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OCR D2 2007 June Q4
16 marks Moderate -0.5
4 Answer this question on the insert provided. The table shows a partially completed dynamic programming tabulation for solving a minimax problem.
StageStateA ctionWorkingM inimax
\multirow{3}{*}{1}0044
1033
2022
\multirow{9}{*}{2}\multirow{3}{*}{0}0\(\max ( 6,4 ) = 6\)\multirow{3}{*}{3}
1\(\max ( 2,3 ) = 3\)
2\(\max ( 3,2 ) = 3\)
\multirow{3}{*}{1}0\(\max ( 2,4 ) =\)\multirow{3}{*}{}
1\(\max ( 4,3 ) =\)
2\(\max ( 5,2 ) =\)
\multirow{3}{*}{2}0max(2,\multirow{3}{*}{}
1max(3,
2max(4,
\multirow{3}{*}{3}\multirow{3}{*}{0}0max(5,\multirow{3}{*}{}
1max(5,
2max(2,
  1. On the insert, complete the last two columns of the table.
  2. State the minimax value and write down the minimax route.
  3. Complete the diagram on the insert to show the network that is represented by the table.
OCR D2 2007 June Q5
13 marks Moderate -0.5
5 Answer this question on the insert provided. The network represents a system of pipes through which fluid can flow from a source, S , to a sink, T . \includegraphics[max width=\textwidth, alt={}, center]{09d4aacd-026b-4d81-a826-3d3f29f9c105-5_1310_1301_447_424} The arrows are labelled to show excess capacities and potential backflows (how much more and how much less could flow in each pipe). The excess capacities and potential backflows are measured in litres per second. Currently the flow is 6 litres per second, all flowing along a single route through the system.
  1. Write down the route of the 6 litres per second that is flowing from \(S\) to \(T\).
  2. What is the capacity of the pipe AG and in which direction can fluid flow along this pipe?
  3. Calculate the capacity of the \(\operatorname { cut } \mathrm { X } = \{ \mathrm { S } , \mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D } , \mathrm { E } \} , \mathrm { Y } = \{ \mathrm { F } , \mathrm { G } , \mathrm { H } , \mathrm { I } , \mathrm { T } \}\).
  4. Describe how a further 7 litres per second can flow from S to T and update the labels on the arrows to show your flow. Explain how you know that this is the maximum flow. \href{http://physicsandmathstutor.com}{physicsandmathstutor.com}
OCR Further Pure Core 2 2023 June Q4
4 marks Challenging +1.2
4 In this question you must show detailed reasoning. The region \(R\) is bounded by the curve with equation \(\mathrm { y } = \frac { 1 } { \sqrt { 3 \mathrm { x } ^ { 2 } - 3 \mathrm { x } + 1 } }\), the \(x\)-axis and the lines with equations \(x = \frac { 1 } { 2 }\) and \(x = 1\) (see diagram). The units of the axes are cm . \includegraphics[max width=\textwidth, alt={}, center]{7b2bfb4e-524f-4d1c-ae98-075c7fb404f9-3_778_1241_497_242} A pendant is to be made out of a precious metal. The shape of the pendant is modelled as the shape formed when \(R\) is rotated by \(2 \pi\) radians about the \(x\)-axis. Find the exact value of the volume of precious metal required to make the pendant, according to the model.
OCR Further Pure Core 2 2023 June Q5
7 marks Challenging +1.2
5 In this question you must show detailed reasoning.
  1. Using the definitions of \(\sinh x\) and \(\cosh x\) in terms of exponentials, show that \(\sinh 2 x \equiv 2 \sinh x \cosh x\).
