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OCR Further Pure Core AS 2021 November Q4
4
  1. A locus \(C _ { 1 }\) is defined by \(C _ { 1 } = \{ \mathrm { z } : | \mathrm { z } + \mathrm { i } | \leqslant \mid \mathrm { z } - 2 \}\).
    1. Indicate by shading on the Argand diagram in the Printed Answer Booklet the region representing \(C _ { 1 }\).
    2. Find the cartesian equation of the boundary line of the region representing \(C _ { 1 }\), giving your answer in the form \(a x + b y + c = 0\).
  2. A locus \(C _ { 2 }\) is defined by \(C _ { 2 } = \{ \mathrm { z } : | \mathrm { z } + 1 | \leqslant 3 \} \cap \{ \mathrm { z } : | \mathrm { z } - 2 \mathrm { i } | \geqslant 2 \}\). Indicate by shading on the Argand diagram in the Printed Answer Booklet the region representing \(C _ { 2 }\).
OCR Further Pure Core AS 2021 November Q5
5 Matrices \(\mathbf { A }\) and \(\mathbf { B }\) are given by \(\mathbf { A } = \left( \begin{array} { r l } - 1 & 0
0 & 1 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { c c } \frac { 5 } { 13 } & - \frac { 12 } { 13 }
\frac { 12 } { 13 } & \frac { 5 } { 13 } \end{array} \right)\).
  1. Use \(\mathbf { A }\) and \(\mathbf { B }\) to disprove the proposition: "Matrix multiplication is commutative". Matrix \(\mathbf { B }\) represents the transformation \(\mathrm { T } _ { \mathrm { B } }\).
  2. Describe the transformation \(\mathrm { T } _ { \mathrm { B } }\).
  3. By considering the inverse transformation of \(\mathrm { T } _ { \mathrm { B } }\), determine \(\mathbf { B } ^ { - 1 }\). Matrix \(\mathbf { C }\) is given by \(\mathbf { C } = \left( \begin{array} { r r } 1 & 0
    0 & - 3 \end{array} \right)\) and represents the transformation \(\mathrm { T } _ { \mathrm { C } }\).
    The transformation \(\mathrm { T } _ { \mathrm { BC } }\) is transformation \(\mathrm { T } _ { \mathrm { C } }\) followed by transformation \(\mathrm { T } _ { \mathrm { B } }\).
    An object shape of area 5 is transformed by \(\mathrm { T } _ { \mathrm { BC } }\) to an image shape \(N\).
  4. Determine the area of \(N\).
OCR Further Pure Core AS 2021 November Q6
6 In this question you must show detailed reasoning.
  1. Solve the equation \(2 z ^ { 2 } - 10 z + 25 = 0\) giving your answers in the form \(\mathrm { a } + \mathrm { bi }\).
  2. Solve the equation \(3 \omega - 2 = \mathrm { i } ( 5 + 2 \omega )\) giving your answer in the form \(\mathrm { a } + \mathrm { bi }\).
OCR Further Pure Core AS 2021 November Q7
7 Prove that \(2 ^ { 3 n } - 3 ^ { n }\) is divisible by 5 for all integers \(n \geqslant 1\).
OCR Further Pure Core AS 2021 November Q8
8 The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { c c c } t - 1 & t - 1 & t - 1
1 - t & 6 & t
2 - 2 t & 2 - 2 t & 1 \end{array} \right)\).
  1. Find, in fully factorised form, an expression for \(\operatorname { det } \mathbf { A }\) in terms of \(t\).
  2. State the values of \(t\) for which \(\mathbf { A }\) is singular. You are given the following system of equations in \(x , y\) and \(z\), where \(b\) is a real number. $$\begin{aligned} \left( b ^ { 2 } + 1 \right) x + \left( b ^ { 2 } + 1 \right) y + \left( b ^ { 2 } + 1 \right) z & = 5
    \left( - b ^ { 2 } - 1 \right) x + \quad 6 y + \left( b ^ { 2 } + 2 \right) z & = 10
    \left( - 2 b ^ { 2 } - 2 \right) x + \left( - 2 b ^ { 2 } - 2 \right) y + \quad z & = 15 \end{aligned}$$
  3. Determine which one of the following statements about the solution of the equations is true.
    • There is a unique solution for all values of \(b\).
