| Exam Board | OCR |
|---|---|
| Module | D1 (Decision Mathematics 1) |
| Year | 2009 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear Programming |
| Type | Parametric objective analysis |
| Difficulty | Moderate -0.8 This is a standard linear programming question requiring reading inequalities from a graph, identifying vertices, evaluating an objective function at vertices, and finding when the optimal vertex changes. All techniques are routine D1 procedures with no novel problem-solving required, making it easier than average but not trivial due to the multi-part nature and the gradient comparison in part (iv). |
| Spec | 7.06d Graphical solution: feasible region, two variables7.06e Sensitivity analysis: effect of changing coefficients |
| Answer | Marks | Guidance |
|---|---|---|
| \(y \geq x\) | M1 | Line \(y = x\) in any form |
| \(x + y \leq 8\) | M1 | Line \(x + y = 8\) in any form |
| \(x \geq 1\) | M1 | Line \(x = 1\) in any form |
| All inequalities correct | A1 | All inequalities correct [Ignore extra inequalities that do not affect the feasible region] |
| [4] |
| Answer | Marks | Guidance |
|---|---|---|
| \((1, 1), (1, 7), (4, 4)\) | M1 | Any two correct coordinates |
| A1 | All three correct [Extra coordinates given \(\Rightarrow\) M1, A0] | |
| [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Maximum value = 23 | M1 | Follow through if possible. Testing vertices or using a line of constant profit (may be implied) |
| A1 | Accept \((1, 7)\) identified | |
| A1 | 23 identified | |
| [3] |
| Answer | Marks | Guidance |
|---|---|---|
| \(k \geq 2\) | M1 | \(2 + 7k\) or implied, or using line of gradient \(-\frac{7}{k}\) |
| A1 | Greater than or equal to 2 (cao) [\(k > 2 \Rightarrow\) M1, A0] | |
| [2] |
## (i)
$y \geq x$ | M1 | Line $y = x$ in any form
$x + y \leq 8$ | M1 | Line $x + y = 8$ in any form
$x \geq 1$ | M1 | Line $x = 1$ in any form
All inequalities correct | A1 | All inequalities correct [Ignore extra inequalities that do not affect the feasible region]
| [4] |
## (ii)
$(1, 1), (1, 7), (4, 4)$ | M1 | Any two correct coordinates
| A1 | All three correct [Extra coordinates given $\Rightarrow$ M1, A0]
| [2] |
## (iii)
$(1, 7)$ 23
$(4, 4)$ 20
At optimum, $x = 1$ and $y = 7$
Maximum value = 23 | M1 | Follow through if possible. Testing vertices or using a line of constant profit (may be implied)
| A1 | Accept $(1, 7)$ identified
| A1 | 23 identified
| [3] |
## (iv)
$2x1 + kx7 > 2x4 + kx4$
$k \geq 2$ | M1 | $2 + 7k$ or implied, or using line of gradient $-\frac{7}{k}$
| A1 | Greater than or equal to 2 (cao) [$k > 2 \Rightarrow$ M1, A0]
| [2] |
---The constraints of a linear programming problem are represented by the graph below. The feasible region is the unshaded region, including its boundaries.
\includegraphics{figure_3}
\begin{enumerate}[label=(\roman*)]
\item Write down the inequalities that define the feasible region. [4]
\item Write down the coordinates of the three vertices of the feasible region. [2]
\end{enumerate}
The objective is to maximise $2x + 3y$.
\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{2}
\item Find the values of $x$ and $y$ at the optimal point, and the corresponding maximum value of $2x + 3y$. [3]
\end{enumerate}
The objective is changed to maximise $2x + ky$, where $k$ is positive.
\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{3}
\item Find the range of values of $k$ for which the optimal point is the same as in part (iii). [2]
\end{enumerate}
\hfill \mbox{\textit{OCR D1 2009 Q3 [11]}}