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OCR Further Statistics 2018 September Q9
11 marks Standard +0.3
9 The continuous random variable \(C\) has the distribution \(\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)\). The sum of a random sample of 16 observations of \(C\) is 224.0 .
  1. Find an unbiased estimate of \(\mu\).
  2. It is given that an unbiased estimate of \(\sigma ^ { 2 }\) is 0.24. Find the value of \(\Sigma c ^ { 2 }\). \(D\) is the sum of 10 independent observations of \(C\).
  3. Explain whether \(D\) has a normal distribution. The continuous random variable \(F\) is normally distributed with mean 15.0, and it is known that \(\mathrm { P } ( F < 13.2 ) = 0.115\).
  4. Use the unbiased estimates of \(\mu\) and \(\sigma ^ { 2 }\) to find \(\mathrm { P } ( D + F > 157.0 )\). \section*{OCR} \section*{Oxford Cambridge and RSA}
OCR Further Mechanics 2018 September Q1
5 marks Standard +0.3
1 A car of mass 850 kg is being driven uphill along a straight road inclined at \(7 ^ { \circ }\) to the horizontal. The resistance to motion is modelled as a constant force of magnitude 140 N . At a certain instant the car's speed is \(12 \mathrm {~ms} ^ { - 1 }\) and its acceleration is \(0.4 \mathrm {~ms} ^ { - 2 }\).
  1. Calculate the power of the car's engine at this instant.
  2. Find the constant speed at which the car could travel up the hill with the engine generating this power.
OCR Further Mechanics 2018 September Q2
6 marks Standard +0.3
2 A particle of mass 0.8 kg is moving in a straight line on a smooth horizontal surface with constant speed \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it is struck by a horizontal impulse. Immediately after the impulse acts, the particle is moving with speed \(9 \mathrm {~ms} ^ { - 1 }\) at an angle of \(50 ^ { \circ }\) to its original direction of motion (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{19c3a9d0-15b6-4dd0-a00b-577c3fd2cf52-2_293_597_989_735} Find
  1. the magnitude of the impulse,
  2. the angle that the impulse makes with the original direction of motion of the particle.
OCR Further Mechanics 2018 September Q3
6 marks Standard +0.3
3 Assume that the earth moves round the sun in a circle of radius \(1.50 \times 10 ^ { 8 } \mathrm {~km}\) at constant speed, with one complete orbit taking 365 days. Given that the mass of the earth is \(5.97 \times 10 ^ { 24 } \mathrm {~kg}\),
  1. calculate the magnitude of the force exerted by the sun on the earth, giving your answer in newtons,
  2. state the direction in which this force acts. \(4 \quad A\) and \(B\) are two points a distance of 5 m apart on a horizontal ceiling. A particle \(P\) of mass \(m \mathrm {~kg}\) is attached to \(A\) and \(B\) by light elastic strings. The particle hangs in equilibrium at a distance of 4 m from \(A\) and 3 m from \(B\) so that angle \(A P B = 90 ^ { \circ }\) (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{19c3a9d0-15b6-4dd0-a00b-577c3fd2cf52-3_286_745_402_660} The string joining \(P\) to \(A\) has natural length 2 m and modulus of elasticity \(\lambda _ { A } \mathrm {~N}\). The string joining \(P\) to \(B\) also has natural length 2 m but has modulus of elasticity \(\lambda _ { B } \mathrm {~N}\).
  3. (a) Show that \(\lambda _ { B } = \frac { 8 } { 3 } \lambda _ { A }\).
    (b) Find an expression for \(\lambda _ { A }\) in terms of \(m\) and \(g\).
  4. Find, in terms of \(m\) and \(g\), the total elastic potential energy stored in the strings. The string joining \(P\) to \(A\) is detached from \(A\) and a second particle, \(Q\), of mass \(0.3 m \mathrm {~kg}\) is attached to the free end of the string. \(Q\) is then gently lowered into a position where the system hangs vertically in equilibrium.
  5. Find the distance of \(Q\) below \(B\) in this equilibrium position.
OCR Further Mechanics 2018 September Q5
10 marks Standard +0.3
5 One end of a non-uniform rod is freely hinged to a fixed point so that the rod can rotate about the point. When the rod rotates with angular velocity \(\omega\) it can be shown that the kinetic energy \(E\) of the rod is given by \(E = \frac { 1 } { 2 } I \omega ^ { 2 }\), where \(I\) is a quantity called the moment of inertia of the rod.