    • There is a unique solution for some, but not all, values of \(b\).
    • There is no unique solution for any value of \(b\).
OCR Further Pure Core AS 2021 November Q9
9 The points \(P ( 3,5 , - 21 )\) and \(Q ( - 1,3 , - 16 )\) are on the ceiling of a long straight underground tunnel. A ventilation shaft must be dug from the point \(M\) on the ceiling of the tunnel midway between \(P\) and \(Q\) to horizontal ground level (where the \(z\)-coordinate is 0 ). The ventilation shaft must be perpendicular to the tunnel. The path of the ventilation shaft is modelled by the vector equation \(\mathbf { r } = \mathbf { a } + \lambda \mathbf { b }\), where \(\mathbf { a }\) is the position vector of \(M\). You are given that \(\mathbf { b } = \left( \begin{array} { l } 1
\mathrm {~s}
\mathrm { t } \end{array} \right)\) where \(s\) and \(t\) are real numbers.
  1. Show that \(\mathrm { S } = 2.5 \mathrm { t } - 2\).
  2. Show that at the point where the ventilation shaft reaches the ground \(\lambda = \frac { \mathrm { C } } { \mathrm { t } }\), where \(c\) is a constant to be determined.
  3. Using the results in parts (a) and (b), determine the shortest possible length of the ventilation shaft.
  4. Explain what the fact that \(\mathbf { b } \times \left( \begin{array} { l } 0
    0
    1 \end{array} \right) \neq \mathbf { O }\) means about the direction of the ventilation shaft.
OCR Further Discrete AS 2021 November Q1
1 A set consists of five distinct non-integer values, \(\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D }\) and E . The set is partitioned into non-empty subsets and there are at least two subsets in each partition.
  1. Show that there are 15 different partitions into two subsets.
  2. Show that there are 25 different partitions into three subsets.
  3. Calculate the total number of different partitions. The numbers 12, 24, 36, 48, 60, 72, 84 and 96 are marked on a number line. The number line is then cut into pieces by making cuts at \(\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D }\) and E , where \(0 < \mathrm { A } < \mathrm { B } < \mathrm { C } < \mathrm { D } < \mathrm { E } < 100\).
  4. Explain why there must be at least one piece with two or more of the numbers 12, 24, 36, 48, 60, 72, 84 and 96.
OCR Further Discrete AS 2021 November Q2
2 Seven items need to be packed into bins. Each bin has capacity 30 kg . The sizes of the items, in kg, in the order that they are received, are as follows.
12
23
15
18
8
7
5
  1. Find the packing that results using each of these algorithms.
    1. The next-fit method
    2. The first-fit method
    3. The first-fit decreasing method
  2. A student claims that all three methods from part (a) can be used for both 'online' and 'offline' lists. Explain why the student is wrong. The bins of capacity 30 kg are replaced with bins of capacity \(M \mathrm {~kg}\), where \(M\) is an integer.
    The item of size 23 kg can be split into two items, of sizes \(x \mathrm {~kg}\) and \(( 23 - x ) \mathrm { kg }\), where \(x\) may be any integer value you choose from 1 to 11. No other item can be split.
  3. Determine the smallest value of \(M\) for which four bins are needed to pack these eight items. Explain your reasoning carefully.
OCR Further Discrete AS 2021 November Q3
3 The diagram shows a simplified map of the main streets in a small town.
\includegraphics[max width=\textwidth, alt={}, center]{4db3fbc5-e664-4803-a5f2-05af376d2591-3_531_1127_301_246} Some of the junctions have traffic lights, these junctions are labelled A to F .
There are no traffic lights at junctions X and Y .
