| Exam Board | OCR |
|---|---|
| Module | D1 (Decision Mathematics 1) |
| Year | 2011 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear Programming |
| Type | Parametric objective analysis |
| Difficulty | Standard +0.8 This question requires understanding of parametric linear programming where the objective function coefficient varies. While part (i) is routine, parts (ii) and (iii) require students to analyze how changing the gradient of the objective line affects which vertex is optimal—a concept requiring geometric insight beyond standard textbook exercises. The parametric analysis in part (iii) demands understanding of when objective line gradients match constraint gradients, which is more sophisticated than typical D1 questions. |
| Spec | 7.06a LP formulation: variables, constraints, objective function7.06d Graphical solution: feasible region, two variables7.06e Sensitivity analysis: effect of changing coefficients |
| Answer | Marks | Guidance |
|---|---|---|
| \(y = x + y \le 8\) or \(y \le 8 - x\) | M1 | Allow = or wrong inequality sign for method |
| Coordinates of point A seen, can fractions 2.6 to 2.7 and 5.3 to 5.4 | B1 | |
| Condone \(>1\) or \(<1\) | B1 | Do not accept 1, 2, 3, ... (integer values) and condone \(m = 1\) is not enough for either mark |
$y = x + y \le 8$ or $y \le 8 - x$ | M1 | Allow = or wrong inequality sign for method
Coordinates of point A seen, can fractions 2.6 to 2.7 and 5.3 to 5.4 | B1 |
Condone $>1$ or $<1$ | B1 | Do not accept 1, 2, 3, ... (integer values) and condone $m = 1$ is not enough for either mark
---1 The constraints of a linear programming problem are represented by the graph below. The feasible region is the unshaded region, including its boundaries.\\
\includegraphics[max width=\textwidth, alt={}, center]{cec8d4db-4a72-43a3-88f3-ff9df2a11d2c-2_885_873_388_635}
\begin{enumerate}[label=(\roman*)]
\item Write down the inequalities that define the feasible region.
The objective is to maximise $P _ { m } = x + m y$, where $m$ is a positive, real-valued constant.
\item In the case when $m = 2$, calculate the values of $x$ and $y$ at the optimal point, and the corresponding value of $P _ { 2 }$.
\item (a) Write down the values of $m$ for which point $A$ is optimal.\\
(b) Write down the values of $m$ for which point $B$ is optimal.
\end{enumerate}
\hfill \mbox{\textit{OCR D1 2011 Q1 [6]}}