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OCR FM1 AS 2018 March Q3
12 marks Standard +0.3
3 A student is investigating fluid flowing through a pipe.
In her first model she assumes a relationship of the form \(P = S \rho ^ { \alpha } g ^ { \beta } h ^ { \gamma }\) where \(\rho\) is the density of the fluid, \(h\) is the length of the pipe, \(P\) is the pressure difference between the ends of the pipe, \(g\) is the acceleration due to gravity and \(S\) is a dimensionless constant. You are given that \(\rho\) is measured in \(\mathrm { kg } \mathrm { m } ^ { - 3 }\).
  1. Use the fact that pressure is force per unit area to show that \([ P ] = \mathrm { ML } ^ { - 1 } \mathrm {~T} ^ { - 2 }\).
  2. Find the values of \(\alpha , \beta\) and \(\gamma\). The density of the fluid the student is using is \(540 \mathrm {~kg} \mathrm {~m} ^ { - 3 }\). In her experiment she finds that when the length of the pipe is 1.40 m the pressure difference between the ends of the pipe is \(3.25 \mathrm { Nm } ^ { - 2 }\).
  3. Find the length of the pipe for which her first model would predict a pressure difference between the ends of the pipe of \(4.65 \mathrm { Nm } ^ { - 2 }\). In an alternative model the student suggests a modified relationship of the form \(P = S \rho ^ { \alpha } g ^ { \beta } h ^ { \gamma } + \frac { 1 } { 2 } h v ^ { 2 }\), where \(v\) is the average velocity of the fluid in the pipe.
  4. Use dimensional analysis to assess the validity of her alternative model.
OCR FM1 AS 2018 March Q4
16 marks Standard +0.8
4 A car has a mass of 850 kg and its engine can generate a maximum power of 35 kW . The total resistance to motion of the car is modelled as \(k v \mathrm {~N}\) where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed of the car and \(k\) is a constant. When the car is moving in a straight line on a straight horizontal road, the greatest constant speed that it can attain is \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Show that \(k = 56\).
  2. Find the greatest possible acceleration of the car on the road at an instant when it is moving with a speed of \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). A trailer of mass 240 kg is attached to the car by means of a light inextensible tow bar which is parallel to the surface of the road. The resistance to motion of the trailer is modelled as a constant force of magnitude 350 N . The car and trailer move on the horizontal road. At a certain instant the car's engine is working at a rate of 30 kW and the acceleration of the car is \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  3. (a) Find the speed of the car at this instant.
    (b) Find the magnitude of the tension in the tow bar at this instant. The car and trailer now move in a straight line on a straight road inclined at \(8 ^ { \circ }\) to the horizontal.
  4. Find the difference between their greatest possible constant speed travelling up the slope and their greatest possible constant speed travelling down the slope.
OCR FM1 AS 2018 March Q5
11 marks Standard +0.8
5 Two particles \(A\) and \(B\) are on a smooth horizontal floor with \(B\) between \(A\) and a vertical wall. The masses of \(A\) and \(B\) are 4 kg and 11 kg respectively. Initially, \(B\) is at rest and \(A\) is moving towards \(B\) with a speed of \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) (see diagram). \(A\) collides directly with \(B\). The coefficient of restitution between \(A\) and \(B\) is \(e\). \includegraphics[max width=\textwidth, alt={}, center]{bf86ac88-0fd1-4d49-a705-9b8d06fbac2a-3_209_803_1658_630}
  1. Show that immediately after the collision the speed of \(B\) is \(\frac { 4 } { 15 } u ( 1 + e )\). After the collision between \(A\) and \(B\) the direction of motion of \(A\) is reversed. \(B\) subsequently collides directly with the vertical wall. The coefficient of restitution between \(B\) and the wall is \(\frac { 1 } { 2 } e\).