The numbers show distances, in km, between junctions. Alex needs to check that the traffic lights at junctions A to F are working correctly.
  1. Find a route from A to E that has length 2.8 km . Alex has started to construct a table of shortest distances between junctions A to F.
    \includegraphics[max width=\textwidth, alt={}, center]{4db3fbc5-e664-4803-a5f2-05af376d2591-3_543_1173_1420_246} For example, the shortest route from C to B has length 1.7 km , the shortest route from C to D has length 2.5 km and the shortest route from C to E has length 1.8 km .
  2. Complete the copy of the table in the Printed Answer Booklet.
  3. Use your table from part (b) to construct a minimum spanning tree for the complete graph on the six vertices A to F .
    • Write down the total length of the minimum spanning tree.
    • List which arcs of the original network correspond to the arcs used in your minimum spanning tree.
    Beth starts from junction B and travels through every junction, including X and Y . Her route has length 5.1 km .
  4. Write down the junctions in the order that Beth visited them. Do not draw on your answer from part (c).
OCR Further Discrete AS 2021 November Q5
5
  1. Find the packing that results using each of these algorithms.
    1. The next-fit method
    2. The first-fit method
    3. The first-fit decreasing method
  2. A student claims that all three methods from part (a) can be used for both 'online' and 'offline' lists. Explain why the student is wrong. The bins of capacity 30 kg are replaced with bins of capacity \(M \mathrm {~kg}\), where \(M\) is an integer.
    The item of size 23 kg can be split into two items, of sizes \(x \mathrm {~kg}\) and \(( 23 - x ) \mathrm { kg }\), where \(x\) may be any integer value you choose from 1 to 11. No other item can be split.
  3. Determine the smallest value of \(M\) for which four bins are needed to pack these eight items. Explain your reasoning carefully. 3 The diagram shows a simplified map of the main streets in a small town.
    \includegraphics[max width=\textwidth, alt={}, center]{4db3fbc5-e664-4803-a5f2-05af376d2591-3_531_1127_301_246} Some of the junctions have traffic lights, these junctions are labelled A to F .
    There are no traffic lights at junctions X and Y .
    The numbers show distances, in km, between junctions. Alex needs to check that the traffic lights at junctions A to F are working correctly.
  4. Find a route from A to E that has length 2.8 km . Alex has started to construct a table of shortest distances between junctions A to F.
    \includegraphics[max width=\textwidth, alt={}, center]{4db3fbc5-e664-4803-a5f2-05af376d2591-3_543_1173_1420_246} For example, the shortest route from C to B has length 1.7 km , the shortest route from C to D has length 2.5 km and the shortest route from C to E has length 1.8 km .
  5. Complete the copy of the table in the Printed Answer Booklet.
  6. Use your table from part (b) to construct a minimum spanning tree for the complete graph on the six vertices A to F .
    • Write down the total length of the minimum spanning tree.
    • List which arcs of the original network correspond to the arcs used in your minimum spanning tree.
    Beth starts from junction B and travels through every junction, including X and Y . Her route has length 5.1 km .
  7. Write down the junctions in the order that Beth visited them. Do not draw on your answer from part (c). 4 Li and Mia play a game in which they simultaneously play one of the strategies \(\mathrm { X } , \mathrm { Y }\) and Z . The tables show the points won by each player for each combination of strategies.
    A negative entry means that the player loses that number of points.
    Mia
    XYZ
    \multirow{3}{*}{Li}X5- 60
    \cline { 2 - 5 }Y- 234
    \cline { 2 - 5 }Z- 148
    \cline { 2 - 5 }
    Mia
    XYZ
    \multirow{2}{*}{Li}X4
    \cline { 2 - 5 }Y115
    \cline { 2 - 5 }Z1051
    \cline { 2 - 5 }
    The game can be converted into a zero-sum game, this means that the total number of points won by Li and Mia is the same for each combination of strategies.
