Moderate -0.8 This is a standard textbook linear programming question requiring routine application of well-practiced techniques: translating constraints into inequalities, graphing lines, identifying the feasible region, and using the vertex method. While multi-part with several steps, each component is straightforward with no novel problem-solving required—significantly easier than average A-level maths questions.
5. A school awards two types of prize, junior and senior.
The school decides that it will award at least 25 junior prizes and at most 60 senior prizes.
Let \(x\) be the number of junior prizes that the school awards and let \(y\) be the number of senior prizes that the school awards.
Write down two inequalities to model these constraints.
(2)
Two further constraints are
$$\begin{aligned}
& 2 x + 5 y \geqslant 250 \\
& 5 x - 3 y \leqslant 150
\end{aligned}$$
Add lines and shading to Diagram 1 in the answer book to represent all four of these constraints. Hence determine the feasible region and label it \(R\).
The cost of a senior prize is three times the cost of a junior prize. The school wishes to minimise the cost of the prizes.
State the objective function, giving your answer in terms of \(x\) and \(y\).
Determine the exact coordinates of the vertices of the feasible region. Hence use the vertex method to find the number of junior prizes and the number of senior prizes that the school should award. You should make your working clear.
(d) \(A = (25, 40), B = (25, 60), C = (66, 60)\); \(D = \left(\frac{1500}{31}, \frac{950}{31}\right)\) or \(\left(48\frac{12}{31}, 30\frac{20}{31}\right)\); At A, C = 145; At B, C = 205; At C, C = 246; At D, \(C = \frac{4350}{31}\) or \(140\frac{10}{31}(= 140.322...)\); So D is the optimal point; Testing integer solutions around D, \(x = 48\) and \(y = 31\) is optimal integer solution, so they should have 48 junior prizes and 31 senior prizes
M1 A1 M1 A1 M1 A1
(8)
15 marks
Notes for Question 5:
a1B1: CAO (\(x \geq 25\))
a2B1: CAO (\(y \leq 60\))
In (b) lines must be long enough to define the correct feasible region and pass through one small 'square' of the points stated in either the horizontal or vertical direction e.g. for (25, 60) the line must pass through a point in the interval [23,27] for x or [59,61] for y:
- \(x = 25\) from (25, 10) to (25, 62)
- \(y = 60\) from (0, 60) to (100, 60)
- \(5x – 3y = 150\) from (36, 10) to (69, 65)
- \(5y + 2x = 250\) from (0, 50) to (125, 0)
b1B1: Any two lines correctly drawn
b2B1: Any three lines correctly drawn
b3B1: All four lines correctly drawn
b4B1: Region, R, correctly labelled – not just implied by shading – dependent on scoring the first three marks in this part
c1B1: CAO – correct expression in the form \(k(x + 3y)\) for any positive real number \(k\) (but not \(k\))
d1B1: Any two of (25, 40), (25, 60), (66, 60) stated correctly – accept \(x = 25, y = 40\), etc. throughout (d)
d2B1: All three integer coordinates stated correctly
d1M1: Using simultaneous equations to find the non-integer vertex – must get to \(x = ...\) and \(y = ...\). Must be a correct method to solve simultaneous equations but allow slips/errors. If no working present then this mark can be awarded for an awrt (48.4, 30.6)
d1A1: CAO \(\left(\frac{1500}{31}, \frac{950}{31}\right)\) or \(\left(48\frac{12}{31}, 30\frac{20}{31}\right)\) – must be exact (condone correct recurring decimal notation). If correct answer seen with no working then award M1 A1 in this part. ISW if correct exact answer seen which is then given in non-exact form
d2M1: Evaluating their objective function at at least three of their vertices for their feasible region.
