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AQA D2 2015 June Q5
11 marks Moderate -0.5
5 Tom is going on a driving holiday and wishes to drive from \(A\) to \(K\).
The network below shows a system of roads. The number on each edge represents the maximum altitude of the road, in hundreds of metres above sea level. Tom wants to ensure that the maximum altitude of any road along the route from \(A\) to \(K\) is minimised. \includegraphics[max width=\textwidth, alt={}, center]{b0f9523e-51dd-495f-99ec-4724243b5619-14_1522_1363_660_342}
  1. Working backwards from \(\boldsymbol { K }\), use dynamic programming to find the optimal route when driving from \(A\) to \(K\). You must complete the table opposite as your solution.
  2. Tom finds that the road \(C F\) is blocked. Find Tom's new optimal route and the maximum altitude of any road on this route.
    [0pt] [2 marks] \section*{Answer space for question 5}
    StageStateFromValue
    1H\(K\)
    I\(K\)
    \(J\)\(K\)
    2
    1. Optimal route is \(\_\_\_\_\)
    2. Tom's route is \(\_\_\_\_\) Maximum altitude is \(\_\_\_\_\) Figure 4 below shows a network of pipes.
      The capacity of each pipe is given by the number not circled on each edge. The numbers in circles represent an initial flow. \begin{figure}[h]
      \captionsetup{labelformat=empty} \caption{Figure 4} \includegraphics[alt={},max width=\textwidth]{b0f9523e-51dd-495f-99ec-4724243b5619-18_1039_1623_561_191}
      \end{figure}
    3. Find the value of the initial flow.
      1. Use the initial flow and the labelling procedure on Figure 5 to find the maximum flow through the network. You should indicate any flow-augmenting routes in the table and modify the potential increases and decreases of the flow on the network.
      2. State the value of the maximum flow and, on Figure 6, illustrate a possible flow along each edge corresponding to this maximum flow.
      3. Confirm that you have a maximum flow by finding a cut of the same value. List the edges of your cut.
      4. On a particular day, there is a restriction at vertex \(G\) which allows a maximum flow through \(G\) of 30 . Find, by inspection, the maximum flow through the network on this day.
    5. Initial flow \(=\) \(\_\_\_\_\)
      1. Figure 5 \includegraphics[max width=\textwidth, alt={}, center]{b0f9523e-51dd-495f-99ec-4724243b5619-19_2158_1559_543_296} \(7 \quad\) Arsene and Jose play a zero-sum game. The game is represented by the following pay-off matrix for Arsene, where \(x\) is a constant. The value of the game is 2.5 .
        Jose
        \cline { 2 - 4 }StrategyCD
        \cline { 2 - 4 } ArseneA\(x + 3\)1
        \cline { 2 - 4 }B\(x + 1\)3
        \cline { 2 - 4 }
        \cline { 2 - 4 }
    7. Find the optimal mixed strategy for Arsene.
    8. Find the value of \(x\).
      \includegraphics[max width=\textwidth, alt={}]{b0f9523e-51dd-495f-99ec-4724243b5619-22_1636_1707_1071_153}
      \includegraphics[max width=\textwidth, alt={}]{b0f9523e-51dd-495f-99ec-4724243b5619-24_2288_1705_221_155}
AQA D2 2016 June Q1
12 marks Moderate -0.5
1
Figure 1 below shows an activity diagram for a project. Each activity requires one worker. The duration required for each activity is given in hours.
  1. Find the earliest start time and the latest finish time for each activity and insert these values on Figure 1.
    1. Find the critical path.
    2. Find the float time of activity \(F\).
  3. Using Figure 2 on page 3, draw a resource histogram to illustrate how the project can be completed in the minimum time, assuming that each activity is to start as early as possible.
    1. Given that there are two workers available for the project, find the minimum completion time for the project.
    2. Write down an allocation of tasks to the two workers that corresponds to your answer in part (d)(i). \section*{Answer space for question 1} \begin{figure}[h]
      \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{34de3f03-a275-44fb-88b2-b88038bcec97-02_687_1655_1941_189}
      \end{figure} \section*{Answer space for question 1} \begin{figure}[h]
      \captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{34de3f03-a275-44fb-88b2-b88038bcec97-03_1115_1575_434_283}
      \end{figure}
      \includegraphics[max width=\textwidth, alt={}]{34de3f03-a275-44fb-88b2-b88038bcec97-03_1024_1593_1683_267}
AQA D2 2016 June Q2
10 marks Standard +0.3
2 Alan, Beth, Callum, Diane and Ethan work for a restaurant chain. The costs, in pounds, for the five people to travel to each of five different restaurants are recorded in the table below. Alan cannot travel to restaurant 1 and Beth cannot travel to restaurants 3 and 5, as indicated by the asterisks in the table.