  1. Deduce the dimensions of \(I\).
  2. Given that the rod has mass \(m\) and length \(r\), suggest an expression for \(I\), explaining any additional symbols that you use. A student notices that the formula \(E = \frac { 1 } { 2 } I \omega ^ { 2 }\) looks similar to the formula \(E = \frac { 1 } { 2 } m v ^ { 2 }\) for the kinetic energy of a particle, with angular velocity for the rod corresponding to velocity for the particle, and moment of inertia corresponding to mass. Assuming a similar correspondence between angular acceleration (i.e. \(\frac { \mathrm { d } \omega } { \mathrm { d } t }\) ) and acceleration, the student thinks that an equation for angular motion of the rod corresponding to Newton's second law for the particle should be \(F = I \alpha\), where \(F\) is the force applied to the rod and \(\alpha\) is the resulting angular acceleration.
  3. Use dimensional analysis to show that the student's suggestion is incorrect.
  4. State the dimensions of a quantity \(x\) for which the equation \(F x = I \alpha\) would be dimensionally consistent.
  5. Explain why the fact that the equation in part (iv) is dimensionally consistent does not necessarily mean that it is correct.
OCR Further Mechanics 2018 September Q6
10 marks Challenging +1.2
6 A particle \(P\) of mass \(m\) moves along the positive \(x\)-axis. When its displacement from the origin \(O\) is \(x\) its velocity is \(v\), where \(v \geqslant 0\). It is subject to two forces: a constant force \(T\) in the positive \(x\) direction, and a resistive force which is proportional to \(v ^ { 2 }\).
  1. Show that \(v ^ { 2 } = \frac { 1 } { k } \left( T - A \mathrm { e } ^ { - \frac { 2 k x } { m } } \right)\) where \(A\) and \(k\) are constants. \(P\) starts from rest at \(O\).
  2. Find an expression for the work done against the resistance to motion as \(P\) moves from \(O\) to the point where \(x = 1\).
  3. Find an expression for the limiting value of the velocity of \(P\) as \(x\) increases.
OCR Further Mechanics 2018 September Q7
9 marks Standard +0.3
7 A uniform solid hemisphere has radius 0.4 m . A uniform solid cone, made of the same material, has base radius 0.4 m and height 1.2 m . A solid, \(S\), is formed by joining the hemisphere and the cone so that their circular faces coincide. \(O\) is the centre of the joint circular face and \(V\) is the vertex of the cone. \(G\) is the centre of mass of \(S\).
  1. Explain briefly why \(G\) lies on the line through \(O\) and \(V\).
  2. Show that the distance of \(G\) from \(O\) is 0.12 m .
    (The volumes of a hemisphere and cone are \(\frac { 2 } { 3 } \pi r ^ { 3 }\) and \(\frac { 1 } { 3 } \pi r ^ { 2 } h\) respectively.) \includegraphics[max width=\textwidth, alt={}, center]{19c3a9d0-15b6-4dd0-a00b-577c3fd2cf52-4_474_719_1347_664} \(S\) is suspended from two light vertical strings, one attached to \(V\) and the other attached to a point on the circumference of the joint circular face, and hangs in equilibrium with OV horizontal (see diagram).
  3. The weight of \(S\) is \(W\). Find the magnitudes of the tensions in the strings in terms of \(W\).
OCR Further Mechanics 2018 September Q8
16 marks Hard +2.3
8 A point \(O\) is situated a distance \(h\) above a smooth horizontal plane, and a particle \(A\) of mass \(m\) is attached to \(O\) by a light inextensible string of length \(h\). A particle \(B\) of mass \(2 m\) is at rest on the plane, directly below \(O\), and is attached to a point \(C\) on the plane, where \(B C = l\), by a light inextensible string of length \(l . A\) is released from rest with the string \(O A\) taut and making an acute angle \(\theta\) with the downward vertical (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{19c3a9d0-15b6-4dd0-a00b-577c3fd2cf52-5_604_1137_486_552} \(A\) moves in a vertical plane perpendicular to \(C B\) and collides directly with \(B\). As a result of this collision, \(A\) is brought to rest and \(B\) moves on the plane in a horizontal circle with centre \(C\). After \(B\) has made one complete revolution the particles collide again.
  1. Show that, on the next occasion that \(A\) comes to rest, the string \(O A\) makes an angle \(\phi\) with the downward vertical through \(O\), where \(\cos \phi = \frac { 3 + \cos \theta } { 4 }\). \(A\) and \(B\) collide again when \(A O\) is next vertical.
