# Question 6:

## Part (a)
$P - x - 2y - 6z = 0$ | B1 | (1)

## Part (b) – First Tableau pivot:
Correct pivot located, attempt to divide row | M1 |
pivot row correct including change of b.v. | A1 |
Correct row operations used at least once or stated correctly | M1 |
Looking at non zero-and-one columns, one column ft correct | A1ft |
cao | A1 | (5)

Second tableau:
$$\begin{array}{c|ccccccc}
\text{b.v.} & x & y & z & r & s & t & \text{Value} \\ \hline
r & -1 & \frac{1}{2} & 0 & 1 & -\frac{1}{2} & 0 & 10 \\
z & \frac{1}{2} & \frac{1}{4} & 1 & 0 & \frac{1}{4} & 0 & 7 \\
t & -\frac{5}{2} & -\frac{1}{4} & 0 & 0 & -\frac{3}{4} & 1 & 1 \\
P & 2 & -\frac{1}{2} & 0 & 0 & \frac{3}{2} & 0 & 42
\end{array}$$
Row ops: $R_1 - 2R_2$, $R_2 \div 4$, $R_3 - 3R_2$, $R_4 + 6R_2$ | M1, A1ft, A1 |

(ft) Correct pivot identified – negative pivot gets M0 M0 | M1 |
ft pivot row correct including change of bv | A1 | (4)

Third tableau:
$$\begin{array}{c|ccccccc}
\text{b.v.} & x & y & z & r & s & t & \text{Value} \\ \hline
y & -2 & 1 & 0 & 2 & -1 & 0 & 20 \\
z & 1 & 0 & 1 & -\frac{1}{2} & \frac{1}{2} & 0 & 2 \\
t & -3 & 0 & 0 & \frac{1}{2} & -1 & 1 & 6 \\
P & 1 & 0 & 0 & 1 & 1 & 0 & 52
\end{array}$$
Row ops: $R_1 \div \frac{1}{2}$, $R_2 - \frac{1}{4}R_1$, $R_3 + \frac{1}{4}R_1$, $R_4 + \frac{1}{2}R_1$ | M1, A1ft, M1, A1 |

## Part (c)
$P=52$, $x=0$, $y=20$, $z=2$, $r=0$, $s=0$, $t=6$ | M1, A1ft, A1 | (3) At least 4 values stated; no negatives

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