    1. Complete the table in the Printed Answer Booklet to show the points won by Mia.
    2. Convert the game into a zero-sum game, giving the pay-offs for Li .
  9. Use dominance to reduce the pay-off matrix for the game to a \(2 \times 2\) table. You do not need to explain the dominance. Mia knows that Li will choose his play-safe strategy.
  10. Determine which strategy Mia should choose to maximise her points. 5 A linear programming problem is formulated as below. Maximise \(\quad \mathrm { P } = 4 \mathrm { x } - \mathrm { y }\)
    subject to \(2 x + 3 y \geqslant 12\)
    \(x + y \leqslant 10\)
    \(5 x + 2 y \leqslant 30\)
    \(x \geqslant 0 , y \geqslant 0\)
    1. Identify the feasible region by representing the constraints graphically and shading the regions where the inequalities are not satisfied.
    2. Hence determine the maximum value of the objective. The constraint \(x + y \leqslant 10\) is changed to \(x + y \leqslant k\), the other constraints are unchanged.
  12. Determine, algebraically, the value of \(k\) for which the maximum value of \(P\) is 3 . Do not draw on the graph from part (a) and do not use the spare grid.
  13. Determine, algebraically, the other value of \(k\) for which there is a (non-optimal) vertex of the feasible region at which \(P = 3\).
    Do not draw on the graph from part (a) and do not use the spare grid.
    14. OCR Further Discrete AS 2021 November Q18
    1 marks
    18
    8
    7
    5
    1. Find the packing that results using each of these algorithms.
      1. The next-fit method
      2. The first-fit method
      3. The first-fit decreasing method
    2. A student claims that all three methods from part (a) can be used for both 'online' and 'offline' lists. Explain why the student is wrong. The bins of capacity 30 kg are replaced with bins of capacity \(M \mathrm {~kg}\), where \(M\) is an integer.
      The item of size 23 kg can be split into two items, of sizes \(x \mathrm {~kg}\) and \(( 23 - x ) \mathrm { kg }\), where \(x\) may be any integer value you choose from 1 to 11. No other item can be split.
    3. Determine the smallest value of \(M\) for which four bins are needed to pack these eight items. Explain your reasoning carefully. 3 The diagram shows a simplified map of the main streets in a small town.
      \includegraphics[max width=\textwidth, alt={}, center]{4db3fbc5-e664-4803-a5f2-05af376d2591-3_531_1127_301_246} Some of the junctions have traffic lights, these junctions are labelled A to F .
      There are no traffic lights at junctions X and Y .
      The numbers show distances, in km, between junctions. Alex needs to check that the traffic lights at junctions A to F are working correctly.
    4. Find a route from A to E that has length 2.8 km . Alex has started to construct a table of shortest distances between junctions A to F.
      \includegraphics[max width=\textwidth, alt={}, center]{4db3fbc5-e664-4803-a5f2-05af376d2591-3_543_1173_1420_246} For example, the shortest route from C to B has length 1.7 km , the shortest route from C to D has length 2.5 km and the shortest route from C to E has length 1.8 km .
    5. Complete the copy of the table in the Printed Answer Booklet.
    6. Use your table from part (b) to construct a minimum spanning tree for the complete graph on the six vertices A to F .
      • Write down the total length of the minimum spanning tree.
      • List which arcs of the original network correspond to the arcs used in your minimum spanning tree.
      Beth starts from junction B and travels through every junction, including X and Y . Her route has length 5.1 km .
    7. Write down the junctions in the order that Beth visited them. Do not draw on your answer from part (c). 4 Li and Mia play a game in which they simultaneously play one of the strategies \(\mathrm { X } , \mathrm { Y }\) and Z . The tables show the points won by each player for each combination of strategies.
      A negative entry means that the player loses that number of points.
      Mia
      XYZ
      \multirow{3}{*}{Li}X5- 60
      \cline { 2 - 5 }Y- 234
      \cline { 2 - 5 }Z- 148
      \cline { 2 - 5 }
      Mia
      XYZ
      \multirow{2}{*}{Li}X4
      \cline { 2 - 5 }Y115
      \cline { 2 - 5 }Z1051
      \cline { 2 - 5 }
      The game can be converted into a zero-sum game, this means that the total number of points won by Li and Mia is the same for each combination of strategies.
