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OCR Further Mechanics 2018 March Q8
11 marks Challenging +1.2
8 A piston of mass 1.5 kg moves in a straight line inside a long straight horizontal cylinder. At time \(t \mathrm {~s}\) the displacement of the piston from its initial position at one end of the cylinder is \(x \mathrm {~m}\) and its velocity is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{a8c9d007-e67f-4637-9e74-630ba9a91442-5_168_805_1726_630} The piston starts moving when \(t = 2\) and is brought to rest when it reaches the other end of the cylinder. While the piston is in motion it is acted on by a force of magnitude \(\frac { 6 } { t ^ { 2 } } \mathrm {~N}\) in the positive \(x\) direction, and also by a force of magnitude \(\frac { 3 v } { t } \mathrm {~N}\) resisting the motion.
  1. Show that, while the piston is in motion, \(\frac { \mathrm { d } v } { \mathrm {~d} t } + \frac { 2 v } { t } = \frac { 4 } { t ^ { 2 } }\). The piston reaches the other end of the cylinder when \(t = 20\).
  2. Find the speed of the piston immediately before it is brought to rest.
  3. Show that the piston travels a distance of 5.61 m , correct to 3 significant figures. \section*{OCR} \section*{Oxford Cambridge and RSA}
OCR Further Discrete 2018 March Q1
6 marks Moderate -0.5
1 The masses, in kg , of ten bags are given below. $$\begin{array} { l l l l l l l l l l } 8 & 10 & 10 & 12 & 12 & 12 & 13 & 15 & 18 & 18 \end{array}$$
  1. Use first-fit decreasing to pack the bags into crates that can hold a maximum of 50 kg each. Only two crates are available, so only some of the bags will be packed. Each bag is given a value.
    BagABCDEFGHIJ
    Mass (kg)8101012121213151818
    Value6332454644
  2. Find a packing into two crates so that the total value of the bags in the crates is at least 32 .
OCR Further Discrete 2018 March Q2
14 marks Challenging +1.2
2 A linear programming problem is $$\begin{array} { l l } \text { Maximise } & P = 4 x - y - 2 z \\ \text { subject to } & x + 5 y + 3 z \leqslant 60 \\ & 2 x - 5 y \leqslant 80 \\ & 2 y + z \leqslant 10 \\ & x \geqslant 0 , y \geqslant 0 , z \geqslant 0 \end{array}$$
  1. Use the simplex algorithm to solve the problem. In the case when \(z = 0\) the feasible region can be represented graphically. \includegraphics[max width=\textwidth, alt={}, center]{9fe422a0-c498-4ad5-bdfc-f70482960d39-2_636_1619_1564_230} The vertices of the feasible region are \(( 0,0 ) , ( 40,0 ) , ( 46.67,2.67 ) , ( 35,5 )\) and \(( 0,5 )\), where non-integer values are given to 2 decimal places. The linear programming problem is given the additional constraint that \(x\) and \(y\) are integers.
  2. Use branch-and-bound, branching on \(x\) first, to show that the optimum solution with this additional constraint is \(x = 45 , y = 2\). 350 people are at a TV game show. 21 of the 50 are there to take part in the game show and the others are friends who are in the audience, 22 are women and 20 are from London, 2 are women from London who are there to take part in the game show and 15 are men who are not from London and are friends who are in the audience.
  3. Deduce how many of the 50 people are in two of the categories 'there to take part in the game show', 'is a woman' and 'is from London', but are not in all three categories. The 21 people who are there to take part in the game show are moved to the stage where they are seated in two rows of seats with 20 seats in each row. Some of the seats are empty.
  4. Show how the pigeonhole principle can be used to show that there must be at least one pair of these 21 people with no empty chair between them. The 21 people are split into three sets of 7 . In each round of the game show, three of the people are chosen. The three people must all be from the same set of 7 but once two people have played in the same round they cannot play together in another round. For example, if A plays with B and C in round 1 then A cannot play with B or with C in any other round.
  5. By first considering how many different rounds can be formed using the first set of seven people, deduce how many rounds there can be altogether.
