| Answer | Marks | Guidance |
|--------|-------|----------|
| (a) Minimise $P = 8x + 10y + 14z$ subject to $x + y + z \geq 200$; $3y \leq x$; $\frac{7}{10}(x + y + z) \geq x$ or $\frac{1}{5}(x + y + z) \leq y$; $x + z \leq 4y$; $3x \leq 7y + 7z$; $(x, y, z \geq 0)$ | B1 B1 B1 M1 A1 A1 (6) | a1B1: Expression correct together with 'minimise' or 'min' but not 'minimum' – isw if coefficients are subsequently simplified but $8x + 10y + 14z$ must be seen at some point for this mark to be awarded. The 'min' must appear beside (or near to) the correct expression. a2B1: CAO ($x + y + z \geq 200$) oe. a3B1: CAO ($3y \leq x$) oe. a1M1: $\frac{7}{10}(x + y + z) \square x$ oe or $\frac{1}{5}(x + y + z) \square y$ oe where $\square$ is any inequality or equals – allow 0.7 and 0.2 but not 70% or 20% unless recovered to a fraction or decimal later. a1A1: Either $x + z \leq 4y$ or $3x \leq 7y + 7z$ or both correct but not simplified or with integer coefficients (so allow 2x + 2z − 8y ≤ 0). a2A1: Both correct – must be simplified (e.g. only one term in each variable) and integer coefficients but allow positive integer multiples (e.g. 2x + 2z − 8y ≤ 0). |
| (b)(i) $z = 200 – x – y$ substituted into constraints gives $x \leq 140, y \geq 40$ | M1 A1 dM1 A1 (5) | b1M1: Using $x + y + z = 200$ to obtain either an inequality (or value) for either $x$ or $y$. b1A1: Any one of $x \leq 140$ or $y \geq 40$ correct (this mark can be implied for either $x = 140$ or $y = 40$ seen provided not from incorrect working). b1d2M1: Using their least value of $y$ and greatest value of $x$ to find $z$ (so not from the inequality $3x \leq x$) – the total of $x$, $y$ and $z$ must be 200 (and dependent on first M mark). b1d2A1: All three types of doughnuts correct (in context) – so not just in terms of $x$, $y$ and $z$ but values stated explicitly and not in terms of x, y and z. |
| | 140 ring doughnuts, 40 jam doughnuts and 20 custard doughnuts; £18 or 1800 | A1 | b1i1A1: CAO for cost – if 1800 given in this is fine (without pence) but if 18 then must be £. |
| (b)(ii) | | 11 marks |  |

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# Notes for Question 8:

a1B1: Expression correct together with 'minimise' or 'min' but not 'minimum' – isw if coefficients are subsequently simplified but $8x + 10y + 14z$ must be seen at some point for this mark to be awarded. The 'min' must appear beside (or near to) the correct expression.

a2B1: CAO ($x + y + z \geq 200$) oe

a3B1: CAO ($3y \leq x$) oe

a1M1: $\frac{7}{10}(x + y + z) \square x$ oe or $\frac{1}{5}(x + y + z) \square y$ oe where $\square$ is any inequality or equals – allow 0.7 and 0.2 but not 70% or 20% unless recovered to a fraction or decimal later.

a1A1: Either $x + z \leq 4y$ or $3x \leq 7y + 7z$ or both correct but not simplified or with integer coefficients (so allow $2x + 2z − 8y \leq 0$).

a2A1: Both correct – must be simplified (e.g. only one term in each variable) and integer coefficients but allow positive integer multiples (e.g. $2x + 2z − 8y \leq 0$).

b1M1: Using $x + y + z = 200$ to obtain either an inequality (or value) for either $x$ or $y$.

b1A1: Any one of $x \leq 140$ or $y \geq 40$ correct (this mark can be implied for either $x = 140$ or $y = 40$ seen provided not from incorrect working).

b1d2M1: Using their least value of $y$ and greatest value of $x$ to find $z$ (so not from the inequality $3x \leq x$) – the total of $x$, $y$ and $z$ must be 200 (and dependent on first M mark).

b1d2A1: All three types of doughnuts correct (in context) – so not just in terms of $x$, $y$ and $z$ but values stated explicitly and not in terms of x, y and z.

b1i1A1: CAO for cost – if 1800 given in this is fine (without pence) but if 18 then must be £.

**SC in (b):** If correct answers from either no working or from explicitly solving three correct equations e.g. $x + y + z = 200$; $3y = x$; $3x − 7y − 7z = 0$ with no algebraic working then award M0A0 M1 (for correct values of $x$, $y$ and $z$) A1 (for $x + y + z = 200$), A1 (for answers in context) and A1 for 1800. If solving these three equations with algebraic working then full marks can be awarded. If no marks awarded according to the notes above and solving any other (e.g. incorrect) sets of equations then no marks in (b).