  2. Find the percentage of the original energy of the system that remains immediately after this collision.
  3. Explain why the total momentum of the particles immediately before the first collision is the same as the total momentum of the particles immediately after the second collision.
  4. Explain why the total momentum of the particles immediately before the first collision is different from the total momentum of the particles immediately after the third collision. \section*{OCR} \section*{Oxford Cambridge and RSA}
OCR Further Discrete 2018 September Q1
7 marks Standard +0.3
1 The design for the lines on a playing area for a game is shown below. The letters are not part of the design. \includegraphics[max width=\textwidth, alt={}, center]{22571082-016b-409b-bfeb-e7ebf48ccac7-2_350_855_388_605} Priya paints the lines by pushing a machine. When she is pushing the machine she is about a metre behind the point being painted. She must not duplicate any line by painting it twice.
  • To relocate the machine, it must be stopped and then started again to continue painting the lines.
  • When the machine is being relocated it must still be pushed along the lines of the design, and not 'cut across' on a diagonal for example.
  • The machine can be turned through \(90 ^ { \circ }\) without having to be stopped.
    1. What is the minimum number of times that the machine will need to be started to paint the design?
The design is horizontally and vertically symmetric. $$\mathrm { AB } = 6 \text { metres, } \mathrm { AE } = 26 \text { metres, } \mathrm { AF } = 1.5 \text { metres and } \mathrm { AS } = 9 \text { metres. }$$
  • (a) Find the minimum distance that Priya needs to walk to paint the design. You should show enough working to make your reasoning clear but you do not need to use an algorithmic method.
    (b) Why, in practice, will the distance be greater than this?
    (c) What additional information would you need to calculate a more accurate shortest distance?
    •
    OCR Further Discrete 2018 September Q2
    8 marks Standard +0.8
    2 A list is used to demonstrate how different sorting algorithms work.
    After two passes through shuttle sort the resulting list is $$\begin{array} { l l l l l l l } 17 & 23 & 84 & 21 & 66 & 35 & 12 \end{array}$$
    1. How many different possibilities are there for the original list? Suppose, instead, that the same sort was carried out using bubble sort on the original list.
    2. Write down the list after two passes through bubble sort. The number of comparisons made is used as a measure of the run-time for a sorting algorithm.
    3. For a list of six values, what is the maximum total number of comparisons made in the first two passes of
      (a) shuttle sort
      (b) bubble sort? Steve used both shuttle sort and bubble sort on a list of five values. He says that shuttle sort is more efficient than bubble sort because it made fewer comparisons in the first two passes.
    4. Comment on what Steve said. The number of comparisons made when shuttle sort and bubble sort are used to sort every permutation of a list of four values is shown in the table below.
      Number of comparisons3456
      Shuttle sortNumber of permutations2688
      Bubble sortNumber of permutations10716
    5. Use the information in the table to decide which algorithm you would expect to have the quicker run-time. Justify your answer with calculations.
    OCR Further Discrete 2018 September Q3
    9 marks Challenging +1.2
    3 The pay-off matrix for a zero-sum game is
    XYZ
    \cline { 2 - 4 } A- 210
    \cline { 2 - 4 } B35- 3
    \cline { 2 - 4 } C- 4- 22
    \cline { 2 - 4 } D02- 1
    \cline { 2 - 4 }
    \cline { 2 - 4 }
    1. Show that the game does not have a stable solution.
    2. Use a graphical technique to find the optimal mixed strategy for the player on columns.
    3. Formulate an initial simplex tableau for the problem of finding the optimal mixed strategy for the player on rows.
    OCR Further Discrete 2018 September Q4
    19 marks Moderate -0.3
    4 A project is represented by the activity network below. The times are in days. \includegraphics[max width=\textwidth, alt={}, center]{22571082-016b-409b-bfeb-e7ebf48ccac7-4_384_935_1110_566}
    1. Explain the reason for each dummy activity.
    2. Calculate the early and late event times.
    3. Identify the critical activities.
    4. Calculate the independent float and interfering float on activity A .
    5. (a) Draw a cascade chart to represent the project, using the grid in the Printed Answer Booklet.
      (b) Describe the effect on
      • the project completion
      • the critical activities
        if the duration of activity D is increased by 5 days.
      The number of workers needed for each activity is shown below.