  2. Given that there is a second collision between \(A\) and \(B\), find the range of possible values of \(e\).
OCR FM1 AS 2018 March Q6
9 marks Hard +2.3
6 A fairground game involves a player kicking a ball, \(B\), from rest so as to project it with a horizontal velocity of magnitude \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The ball is attached to one end of a light rod of length \(l \mathrm {~m}\). The other end of the rod is smoothly hinged at a fixed point \(O\) so that \(B\) can only move in the vertical plane which contains \(O\), a fixed barrier and a bell which is fixed \(l \mathrm {~m}\) vertically above \(O\). Initially \(B\) is vertically below \(O\). The barrier is positioned so that when \(B\) collides directly with the barrier, \(O B\) makes an angle \(\theta\) with the downwards vertical through \(O\) (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{bf86ac88-0fd1-4d49-a705-9b8d06fbac2a-4_643_659_584_724} The coefficient of restitution between \(B\) and the barrier is \(e . B\) rebounds from the barrier, passes through its original position and continues on a circular path towards the bell. The bell will only ring if the ball strikes it with a speed of at least \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The player wins the game if the player causes the bell to ring having kicked \(B\) so that it first collides with the barrier. You may assume that \(B\) and the bell are small and that the barrier has negligible thickness. Show that, whatever the position of the barrier, the player cannot win the game if \(u ^ { 2 } < 4 g l + \frac { V ^ { 2 } } { e ^ { 2 } }\). \section*{END OF QUESTION PAPER}
OCR FD1 AS 2018 March Q1
10 marks Standard +0.3
1
  1. (a) Show that the number of arrangements of 25 distinct objects is an integer multiple of \(5 ^ { 6 }\).
    (b) Explain how this shows that the number of arrangements of 25 distinct objects is a whole number of millions.
  2. (a) Calculate the values of
OCR FD1 AS 2018 March Q2
8 marks Standard +0.8
2 The diagram shows an incomplete solution to the problem of using Dijkstra's algorithm to find a shortest path from \(A\) to \(F\). Any cell that has values in it is complete. \includegraphics[max width=\textwidth, alt={}, center]{a51b112d-1f3f-4214-94c1-8b9cd7eb831c-2_650_1246_1107_411}
  1. (a) Find the missing weight on \(\operatorname { arc } B E\).
    (b) What can you deduce about the missing weight on arc \(C D\) ? You are now given that the weight of arc \(C E\) is not 3 .
  2. (a) What can you deduce about the missing weight on arc \(C E\) ?
    (b) Complete the labelling of the boxes at \(E\) and \(F\) on the diagram in the Printed Answer Booklet. [2] Suppose that there are two shortest routes from \(A\) to \(F\).
  3. Show how trace back is used to find the shortest routes from \(A\) to \(F\).
OCR FD1 AS 2018 March Q3
9 marks Standard +0.3
3 Lee and Maria are playing a strategy game. The tables below show the points scored by Lee and the points scored by Maria for each combination of strategies. Points scored by Lee Lee's choice \begin{table}[h]
\captionsetup{labelformat=empty} \caption{Maria's choice}
WXYZ
P5834
Q4275
R2153
\end{table} Points scored by Maria Lee's choice \includegraphics[max width=\textwidth, alt={}, center]{a51b112d-1f3f-4214-94c1-8b9cd7eb831c-3_335_481_392_1139}
  1. Show how this game can be reformulated as a zero-sum game.
  2. By first using dominance to eliminate one of Lee's choices, use a graphical method to find the optimal mixed strategy for Lee.
OCR FD1 AS 2018 March Q4
9 marks Moderate -0.8
4 Deva is having some work done on his house. The table shows the activities involved, their durations and their immediate predecessors.
ActivityImmediate predecessorsDuration (hours)
A Have skip delivered-3
B Remodel wallsA3
C Buy new fittings-2
D Fit electricsB2
E Fit plumbingB2
F Install fittingsC, E3
G PlasteringD,E2
H DecoratingF, G3
  1. Model this information as an activity network.
  2. Find the minimum time in which the work can be completed.
  3. Describe the effect on the minimum project completion time of each of the following happening individually.
    1. The duration of activity A is increased to 3.5 hours.
    2. The duration of activity D is increased to 4 hours.
    3. The duration of activity F is decreased to 2 hours. The decorators working on activity H cannot work for 3 hours without a break.