AQA D2 2016 June Q3
13 marks Standard +0.8
3
Maximise \(\quad P = 2 x - 3 y + 4 z\) subject to \(\quad x + 2 y + z \leqslant 20\) \(x - y + 3 z \leqslant 24\) \(3 x - 2 y + 2 z \leqslant 30\) and \(\quad x \geqslant 0 , y \geqslant 0 , z \geqslant 0\).
  1. Display the linear programming problem in a Simplex tableau.
    1. The first pivot to be chosen is from the \(z\)-column. Identify the pivot and explain why this particular value is chosen.
    2. Perform one iteration of the Simplex method.
    3. Perform one further iteration.
  3. Interpret your final tableau and state the values of your slack variables.
    [0pt] [3 marks]
AQA D2 2016 June Q4
15 marks Standard +0.8
4 Monica and Vladimir play a zero-sum game. The game is represented by the following pay-off matrix for Monica.
AQA D2 2016 June Q5
11 marks Moderate -0.5
5 Robert is planning to renovate four houses, \(A , B , C\) and \(D\), at the rate of one per month. The houses can be renovated in any order but the costs will vary because some of the materials left over from renovating one house can be used for the next one. The expected profits, in hundreds of pounds, are given in the table below.
AQA D2 2016 June Q6
14 marks Standard +0.3
6 The network shows a system of pipes with lower and upper capacities for each pipe in litres per second. \includegraphics[max width=\textwidth, alt={}, center]{34de3f03-a275-44fb-88b2-b88038bcec97-22_817_744_397_648}
    1. Find the value of the cut \(X\).
    2. Hence state what can be deduced about the maximum flow from \(A\) to \(H\).
  2. Figure 3 shows a partially completed diagram for a feasible flow of 28 litres per second from \(A\) to \(H\). Indicate, on Figure 3, the flows along the edges \(B D , B E\) and \(C D\).
    1. Using your feasible flow from part (b) as an initial flow, indicate potential increases and decreases of the flow along each edge on Figure 4.
    2. Use flow augmentation on Figure 4 to find the maximum flow from \(A\) to \(H\). You should indicate any flow augmenting paths in the table and modify the potential increases and decreases of the flow on the network.
    3. State the maximum flow and indicate a maximum flow on Figure 5. \section*{Answer space for question 6} \begin{figure}[h]
      \captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{34de3f03-a275-44fb-88b2-b88038bcec97-23_682_689_312_397}
      \end{figure} \begin{figure}[h]
      \captionsetup{labelformat=empty} \caption{Figure 4} \includegraphics[alt={},max width=\textwidth]{34de3f03-a275-44fb-88b2-b88038bcec97-23_935_1477_1037_365}
      \end{figure} Figure 5
      \includegraphics[max width=\textwidth, alt={}]{34de3f03-a275-44fb-88b2-b88038bcec97-24_2032_1707_219_153}
AQA AS Paper 2 2019 June Q1
1 marks Easy -1.8
1 Find the gradient of the curve \(y = \mathrm { e } ^ { - 3 x }\) at the point where it crosses the \(y\)-axis. Circle your answer. \(\begin{array} { l l l } - 3 & - 1 & 1 \end{array}\)
AQA AS Paper 2 2019 June Q2
1 marks Easy -1.3
2 Find the centre of the circle \(x ^ { 2 } + y ^ { 2 } + 4 x - 6 y = 12\) Tick ( \(\checkmark\) ) one box.
(-2, -3) □
(-2, 3) □ \(( 2 , - 3 )\) □ \(( 2,3 )\) □
AQA AS Paper 2 2019 June Q3
2 marks Moderate -0.8
3 It is given that \(\sin \theta = - 0.1\) and \(180 ^ { \circ } < \theta < 270 ^ { \circ }\) Find the exact value of \(\cos \theta\)
AQA AS Paper 2 2019 June Q4
4 marks Moderate -0.8
4 Show that, for \(x > 0\) $$\log _ { 10 } \frac { x ^ { 4 } } { 100 } + \log _ { 10 } 9 x - \log _ { 10 } x ^ { 3 } \equiv 2 \left( - 1 + \log _ { 10 } 3 x \right)$$
AQA AS Paper 2 2019 June Q5
4 marks Moderate -0.3
5 A triangular prism has a cross section \(A B C\) as shown in the diagram below. Angle \(A B C = 25 ^ { \circ }\) Angle \(A C B = 30 ^ { \circ }\) \(B C = 40\) millimetres. The length of the prism is 300 millimetres.
