# Question 5:

## Part (i)
| Answer | Mark | Guidance |
|--------|------|----------|
| $x \geq 0,\ y \geq 0$ | B1 | For both trivial constraints; allow $>$ |
| $y \leq 2x + 1$ | B1 | For this inequality or equivalent; allow $<$ |
| $4x + 3y \leq 12$ | B1 | For this inequality or equivalent; allow $<$ |

## Part (ii)
| Answer | Mark | Guidance |
|--------|------|----------|
| $(0,0)$, $(3,0)$, $(0,1)$ | B2 | For these three vertices, any two correct $\Rightarrow$ B1 |
| $(0.9, 2.8)$ | B1 | For this vertex exact, in decimals or fractions |
| $(0,0) \to P=0$; $(0,1) \to P=3$; $(0.9, 2.8) \to P=12.9$; $(3,0) \to P=15$ | M1 | For calculating $P = 5x + 3y$ for at least one of their vertices or clear evidence of using an appropriate line of constant profit |
| $x = 3$ and $y = 0$ | A1 | For the correct values of $x$ and $y$ clearly identified |
| $P = 15$ | A1 | For 15 clearly identified as the optimum value |

## Part (iii)
| Answer | Mark | Guidance |
|--------|------|----------|
| Either consider the gradient of the profit line $(-\frac{1}{3}a)$ and the gradients of the boundary lines $(2$ and $-1\frac{1}{2})$ | M1 | One method mark for each appropriate gradient calculated correctly |
| Or calculate $Q$ at vertices $\Rightarrow 3,\ 0.9a+8.4,\ 3a$ | M1 | Or for each appropriate value of $Q$ calculated correctly |
| Hence require $a \leq -6$ | M1 A1 | For the correct set of values identified. $[a = -6$ or any valid proper subset of the correct answer with no method shown $\Rightarrow$ B1 only$]$ |

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