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AQA Further Paper 3 Discrete Specimen Q5
10 marks Standard +0.8
5 The binary operation * is defined as $$a * b = a + b + 4 ( \bmod 6 )$$ where \(a , b \in \mathbb { Z }\). 5
  1. Show that the set \(\{ 0,1,2,3,4,5 \}\) forms a group \(G\) under *.
    5
  2. Find the proper subgroups of the group \(G\) in part (a).
    5
  3. Determine whether or not the group \(G\) in part (a) is isomorphic to the group \(K = \left( \langle 3 \rangle , \times _ { 14 } \right)\) [0pt] [3 marks]
AQA Further Paper 3 Discrete Specimen Q6
11 marks Challenging +1.2
6
The network shows a system of pipes, where \(S\) is the source and \(T\) is the sink.
The lower and upper capacities, in litres per second, of each pipe are shown on each arc. \includegraphics[max width=\textwidth, alt={}, center]{88669bc0-9d3f-431a-8939-8aef2682412b-09_649_1399_580_424} 6
  1. There is a feasible flow from \(S\) to \(T\). 6
      1. Explain why arc \(A D\) must be at its lower capacity.
        [0pt] [1 mark] 6
      2. Explain why arc \(B E\) must be at its upper capacity.
        [0pt] [1 mark] 6
      3. Explain why a flow of 11 litres per second through the network is impossible.
        [0pt] [1 mark] 6
      4. The network in Figure 2 shows a second system of pipes, where \(S\) is the source and \(T\) is the sink. The lower and upper capacities, in litres per second, of each pipe are shown on each edge. \begin{figure}[h]
        \captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{88669bc0-9d3f-431a-8939-8aef2682412b-10_760_1372_680_470}
        \end{figure} Figure 3 shows a feasible flow of 17 litres per second through the system of pipes. \begin{figure}[h]
        \captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{88669bc0-9d3f-431a-8939-8aef2682412b-10_750_1371_1811_466}
        \end{figure}
      6
      1. Using Figures 2 and 3, indicate on Figure 4 potential increases and decreases in the flow along each arc.
        [0pt] [2 marks] \begin{figure}[h]
        \captionsetup{labelformat=empty} \caption{Figure 4} \includegraphics[alt={},max width=\textwidth]{88669bc0-9d3f-431a-8939-8aef2682412b-11_749_1384_457_426}
        \end{figure} 6
      2. Use flow augmentation on Figure 4 to find the maximum flow from \(S\) to \(T\).
        You should indicate any flow augmenting paths clearly in the table below and modify the potential increases and decreases of the flow on Figure 4.
        [0pt] [3 marks]
        Augmenting PathFlow
        6
      3. Prove the flow found in part
      (d) (ii) is maximum.
      6
    3. (iv) Due to maintenance work, the flow through node \(E\) is restricted to 9 litres per second.
      [0pt] Interpret the impact of this restriction on the maximum flow through the system of pipes. [2 marks]
AQA Further Paper 3 Discrete Specimen Q7
11 marks Challenging +1.2
7 A company repairs and sells computer hardware, including monitors, hard drives and keyboards. Each monitor takes 3 hours to repair and the cost of components is \(\pounds 40\). Each hard drive takes 2 hours to repair and the cost of components is \(\pounds 20\). Each keyboard takes 1 hour to repair and the cost of components is \(\pounds 5\). Each month, the business has 360 hours available for repairs and \(\pounds 2500\) available to buy components. Each month, the company sells all of its repaired hardware to a local computer shop. Each monitor, hard drive and keyboard sold gives the company a profit of \(\pounds 80 , \pounds 35\) and \(\pounds 15\) respectively. The company repairs and sells \(x\) monitors, \(y\) hard drives and \(z\) keyboards each month. The company wishes to maximise its total profit. 7
  1. Find five inequalities involving \(x , y\) and \(z\) for the company's problem.
    [0pt] [3 marks]
    7
  2. (i) Find how many of each type of computer hardware the company should repair and sell each month.
    7 (b) (ii) Explain how you know that you had reached the optimal solution in part (b) (i).
    7 (b) (iii) The local computer shop complains that they are not receiving one of the types of computer hardware that the company repairs and sells. Using your answer to part (b) (i), suggest a way in which the company's problem can be modified to address the complaint.
    [0pt] [1 mark]
AQA Further Paper 3 Discrete Specimen Q8
6 marks Challenging +1.2
8 John and Danielle play a zero-sum game which does not have a stable solution. The game is represented by the following pay-off matrix for John.