OCR Further Discrete 2018 March Q4
12 marks Challenging +1.8
4 The graph below connects nine vertices A, B, ..., H, I. \includegraphics[max width=\textwidth, alt={}, center]{9fe422a0-c498-4ad5-bdfc-f70482960d39-3_543_693_1347_680}
  1. (a) Show that the minimum sum of the degrees of each pair of non-adjacent vertices is 9 .
    (b) Explain what you can deduce from the result in part (a).
  2. Use Kuratowski's theorem to prove that the graph is non-planar.
  3. Prove that there is no subgraph of the graph that is isomorphic to \(\mathrm { K } _ { 4 }\), without using subdivision or contraction.
OCR Further Discrete 2018 March Q5
12 marks Moderate -0.5
5 The diagram represents a map of seven locations (A to G) and the direct road distances (km) between some of them. \includegraphics[max width=\textwidth, alt={}, center]{9fe422a0-c498-4ad5-bdfc-f70482960d39-4_382_771_356_644} A delivery driver needs to start from his depot at D , make deliveries at each of \(\mathrm { A } , \mathrm { B } , \mathrm { F }\) and G , and finish at D .
  1. Write down a route from A to G of length 70 km . The table shows the length of the shortest path between some pairs of places.
    DABFG
    D-
    A-70
    B-84
    F84-
    G70-
  2. (a) Complete the table.
    (b) Use the nearest neighbour method on the table, starting at D , to find the length of a cycle through \(\mathrm { D } , \mathrm { A } , \mathrm { B } , \mathrm { F }\) and G , ignoring possibly repeating E and C .
  3. By first considering the table with the row and column for D removed, find a lower bound for the distance that the driver must travel.
  4. What can you conclude from your previous answers about the distance that the driver must travel? A new road is constructed between D and F . Using this road the driver starts from D , makes the deliveries and returns to D having travelled just 172 km .
  5. Find the length of the new road if
    (a) the driver does not return to D until all the deliveries have been made,
    (b) the driver uses the new road twice in making the deliveries.
OCR Further Discrete 2018 March Q6
15 marks Standard +0.8
6 The activities involved in a project, their durations, immediate predecessors and the number of workers required for each activity are shown in the table.
ActivityDuration (hours)Immediate predecessorsNumber of workers
A6-2
B4-1
C4-1
D2A2
E3A, B1
F4C1
G3D1
H3E, F2
  1. Model the project using an activity network.
    • Calculate the early and late event times.
    • Calculate the independent and interfering float for each activity.
    • Draw a cascade chart for the project, showing each activity starting at its earliest possible start time.
    • Construct a schedule to show how three workers can complete the project in the minimum possible time.
OCR Further Discrete 2018 March Q7
8 marks Challenging +1.8
7 Each day Alix and Ben play a game. They each choose a card and use the table below to find the number of points they win. The table shows the cards available to each player. The entries in the cells are of the form ( \(a , b\) ), where \(a =\) points won by Alix and \(b =\) points won be Ben. Each is trying to maximise the points they win. \begin{table}[h]
\captionsetup{labelformat=empty} \caption{Ben}
Card XCard YCard Z
Card P\(( 4,4 )\)\(( 5,9 )\)\(( 1,7 )\)
\multirow[t]{2}{*}{Alix}Card Q\(( 3,5 )\)\(( 4,1 )\)\(( 8,2 )\)
Card R\(( x , y )\)\(( 2,2 )\)\(( 9,4 )\)
\end{table}
  1. Explain why the table cannot be reduced through dominance no matter what values \(x\) and \(y\) have.
  2. Show that the game is not stable no matter what values \(x\) and \(y\) have.
  3. Find the Nash equilibrium solutions for the various values that \(x\) and \(y\) can have. \section*{OCR} \section*{Oxford Cambridge and RSA}
OCR Further Additional Pure 2018 March Q1
5 marks Standard +0.3
1 Determine the solution of the simultaneous linear congruences $$x \equiv 4 ( \bmod 7 ) , \quad x \equiv 25 ( \bmod 41 ) .$$
OCR Further Additional Pure 2018 March Q2
5 marks Standard +0.8
2 Four points \(A , B , C\) and \(D\) have coordinates \(( 1,2,5 ) , ( 3,4 , - 4 ) , ( 6,2,3 )\) and \(( 0,3,7 )\) respectively. Find the volume of tetrahedron \(A B C D\).