      ActivityABCDEFGH
      Workers21121111
      The project needs to be completed in at most 3 weeks ( 21 days).
      The duration of activity D is 9 days.
    6. Find the minimum number of workers needed. You should explain your reasoning carefully.
    OCR Further Discrete 2018 September Q5
    10 marks Moderate -0.8
    5 Consider the problem given below: $$\begin{array} { l l } \text { Minimise } & 4 \mathrm { AB } + 7 \mathrm { AC } + 8 \mathrm { BD } + 5 \mathrm { CD } + 5 \mathrm { CE } + 6 \mathrm { DF } + 3 \mathrm { EF } \\ \text { subject to } & \mathrm { AB } , \mathrm { AC } , \mathrm { BD } , \mathrm { CD } , \mathrm { CE } , \mathrm { DF } \text { and } \mathrm { EF } \text { are each either } 0 \text { or } 1 \\ & \mathrm { AB } + \mathrm { AC } + \mathrm { BD } + \mathrm { CD } + \mathrm { CE } + \mathrm { DF } + \mathrm { EF } = 5 \\ & \mathrm { AB } + \mathrm { AC } \geqslant 1 , \quad \mathrm { AB } + \mathrm { BD } \geqslant 1 , \quad \mathrm { AC } + \mathrm { CD } + \mathrm { CE } \geqslant 1 , \\ & \mathrm { BD } + \mathrm { CD } + \mathrm { DF } \geqslant 1 , \quad \mathrm { CE } + \mathrm { EF } \geqslant 1 , \quad \mathrm { DF } + \mathrm { EF } \geqslant 1 \end{array}$$
    1. Explain why this is not a standard LP formulation that could be set up as a Simplex tabulation. The variables \(\mathrm { AB } , \mathrm { AC } , \ldots\) correspond to arcs in a network. The weight on each arc is the coefficient of the corresponding variable in the objective function.
    2. Draw the network on the vertices in the Printed Answer Booklet. A variable that takes the value 1 corresponds to an arc that is used in the solution and a variable with the value 0 corresponds to an arc that is not used in the solution.
    3. Explain what is ensured by the constraint \(\mathrm { AB } + \mathrm { AC } \geqslant 1\). Julie claims that the solution to the problem will give the minimum spanning tree for the network.
    4. Find the minimum spanning tree for the network.
      • State the algorithm you have used.
      • Show your method clearly.
      • Draw the tree.
      • State the total weight of the tree.
      • Find the solution to the problem given at the start of the question.
      • You do not need to use a formal method.
      • Draw the arcs in the solution.
      • State the minimum value of the objective function.
      Kim has a different network, exactly one of the arcs in this network is a directed arc.
      Kim wants to find a minimum weight set of arcs such that it is possible to get from any vertex to any other vertex.
    5. Explain why, if Kim's problem has a solution, the directed arc cannot be part of it.
    OCR Further Discrete 2018 September Q6
    8 marks Standard +0.3
    6 Kai mixes hot drinks using coffee and steamed milk.
    The amounts ( ml ) needed and profit ( \(\pounds\) ) for a standard sized cup of four different drinks are given in the table. The table also shows the amount of the ingredients available.
    Type of drinkCoffeeFoamed milkProfit
    w Americano8001.20
    \(x\) Cappuccino60120X
    \(y\) Flat White601001.40
    \(z\) Latte401201.50
    Available9001500
    Kai makes the equivalent of \(w\) standard sized americanos, \(x\) standard sized cappuccinos, \(y\) standard sized flat whites and \(z\) standard sized lattes. He can make different sized drinks so \(w , x , y , z\) need not be integers. Kai wants to find the maximum profit that he can make, assuming that the customers want to buy the drinks he has made.
    1. What is the minimum value of X for it to be worthwhile for Kai to make cappuccinos? Kai makes no cappuccinos.
    2. Use the simplex algorithm to solve Kai's problem. The grids in the Printed Answer Booklet should have at least enough rows and columns and there should be at least enough grids to show all the iterations needed. Only record the output from each iteration, not any intermediate stages.
      Interpret the solution and state the maximum profit that Kai can make.
    OCR Further Discrete 2018 September Q7
    13 marks Challenging +1.8
    7 A simply connected graph has 6 vertices and 10 arcs.
    1. What is the maximum vertex degree? You are now given that the graph is also Eulerian.
    2. Explaining your reasoning carefully, show that exactly two of the vertices have degree 2 .
    3. Prove that the vertices of degree 2 cannot be adjacent to one another.
    4. Use Kuratowski's theorem to show that the graph is planar.
    5. Show that it is possible to make a non-planar graph by the addition of one more arc. A digraph is created from a simply connected graph with 6 vertices and \(10 \operatorname { arcs }\) by making each arc into a single directed arc.
