# Question 6:

## Part (i)

| Answer | Mark | Guidance |
|--------|------|----------|
| Line through $(4, 4.5)$ and $(6, 3)$ | M1 | $\pm 0.2$ on each coordinate |
| Line through $(4, 0)$ and $(6, 3)$ | M1 | $\pm 0.2$ on each coordinate |
| Line through $(0, 3)$ and $(4, 4.5)$ | M1 | $\pm 0.2$ on each coordinate |
| Shading correct, feasible region (FR) need not be labelled (dependent on all three M marks); need not show shading out $x < 0$, $y < 0$ | A1 | Scroll down to check spare copy; visually impaired candidates have tolerance $\pm 0.5$ throughout |
| **[4]** | | |

## Part (ii)

| Answer | Mark | Guidance |
|--------|------|----------|
| $(4, 0)$, $(6, 3)$, $(4, 4.5)$ and $(0, 3)$ | M1 | Any two vertices of their FR (other than origin), as coordinates or equivalent (from their graph to within $\pm 0.2$); or two of the correct vertices (new start) |
| These four coordinates correct, ignore any extra points | A1 | |
| $(6, 3) \to 54$ and $(4, 4.5) \to 56$ (both seen, not implied); or $P = 5x + 8y$ calculated for two of their vertices with $x > 0$, $y > 0$ | M1 | |
| Optimal when $x = 4$, $y = 4.5$, $P = 56$ | A1 | $(4, 4.5)$ indicated (e.g. arrowed) and $(4, 4.5) \to 56$ seen (cao) |
| **[4]** | | |

## Part (iii)

| Answer | Mark | Guidance |
|--------|------|----------|
| Four ($x$, max $y$) pairs, in any form (follow through their FR) | M1 | |
| All 7 pairs correct, and no extras (cao, not ft) | A1 | |
| Calculating $P = 5x + 8y$ correctly for at least three of their listed integer-valued points (not implied from answer $(6, 3)$) | M1 | |
| Optimal when $x = 6$, $y = 3$ | A1 | $(6, 3)$ written down, in any form (cao) |
| **[4]** | | |

## Part (iv)

| Answer | Mark | Guidance |
|--------|------|----------|
| So that a slack variable can be added to form an equality, or RHS (value) needs to be non-negative | B1 | Allow 'constraints must be $\leq$' or 'third constraint was $\geq$'; do not allow 'positive' for 'non-negative' |
| **[1]** | | |

## Part (v)

| Answer | Mark | Guidance |
|--------|------|----------|
| $4 \times 7$ table with three constraint rows correct (including $P$ column, although columns need not be labelled) | M1 | Order of rows and columns may be changed, need not be labelled |
| Objective row correct; assume blanks mean zero | A1 | |
| **[2]** | May credit initial tableau seen in (vi) if no response seen in (v) |

## Part (vi)

| Answer | Mark | Guidance |
|--------|------|----------|
| Least non-negative ratio in $y$ column is $24 \div 8$; pivot on 8 in row 4 of $y$ column | B1 | Correct pivot choice for $y$ col of their tableau (may be implied) |
| Row 4 (pr) $= \text{r4} \div 8$; Row 1 $= \text{r1} + 8\text{pr}$ or $\text{r1} + \text{r4}$; Row 2 $= \text{r2} + 2\text{pr}$, i.e. $\text{r2} + \frac{1}{4}\text{r4}$; Row 3 $= \text{r3} - 4\text{pr}$, i.e. $\text{r3} - \frac{1}{2}\text{r4}$ | B1 | Correct methods for rows for their positive pivot choice; condone pivot row method not shown; accept reasonable abbreviations, e.g. $+8\text{pr}$ or $a+8h$ but not just $+8$; ft their tableau; condone (r4) for (pr) provided consistent |
| Augmented tableau with $P$ value increased and all RHS entries $\geq 0$; $x = 0$, $y = 3$, $P = 24$ | M1, A1 | Values of $x$, $y$, $P$ written, ft their tableau for non-negative $x$, $y$ and positive $P$ |
| Second augmented tableau with $P$ value increased and all RHS entries $\geq 0$ | M1 | |
| Basis columns correspond to $P = 56$, $x = 4$, $y = 4.5$ and a slack variable $= 9$ (values do not need to be written down); tableau does not need to be correct but does need to correspond to $(4, 4.5)$, 56 and 9 from basis cols with 0's and 1 in them | A1 | |
| **[6]** | | |