| Exam Board | OCR |
|---|---|
| Module | D1 (Decision Mathematics 1) |
| Year | 2010 |
| Session | June |
| Marks | 10 |
| 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 standard textbook exercises by introducing a parametric objective function. Parts (i)-(ii) are routine D1 content, but parts (iii)-(iv) require understanding how changing the gradient of the objective function affects which vertex is optimal—a conceptual step requiring geometric insight about when objective lines become parallel to constraint boundaries, not just mechanical application of corner-point method. |
| Spec | 7.06d Graphical solution: feasible region, two variables7.06e Sensitivity analysis: effect of changing coefficients |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(y \leq x + 2\) | B1 | |
| \(x + y \leq 7\) | B1 | |
| \(y \geq \frac{1}{2}x\) (or \(2y \geq x\)) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Vertices of feasible region evaluated or objective line used | M1 | |
| Optimal at \((1, 6)\)... check intersection points | M1 | |
| \(x=1, y=6\)... but checking: intersection of \(y=x+2\) and \(x+y=7\): \(x=2.5, y=4.5\) | A1 | |
| \(P_1 = 1 + 6(6) = 37\) or correct value at correct point | A1 | ft |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Different optimal point identified (other vertex of feasible region) | M1 | |
| Coordinates and \(P_k\) value stated | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Range of \(k\) found by considering gradient of objective vs boundary lines | M1 | |
| Correct range stated | A1 |
# Question 3:
## Part (i)
| Answer | Mark | Guidance |
|--------|------|----------|
| $y \leq x + 2$ | B1 | |
| $x + y \leq 7$ | B1 | |
| $y \geq \frac{1}{2}x$ (or $2y \geq x$) | B1 | |
## Part (ii)
| Answer | Mark | Guidance |
|--------|------|----------|
| Vertices of feasible region evaluated or objective line used | M1 | |
| Optimal at $(1, 6)$... check intersection points | M1 | |
| $x=1, y=6$... but checking: intersection of $y=x+2$ and $x+y=7$: $x=2.5, y=4.5$ | A1 | |
| $P_1 = 1 + 6(6) = 37$ or correct value at correct point | A1 | ft |
## Part (iii)
| Answer | Mark | Guidance |
|--------|------|----------|
| Different optimal point identified (other vertex of feasible region) | M1 | |
| Coordinates and $P_k$ value stated | A1 | |
## Part (iv)
| Answer | Mark | Guidance |
|--------|------|----------|
| Range of $k$ found by considering gradient of objective vs boundary lines | M1 | |
| Correct range stated | A1 | |
---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]{7ca6d572-d776-4ad7-a0ed-9ec43c975585-03_908_915_392_614}\\
(i) Write down the inequalities that define the feasible region.
The objective is to maximise $P _ { 1 } = x + 6 y$.\\
(ii) Find the values of $x$ and $y$ at the optimal point, and the corresponding value of $P _ { 1 }$.
The objective is changed to maximise $P _ { k } = k x + 6 y$, where $k$ is positive.\\
(iii) Calculate the coordinates of the optimal point, and the corresponding value of $P _ { k }$ when the optimal point is not the same as in part (ii).\\
(iv) Find the range of values of $k$ for which the point identified in part (ii) is still optimal.
\hfill \mbox{\textit{OCR D1 2010 Q3 [10]}}