    6. What can be deduced about the indegrees and outdegrees?
    7. If a Hamiltonian cycle exists on the digraph, what can be deduced about the indegrees and outdegrees? \section*{OCR} \section*{Oxford Cambridge and RSA}
    OCR Further Additional Pure 2018 September Q1
    5 marks Challenging +1.2
    1
    1. Write the number \(100011 _ { n }\), where \(n \geqslant 2\), as a polynomial in \(n\).
    2. Show that \(n ^ { 2 } + n + 1\) is a factor of this expression.
    3. Hence show that \(100011 _ { n }\) is composite in any number base \(n \geqslant 2\).
    OCR Further Additional Pure 2018 September Q2
    10 marks Challenging +1.2
    2 In this question, you must show detailed reasoning.
    A curve is defined parametrically by \(x = t ^ { 3 } - 3 t + 1 , y = 3 t ^ { 2 } - 1\), for \(0 \leqslant t \leqslant 5\). Find, in exact form,
    1. the length of the curve,
    2. the area of the surface generated when the curve is rotated completely about the \(x\)-axis.
    OCR Further Additional Pure 2018 September Q3
    11 marks Challenging +1.8
    3 The function \(w = \mathrm { f } ( x , y , z )\) is given by \(\mathrm { f } ( x , y , z ) = x ^ { 2 } y z + 2 x y ^ { 2 } z + 3 x y z ^ { 2 } - 24 x y z\), for \(x , y , z \neq 0\).
    1. (a) Find
      • \(\mathrm { f } _ { x }\),
      • \(\mathrm { f } _ { y }\),
      • \(\mathrm { f } _ { z }\).
        (b) Hence find the values of \(a , b , c\) and \(d\) for which \(w\) has a stationary value when \(d = \mathrm { f } ( a , b , c )\).
      • You are given that this stationary value is a local minimum of \(w\). Find values of \(x , y\) and \(z\) which show that it is not a global minimum of \(w\).
    OCR Further Additional Pure 2018 September Q4
    12 marks Standard +0.3
    4 The points \(A , B , C\) and \(P\) have coordinates ( \(a , 0,0\) ), ( \(0 , b , 0\) ), ( \(0,0 , c\) ) and ( \(a , b , c\) ) respectively, where \(a , b\) and \(c\) are positive constants.
    The plane \(\Pi\) contains \(A , B\) and \(C\).
    1. (a) Use the scalar triple product to determine
      • the volume of tetrahedron \(O A B C\),
      • the volume of tetrahedron PABC.
        (b) Hence show that the distance from \(P\) to \(\Pi\) is twice the distance from \(O\) to \(\Pi\).
      • (a) Determine a vector which is normal to \(\Pi\).
        (b) Hence determine, in terms of \(a , b\) and \(c\) only, the distance from \(P\) to \(\Pi\). consisting of the powers of a particular, non-singular \(2 \times 2\) real matrix \(\mathbf { M } = \left( \begin{array} { l l } a & b \\ c & d \end{array} \right)\), under the operation of
        matrix multiplication. matrix multiplication.
      • Explain why such a group is only possible if \(\operatorname { det } ( \mathbf { M } ) = 1\) or - 1 .
      • Write down values of \(a , b , c\) and \(d\) that would give a suitable matrix \(\mathbf { M }\) for which \(\mathbf { M } ^ { 6 } = \mathbf { I }\) and
      Student Q observes that their class has already found a group of order 6 in a previous task; a group
      Student Q observes that their class has already found a group of order 6 in a previous task; a group consisting of the powers of a particular, non-singular \(2 \times 2\) real matrix \(\mathbf { M } = \left( \begin{array} { l l } a & b \\ c & d \end{array} \right)\), under the operation of
      matrix multiplication. \(\operatorname { det } ( \mathbf { M } ) = 1\).
      Explain why such a group is only possible if \(\operatorname { det } ( \mathbf { M } ) = 1\) or - 1 . Student Q believes that it is possible to construct a rational function f in the form \(\mathrm { f } ( x ) = \frac { a x + b } { c x + d }\) so that the group of functions is isomorphic to the matrix group which is generated by the matrix \(\mathbf { M }\) of part (iii).