    4. How would you adapt your model to incorporate the break?
OCR FD1 AS 2018 March Q5
8 marks Challenging +1.2
5
  1. How many arcs does the complete bipartite graph \(K _ { 5,5 }\) have? A subgraph of \(K _ { 5,5 }\) contains 5 arcs joining each of the elements of the set \(\{ 1,2,3,4,5 \}\) to an element in a permutation of the set \(\{ 1,2,3,4,5 \}\). Suppose that \(r\) is connected to \(p ( r )\) for \(r = 1,2,3,4,5\).
  2. How many permutations would have \(p ( 1 ) \neq 1\) ?
  3. Using the pigeonhole principle, show that for every permutation of \(\{ 1,2,3,4,5 \}\), the product \(\Pi _ { r = 1 } ^ { 5 } ( r - p ( r ) )\) is even (i.e. an integer multiple of 2, including 0 ).
  4. Is the result in part (iii) true when the permutation is of the set \(\{ 1,2,3,4,5,6 \}\) ? Give a reason for your answer.
OCR FD1 AS 2018 March Q6
16 marks Moderate -0.3
6 An online magazine consists of an editorial, articles, reviews and advertisements.
The magazine must have a total of at least 12 pages. The editorial always takes up exactly half a page. There must be at least 3 pages of articles and at most 1.5 pages of reviews. At least a quarter but fewer than half of the pages in the magazine must be used for advertisements. Let \(x\) be the number of pages used for articles, \(y\) be the number of pages used for reviews and \(z\) be the number of pages used for advertisements. The constraints on the values of \(x , y\) and \(z\) are: $$\begin{aligned} & x + y + z \geqslant 11.5 \\ & x \geqslant 3 \\ & y \leqslant 1.5 \\ & 2 x + 2 y - 2 z + 1 \geqslant 0 \\ & 2 x + 2 y - 6 z + 1 \leqslant 0 \\ & y \geqslant 0 \end{aligned}$$
  1. (a) Explain why \(x + y + z \geqslant 11.5\).
    (b) Explain why only one non-negativity constraint is needed.
    (c) Show that the requirement that at least one quarter of the pages in the magazine must be used for advertisements leads to the constraint \(2 x + 2 y - 6 z + 1 \leqslant 0\). Advertisements bring in money but are not popular with the subscribers. The editor decides to limit the number of pages of advertisements to at most four.
  2. Graph the feasible region in the case when \(z = 4\) using the axes in the Printed Answer Booklet. To be successful the magazine needs to maximise the number of subscribers.
    The editor has found that when \(z \leqslant 4\) the expected number of subscribers is given by \(P = 300 x + 400 y\).
  3. (a) What is the maximum expected number of subscribers when \(z = 4\) ?
    (b) By first considering the feasible region for \(z = k\), where \(k \leqslant 4\), find an expression for the maximum number of subscribers in terms of \(k\). \section*{END OF QUESTION PAPER} \section*{OCR} \section*{Oxford Cambridge and RSA}
OCR Further Additional Pure AS 2018 March Q1
4 marks Moderate -0.8
1 Use standard divisibility tests to show that the number $$N = 91039173588$$
  • is divisible by 9
  • is divisible by 11
  • is not divisible by 8 .
OCR Further Additional Pure AS 2018 March Q2
13 marks Standard +0.8
2 The surface \(S\) has equation \(z = x ^ { 2 } y - 8 x y ^ { 2 } + \frac { x } { y }\) for \(y \neq 0\).
  1. (a) Find the following.
OCR Further Additional Pure AS 2018 March Q3
5 marks Standard +0.3
3 In this question you must show detailed reasoning.
  1. The sequence \(\left\{ A _ { n } \right\}\) is given by \(A _ { 1 } = \sqrt { 3 }\) and \(A _ { n + 1 } = ( \sqrt { 3 } + 1 ) A _ { n }\) for \(n \geqslant 1\). Find an expression for \(A _ { n }\) in terms of \(n\).
  2. The sequence \(\left\{ B _ { n } \right\}\) is given by the formula $$B _ { n } = \frac { 1 } { \sqrt { 3 } } \left( ( \sqrt { 3 } + 1 ) ^ { n } - ( \sqrt { 3 } - 1 ) ^ { n } \right) \text { for } n \geqslant 1 .$$ Explain why \(B _ { n } \rightarrow \frac { 1 } { \sqrt { 3 } } ( \sqrt { 3 } + 1 ) ^ { n }\) as \(n \rightarrow \infty\).
