5 10 marks

Exam BoardAQA
ModuleD1 (Decision Mathematics 1)
Year2012
SessionJanuary
Marks10
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicLinear Programming
TypeDual objective optimization
DifficultyStandard +0.3 This is a standard D1 linear programming question requiring plotting constraints, identifying the feasible region, and evaluating an objective function at vertices. While it involves multiple constraints and two objective functions, the techniques are routine and well-practiced. The dual objectives add slight complexity but this remains a textbook exercise with no novel insight required, making it slightly easier than average.
Spec7.06d Graphical solution: feasible region, two variables7.06e Sensitivity analysis: effect of changing coefficients
5 The feasible region of a linear programming problem is determined by the following: $$\begin{aligned} y & \geqslant 20 \\ x + y & \geqslant 25 \\ 5 x + 2 y & \leqslant 100 \\ y & \leqslant 4 x \\ y & \geqslant 2 x \end{aligned}$$
  1. On Figure 1 opposite, draw a suitable diagram to represent the inequalities and indicate the feasible region.
  2. Use your diagram to find the minimum value of \(P\), on the feasible region, in the case where:
    1. \(P = x + 2 y\);
    2. \(P = - x + y\). In each case, state the corresponding values of \(x\) and \(y\).
Question 5:
Part (a):
AnswerMarks Guidance
AnswerMark Guidance
\(y = 20\) drawn correctlyB1 Horizontal line
\(x + y = 25\) drawn correctlyB1
\(5x + 2y = 100\) drawn correctlyB1
\(y = 4x\) drawn correctlyB1
\(y = 2x\) drawn correctlyB1
Feasible region correctly identified/shadedB1 All five lines must be correct
Part (b)(i):
AnswerMarks Guidance
AnswerMark Guidance
Correct method using objective line or verticesM1
Minimum \(P = x + 2y\) identified at correct vertexA1
Correct values of \(x\) and \(y\) statedA1 ft e.g. \(x = 5, y = 20\), \(P = 45\)
Part (b)(ii):
AnswerMarks Guidance
AnswerMark Guidance
Correct method using objective line or verticesM1
Minimum \(P = -x + y\) identified at correct vertexA1
Correct values of \(x\) and \(y\) statedA1 ft cao 
## Question 5:

### Part (a):

| Answer | Mark | Guidance |
|--------|------|----------|
| $y = 20$ drawn correctly | B1 | Horizontal line |
| $x + y = 25$ drawn correctly | B1 | |
| $5x + 2y = 100$ drawn correctly | B1 | |
| $y = 4x$ drawn correctly | B1 | |
| $y = 2x$ drawn correctly | B1 | |
| Feasible region correctly identified/shaded | B1 | All five lines must be correct |

### Part (b)(i):

| Answer | Mark | Guidance |
|--------|------|----------|
| Correct method using objective line or vertices | M1 | |
| Minimum $P = x + 2y$ identified at correct vertex | A1 | |
| Correct values of $x$ and $y$ stated | A1 ft | e.g. $x = 5, y = 20$, $P = 45$ |

### Part (b)(ii):

| Answer | Mark | Guidance |
|--------|------|----------|
| Correct method using objective line or vertices | M1 | |
| Minimum $P = -x + y$ identified at correct vertex | A1 | |
| Correct values of $x$ and $y$ stated | A1 ft | cao |
5 The feasible region of a linear programming problem is determined by the following:

$$\begin{aligned}
y & \geqslant 20 \\
x + y & \geqslant 25 \\
5 x + 2 y & \leqslant 100 \\
y & \leqslant 4 x \\
y & \geqslant 2 x
\end{aligned}$$
\begin{enumerate}[label=(\alph*)]
\item On Figure 1 opposite, draw a suitable diagram to represent the inequalities and indicate the feasible region.
\item Use your diagram to find the minimum value of $P$, on the feasible region, in the case where:
\begin{enumerate}[label=(\roman*)]
\item $P = x + 2 y$;
\item $P = - x + y$.

In each case, state the corresponding values of $x$ and $y$.
\end{enumerate}\end{enumerate}

\hfill \mbox{\textit{AQA D1 2012 Q5 [10]}}
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