      1. Complete the table in the Printed Answer Booklet to show the points won by Mia.
      2. Convert the game into a zero-sum game, giving the pay-offs for Li .
    9. Use dominance to reduce the pay-off matrix for the game to a \(2 \times 2\) table. You do not need to explain the dominance. Mia knows that Li will choose his play-safe strategy.
    10. Determine which strategy Mia should choose to maximise her points. 5 A linear programming problem is formulated as below. Maximise \(\quad \mathrm { P } = 4 \mathrm { x } - \mathrm { y }\)
      subject to \(2 x + 3 y \geqslant 12\)
      \(x + y \leqslant 10\)
      \(5 x + 2 y \leqslant 30\)
      \(x \geqslant 0 , y \geqslant 0\)
      1. Identify the feasible region by representing the constraints graphically and shading the regions where the inequalities are not satisfied.
      2. Hence determine the maximum value of the objective. The constraint \(x + y \leqslant 10\) is changed to \(x + y \leqslant k\), the other constraints are unchanged.
    12. Determine, algebraically, the value of \(k\) for which the maximum value of \(P\) is 3 . Do not draw on the graph from part (a) and do not use the spare grid.
    13. Determine, algebraically, the other value of \(k\) for which there is a (non-optimal) vertex of the feasible region at which \(P = 3\).
      Do not draw on the graph from part (a) and do not use the spare grid. 6 Sarah is having some work done on her garden.
      The table below shows the activities involved, their durations and their immediate predecessors. These durations and immediate predecessors are known to be correct.
      ActivityImmediate predecessorsDuration (hours)
      A Clear site-4
      B Mark out new designA1
      C Buy materials, turf, plants and trees-3
      D Lay pathsB, C1
      E Build patioB, C2
      F Plant treesD1
      G Lay turfD, E1
      H Finish plantingF, G1
      1. Use a suitable model to determine the following.
        • The minimum time in which the work can be completed
    15. The activities with zero float
      [0pt] (ii) State one practical issue that could affect the minimum completion time in part (a)(i). [1]
      16. Sarah needs the work to be completed as quickly as possible. There will be at least one activity happening at all times, but it may not always be possible to do all the activities that are needed at the same time.
    16. Determine the earliest and latest times at which building the patio (activity E) could start. There needs to be a 2-hour break after laying the paths (activity D). During this time other activities that do not depend on activity D can still take place.
    17. Describe how you would adapt your model to incorporate the 2-hour break.
      18. OCR Further Additional Pure AS 2024 June Q1
      1 In this question you must show detailed reasoning. The number \(N\) is written as 28 A 3 B in base-12 form. Express \(N\) in decimal (base-10) form.
      OCR Further Additional Pure AS 2024 June Q2
      2 The points \(A\) and \(B\) have position vectors \(\mathbf { a }\) and \(\mathbf { b }\) relative to the origin \(O\). It is given that \(\mathbf { a } = \left( \begin{array} { c } 2
      4
      3 \lambda \end{array} \right)\) and \(\mathbf { b } = \left( \begin{array} { r } \lambda
      - 4
      6 \end{array} \right)\), where \(\lambda\) is a real parameter.
      1. In the case when \(\lambda = 3\), determine the area of triangle \(O A B\).
      2. Determine the value of \(\lambda\) for which \(\mathbf { a } \times \mathbf { b } = \mathbf { 0 }\).
      OCR Further Additional Pure AS 2024 June Q3
      3 The surface \(S\) has equation \(z = f ( x , y )\), where \(f ( x , y ) = 4 x ^ { 2 } y - 6 x y ^ { 2 } - \frac { 1 } { 12 } x ^ { 4 }\) for all real values of \(x\) and \(y\). You are given that \(S\) has a stationary point at the origin, \(O\), and a second stationary point at the point \(P ( a , b , c )\), where \(\mathrm { c } = \mathrm { f } ( \mathrm { a } , \mathrm { b } )\).