Calculate the volume of the prism, giving your answer to three significant figures.
AQA AS Paper 2 2019 June Q6
5 marks Moderate -0.3
6 A curve has equation \(y = \frac { 2 } { x \sqrt { x } }\) \includegraphics[max width=\textwidth, alt={}, center]{b45dc98e-1699-47c9-9228-5abe0e5c9195-05_508_549_420_744} The region enclosed between the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = a\) has area 3 units. Given that \(a > 1\), find the value of \(a\).
Fully justify your answer.
AQA AS Paper 2 2019 June Q7
6 marks Moderate -0.3
7 The points \(A ( a , 3 )\) and \(B ( 10,6 )\) lie on a circle. \(A B\) is a diameter of the circle and passes through the point ( 2,4 )
The circle has equation $$( x - c ) ^ { 2 } + ( y - d ) ^ { 2 } = e$$ where \(c , d\) and \(e\) are rational numbers. Find the values of \(a , c , d\) and \(e\).
AQA AS Paper 2 2019 June Q8
10 marks Standard +0.3
8 A curve has equation $$y = x ^ { 3 } + p x ^ { 2 } + q x - 45$$ The curve passes through point \(R ( 2,3 )\) The gradient of the curve at \(R\) is 8
8
  1. Find the value of \(p\) and the value of \(q\).
    8
  2. Calculate the area enclosed between the normal to the curve at \(R\) and the coordinate 8 (b) axes. \(9 \quad\) A curve \(C\) has equation \(y = \mathrm { f } ( x )\) where $$f ( x ) = ( x - 2 ) ( x - 3 ) ^ { 2 }$$
AQA AS Paper 2 2019 June Q9
10 marks Moderate -0.3
9
  1. Find the exact coordinates of the turning points of \(C\).
    Determine the nature of each turning point.
    Fully justify your answer.
    9
  2. State the coordinates of the turning points of the curve $$y = \mathrm { f } ( x + 1 ) - 4$$
AQA AS Paper 2 2019 June Q10
10 marks Moderate -0.3
10 As part of an experiment, Zena puts a bucket of hot water outside on a day when the outside temperature is \(0 ^ { \circ } \mathrm { C }\). She measures the temperature of the water after 10 minutes and after 20 minutes. Her results are shown below.
Time (minutes)1020
Temperature (degrees Celsius)3012
Zena models the relationship between \(\theta\), the temperature of the water in \({ } ^ { \circ } \mathrm { C }\), and \(t\), the time in minutes, by $$\theta = A \times 10 ^ { - k t }$$ where \(A\) and \(k\) are constants. 10
  1. Using \(t = 0\), explain how the value of \(A\) relates to the experiment. 10
  2. Show that $$\log _ { 10 } \theta = \log _ { 10 } A - k t$$ 10
  3. Using Zena's results, calculate the values of \(A\) and \(k\).
    10
  4. Zena states that the temperature of the water will be less than \(1 ^ { \circ } \mathrm { C }\) after 45 minutes. Determine whether the model supports this statement.
    10
  5. Explain why Zena's model is unlikely to accurately give the value of \(\theta\) after 45 minutes.
AQA AS Paper 2 2019 June Q11
1 marks Easy -2.0
11 A survey is undertaken to find out the most popular political party in London.
The first 1100 available people from London are surveyed.
Identify the name of this type of sampling.
Circle your answer.
simple random
opportunity
stratified
quota
AQA AS Paper 2 2019 June Q12
1 marks Easy -1.8
12 Manny is studying the price and number of pages of a random sample of books.
He calculates the value of the product moment correlation coefficient between the price and number of pages in each book as 1.05 Which of the following best describes the value 1.05 ?
