| Exam Board | OCR |
|---|---|
| Module | D1 (Decision Mathematics 1) |
| Year | 2015 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear Programming |
| Type | Parametric objective analysis |
| Difficulty | Standard +0.8 This is a multi-part linear programming question that goes beyond routine vertex evaluation. Parts (i)-(ii) are standard (evaluate objective at vertices, consider integer constraints), but parts (iii)-(iv) require conceptual understanding of how objective function gradient relates to feasible region geometry and determining parameter ranges for optimality—this parametric analysis requires deeper insight than typical D1 questions. |
| Spec | 7.06d Graphical solution: feasible region, two variables7.06e Sensitivity analysis: effect of changing coefficients |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Evaluate \(P = x + 3y\) at each vertex | M1 | |
| \(B(1.5, 3): P = 1.5 + 9 = 10.5\) is maximum | A1 | |
| Optimum vertex is \(B\), \(P = 10.5\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \((2, 3)\), \(P = 11\) | B1 | Must be integer point in feasible region |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(C\) and \(D\) have smaller \(x\) and \(y\) values than \(A\), so \(P\) is smaller at \(C\) and \(D\) for positive \(k\) | B1 | Or equivalent argument that \(A\) dominates \(C\) and \(D\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| At \(A\) and \(B\): \(3.5 + 2k = 1.5 + 3k \Rightarrow k = 2\) | M1 | Setting \(P_A = P_B\) |
| At \(A\) and \(D\): gradient of \(AD\) gives lower bound | M1 | |
| \(k \leq 2\) and \(k \geq -\frac{1}{2}\) (from slope of \(AD\)) | A1A1 | \(-\frac{1}{2} \leq k \leq 2\) |
# Question 3:
## Part (i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Evaluate $P = x + 3y$ at each vertex | M1 | |
| $B(1.5, 3): P = 1.5 + 9 = 10.5$ is maximum | A1 | |
| Optimum vertex is $B$, $P = 10.5$ | A1 | |
## Part (ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $(2, 3)$, $P = 11$ | B1 | Must be integer point in feasible region |
## Part (iii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $C$ and $D$ have smaller $x$ and $y$ values than $A$, so $P$ is smaller at $C$ and $D$ for positive $k$ | B1 | Or equivalent argument that $A$ dominates $C$ and $D$ |
## Part (iv)
| Answer | Marks | Guidance |
|--------|-------|----------|
| At $A$ and $B$: $3.5 + 2k = 1.5 + 3k \Rightarrow k = 2$ | M1 | Setting $P_A = P_B$ |
| At $A$ and $D$: gradient of $AD$ gives lower bound | M1 | |
| $k \leq 2$ and $k \geq -\frac{1}{2}$ (from slope of $AD$) | A1A1 | $-\frac{1}{2} \leq k \leq 2$ |
---3 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]{372c062a-793f-4fb8-a769-957479f5fce7-05_846_833_365_614}
The vertices of the feasible region are $A ( 3.5,2 ) , B ( 1.5,3 ) , C ( 0.5,1.5 ) , D ( 1,0.5 )$.\\
The objective is to maximise $P = x + 3 y$.\\
\begin{enumerate}[label=(\roman*)]
\item Find the coordinates of the optimum vertex and the corresponding value of $P$.
\item Find the optimum point if $x$ and $y$ must both have integer values.
The objective is changed to maximise $P = x + k y$.
\item If $k$ is positive, explain why the optimum point cannot be at $C$ or $D$.
\item If $k$ can take any value, find the range of values of $k$ for which $A$ is the optimum point.
\end{enumerate}
\hfill \mbox{\textit{OCR D1 2015 Q3 [9]}}