  3. The sequence \(\left\{ C _ { n } \right\}\) converges and is defined by \(C _ { n } = \frac { A _ { n } } { B _ { n } }\) for \(n \geqslant 1\). Identify the limit of \(C n\) as \(n \rightarrow \infty\).
OCR Further Additional Pure AS 2018 March Q4
11 marks Standard +0.8
4 The group \(G\) consists of the symmetries of the equilateral triangle \(A B C\) under the operation of composition of transformations (which may be assumed to be associative). Three elements of \(G\) are
  • \(\boldsymbol { i }\), the identity
  • \(\boldsymbol { j }\), the reflection in the vertical line of symmetry of the triangle
  • \(\boldsymbol { k }\), the anticlockwise rotation of \(120 ^ { \circ }\) about the centre of the triangle.
These are shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{0b4458dc-4f82-40e4-adcf-cbffca088389-3_204_531_735_772} \includegraphics[max width=\textwidth, alt={}, center]{0b4458dc-4f82-40e4-adcf-cbffca088389-3_211_543_975_762} \includegraphics[max width=\textwidth, alt={}, center]{0b4458dc-4f82-40e4-adcf-cbffca088389-3_216_543_1215_762}
  1. Explain why the order of \(G\) is 6 .
  2. Determine
OCR Further Additional Pure AS 2018 March Q5
7 marks Challenging +1.2
5 The points \(A\) and \(B\) have position vectors \(\mathbf { a }\) and \(\mathbf { b }\) respectively, relative to a fixed origin \(O\).
  1. (a) Prove that \(\mathbf { a } \times ( \mathbf { b } - \mathbf { a } ) = \mathbf { a } \times \mathbf { b }\).
    (b) Determine the relationship between \(\mathbf { a } \times ( \mathbf { b } - \mathbf { a } )\) and \(\mathbf { b } \times ( \mathbf { b } - \mathbf { a } )\).
  2. The point \(D\) is on the line \(A B\). \(O D\) is perpendicular to \(A B\). By considering the area of triangle \(O A B\), show
    that \(| O D | = \frac { | \mathbf { a } \times \mathbf { b } | } { | \mathbf { b } - \mathbf { a } | }\).
OCR Further Additional Pure AS 2018 March Q6
8 marks Challenging +1.2
6 You are given that \(n\) is an integer.
  1. (a) Show that \(\operatorname { hcf } ( 2 n + 1,3 n + 2 ) = 1\).
    (b) Hence prove that, if \(( 2 n + 1 )\) divides \(\left( 36 n ^ { 2 } + 3 n - 14 \right)\), then \(( 2 n + 1 )\) divides \(( 12 n - 7 )\).
  2. Use the results of part (i) to find all integers \(n\) for which \(\frac { 36 n ^ { 2 } + 3 n - 14 } { 2 n + 1 }\) is also an integer.
OCR Further Additional Pure AS 2018 March Q7
12 marks Challenging +1.2
7 Irrational numbers can be modelled by sequences \(\left\{ u _ { n } \right\}\) of rational numbers of the form $$u _ { 0 } = 1 \text { and } u _ { n + 1 } = a + \frac { 1 } { b + u _ { n } } \text { for } n \geqslant 0 \text {, }$$ where \(a\) and \(b\) are non-negative integer constants.
  1. (a) The constants \(a = 1\) and \(b = 0\) produce the irrational number \(\omega\). State the value of \(\omega\) correct to six decimal places.
    (b) By setting \(u _ { n + 1 }\) and \(u _ { n }\) equal to \(\omega\), determine the exact value of \(\omega\).
  2. Use the method of part (i) (b) to find the exact value of the irrational number produced by taking \(a = 0\) and \(b = 2\).
  3. Find positive integers \(a\) and \(b\) which would produce the irrational number \(2 + \sqrt { 10 }\). \section*{END OF QUESTION PAPER}
OCR Further Pure Core 1 2018 September Q1
7 marks Moderate -0.8
1 In this question you must show detailed reasoning.
For the complex number \(z\) it is given that \(| z | = 2\) and \(\arg z = \frac { 1 } { 6 } \pi\).