Tick ( \(\checkmark\) ) one box.
definitely correct □
probably correct □
probably incorrect □
definitely incorrect □ \includegraphics[max width=\textwidth, alt={}, center]{b45dc98e-1699-47c9-9228-5abe0e5c9195-15_2488_1716_219_153}
AQA AS Paper 2 2019 June Q13
6 marks Easy -1.2
13 Denzel wants to buy a car with a propulsion type other than petrol or diesel.
He takes a sample, from the Large Data Set, of the CO2 emissions, in \(\mathrm { g } / \mathrm { km }\), of cars with one particular propulsion type. The sample is as follows $$\begin{array} { l l l l l l l l } 82 & 13 & 96 & 49 & 96 & 92 & 70 & 81 \end{array}$$ 13
  1. Using your knowledge of the Large Data Set, state which propulsion type this sample is for, giving a reason for your answer.
    13
  2. Calculate the mean of the sample.
    13
  3. Calculate the standard deviation of the sample.
    13
  4. Denzel claims that the value 13 is an outlier. 13 (d) (i) Any value more than 2 standard deviations from the mean can be regarded as an outlier. Verify that Denzel's claim is correct.
    13 (d) (ii) State what effect, if any, removing the value 13 from the sample would have on the standard deviation.
AQA AS Paper 2 2019 June Q14
4 marks Easy -1.2
14 A probability distribution is given by $$\mathrm { P } ( X = x ) = c ( 4 - x ) , \text { for } x = 0,1,2,3$$ where \(c\) is a constant.
14
  1. Show that \(c = \frac { 1 } { 10 }\) 14
  2. Calculate \(\mathrm { P } ( X \geq 1 )\)
AQA AS Paper 2 2019 June Q15
6 marks Moderate -0.3
15 Two independent events, \(A\) and \(B\), are such that $$\begin{aligned} \mathrm { P } ( A ) & = 0.2 \\ \mathrm { P } ( A \cup B ) & = 0.8 \end{aligned}$$ 15
    1. Find \(\mathrm { P } ( B )\) 15
      1. (ii) Find \(\mathrm { P } ( A \cap B )\) 15
    2. State, with a reason, whether or not the events \(A\) and \(B\) are mutually exclusive.
AQA AS Paper 2 2019 June Q16
9 marks Moderate -0.3
16
16
Andrea is the manager of a company which makes mobile phone chargers.
In the past, she had found that \(12 \%\) of all chargers are faulty.
Andrea decides to move the manufacture of chargers to a different factory.
Andrea tests 60 of the new chargers and finds that 4 chargers are faulty.
Investigate, at the \(10 \%\) level of significance, whether the proportion of faulty chargers has reduced.
[7 marks] \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)
16
  • State, in context, two assumptions that are necessary for the distribution that you have used in part (a) to be valid.
    •
    AQA AS Paper 2 2021 June Q1
    1 marks Easy -1.8
    1 Express as a single power of \(a\) $$\frac { a ^ { 2 } } { \sqrt { a } }$$ where \(a \neq 0\) Circle your answer. \(a ^ { 1 }\) \(a ^ { \frac { 3 } { 2 } }\) \(a ^ { \frac { 5 } { 2 } }\) \(a ^ { 4 }\)
    AQA AS Paper 2 2021 June Q2
    1 marks Easy -1.3
    2 One of the diagrams below shows the graph of \(y = \sin \left( x + 90 ^ { \circ } \right)\) for \(0 ^ { \circ } \leq x \leq 360 ^ { \circ }\) Identify the correct graph. Tick ( \(\checkmark\) ) one box. \includegraphics[max width=\textwidth, alt={}, center]{f87d1b36-26db-4a0b-b9ec-d7d82a396aba-03_451_465_568_497} \includegraphics[max width=\textwidth, alt={}, center]{f87d1b36-26db-4a0b-b9ec-d7d82a396aba-03_124_154_724_1073} \includegraphics[max width=\textwidth, alt={}, center]{f87d1b36-26db-4a0b-b9ec-d7d82a396aba-03_458_472_1105_495}
    □ \includegraphics[max width=\textwidth, alt={}, center]{f87d1b36-26db-4a0b-b9ec-d7d82a396aba-03_453_468_1647_497} \includegraphics[max width=\textwidth, alt={}, center]{f87d1b36-26db-4a0b-b9ec-d7d82a396aba-03_117_132_1809_1091} \includegraphics[max width=\textwidth, alt={}, center]{f87d1b36-26db-4a0b-b9ec-d7d82a396aba-03_461_479_2183_488}