  2. Solve the equation \(15 \sinh x + 16 \cosh x - 6 \sinh 2 x = 20\), giving all your answers in logarithmic form.
OCR Further Pure Core 2 2023 June Q10
7 marks Challenging +1.2
10 In this question you must show detailed reasoning. A region, \(R\), of the floor of an art gallery is to be painted for the purposes of an art installation. A suitable polar coordinate system is set up on the floor of the gallery with units in metres and radians. \(R\) is modelled as being the region enclosed by two curves, \(C _ { 1 }\) and \(C _ { 2 }\). The polar equations of \(C _ { 1 }\) and \(C _ { 2 }\) are $$\begin{array} { l l } C _ { 1 } : r = 5 , & - \frac { 1 } { 2 } \pi \leqslant \theta \leqslant \frac { 1 } { 2 } \pi \\ C _ { 2 } : r = 3 \cosh \theta , & - \frac { 1 } { 2 } \pi \leqslant \theta \leqslant \frac { 1 } { 2 } \pi \end{array}$$ Both curves are shown in the diagram, with \(R\) indicated. \includegraphics[max width=\textwidth, alt={}, center]{7b2bfb4e-524f-4d1c-ae98-075c7fb404f9-6_1481_821_836_251} The gallery must buy tins of paint to paint \(R\). Each tin of paint can cover an area of \(0.5 \mathrm {~m} ^ { 2 }\).
Determine the smallest number of tins of paint that the gallery must buy in order to be able to paint \(R\) completely.
OCR Further Pure Core 2 2020 November Q8
9 marks Standard +0.8
8 In this question you must show detailed reasoning. The complex number \(- 4 + i \sqrt { 48 }\) is denoted by \(z\).
  1. Determine the cube roots of \(z\), giving the roots in exponential form. The points which represent the cube roots of \(z\) are denoted by \(A , B\) and \(C\) and these form a triangle in an Argand diagram.
  2. Write down the angles that any lines of symmetry of triangle \(A B C\) make with the positive real axis, justifying your answer.
OCR MEI D2 2006 June Q2
16 marks Standard +0.3
2 Answer this question on the insert provided. Fig. 2 shows a network in which the weights on the arcs represent distances. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{9716cf3f-afa5-44a4-a8cd-f7511449d06b-2_405_497_1046_776} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure}
  1. Apply Floyd's algorithm on the insert provided to find the complete network of shortest distances.
  2. Show how to use your final matrices to find the shortest route from vertex \(\mathbf { 1 }\) to vertex 3, together with the length of that route.
  3. Use the nearest neighbour algorithm, starting at vertex 1, to find a Hamilton cycle in the complete network of shortest distances. Give the corresponding cycle in the original network, together with its length.
OCR MEI Further Pure Core AS 2019 June Q8
11 marks Standard +0.3
8 In this question you must show detailed reasoning. You are given that i is a root of the equation \(z ^ { 4 } - 2 z ^ { 3 } + 3 z ^ { 2 } + a z + b = 0\), where \(a\) and \(b\) are real constants.
  1. Show that \(a = - 2\) and \(b = 2\).
  2. Find the other roots of this equation.
OCR MEI Further Pure Core AS 2022 June Q4
6 marks Standard +0.8
4 In this question you must show detailed reasoning. The equation \(z ^ { 3 } + 2 z ^ { 2 } + k z + 3 = 0\), where \(k\) is a constant, has roots \(\alpha , \frac { 1 } { \alpha }\) and \(\beta\).
Determine the roots in exact form.
OCR MEI C1 2007 January Q11
12 marks Moderate -0.3
11 There is an insert for use in this question. The graph of \(y = x + \frac { 1 } { x }\) is shown on the insert. The lowest point on one branch is \(( 1,2 )\). The highest point on the other branch is \(( - 1 , - 2 )\).
  1. Use the graph to solve the following equations, showing your method clearly. $$\text { (A) } x + \frac { 1 } { x } = 4$$ $$\text { (B) } 2 x + \frac { 1 } { x } = 4$$
  2. The equation \(( x - 1 ) ^ { 2 } + y ^ { 2 } = 4\) represents a circle. Find in exact form the coordinates of the points of intersection of this circle with the \(y\)-axis.
  3. State the radius and the coordinates of the centre of this circle. Explain how these can be used to deduce from the graph that this circle touches one branch of the curve \(y = x + \frac { 1 } { x }\) but does not intersect with the other.
OCR MEI C1 2009 January Q13
11 marks Moderate -0.3
13 Answer part (i) of this question on the insert provided. The insert shows the graph of \(y = \frac { 1 } { x }\).
  1. On the insert, on the same axes, plot the graph of \(y = x ^ { 2 } - 5 x + 5\) for \(0 \leqslant x \leqslant 5\).
  2. Show algebraically that the \(x\)-coordinates of the points of intersection of the curves \(y = \frac { 1 } { x }\) and \(y = x ^ { 2 } - 5 x + 5\) satisfy the equation \(x ^ { 3 } - 5 x ^ { 2 } + 5 x - 1 = 0\).