      1. Determine the values of \(a , b\) and \(c\).
      2. Throughout this part, take the values of \(a\) and \(b\) to be those found in part (a).
        1. Evaluate \(\mathrm { f } _ { x }\) at the points \(\mathrm { U } _ { 1 } ( \mathrm { a } - 0.1 , \mathrm {~b} , \mathrm { f } ( \mathrm { a } - 0.1 , \mathrm {~b} ) )\) and \(\mathrm { U } _ { 2 } ( \mathrm { a } + 0.1 , \mathrm {~b} , \mathrm { f } ( \mathrm { a } + 0.1 , \mathrm {~b} ) )\).
        2. Evaluate \(\mathrm { f } _ { y }\) at the points \(\mathrm { V } _ { 1 } ( \mathrm { a } , \mathrm { b } - 0.1 , \mathrm { f } ( \mathrm { a } , \mathrm { b } - 0.1 ) )\) and \(\mathrm { V } _ { 2 } ( \mathrm { a } , \mathrm { b } + 0.1 , \mathrm { f } ( \mathrm { a } , \mathrm { b } + 0.1 ) )\).
        3. Use the answers to parts (b)(i) and (b)(ii) to sketch the portions of the sections of \(S\), given by
          • \(z = f ( x , b )\), for \(| x - a | \leqslant 0.1\),
      3. \(z = f ( a , y )\), for \(| y - b | \leqslant 0.1\).
      OCR Further Additional Pure AS 2024 June Q4
      4 The first five terms of the Fibonacci sequence, \(\left\{ \mathrm { F } _ { \mathrm { n } } \right\}\), where \(n \geqslant 1\), are \(F _ { 1 } = 1 , F _ { 2 } = 1 , F _ { 3 } = 2 , F _ { 4 } = 3\) and \(F _ { 5 } = 5\).
      1. Use the recurrence definition of the Fibonacci sequence, \(\mathrm { F } _ { \mathrm { n } + 1 } = \mathrm { F } _ { \mathrm { n } } + \mathrm { F } _ { \mathrm { n } - 1 }\), to express \(\mathrm { F } _ { \mathrm { n } + 4 }\) in terms of \(\mathrm { F } _ { \mathrm { n } }\) and \(\mathrm { F } _ { \mathrm { n } - 1 }\).
      2. Hence prove by induction that \(\mathrm { F } _ { \mathrm { n } }\) is a multiple of 3 when \(n\) is a multiple of 4 .
      OCR Further Additional Pure AS 2024 June Q5
      5 The set \(S\) consists of all \(2 \times 2\) matrices having determinant 1 or - 1 . For instance, the matrices \(\mathbf { P } = \left( \begin{array} { c c } \frac { 1 } { 2 } & \frac { \sqrt { 3 } } { 2 }
      - \frac { \sqrt { 3 } } { 2 } & \frac { 1 } { 2 } \end{array} \right) , \mathbf { Q } = - \left( \begin{array} { c c } \frac { 1 } { 2 } & \frac { \sqrt { 3 } } { 2 }
      - \frac { \sqrt { 3 } } { 2 } & \frac { 1 } { 2 } \end{array} \right)\) and \(\mathbf { R } = \left( \begin{array} { r r } 1 & 0
      0 & - 1 \end{array} \right)\) are elements of \(S\). It is given that \(\times _ { \mathbf { M } }\) is the operation of matrix multiplication.
      1. State the identity element of \(S\) under \(\times _ { \mathbf { M } }\). The group \(G\) is generated by \(\mathbf { P }\), under \(\times _ { \mathbf { M } }\).
      2. Determine the order of \(G\). The group \(H\) is generated by \(\mathbf { Q }\) and \(\mathbf { R }\), also under \(\times _ { \mathbf { M } }\).
