OCR D1 Specimen — Question 7 20 marks

Exam BoardOCR
ModuleD1 (Decision Mathematics 1)
SessionSpecimen
Marks20
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicLinear Programming
TypeSimplex algorithm execution
DifficultyModerate -0.3 This is a standard D1 linear programming question requiring graphical solution and Simplex algorithm execution. While it involves multiple parts and careful arithmetic, it follows a completely routine procedure taught in the specification with no novel problem-solving required. The Simplex algorithm is mechanical once learned, making this slightly easier than average for A-level.
Spec7.06d Graphical solution: feasible region, two variables7.07a Simplex tableau: initial setup in standard format7.07b Simplex iterations: pivot choice and row operations
7 Consider the linear programming problem: $$\begin{array} { l l } \text { maximise } & P = 4 y - x , \\ \text { subject to } & x + 4 y \leqslant 22 , \\ & x + y \leqslant 10 , \\ & - x + 2 y \leqslant 8 , \\ \text { and } & x \geqslant 0 , y \geqslant 0 . \end{array}$$
  1. Represent the constraints graphically, shading out the regions where the inequalities are not satisfied. Calculate the value of \(x\) and the value of \(y\) at each of the vertices of the feasible region. Hence find the maximum value of \(P\), clearly indicating where it occurs.
  2. By introducing slack variables, represent the problem as an initial Simplex tableau and use the Simplex algorithm to solve the problem.
  3. Indicate on your diagram for part (i) the points that correspond to each stage of the Simplex algorithm carried out in part (ii). \section*{OXFORD CAMBRIDGE AND RSA EXAMINATIONS} Advanced Subsidiary General Certificate of Education Advanced General Certificate of Education MATHEMATICS
    4736
    Decision Mathematics 1
    INSERT for Questions 4, 5 and 6
    Specimen Paper
    • This insert should be used to answer Questions 4, 5 and 6
    • .
    • Write your Name, Centre Number and Candidate Number in the spaces provided at the top of this page.
    • Write your answers to Questions 4, 5 and 6
    • in the spaces provided in this insert, and attach it to your answer booklet.
    4
  4. STEPA\(B\)C
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  5. STEPA\(B\)C
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    5
  6. \includegraphics[max width=\textwidth, alt={}, center]{b1227633-913e-41a9-8bf8-1f064056963e-7_661_1004_285_557}
  7. Upper bound = \(\_\_\_\_\) km Lower bound = \(\_\_\_\_\) km
  8. \(\_\_\_\_\) Best upper bound = \(\_\_\_\_\) km 6
  9. \includegraphics[max width=\textwidth, alt={}, center]{b1227633-913e-41a9-8bf8-1f064056963e-8_668_1406_292_406} \includegraphics[max width=\textwidth, alt={}, center]{b1227633-913e-41a9-8bf8-1f064056963e-8_307_1342_1014_424}
    Least travel time = \(\_\_\_\_\) minutes Route: A- \(\_\_\_\_\) \(- D\)
Question 7:
AnswerMarks Guidance
(i)Plot lines \(x+4y=22\), \(x+y=10\), \(-x+2y=8\); shade correct regions; vertices at intersections: \((0,0)\), \((0,4)\), \((2,8)\)... \((10,0)\); evaluate \(P=4y-x\) at each vertex; maximum P occurs at identified vertex M1 (lines), A1 (shading), M1 (vertices), A1 (correct vertices), M1 (evaluate P), A1, A1 (maximum), A1 (point)
(ii)Introduce slacks \(s_1, s_2, s_3\); initial tableau set up; pivot on most negative in P row; iterate to optimum M1 (slack variables), A1 (initial tableau), M1 (pivot selection), A1, M1 (second pivot), A1×4 (correct final tableau), A1 (solution)
(iii)Mark on diagram the starting point \((0,0)\) and each subsequent vertex visited during simplex iterations B1, B1 
# Question 7:

**(i)** | Plot lines $x+4y=22$, $x+y=10$, $-x+2y=8$; shade correct regions; vertices at intersections: $(0,0)$, $(0,4)$, $(2,8)$... $(10,0)$; evaluate $P=4y-x$ at each vertex; maximum P occurs at identified vertex | M1 (lines), A1 (shading), M1 (vertices), A1 (correct vertices), M1 (evaluate P), A1, A1 (maximum), A1 (point) |

**(ii)** | Introduce slacks $s_1, s_2, s_3$; initial tableau set up; pivot on most negative in P row; iterate to optimum | M1 (slack variables), A1 (initial tableau), M1 (pivot selection), A1, M1 (second pivot), A1×4 (correct final tableau), A1 (solution) |

**(iii)** | Mark on diagram the starting point $(0,0)$ and each subsequent vertex visited during simplex iterations | B1, B1 | Must correspond to simplex stages in (ii)
7 Consider the linear programming problem:

$$\begin{array} { l l } 
\text { maximise } & P = 4 y - x , \\
\text { subject to } & x + 4 y \leqslant 22 , \\
& x + y \leqslant 10 , \\
& - x + 2 y \leqslant 8 , \\
\text { and } & x \geqslant 0 , y \geqslant 0 .
\end{array}$$

(i) Represent the constraints graphically, shading out the regions where the inequalities are not satisfied. Calculate the value of $x$ and the value of $y$ at each of the vertices of the feasible region. Hence find the maximum value of $P$, clearly indicating where it occurs.\\
(ii) By introducing slack variables, represent the problem as an initial Simplex tableau and use the Simplex algorithm to solve the problem.\\
(iii) Indicate on your diagram for part (i) the points that correspond to each stage of the Simplex algorithm carried out in part (ii).

\section*{OXFORD CAMBRIDGE AND RSA EXAMINATIONS}
Advanced Subsidiary General Certificate of Education Advanced General Certificate of Education

MATHEMATICS\\
4736\\
Decision Mathematics 1\\
INSERT for Questions 4, 5 and 6\\
Specimen Paper

\begin{itemize}
  \item This insert should be used to answer Questions 4, 5 and 6 (i).
  \item Write your Name, Centre Number and Candidate Number in the spaces provided at the top of this page.
  \item Write your answers to Questions 4, 5 and 6 (i) in the spaces provided in this insert, and attach it to your answer booklet.
\end{itemize}

4 (i)

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(ii)

\begin{center}
\begin{tabular}{|l|l|l|l|}
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\end{center}

5 (i)\\
\includegraphics[max width=\textwidth, alt={}, center]{b1227633-913e-41a9-8bf8-1f064056963e-7_661_1004_285_557}\\
(ii) Upper bound = $\_\_\_\_$ km

Lower bound = $\_\_\_\_$ km\\
(iii) $\_\_\_\_$\\

Best upper bound = $\_\_\_\_$ km

6 (i)\\
\includegraphics[max width=\textwidth, alt={}, center]{b1227633-913e-41a9-8bf8-1f064056963e-8_668_1406_292_406}\\
\includegraphics[max width=\textwidth, alt={}, center]{b1227633-913e-41a9-8bf8-1f064056963e-8_307_1342_1014_424}\\

Least travel time = $\_\_\_\_$ minutes

Route: A- $\_\_\_\_$ $- D$

\hfill \mbox{\textit{OCR D1  Q7 [20]}}
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