  3. Given that \(x = 1\) at one of the points of intersection of the curves, factorise \(x ^ { 3 } - 5 x ^ { 2 } + 5 x - 1\) into a linear and a quadratic factor. Show that only one of the three roots of \(x ^ { 3 } - 5 x ^ { 2 } + 5 x - 1 = 0\) is rational.
OCR MEI C1 Q6
12 marks Moderate -0.8
6 Answer the whole of this question on the insert provided. The insert shows the graph of \(y = \frac { 1 } { x } , x \neq 0\).
  1. Use the graph to find approximate roots of the equation \(\frac { 1 } { x } = 2 x + 3\), showing your method clearly.
  2. Rearrange the equation \(\frac { 1 } { x } = 2 x + 3\) to form a quadratic equation. Solve the resulting equation, leaving your answers in the form \(\frac { p \pm \sqrt { q } } { r }\).
  3. Draw the graph of \(y = \frac { 1 } { x } + 2 , x \neq 0\), on the grid used for part (i).
  4. Write down the values of \(x\) which satisfy the equation \(\frac { 1 } { x } + 2 = 2 x + 3\).
OCR MEI C1 Q4
12 marks Moderate -0.8
4 There is an insert for use in this question. The graph of \(y = x + \frac { 1 } { x }\) is shown on the insert. The lowest point on one branch is \(( 1,2 )\). The highest point on the other branch is \(( - 1 , - 2 )\).
  1. Use the graph to solve the following equations, showing your method clearly.
    (A) \(x + \frac { 1 } { x } = 4\) (B) \(2 x + \frac { 1 } { x } = 4\)
  2. The equation \(( x - 1 ) ^ { 2 } + y ^ { 2 } = 4\) represents a circle. Find in exact form the coordinates of the points of intersection of this circle with the \(y\)-axis.
  3. State the radius and the coordinates of the centre of this circle. Explain how these can be used to deduce from the graph that this circle touches one branch of the curve \(y = x + \frac { 1 } { x }\) but does not intersect with the other.
OCR MEI C1 Q4
12 marks Moderate -0.8
4 Answer the whole of this question on the insert provided. The insert shows the graph of \(y = \frac { 1 } { x } , x \neq 0\).
  1. Use the graph to find approximate roots of the equation \(\frac { 1 } { x } = 2 x + 3\), showing your method clearly.
  2. Rearrange the equation \(\frac { 1 } { x } = 2 x + 3\) to form a quadratic equation. Solve the resulting equation, leaving your answers in the form \(\frac { p \pm \sqrt { q } } { r }\).
  3. Draw the graph of \(y = \frac { 1 } { x } + 2 , x \neq 0\), on the grid used for part (i).
  4. Write down the values of \(x\) which satisfy the equation \(\frac { 1 } { x } + 2 = 2 x + 3\).
OCR MEI C2 2007 January Q13
12 marks Moderate -0.3
13 Answer part (ii) of this question on the insert provided. The table gives a firm's monthly profits for the first few months after the start of its business, rounded to the nearest \(\pounds 100\).
Number of months after start-up \(( x )\)123456
Profit for this month \(( \pounds y )\)5008001200190030004800
The firm's profits, \(\pounds y\), for the \(x\) th month after start-up are modelled by $$y = k \times 10 ^ { a x }$$ where \(a\) and \(k\) are constants.
  1. Show that, according to this model, a graph of \(\log _ { 10 } y\) against \(x\) gives a straight line of gradient \(a\) and intercept \(\log _ { 10 } k\).
  2. On the insert, complete the table and plot \(\log _ { 10 } y\) against \(x\), drawing by eye a line of best fit.
  3. Use your graph to find an equation for \(y\) in terms of \(x\) for this model.
  4. For which month after start-up does this model predict profits of about \(\pounds 75000\) ?
  5. State one way in which this model is unrealistic.
OCR MEI C2 2009 January Q5
4 marks Moderate -0.8
5 Answer this question on the insert provided. Fig. 5 shows the graph of \(y = \mathrm { f } ( x )\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{838d6b15-69a9-4e67-bc36-5bf60254a767-3_979_1077_422_536} \captionsetup{labelformat=empty} \caption{Fig. 5}
\end{figure} On the insert, draw the graph of
  1. \(y = \mathrm { f } ( x - 2 )\),
  2. \(y = 3 \mathrm { f } ( x )\).