OCR Further Additional Pure 2018 March Q3
10 marks Challenging +1.2
3 The surface \(S\) has equation \(z = \frac { x } { y } \sin y + \frac { y } { x } \cos x\) where \(0 < x \leqslant \pi\) and \(0 < y \leqslant \pi\).
  1. Find
    • \(\frac { \partial z } { \partial x }\),
    • \(\frac { \partial z } { \partial y }\).
    • Determine the equation of the tangent plane to \(S\) at the point \(A\) where \(x = y = \frac { 1 } { 4 } \pi\). Give your answer in the form \(a x + b y + c z = d\) where \(a , b , c\) and \(d\) are exact constants.
    • Write down a normal vector to \(S\) at \(A\).
OCR Further Additional Pure 2018 March Q4
12 marks Hard +2.3
4
  1. (a) Find all the quadratic residues modulo 11.
    (b) Prove that the equation \(y ^ { 5 } = x ^ { 2 } + 2017\) has no solution in integers \(x\) and \(y\).
  2. In this question you must show detailed reasoning. The numbers \(M\) and \(N\) are given by $$M = 11 ^ { 12 } - 1 \text { and } N = 3 ^ { 2 } \times 5 \times 7 \times 13 \times 61$$ Prove that \(M\) is divisible by \(N\).
OCR Further Additional Pure 2018 March Q5
15 marks Challenging +1.8
5
  1. (a) Solve the recurrence relation $$X _ { n + 2 } = 1.3 X _ { n + 1 } + 0.3 X _ { n } \text { for } n \geqslant 0$$ given that \(X _ { 0 } = 12\) and \(X _ { 1 } = 1\).
    (b) Show that the sequence \(\left\{ X _ { n } \right\}\) approaches a geometric sequence as \(n\) increases. The recurrence relation in part (i) models the projected annual profit for an investment company, so that \(X _ { n }\) represents the profit (in \(\pounds\) ) at the end of year \(n\).
  2. (a) Determine the number of years taken for the projected profit to exceed one million pounds.
    (b) Compare your answer to part (ii)(a) with the corresponding figure given by the geometric sequence of part (i)(b).
  3. (a) In a modified model, any non-integer values obtained are rounded down to the nearest integer at each step of the process. Write down the recurrence relation for this model.
    (b) Write down the recurrence relation for the model in which any non-integer values obtained are rounded up to the nearest integer at each step of the process.
    (c) Describe a situation that might arise in the implementation of part (iii)(b) that would result in an incorrect value for the next \(X _ { n }\) in the process.
OCR Further Additional Pure 2018 March Q6
14 marks Challenging +1.8
6 In this question you must show detailed reasoning. It is given that \(I _ { n } = \int _ { 0 } ^ { \sqrt { 3 } } t ^ { n } \sqrt { 1 + t ^ { 2 } } \mathrm {~d} t\) for integers \(n \geqslant 0\).
  1. Show that \(I _ { 1 } = \frac { 7 } { 3 }\).
  2. Prove that, for \(n \geqslant 2 , ( n + 2 ) I _ { n } = 8 ( \sqrt { 3 } ) ^ { n - 1 } - ( n - 1 ) I _ { n - 2 }\). The curve \(C\) is defined parametrically by $$x = 10 t ^ { 3 } , y = 15 t ^ { 2 } \text { for } 0 \leqslant t \leqslant \sqrt { 3 }$$ When the curve \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis, a surface of revolution is formed with surface area \(A\).
  3. Determine
    • the values of the integers \(k\) and \(m\) such that \(A = k \pi I _ { m }\),
    • the exact value of \(A\).