        1. By finding each element of \(H\), determine the order of \(H\).
        2. List all the proper subgroups of \(H\).
      4. State whether each of the following statements is true or false. Give a reason for each of your answers.
        • \(G\) is abelian
        • \(G\) is cyclic
        • \(H\) is abelian
        • \(H\) is cyclic
      OCR Further Additional Pure AS 2024 June Q6
      6 For positive integers \(n\), let \(f ( n ) = 1 + 2 ^ { n } + 4 ^ { n }\).
        1. Given that \(n\) is a multiple of 3 , but not of 9 , use the division algorithm to write down the two possible forms that \(n\) can take.
        2. Show that when \(n\) is a multiple of 3 , but not of 9 , \(f ( n )\) is a multiple of 73 .
      2. Determine the value of \(\mathrm { f } ( n )\), modulo 73 , in the case when \(n\) is a multiple of 9 .
      OCR Further Additional Pure AS 2024 June Q7
      7 In a long-running biochemical experiment, an initial amount of 1200 mg of an enzyme is placed into a mixture. The model for the amount of enzyme present in the mixture suggests that, at the end of each hour, one-eighth of the amount of enzyme that was present at the start of that hour is used up due to chemical reactions within the mixture. To compensate for this, at the end of each six-hour period of time, a further 500 mg of the enzyme is added to the mixture.
      1. Let \(n\) be the number of six-hour periods that have elapsed since the experiment began. Explain how the amount of enzyme, \(\mathrm { E } _ { \mathrm { n } } \mathrm { mg }\), in the mixture is given by the recurrence system \(E _ { 0 } = 1200\) and \(E _ { n + 1 } = \left( \frac { 7 } { 8 } \right) ^ { 6 } E _ { n } + 500\) for \(n \geqslant 0\).
      2. Solve the recurrence system given in part (a) to obtain an exact expression for \(\mathrm { E } _ { \mathrm { n } }\) in terms of \(n\).
      3. Hence determine, in the long term, the amount of enzyme in the mixture. Give your answer correct to \(\mathbf { 3 }\) significant figures.
      4. In this question you must show detailed reasoning. The long-running experiment is then repeated. This time a new requirement is added that the amount of enzyme present in the mixture must always be at least 500 mg . Show that the new requirement ceases to be satisfied before 12 hours have elapsed. \section*{END OF QUESTION PAPER} OCR is committed to seeking permission to reproduce all third-party content that it uses in its assessment materials. OCR has attempted to identify and contact all copyright holders whose work is used in this paper. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced in the OCR Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download from our public website (\href{http://www.ocr.org.uk}{www.ocr.org.uk}) after the live examination series.
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      OCR Further Additional Pure AS 2021 November Q1
      1 The points \(A , B\) and \(C\) have position vectors \(\mathbf { a } = \left( \begin{array} { l } 3
      0
      0 \end{array} \right) , \mathbf { b } = \left( \begin{array} { l } 0
      4
      0 \end{array} \right)\) and \(\mathbf { c } = \left( \begin{array} { l } 0
      0
      1 \end{array} \right)\) respectively, relative to the origin \(O\).
        1. Calculate \(\mathbf { a } \times \mathbf { b }\), giving your answer as a multiple of \(\mathbf { c }\).
        2. Explain, geometrically, why \(\mathbf { a } \times \mathbf { b }\) must be a multiple of \(\mathbf { c }\).
      2. Use a vector product method to calculate the area of triangle \(A B C\).
      OCR Further Additional Pure AS 2021 November Q2
      2 The surface \(S\) is given by \(z = x ^ { 2 } + 4 x y\) for \(- 6 \leqslant x \leqslant 6\) and \(- 2 \leqslant y \leqslant 2\).