OCR MEI C2 2009 January Q12
12 marks Moderate -0.3
12 Answer part (ii) of this question on the insert provided. The proposal for a major building project was accepted, but actual construction was delayed. Each year a new estimate of the cost was made. The table shows the estimated cost, \(\pounds y\) million, of the project \(t\) years after the project was first accepted.
Years after proposal accepted \(( t )\)12345
Cost \(( \pounds y\) million \()\)250300360440530
The relationship between \(y\) and \(t\) is modelled by \(y = a b ^ { t }\), where \(a\) and \(b\) are constants.
  1. Show that \(y = a b ^ { t }\) may be written as $$\log _ { 10 } y = \log _ { 10 } a + t \log _ { 10 } b$$
  2. On the insert, complete the table and plot \(\log _ { 10 } y\) against \(t\), drawing by eye a line of best fit.
  3. Use your graph and the results of part (i) to find the values of \(\log _ { 10 } a\) and \(\log _ { 10 } b\) and hence \(a\) and \(b\).
  4. According to this model, what was the estimated cost of the project when it was first accepted?
  5. Find the value of \(t\) given by this model when the estimated cost is \(\pounds 1000\) million. Give your answer rounded to 1 decimal place.
OCR MEI C2 2010 January Q12
13 marks Moderate -0.8
12 Answer part (ii) of this question on the insert provided. Since 1945 the populations of many countries have been growing. The table shows the estimated population of 15- to 59-year-olds in Africa during the period 1955 to 2005.
Year195519651975198519952005
Population (millions)131161209277372492
Source: United Nations Such estimates are used to model future population growth and world needs of resources. One model is \(P = a 10 ^ { b t }\), where the population is \(P\) millions, \(t\) is the number of years after 1945 and \(a\) and \(b\) are constants.
  1. Show that, using this model, the graph of \(\log _ { 10 } P\) against \(t\) is a straight line of gradient \(b\). State the intercept of this line on the vertical axis.
  2. On the insert, complete the table, giving values correct to 2 decimal places, and plot the graph of \(\log _ { 10 } P\) against \(t\). Draw, by eye, a line of best fit on your graph.
  3. Use your graph to find the equation for \(P\) in terms of \(t\).
  4. Use your results to estimate the population of 15- to 59-year-olds in Africa in 2050. Comment, with a reason, on the reliability of this estimate.
OCR MEI C2 2006 June Q12
12 marks Moderate -0.8
12 Answer the whole of this question on the insert provided. A colony of bats is increasing. The population, \(P\), is modelled by \(P = a \times 10 ^ { b t }\), where \(t\) is the time in years after 2000.
  1. Show that, according to this model, the graph of \(\log _ { 10 } P\) against \(t\) should be a straight line of gradient \(b\). State, in terms of \(a\), the intercept on the vertical axis.
  2. The table gives the data for the population from 2001 to 2005.
    Year20012002200320042005
    \(t\)12345
    \(P\)79008800100001130012800
    Complete the table of values on the insert, and plot \(\log _ { 10 } P\) against \(t\). Draw a line of best fit for the data.
  3. Use your graph to find the equation for \(P\) in terms of \(t\).
  4. Predict the population in 2008 according to this model.
OCR MEI C3 Q9
18 marks Standard +0.3
9 Answer parts (i) and (iii) on the insert provided. Fig. 9 shows a sketch graph of \(y = \mathrm { f } ( x )\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3f8be5ab-d241-4027-af26-c49da9394adc-4_401_799_488_593} \captionsetup{labelformat=empty} \caption{Fig. 9}
\end{figure}
  1. On the Insert sketch graphs of
    (A) \(y = 2 \mathrm { f } ( x )\),
    (B) \(y = \mathrm { f } ( - x )\),
    (C) \(y = \mathrm { f } ( x - 2 )\) In each case describe the transformations.
  2. Explain why the function \(y = \mathrm { f } ( x )\) does not have an inverse function.
  3. The function \(\mathrm { g } ( x )\) is defined as follows: $$\mathrm { g } ( x ) = \mathrm { f } ( x ) \text { for } x \geq 0$$ On the Insert sketch the graph of \(y = \mathrm { g } ^ { - 1 } ( x )\).
