| Exam Board | AQA |
|---|---|
| Module | Further Paper 3 Discrete (Further Paper 3 Discrete) |
| Year | 2019 |
| Session | June |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | The Simplex Algorithm |
| Type | Interpret optimal tableau |
| Difficulty | Standard +0.8 This requires understanding optimality conditions for the Simplex algorithm (all coefficients in objective row must be non-negative) and applying this to solve an inequality involving a quadratic expression. It combines algorithmic understanding with algebraic manipulation, going beyond routine tableau interpretation but remaining within standard Further Maths scope. |
| Spec | 7.07b Simplex iterations: pivot choice and row operations7.07c Interpret simplex: values of variables, slack, and objective |
| \(\boldsymbol { P }\) | \(\boldsymbol { x }\) | \(\boldsymbol { y }\) | \(\boldsymbol { z }\) | \(\boldsymbol { r }\) | \(\boldsymbol { s }\) | value |
| 1 | \(k ^ { 2 } + k - 6\) | 0 | 0 | \(k - 1\) | 1 | 20 |
| 0 | 0 | 0 | 1 | 1.5 | 0 | 6 |
| 0 | 0 | 1 | 0 | 0 | 0.5 | 86 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(k^2 + k - 6 \geq 0\) and \(k - 1 \geq 0\) | M1 | Uses the optimal objective row to find at least one correct inequality in \(k\); condone strict inequality or inequalities |
| Critical values \(k = 2, -3\) | A1 | Calculates both correct critical values of the quadratic inequality; allow equals signs and inequalities |
| \(k \geq 2\) | A1 | Uses requirement that both inequalities must be satisfied simultaneously; do not condone strict inequality |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(s = 0\) | B1 | Writes down correctly the value of the variable \(s\) |
| Total: 4 |
## Question 3:
### Part (a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $k^2 + k - 6 \geq 0$ and $k - 1 \geq 0$ | M1 | Uses the optimal objective row to find at least one correct inequality in $k$; condone strict inequality or inequalities |
| Critical values $k = 2, -3$ | A1 | Calculates both correct critical values of the quadratic inequality; allow equals signs and inequalities |
| $k \geq 2$ | A1 | Uses requirement that both inequalities must be satisfied simultaneously; do not condone strict inequality |
### Part (b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $s = 0$ | B1 | Writes down correctly the value of the variable $s$ |
| **Total: 4** | | |3 The Simplex tableau below is optimal.
\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | c | }
\hline
$\boldsymbol { P }$ & $\boldsymbol { x }$ & $\boldsymbol { y }$ & $\boldsymbol { z }$ & $\boldsymbol { r }$ & $\boldsymbol { s }$ & value \\
\hline
1 & $k ^ { 2 } + k - 6$ & 0 & 0 & $k - 1$ & 1 & 20 \\
\hline
0 & 0 & 0 & 1 & 1.5 & 0 & 6 \\
\hline
0 & 0 & 1 & 0 & 0 & 0.5 & 86 \\
\hline
\end{tabular}
\end{center}
3
\begin{enumerate}[label=(\alph*)]
\item Deduce the range of values that $k$ must satisfy.\\
3
\item Write down the value of the variable $s$ which corresponds to the optimal value of $P$.
\end{enumerate}
\hfill \mbox{\textit{AQA Further Paper 3 Discrete 2019 Q3 [4]}}