        1. Write down the equation of any one section of \(S\) which is parallel to the \(x\)-z plane
        2. Sketch the section of (a)(i) on the axes provided in the Printed Answer Booklet.
      2. Write down the equation of any one contour of \(S\) which does not include the origin.
      OCR Further Additional Pure AS 2021 November Q3
      3 For positive integers \(n\), the sequence of Fibonacci numbers, \(\left\{ \mathrm { F } _ { \mathrm { n } } \right\}\), starts with the terms \(F _ { 1 } = 1 , F _ { 2 } = 1 , F _ { 3 } = 2 , \ldots\) and is given by the recurrence relation \(\mathrm { F } _ { \mathrm { n } } = \mathrm { F } _ { \mathrm { n } - 1 } + \mathrm { F } _ { \mathrm { n } - 2 } ( \mathrm { n } \geqslant 3 )\).
      1. Show that \(\mathrm { F } _ { 3 \mathrm { k } + 3 } = 2 \mathrm {~F} _ { 3 \mathrm { k } + 1 } + \mathrm { F } _ { 3 \mathrm { k } }\), where \(k\) is a positive integer.
      2. Prove by induction that \(\mathrm { F } _ { 3 n }\) is even for all positive integers \(n\).
      OCR Further Additional Pure AS 2021 November Q4
      4
      1. Let \(a = 1071\) and \(b = 67\).
        1. Find the unique integers \(q\) and \(r\) such that \(\mathrm { a } = \mathrm { bq } + \mathrm { r }\), where \(q > 0\) and \(0 \leqslant r < b\).
        2. Hence express the answer to (a)(i) in the form of a linear congruence modulo \(b\).
      2. Use the fact that \(358 \times 715 - 239 \times 1071 = 1\) to prove that 715 and 1071 are co-prime.
      OCR Further Additional Pure AS 2021 November Q5
      5 A trading company deals in two goods. The formula used to estimate \(z\), the total weekly cost to the company of trading the two goods, in tens of thousands of pounds, is
      \(z = 0.9 x + \frac { 0.096 y } { x } - x ^ { 2 } y ^ { 2 }\),
      where \(x\) and \(y\) are the masses, in thousands of tonnes, of the two goods. You are given that \(x > 0\) and \(y > 0\).
      1. In the first week of trading, it was found that the values of \(x\) and \(y\) corresponded to the stationary value of \(z\). Determine the total cost to the company for this week.
      2. For the second week, the company intends to make a small change in either \(x\) or \(y\) in order to reduce the total weekly cost. Determine whether the company should change \(x\) or \(y\). (You are not expected to say by how much the company should reduce its costs.)
      OCR Further Additional Pure AS 2021 November Q6
      6 The set \(S\) consists of the following four complex numbers.
      \(\begin{array} { l l l l } \sqrt { 3 } + \mathrm { i } & - \sqrt { 3 } - \mathrm { i } & 1 - \mathrm { i } \sqrt { 3 } & - 1 + \mathrm { i } \sqrt { 3 } \end{array}\)
      For \(z _ { 1 } , z _ { 2 } \in S\), the binary operation \(\bigcirc\) is defined by \(z _ { 1 } \bigcirc z _ { 2 } = \frac { 1 } { 4 } ( 1 + i \sqrt { 3 } ) z _ { 1 } z _ { 2 }\).
        1. Complete the Cayley table for \(( S , \bigcirc )\) given in the Printed Answer Booklet.
        2. Verify that ( \(S , \bigcirc\) ) is a group.
        3. State the order of each element of \(( S , \bigcirc )\).
      2. Write down the only proper subgroup of ( \(S , \bigcirc\) ).
        1. Explain why ( \(S , \bigcirc\) ) is a cyclic group.
        2. List all possible generators of \(( S , \bigcirc )\).
      OCR Further Additional Pure AS 2021 November Q7
      7
      1. Let \(f ( n ) = 2 ^ { 4 n + 3 } + 3 ^ { 3 n + 1 }\). Use arithmetic modulo 11 to prove that \(\mathrm { f } ( n ) \equiv 0 ( \bmod 11 )\) for all integers \(n \geqslant 0\).
      2. Use the standard test for divisibility by 11 to prove the following statements.
        1. \(10 ^ { 33 } + 1\) is divisible by 11
        2. \(10 ^ { 33 } + 1\) is divisible by 121