  4. You are given that \(\mathrm { f } ( x ) = x ^ { 2 } ( x + 2 )\). Calculate the gradient of the curve \(y = \mathrm { f } ( x )\) at the point \(( 1,3 )\).
    Deduce the gradient of the function \(\mathrm { g } ^ { - 1 } ( x )\) at the point where \(x = 3\).
  5. Show that \(\mathrm { g } ( x )\) and \(\mathrm { g } ^ { - 1 } ( x )\) cross where \(x = - 1 + \sqrt { 2 }\). \section*{Insert for question 9.}
  6. (A) On the axes below sketch the graph of \(y = 2 \mathrm { f } ( x )\). Describe the transformation. \includegraphics[max width=\textwidth, alt={}, center]{3f8be5ab-d241-4027-af26-c49da9394adc-5_563_1102_484_395} Description:
  7. (B) On the axes below sketch the graph of \(y = \mathrm { f } ( - x )\). Describe the transformation. \includegraphics[max width=\textwidth, alt={}, center]{3f8be5ab-d241-4027-af26-c49da9394adc-5_588_1154_1576_404} Description:
  8. (C) On the axes below sketch the graph of \(y = \mathrm { f } ( x - 2 )\). Describe the transformation. \includegraphics[max width=\textwidth, alt={}, center]{3f8be5ab-d241-4027-af26-c49da9394adc-6_615_1230_402_406} Description:
  9. The function \(\mathrm { g } ( x )\) is defined as follows: $$\mathrm { g } ( x ) = \mathrm { f } ( x ) \text { for } x \geq 0$$ On the axes below sketch the graph of \(y = g ^ { - 1 } ( x )\). \includegraphics[max width=\textwidth, alt={}, center]{3f8be5ab-d241-4027-af26-c49da9394adc-6_677_1356_1567_312}
OCR MEI C3 Q9
18 marks Moderate -0.3
9 Answer parts (ii) and (iii) of this question on the Insert provided. The bat population of a colony is being investigated and data are collected of the estimated number of bats in the colony at the beginning of each year. It is thought that the population may be modelled by the formula $$P = P _ { 0 } \mathrm { e } ^ { k t }$$ where \(P _ { 0 }\) and \(k\) are constants, \(P\) is the number of bats and \(t\) is the number of years after the start of the collection of data.
  1. Explain why a graph of \(\ln P\) against \(t\) should give a straight line. State the gradient and intercept of this line.
  2. The data collected are as follows.
    Time \(( t\) years \()\)01234
    Number of bats, \(P\)100170300340360
    Using the first three pairs of data in the table, plot \(\ln P\) against \(t\) on the axes given on the Insert, and hence estimate values for \(P _ { 0 }\) and \(k\).
    (Work to three significant figures.) This model assumes exponential growth, and assumes that once born a bat does not die, continuing to reproduce. This is unrealistic and so a second model is proposed with formula $$P = 150 \arctan ( t - 1 ) + 170$$ (You are reminded that arctan values should be given in radians.)
  3. Plot on a single graph on the Insert the curves \(P = P _ { 0 } \mathrm { e } ^ { k t }\) for your values of \(P _ { 0 }\) and \(k\) and \(P = 150 \arctan ( t - 1 ) + 170\). The data pairs in the table above have been plotted for you.
  4. Using the second model calculate an estimate of the number of years it is before the bat population exceeds 375. \section*{Insert for question 3.}
  5. Sketch the graph of \(y = 2 \mathrm { f } ( x )\) \includegraphics[max width=\textwidth, alt={}, center]{3853d1e7-ae1f-4eca-93c7-96f03b6d31c3-6_641_1431_541_354}
  6. Sketch the graph of \(y = \mathrm { f } ( 2 x )\). \includegraphics[max width=\textwidth, alt={}, center]{3853d1e7-ae1f-4eca-93c7-96f03b6d31c3-6_691_1539_1468_374} \section*{Insert for question 9.}
  7. Plot \(\ln P\) against \(t\). \includegraphics[max width=\textwidth, alt={}, center]{3853d1e7-ae1f-4eca-93c7-96f03b6d31c3-7_704_1442_443_338}
  8. Plot the curves \(P = P _ { 0 } \mathrm { e } ^ { k t }\) and \(P = 150 \arctan ( t - 1 ) + 170\) for your values of \(P _ { 0 }\) and \(k\). The data pairs are plotted on the graph. \includegraphics[max width=\textwidth, alt={}, center]{3853d1e7-ae1f-4eca-93c7-96f03b6d31c3-7_780_1399_1546_333}
OCR MEI C4 Q6
4 marks Moderate -0.8
6 Use the Insert provided for this question. The graph of \(y = \tan x\) is given on the Insert.
On this graph sketch the graph of \(y = \operatorname { cotx }\).