Find the following in the form \(a + \mathrm { i } b\), where \(a\) and \(b\) are exact numbers.
  1. \(z\)
  2. \(z ^ { 2 }\)
  3. \(\frac { z } { z ^ { * } }\)
OCR Further Pure Core 1 2018 September Q2
6 marks Standard +0.3
2 The loci \(C _ { 1 }\) and \(C _ { 2 }\) are given by \(| z - 1 | = 5\) and \(\arg ( z + 4 + 4 \mathrm { i } ) = \frac { 1 } { 4 } \pi\) respectively.
  1. Sketch on a single Argand diagram the loci \(C _ { 1 }\) and \(C _ { 2 }\).
  2. Indicate by shading on your Argand diagram the following set of points. $$\{ z : | z - 1 | \leqslant 5 \} \cap \left\{ z : 0 \leqslant \arg ( z + 4 + 4 i ) \leqslant \frac { 1 } { 4 } \pi \right\}$$
OCR Further Pure Core 1 2018 September Q3
5 marks Standard +0.3
3 A sequence is defined by \(a _ { 1 } = 6\) and \(a _ { n + 1 } = 5 a _ { n } - 2\) for \(n \geqslant 1\).
Prove by induction that for all integers \(n \geqslant 1 , a _ { n } = \frac { 11 \times 5 ^ { n - 1 } + 1 } { 2 }\).
OCR Further Pure Core 1 2018 September Q4
6 marks Standard +0.3
4 In this question you must show detailed reasoning.
Find the exact value of each of the following.
  1. \(\int _ { 1 } ^ { 4 } \frac { 1 } { x ^ { 2 } - 2 x + 10 } \mathrm {~d} x\)
  2. The mean value of \(\frac { 1 } { \sqrt { 1 - x ^ { 2 } } }\) in the interval \([ 0,0.5 ]\)
OCR Further Pure Core 1 2018 September Q5
8 marks Standard +0.3
5 Two planes, \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\), have equations \(3 x + 2 y + z = 4\) and \(2 x + y + z = 3\) respectively.
  1. Find the acute angle between \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\). The line \(L\) has equation \(x = 1 - y = 2 - z\).
  2. Show that \(L\) lies in both planes.
OCR Further Pure Core 1 2018 September Q6
5 marks Standard +0.8
6
  1. Find as a single algebraic fraction an expression for \(\sum _ { r = 1 } ^ { n } \frac { 1 } { ( 2 r - 1 ) ( 2 r + 1 ) }\).
  2. Determine the value of \(\sum _ { r = 1 } ^ { \infty } \frac { 1 } { ( 2 r - 1 ) ( 2 r + 1 ) }\).
OCR Further Pure Core 1 2018 September Q7
6 marks Challenging +1.2
7 In this question you must show detailed reasoning.
Find \(\int _ { 2 } ^ { 3 } \frac { x + 1 } { x ^ { 3 } - x ^ { 2 } + x - 1 } \mathrm {~d} x\), expressing your answer in the form \(a \ln b\) where \(a\) and \(b\) are rational numbers.
OCR Further Pure Core 1 2018 September Q8
13 marks Challenging +1.2
8
  1. Using the definitions of \(\cosh x\) and \(\sinh x\) in terms of \(\mathrm { e } ^ { x }\) and \(\mathrm { e } ^ { - x }\), show that \(\sinh 2 x = 2 \sinh x \cosh x\). You are given the function \(\mathrm { f } ( x ) = a \cosh x - \cosh 2 x\), where \(a\) is a positive constant.
  2. Verify that, for any value of \(a\), the curve \(y = \mathrm { f } ( x )\) has a stationary point on the \(y\)-axis.
  3. Find the coordinates of the stationary point found in part (ii).
  4. Determine the maximum value of \(a\) for which the stationary point found in part (ii) is the only stationary point on the curve \(y = \mathrm { f } ( x )\). You are given that for any value of \(a\) greater than the value found in part (iv) there are three stationary points, the one found in part (ii) and two others, one of which satisfies \(x > 0\).
  5. Find the coordinates of this point when \(a = 6\). Give your answer in the form \(\left( \cosh ^ { - 1 } p , q \right)\).