d2A1: All four correct C values (from a correct objective function) either given exactly or correct to at least 1 dp
d3M1: Testing any two of (48, 30) or (48, 31) or (49, 30) or (49, 31) in a correct objective function or the correct pair of inequalities. Note candidates may reject a point after testing in only one correct inequality which is acceptable – this mark is not dependent on any previous mark
d3A1: CSO (all previous marks must have been awarded in this question) – must have tested (48, 31) in the correct objective function or correct pair of inequalities – accept \(x = 48\) and \(y = 31\) or stated as a pair of coordinates
| Answer/Scheme | Marks | Guidance |
|---|---|---|
| (a) $x \geq 25, y \leq 60$ | B1 B1 | (2) |
| (b) Four lines, each marked B1 | B1 B1 B1 B1 | (4) |
| (c) $(C =) x + 3y$ | B1 | (1) |
| (d) $A = (25, 40), B = (25, 60), C = (66, 60)$; $D = \left(\frac{1500}{31}, \frac{950}{31}\right)$ or $\left(48\frac{12}{31}, 30\frac{20}{31}\right)$; At A, C = 145; At B, C = 205; At C, C = 246; At D, $C = \frac{4350}{31}$ or $140\frac{10}{31}(= 140.322...)$; So D is the optimal point; Testing integer solutions around D, $x = 48$ and $y = 31$ is optimal integer solution, so they should have 48 junior prizes and 31 senior prizes | M1 A1 M1 A1 M1 A1 | (8) |
| | **15 marks** | |
**Notes for Question 5:**
a1B1: CAO ($x \geq 25$)
a2B1: CAO ($y \leq 60$)
In (b) lines must be long enough to define the correct feasible region and pass through one small 'square' of the points stated in either the horizontal or vertical direction e.g. for (25, 60) the line must pass through a point in the interval [23,27] for x or [59,61] for y:
- $x = 25$ from (25, 10) to (25, 62)
- $y = 60$ from (0, 60) to (100, 60)
- $5x – 3y = 150$ from (36, 10) to (69, 65)
- $5y + 2x = 250$ from (0, 50) to (125, 0)
b1B1: Any two lines correctly drawn
b2B1: Any three lines correctly drawn
b3B1: All four lines correctly drawn
b4B1: Region, R, correctly labelled – not just implied by shading – dependent on scoring the first three marks in this part
c1B1: CAO – correct expression in the form $k(x + 3y)$ for any positive real number $k$ (but not $k$)
d1B1: Any two of (25, 40), (25, 60), (66, 60) stated correctly – accept $x = 25, y = 40$, etc. throughout (d)
d2B1: All three integer coordinates stated correctly
d1M1: Using simultaneous equations to find the non-integer vertex – must get to $x = ...$ and $y = ...$. Must be a correct method to solve simultaneous equations but allow slips/errors. If no working present then this mark can be awarded for an awrt (48.4, 30.6)
d1A1: CAO $\left(\frac{1500}{31}, \frac{950}{31}\right)$ or $\left(48\frac{12}{31}, 30\frac{20}{31}\right)$ – must be exact (condone correct recurring decimal notation). If correct answer seen with no working then award M1 A1 in this part. ISW if correct exact answer seen which is then given in non-exact form
d2M1: Evaluating their objective function at at least three of their vertices for their feasible region.
d2A1: All four correct C values (from a correct objective function) either given exactly or correct to at least 1 dp
d3M1: Testing any two of (48, 30) or (48, 31) or (49, 30) or (49, 31) in a correct objective function or the correct pair of inequalities. Note candidates may reject a point after testing in only one correct inequality which is acceptable – this mark is not dependent on any previous mark
d3A1: CSO (all previous marks must have been awarded in this question) – must have tested (48, 31) in the correct objective function or correct pair of inequalities – accept $x = 48$ and $y = 31$ or stated as a pair of coordinates
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5. A school awards two types of prize, junior and senior.
The school decides that it will award at least 25 junior prizes and at most 60 senior prizes.\\
Let $x$ be the number of junior prizes that the school awards and let $y$ be the number of senior prizes that the school awards.
\begin{enumerate}[label=(\alph*)]
\item Write down two inequalities to model these constraints.\\
(2)
Two further constraints are
$$\begin{aligned}
& 2 x + 5 y \geqslant 250 \\
& 5 x - 3 y \leqslant 150
\end{aligned}$$
\item Add lines and shading to Diagram 1 in the answer book to represent all four of these constraints. Hence determine the feasible region and label it $R$.
The cost of a senior prize is three times the cost of a junior prize. The school wishes to minimise the cost of the prizes.
\item State the objective function, giving your answer in terms of $x$ and $y$.
\item Determine the exact coordinates of the vertices of the feasible region. Hence use the vertex method to find the number of junior prizes and the number of senior prizes that the school should award. You should make your working clear.
\end{enumerate}
\hfill \mbox{\textit{Edexcel D1 2017 Q5 [15]}}