\multirow{2}{*}{}Danielle
Strategy\(\boldsymbol { X }\)\(Y\)\(\boldsymbol { Z }\)
\multirow{3}{*}{John}\(A\)21-1
B-3-22
\(\boldsymbol { C }\)-3-41
Find the optimal mixed strategy for John.
AQA C1 2014 June Q7
14 marks Moderate -0.5
  1. Express the equation of the circle in the form $$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = k$$
    1. Write down the coordinates of \(C\).
    2. Show that the circle has radius \(n \sqrt { 5 }\), where \(n\) is an integer.
  3. Find the equation of the tangent to the circle at the point \(A\), giving your answer in the form \(x + p y = q\), where \(p\) and \(q\) are integers.
  4. The point \(B\) lies on the tangent to the circle at \(A\) and the length of \(B C\) is 6. Find the length of \(A B\).
    [0pt] [3 marks]
    \includegraphics[max width=\textwidth, alt={}]{f2124c89-79de-4758-b7b8-ff273345b9dd-8_1421_1709_1286_153}
AQA C1 2015 June Q4
11 marks Standard +0.3
  1. Express this equation in the form $$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = d$$
    1. State the coordinates of \(C\).
    2. Find the radius of the circle, giving your answer in the form \(n \sqrt { 2 }\).
  3. The point \(P\) with coordinates \(( 4 , k )\) lies on the circle. Find the possible values of \(k\).
  4. The points \(Q\) and \(R\) also lie on the circle, and the length of the chord \(Q R\) is 2 . Calculate the shortest distance from \(C\) to the chord \(Q R\).
    [0pt] [2 marks]
AQA C2 2011 January Q7
16 marks Moderate -0.3
  1. Given that \(y = x + 3 + \frac { 8 } { x ^ { 4 } }\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
  2. Find an equation of the tangent at the point on the curve \(C\) where \(x = 1\).
  3. The curve \(C\) has a minimum point \(M\). Find the coordinates of \(M\).
    1. Find \(\int \left( x + 3 + \frac { 8 } { x ^ { 4 } } \right) \mathrm { d } x\).
    2. Hence find the area of the region bounded by the curve \(C\), the \(x\)-axis and the lines \(x = 1\) and \(x = 2\).
  5. The curve \(C\) is translated by \(\left[ \begin{array} { l } 0 \\ k \end{array} \right]\) to give the curve \(y = \mathrm { f } ( x )\). Given that the \(x\)-axis is a tangent to the curve \(y = \mathrm { f } ( x )\), state the value of the constant \(k\).
    (1 mark)
AQA C2 2011 June Q6
10 marks Moderate -0.3
  1. The area of the shaded region is given by \(\int _ { 0 } ^ { 2 } \sin x \mathrm {~d} x\), where \(x\) is in radians. Use the trapezium rule with five ordinates (four strips) to find an approximate value for the area of the shaded region, giving your answer to three significant figures.
  2. Describe the geometrical transformation that maps the graph of \(y = \sin x\) onto the graph of \(y = 2 \sin x\).
  3. Use a trigonometrical identity to solve the equation $$2 \sin x = \cos x$$ in the interval \(0 \leqslant x \leqslant 2 \pi\), giving your solutions in radians to three significant figures.
AQA C4 2014 June Q7
9 marks Moderate -0.3
    1. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
    2. Hence find the exact value of the gradient of the curve at \(A\).
  2. The normal at \(A\) crosses the \(y\)-axis at the point \(B\). Find the exact value of the \(y\)-coordinate of \(B\).
    [0pt] [2 marks]
AQA S2 2011 June Q3
10 marks Standard +0.3
  1. State the null hypothesis that Emily used.
  2. Find the value of the test statistic, \(X ^ { 2 }\), giving your answer to one decimal place.
  3. State, in context, the conclusion that Emily should reach based on the results of her \(\chi ^ { 2 }\) test.
  4. Make one comment on the GCSE performances of 16-year-old students attending Bailey Language School.
  5. Emily's friend, Joanna, used the same data to correctly conduct a \(\chi ^ { 2 }\) test using the \(10 \%\) level of significance. State, with justification, the conclusion that Joanna should reach.
AQA M2 2012 January Q7
11 marks Standard +0.3
  1. Show that \(v ^ { 2 } = u ^ { 2 } - 4 a g\).
  2. The ratio of the tensions in the string when the bead is at the two points \(A\) and \(B\) is \(2 : 5\).
    1. Find \(u\) in terms of \(g\) and \(a\).
    2. Find the ratio \(u : v\).
AQA M2 2010 June Q7
12 marks Standard +0.8
  1. Draw a diagram to show the forces acting on the rod.
  2. Find the magnitude of the normal reaction force between the rod and the ground.
    1. Find the normal reaction acting on the rod at \(C\).
    2. Find the friction force acting on the rod at \(C\).
  4. In this position, the rod is on the point of slipping. Calculate the coefficient of friction between the rod and the peg.
