Questions — OCR (4619 questions)

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OCR FP1 AS 2018 March Q7
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
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
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
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
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.
OCR FS1 AS 2018 March Q4
Moderate -0.3
4 A spinner has edges numbered \(1,2,3,4\) and 5 . When the spinner is spun, the number of the edge on which it lands is the score. The probability distribution of the score, \(N\), is given in the table.
Score, \(N\)12345
Probability0.30.20.2\(x\)\(y\)
It is known that \(\mathrm { E } ( N ) = 2.55\).
  1. Find \(\operatorname { Var } ( N )\).
  2. Find \(\mathrm { E } ( 3 N + 2 )\).
  3. Find \(\operatorname { Var } ( 3 N + 2 )\).
OCR FS1 AS 2018 March Q5
Easy -1.8
5 The speed \(v \mathrm {~ms} ^ { - 1 }\) of a car at time \(t\) seconds after it starts to accelerate was measured at 1 -second intervals. The results are shown in the following diagram.
\includegraphics[max width=\textwidth, alt={}, center]{d5843350-52f9-4fed-adf4-86ceb958033f-3_661_1186_1078_443}
  1. State whether \(t\) or \(v\) or neither is a controlled variable. The value of the product moment correlation coefficient \(r\) for the data is 0.987 correct to 3 significant figures.
  2. The speed of the car is converted to miles per hour and the time to minutes. State the value of \(r\) for the converted data.
  3. State the value of Spearman's rank correlation coefficient \(r _ { s }\) for the data.
  4. What information does \(r\) give about the data that is not given by \(r _ { s }\) ?
OCR FS1 AS 2018 March Q6
7 marks Challenging +1.2
6 The discrete random variable \(R\) has the distribution \(\operatorname { Po } ( \lambda )\).
Use an algebraic method to find the range of values of \(\lambda\) for which the single most likely value of \(R\) is 7. [7]
OCR FS1 AS 2018 March Q7
Standard +0.3
7 The numbers of students taking A levels in three subjects at a school were classified by the year in which they entered the school as follows.
\cline { 2 - 5 } \multicolumn{1}{c|}{}SubjectMathematicsEnglishPhysics
\multirow{3}{*}{
Year of
Entry
}		Year 717167
\cline { 2 - 5 }Year 121325
The Head of the school carries out a significance test at the \(10 \%\) level to test whether subjects taken are independent of year of entry.
  1. Show that in carrying out the test it is necessary to combine columns.
  2. Suggest a reason why it is more sensible to combine the columns for Mathematics and Physics than the columns for Physics and English.
  3. Carry out the test.
  4. State which cell gives the largest contribution to the test statistic.
  5. Interpret your answer to part (iv).
OCR FS1 AS 2018 March Q8
Challenging +1.2
8 In a competition, entrants have to give ranks from 1 to 7 to each of seven resorts. The correct ranks for the resorts are decided by an expert.
  1. One competitor chooses his ranks randomly. By considering all the possible rankings, find the probability that the value of Spearman's rank correlation coefficient \(r _ { s }\) between the competitor's ranks and the expert's ranks is at least \(\frac { 27 } { 28 }\).
  2. Another competitor ranks the seven resorts. A significance test is carried out to test whether there is evidence that this competitor is merely guessing the rank order of the seven resorts. The critical region is \(r _ { s } \geqslant \frac { 27 } { 28 }\). State the significance level of the test. \section*{END OF QUESTION PAPER}
OCR FM1 AS 2018 March Q1
Easy -1.2
1 A particle \(P\) of mass 2.4 kg is attached to one end of a light inextensible string of length 1.4 m . The other end of the string is attached to a fixed point \(O\) on a smooth horizontal table. \(P\) moves on the table at constant speed along a circular path with \(O\) at its centre. The magnitude of the tension in the string is 21 N .
  1. (a) Find the magnitude of the acceleration of \(P\).
    (b) State the direction of the acceleration of \(P\).
  2. Find the speed of \(P\).
  3. Find the time taken for \(P\) to complete a single revolution.
OCR FM1 AS 2018 March Q2
Standard +0.3
2 A pump is pumping still water from the base of a well at a constant rate of 300 kg per minute. The well is 4.5 m deep and water is released from the pump at ground level in a horizontal jet with a speed of \(6.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Ignoring any energy losses due to resistance, calculate the power generated by the pump.
OCR FM1 AS 2018 March Q3
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
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
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
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
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
    • INT(720 \(\div 25\) )
    • INT(720 \(\div 125\) ).
      (b) Deduce the largest power of 10 that is a factor of 720!
    • Use the inclusion-exclusion principle to find the number of integers from 1 to 720 that are not divisible by either 2 or 5 .
OCR FD1 AS 2018 March Q2
2 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
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
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.
    (a) The duration of activity A is increased to 3.5 hours.
    (b) The duration of activity D is increased to 4 hours.
    (c) 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
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
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
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
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.
    • \(\frac { \partial z } { \partial x }\)
    • \(\frac { \partial z } { \partial y }\)
      (b) Find the coordinates of all stationary points of \(S\).
    • Find all four second partial derivatives of \(z\) with respect to \(x\) and/or \(y\).
OCR Further Additional Pure AS 2018 March Q3
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 \(n \rightarrow \infty\).