Show clearly where your graph crosses the graph of \(y = \tan x\) and indicate the asymptotes.
OCR MEI D1 2006 January Q5
16 marks Moderate -0.8
5 Answer this question on the insert provided. Table 5 specifies a road network connecting 7 towns, A, B, \(\ldots\), G. The entries in Table 5 give the distances in miles between towns which are connected directly by roads. \begin{table}[h]
ABCDEFG
A-10---1215
B10-1520--8
C-15-7--11
D-207-20-13
E---20-179
F12---17-13
G1581113913-
\captionsetup{labelformat=empty} \caption{Table 5}
\end{table}
  1. Using the copy of Table 5 in the insert, apply the tabular form of Prim's algorithm to the network, starting at vertex A. Show the order in which you connect the vertices. Draw the resulting tree, give its total length and describe a practical application.
  2. The network in the insert shows the information in Table 5. Apply Dijkstra's algorithm to find the shortest route from A to E. Give your route and its length.
  3. A tunnel is built through a hill between A and B , shortening the distance between A and B to 6 miles. How does this affect your answers to parts (i) and (ii)?
OCR MEI D1 2006 January Q6
16 marks Easy -1.2
6 Answer part (iv) of this question on the insert provided. There are two types of customer who use the shop at a service station. \(70 \%\) buy fuel, the other \(30 \%\) do not. There is only one till in operation.
  1. Give an efficient rule for using one-digit random numbers to simulate the type of customer arriving at the service station. Table 6.1 shows the distribution of time taken at the till by customers who are buying fuel.
    Time taken (mins)11.522.5
    Probability\(\frac { 3 } { 10 }\)\(\frac { 2 } { 5 }\)\(\frac { 1 } { 5 }\)\(\frac { 1 } { 10 }\)
    \section*{Table 6.1}
  2. Specify an efficient rule for using one-digit random numbers to simulate the time taken at the till by customers purchasing fuel. Table 6.2 shows the distribution of time taken at the till by customers who are not buying fuel.
    Time taken (mins)11.522.53
    Probability\(\frac { 1 } { 7 }\)\(\frac { 2 } { 7 }\)\(\frac { 2 } { 7 }\)\(\frac { 1 } { 7 }\)\(\frac { 1 } { 7 }\)
    \section*{Table 6.2}
  3. Specify an efficient rule for using two-digit random numbers to simulate the time taken at the till by customers not buying fuel. What is the advantage in using two-digit random numbers instead of one-digit random numbers in this part of the question? The table in the insert shows a partially completed simulation study of 10 customers arriving at the till.
  4. Complete the table using the random numbers which are provided.
  5. Calculate the mean total time spent queuing and paying.
OCR MEI S1 2005 June Q6
15 marks Standard +0.3
6 Answer part (i) of this question on the insert provided. Mancaster Hockey Club invite prospective new players to take part in a series of three trial games. At the end of each game the performance of each player is assessed as pass or fail. Players who achieve a pass in all three games are invited to join the first team squad. Players who achieve a pass in two games are invited to join the second team squad. Players who fail in two games are asked to leave. This may happen after two games.
  • The probability of passing the first game is 0.9
  • Players who pass any game have probability 0.9 of passing the next game
  • Players who fail any game have probability 0.5 of failing the next game
    1. On the insert, complete the tree diagram which illustrates the information above. \includegraphics[max width=\textwidth, alt={}, center]{668963b4-994d-475a-a1c8-c3e3a252e4e6-4_691_1329_978_397}
    2. Find the probability that a randomly selected player
      (A) is invited to join the first team squad,
      (B) is invited to join the second team squad.
    3. Hence write down the probability that a randomly selected player is asked to leave.
    4. Find the probability that a randomly selected player is asked to leave after two games, given that the player is asked to leave.
Angela, Bryony and Shareen attend the trials at the same time. Assuming their performances are independent, find the probability that
  • at least one of the three is asked to leave,
  • they pass a total of 7 games between them.
    •