    \includegraphics[max width=\textwidth, alt={}]{3ffa0a2b-aa7d-46eb-b92b-3e3ee59f235c-15_2484_1709_223_153}
AQA FP1 2013 January Q7
7 marks Standard +0.8
  1. Show that there is a linear relationship between \(Y\) and \(X\).
  2. The graph of \(Y\) against \(X\) is shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{cf9337b9-b766-4ce5-967c-5d7522e2aa42-4_748_858_849_593} Find the value of \(n\) and the value of \(a\).
AQA FP1 2015 June Q6
5 marks Standard +0.3
  1. Sketch the curve \(C _ { 1 }\), stating the values of its intercepts with the coordinate axes.
  2. The curve \(C _ { 1 }\) is translated by the vector \(\left[ \begin{array} { l } k \\ 0 \end{array} \right]\), where \(k < 0\), to give a curve \(C _ { 2 }\). Given that \(C _ { 2 }\) passes through the origin \(( 0,0 )\), find the equations of the asymptotes of \(C _ { 2 }\).
    [0pt] [3 marks]
AQA C1 2007 January Q1
11 marks Moderate -0.8
1 The polynomial \(\mathrm { p } ( x )\) is given by $$\mathrm { p } ( x ) = x ^ { 3 } - 4 x ^ { 2 } - 7 x + k$$ where \(k\) is a constant.
    1. Given that \(x + 2\) is a factor of \(\mathrm { p } ( x )\), show that \(k = 10\).
    2. Express \(\mathrm { p } ( x )\) as the product of three linear factors.
  2. Use the Remainder Theorem to find the remainder when \(\mathrm { p } ( x )\) is divided by \(x - 3\).
  3. Sketch the curve with equation \(y = x ^ { 3 } - 4 x ^ { 2 } - 7 x + 10\), indicating the values where the curve crosses the \(x\)-axis and the \(y\)-axis. (You are not required to find the coordinates of the stationary points.)
AQA C1 2007 January Q2
11 marks Moderate -0.3
2 The line \(A B\) has equation \(3 x + 5 y = 8\) and the point \(A\) has coordinates (6, -2).
    1. Find the gradient of \(A B\).
    2. Hence find an equation of the straight line which is perpendicular to \(A B\) and which passes through \(A\).
  2. The line \(A B\) intersects the line with equation \(2 x + 3 y = 3\) at the point \(B\). Find the coordinates of \(B\).
  3. The point \(C\) has coordinates \(( 2 , k )\) and the distance from \(A\) to \(C\) is 5 . Find the two possible values of the constant \(k\).
AQA C1 2007 January Q3
8 marks Moderate -0.8
3
  1. Express \(\frac { \sqrt { 5 } + 3 } { \sqrt { 5 } - 2 }\) in the form \(p \sqrt { 5 } + q\), where \(p\) and \(q\) are integers.
    1. Express \(\sqrt { 45 }\) in the form \(n \sqrt { 5 }\), where \(n\) is an integer.
    2. Solve the equation $$x \sqrt { 20 } = 7 \sqrt { 5 } - \sqrt { 45 }$$ giving your answer in its simplest form.
AQA C1 2007 January Q4
14 marks Moderate -0.8
4 A circle with centre \(C\) has equation \(x ^ { 2 } + y ^ { 2 } + 2 x - 12 y + 12 = 0\).
  1. By completing the square, express this equation in the form $$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = r ^ { 2 }$$
  2. Write down:
    1. the coordinates of \(C\);
    2. the radius of the circle.
  3. Show that the circle does not intersect the \(x\)-axis.
  4. The line with equation \(x + y = 4\) intersects the circle at the points \(P\) and \(Q\).
    1. Show that the \(x\)-coordinates of \(P\) and \(Q\) satisfy the equation $$x ^ { 2 } + 3 x - 10 = 0$$
    2. Given that \(P\) has coordinates (2,2), find the coordinates of \(Q\).
    3. Hence find the coordinates of the midpoint of \(P Q\).
AQA C1 2007 January Q5
10 marks Moderate -0.5
5 The diagram shows an open-topped water tank with a horizontal rectangular base and four vertical faces. The base has width \(x\) metres and length \(2 x\) metres, and the height of the tank is \(h\) metres. \includegraphics[max width=\textwidth, alt={}, center]{33da89e2-f74f-4d5a-8bbd-ceaa728b6c34-4_403_410_477_792} The combined internal surface area of the base and four vertical faces is \(54 \mathrm {~m} ^ { 2 }\).