OCR Further Additional Pure 2018 March Q7
14 marks Challenging +1.8
7 The set \(M\) contains all matrices of the form \(\mathbf { X } ^ { n }\), where \(\mathbf { X } = \frac { 1 } { \sqrt { 3 } } \left( \begin{array} { r r } 2 & - 1 \\ 1 & 1 \end{array} \right)\) and \(n\) is a positive integer.
  1. Show that \(M\) contains exactly 12 elements.
  2. Deduce that \(M\), together with the operation of matrix multiplication, form a cyclic group \(G\).
  3. Determine all the proper subgroups of \(G\). \section*{END OF QUESTION PAPER}
OCR FP1 AS 2018 March Q1
6 marks
1
  1. The complex number 3-4i is denoted by \(z _ { 1 }\). Write \(z _ { 1 }\) in modulus-argument form, giving your angle in radians to 3 significant figures.
  2. The complex number \(z _ { 2 }\) has modulus 6 and argument - 2.5 radians. Express \(z _ { 1 } z _ { 2 }\) in modulus-argument form with the angle in radians correct to 3 significant figures.
OCR FP1 AS 2018 March Q2
5 marks Moderate -0.5
2 In this question you must show detailed reasoning.
The quadratic equation \(3 x ^ { 2 } - 7 x + 5 = 0\) has roots \(\alpha\) and \(\beta\).
  1. Write down the values of \(\alpha + \beta\) and \(\alpha \beta\).
  2. Hence find the values of the following expressions.
    (a) \(\frac { 1 } { \alpha } + \frac { 1 } { \beta }\) (b) \(\alpha ^ { 2 } + \beta ^ { 2 }\) \(3 \quad l _ { 1 }\) and \(l _ { 2 }\) are two intersecting straight lines with the following equations. $$\begin{aligned} & l _ { 1 } : \mathbf { r } = \left( \begin{array} { c }
OCR FP1 AS 2018 March Q3
6 marks Standard +0.3
3
3
- 5 \end{array} \right) + \lambda \left( \begin{array} { c } 1
3
- 2 \end{array} \right)
& l _ { 2 } : \mathbf { r } = \left( \begin{array} { l } 1
a
1 \end{array} \right) + \mu \left( \begin{array} { c } 2
2
- 3 \end{array} \right) \end{aligned}$$
  1. Find the position vector of the point of intersection of \(l _ { 1 }\) and \(l _ { 2 }\).
  2. Determine the value of \(a\).
OCR FP1 AS 2018 March Q4
6 marks Standard +0.8
4 Find, in exact form, the area of the region on an Argand diagram which represents the locus of points for which \(| z - 5 - 2 \mathrm { i } | \leqslant \sqrt { 32 }\) and \(\operatorname { Re } ( z ) \geqslant 9\).
OCR FP1 AS 2018 March Q5
7 marks
5 The matrix \(\mathbf { A }\) is given by \(\left( \begin{array} { c c c } 1 & 0 & 0 \\ 0 & a ^ { 2 } & 0 \\ 0 & 0 & 1 \end{array} \right)\) and the matrix \(\mathbf { B }\) is given by \(\left( \begin{array} { c c c } 0.6 & b & 0 \\ - b & 0.6 & 0 \\ 0 & 0 & 1 \end{array} \right)\).
  1. \(\mathbf { A }\) represents a reflection. Write down the value of \(\operatorname { det } \mathbf { A }\).
  2. Hence find the possible values of \(a\).
  3. \(\mathbf { r }\) is the position vector of a point \(R\). Given that \(\mathbf { A r } = \mathbf { r }\) describe the location of \(R\).
  4. \(\mathbf { B }\) represents a rotation. Write down the value of \(\operatorname { det } \mathbf { B }\).
  5. Hence find the possible values of \(b\).
OCR FP1 AS 2018 March Q6
10 marks Standard +0.3
6 The matrix \(\mathbf { A }\) is given by \(\left( \begin{array} { l l } 1 & 2 \\ 1 & a \end{array} \right)\) and the matrix \(\mathbf { B }\) is given by \(\left( \begin{array} { c c } 2 & 1 \\ - 1 & b \end{array} \right)\).