    2. (a) Write down and simplify the function f that, according to Student Q , corresponds to \(\mathbf { M }\).
      (b) By calculating \(\mathbf { M } ^ { 3 }\), show that Student Q's suggestion does not work.
      (c) Find a different function \(f\) that will satisfy the requirements of the task.
    OCR Further Additional Pure 2018 September Q7
    14 marks Challenging +1.8
    7 The members of the family of the sequences \(\left\{ u _ { n } \right\}\) satisfy the recurrence relation $$u _ { n + 1 } = 10 u _ { n } - u _ { n - 1 } \text { for } n \geqslant 1$$
    1. Determine the general solution of (*).
    2. The sequences \(\left\{ a _ { n } \right\}\) and \(\left\{ b _ { n } \right\}\) are members of this family of sequences, corresponding to the initial terms \(a _ { 0 } = 1 , a _ { 1 } = 5\) and \(b _ { 0 } = 0 , b _ { 1 } = 2\) respectively.
      (a) Find the next two terms of each sequence.
      (b) Prove that, for all non-negative integers \(n , \left( a _ { n } \right) ^ { 2 } - 6 \left( b _ { n } \right) ^ { 2 } = 1\).
      (c) Determine \(\lim _ { n \rightarrow \infty } \left( \frac { a _ { n } } { b _ { n } } \right)\). \section*{OCR} Oxford Cambridge and RSA
    OCR Further Pure Core 1 2018 December Q1
    5 marks Standard +0.3
    1 Points \(A , B\) and \(C\) have coordinates \(( 0,1 , - 4 ) , ( 1,1 , - 2 )\) and \(( 3,2,5 )\) respectively.
    1. Find the vector product \(\overrightarrow { A B } \times \overrightarrow { A C }\).
    2. Hence find the equation of the plane \(A B C\) in the form \(a x + b y + c z = d\).
    OCR Further Pure Core 1 2018 December Q2
    9 marks Standard +0.8
    2 The equation of the curve shown on the graph is, in polar coordinates, \(r = 3 \sin 2 \theta\) for \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\). \includegraphics[max width=\textwidth, alt={}, center]{8315a796-0e7d-464f-8604-9fe3ab7af359-2_470_657_913_319}
    1. The greatest value of \(r\) on the curve occurs at the point \(P\).
      1. Show that \(\theta = \frac { 1 } { 4 } \pi\) at the point \(P\).
      2. Find the value of \(r\) at the point \(P\).
      3. Mark the point \(P\) on the copy of the graph in the Printed Answer Booklet.
    2. In this question you must show detailed reasoning. Find the exact area of the region enclosed by the curve.
    OCR Further Pure Core 1 2018 December Q3
    7 marks Standard +0.3
    3 You are given that \(\mathrm { f } ( x ) = \ln ( 2 + x )\).
    1. Determine the exact value of \(\mathrm { f } ^ { \prime } ( 0 )\).
    2. Show that \(\mathrm { f } ^ { \prime \prime } ( 0 ) = - \frac { 1 } { 4 }\).
    3. Hence write down the first three terms of the Maclaurin series for \(\mathrm { f } ( x )\).
    OCR Further Pure Core 1 2018 December Q4
    4 marks Standard +0.3
    4 In this question you must show detailed reasoning.
    You are given that \(z = \sqrt { 3 } + \mathrm { i }\). \(n\) is the smallest positive whole number such that \(z ^ { n }\) is a positive whole number.
    1. Determine the value of \(n\).
    2. Find the value of \(z ^ { n }\).
    OCR Further Pure Core 1 2018 December Q5
    6 marks Standard +0.3
    5 You are given that \(\mathbf { A } = \left( \begin{array} { c c c } 1 & 2 & 1 \\ 2 & 5 & 2 \\ 3 & - 2 & - 1 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { c c c } 1 & 0 & 1 \\ - 8 & 4 & 0 \\ 19 & - 8 & - 1 \end{array} \right)\).
    1. Find \(\mathbf { A B }\).
    2. Hence write down \(\mathbf { A } ^ { - 1 }\).
    3. You are given three simultaneous equations $$\begin{array} { r } x + 2 y + z = 0 \\ 2 x + 5 y + 2 z = 1 \\ 3 x - 2 y - z = 4 \end{array}$$
      1. Explain how you can tell, without solving them, that there is a unique solution to these equations.
      2. Find this unique solution.