    1. Show that \(x ^ { 2 } + 3 x h = 27\).
    2. Hence express \(h\) in terms of \(x\).
    3. Hence show that the volume of water, \(V \mathrm {~m} ^ { 3 }\), that the tank can hold when full is given by $$V = 18 x - \frac { 2 x ^ { 3 } } { 3 }$$
    1. Find \(\frac { \mathrm { d } V } { \mathrm {~d} x }\).
    2. Verify that \(V\) has a stationary value when \(x = 3\).
  3. Find \(\frac { \mathrm { d } ^ { 2 } V } { \mathrm {~d} x ^ { 2 } }\) and hence determine whether \(V\) has a maximum value or a minimum value when \(x = 3\).
    (2 marks)
AQA C1 2007 January Q6
14 marks Moderate -0.8
6 The curve with equation \(y = 3 x ^ { 5 } + 2 x + 5\) is sketched below. \includegraphics[max width=\textwidth, alt={}, center]{33da89e2-f74f-4d5a-8bbd-ceaa728b6c34-5_428_563_372_740} The curve cuts the \(x\)-axis at the point \(A ( - 1,0 )\) and cuts the \(y\)-axis at the point \(B\).
    1. State the coordinates of the point \(B\) and hence find the area of the triangle \(A O B\), where \(O\) is the origin.
    2. Find \(\int \left( 3 x ^ { 5 } + 2 x + 5 \right) \mathrm { d } x\).
    3. Hence find the area of the shaded region bounded by the curve and the line \(A B\).
    1. Find the gradient of the curve with equation \(y = 3 x ^ { 5 } + 2 x + 5\) at the point \(A ( - 1,0 )\).
    2. Hence find an equation of the tangent to the curve at the point \(A\).
AQA C1 2007 January Q7
7 marks Moderate -0.3
7 The quadratic equation \(( k + 1 ) x ^ { 2 } + 12 x + ( k - 4 ) = 0\) has real roots.
  1. Show that \(k ^ { 2 } - 3 k - 40 \leqslant 0\).
  2. Hence find the possible values of \(k\).
AQA C1 2008 January Q1
11 marks Moderate -0.3
1 The triangle \(A B C\) has vertices \(A ( - 2,3 ) , B ( 4,1 )\) and \(C ( 2 , - 5 )\).
  1. Find the coordinates of the mid-point of \(B C\).
    1. Find the gradient of \(A B\), in its simplest form.
    2. Hence find an equation of the line \(A B\), giving your answer in the form \(x + q y = r\), where \(q\) and \(r\) are integers.
    3. Find an equation of the line passing through \(C\) which is parallel to \(A B\).
  3. Prove that angle \(A B C\) is a right angle.
AQA C1 2008 January Q2
11 marks Moderate -0.8
2 The curve with equation \(y = x ^ { 4 } - 32 x + 5\) has a single stationary point, \(M\).
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
  2. Hence find the \(x\)-coordinate of \(M\).
    1. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
    2. Hence, or otherwise, determine whether \(M\) is a maximum or a minimum point.
  4. Determine whether the curve is increasing or decreasing at the point on the curve where \(x = 0\).
AQA C1 2008 January Q3
7 marks Easy -1.2
3
  1. Express \(5 \sqrt { 8 } + \frac { 6 } { \sqrt { 2 } }\) in the form \(n \sqrt { 2 }\), where \(n\) is an integer.
  2. Express \(\frac { \sqrt { 2 } + 2 } { 3 \sqrt { 2 } - 4 }\) in the form \(c \sqrt { 2 } + d\), where \(c\) and \(d\) are integers.
AQA C1 2008 January Q4
11 marks Moderate -0.3
4 A circle with centre \(C\) has equation \(x ^ { 2 } + y ^ { 2 } - 10 y + 20 = 0\).
  1. By completing the square, express this equation in the form $$x ^ { 2 } + ( y - b ) ^ { 2 } = k$$
  2. Write down:
    1. the coordinates of \(C\);
    2. the radius of the circle, leaving your answer in surd form.
  3. A line has equation \(y = 2 x\).
    1. Show that the \(x\)-coordinate of any point of intersection of the line and the circle satisfies the equation \(x ^ { 2 } - 4 x + 4 = 0\).
    2. Hence show that the line is a tangent to the circle and find the coordinates of the point of contact, \(P\).
  4. Prove that the point \(Q ( - 1,4 )\) lies inside the circle.