  1. Find the matrix \(\mathbf { A B }\).
  2. State the conditions on \(a\) and \(b\) for \(\mathbf { A B }\) to be a singular matrix. \(P Q R S\) is a quadrilateral and the vertices \(P , Q , R\) and \(S\) are in clockwise order. A transformation, T , is represented by the matrix \(\mathbf { A B }\).
  3. State the effect on both the area and also the orientation of the image of \(P Q R S\) under T in each of the following cases.
    (a) \(\quad a = 1\) and \(b = 1\) (b) \(\quad a = 2\) and \(b = 3\)
OCR FP1 AS 2018 March Q7
9 marks Standard +0.3
7 In this question you must show detailed reasoning.
  1. Find the square roots of the number \(528 + 46 \mathrm { i }\) giving your answers in the form \(a + b \mathrm { i }\).
  2. \(\quad 3 + 2 \mathrm { i }\) is a root of the equation \(x ^ { 3 } - a x + 78 = 0\), where \(a\) is a real number. Find the value of \(a\).
OCR FP1 AS 2018 March Q8
11 marks Challenging +1.3
8 In this question you must show detailed reasoning. A sequence of vectors \(\mathbf { a } _ { 1 } , \mathbf { a } _ { 2 } , \mathbf { a } _ { 3 } , \ldots\) is defined by
  • \(\mathbf { a } _ { 1 } = \left( \begin{array} { l } 1 \\ 1 \\ 1 \end{array} \right)\)
  • \(\quad \mathbf { a } _ { n + 1 } = \left( \mathbf { a } _ { n } \times \mathbf { b } \right) \times \mathbf { b }\), for integers \(n \geqslant 1\), where \(\mathbf { b }\) is the vector \(\frac { 1 } { 4 } \left( \begin{array} { c } - 3 \\ 1 \\ 2 \end{array} \right)\).
    1. Prove by induction that \(\mathbf { a } _ { n } = \left( - \frac { 7 } { 8 } \right) ^ { n - 1 } \left( \begin{array} { l } 1 \\ 1 \\ 1 \end{array} \right)\). for all integers \(n \geqslant 1\).
    2. Use an algebraic method to find the smallest value of \(n\) such that \(\left| \mathbf { a } _ { n } \right| < 0.001\).
\section*{END OF QUESTION PAPER}
OCR FS1 AS 2018 March Q1
7 marks Easy -1.2
1 A learner driver keeps taking the driving test until she passes. The number of attempts taken, up to and including the pass, is denoted by \(X\).
  1. State two assumptions needed for \(X\) to be well modelled by a geometric distribution. Assume now that \(X \sim \operatorname { Geo } ( 0.4 )\).
  2. Find \(\mathrm { P } ( X < 6 )\).
  3. Find \(\mathrm { E } ( X )\).
  4. Find \(\operatorname { Var } ( X )\).
OCR FS1 AS 2018 March Q2
6 marks Standard +0.3
2 The number of calls received by a customer service department in 30 minutes is denoted by \(W\). It is known that \(\mathrm { E } ( W ) = 6.5\).
  1. It is given that \(W\) has a Poisson distribution.
    (a) Write down the standard deviation of \(W\).
    (b) Find the probability that the total number of calls received in a randomly chosen period of 2 hours is less than 30 .
  2. It is given instead that \(W\) has a uniform distribution on \([ 1 , N ]\). Calculate the value of \(\mathrm { P } ( W > 3 )\).
OCR FS1 AS 2018 March Q3
8 marks Standard +0.8
3 A pack of 40 cards consists of 10 cards in each of four colours: red, yellow, blue and green. The pack is dealt at random into four "hands", each of 10 cards. The hands are labelled North, South, East and West.
  1. Find the probability that West has exactly 3 red cards.
  2. Find the probability that West has exactly 3 red cards, given that East and West have between them a total of exactly 5 red cards.
  3. South has 5 red cards and 5 blue cards. These cards are placed in a row in a random order. Find the probability that the colour of each card is different from the colour of the preceding card.