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OCR MEI S1 2005 June Q5
6 marks Moderate -0.8
  1. On the insert, complete the table giving the lowest common multiples of all pairs of integers between 1 and 6 .
    [0pt] [1]
    \multirow{2}{*}{}Second integer
    123456
    \multirow{6}{*}{First integer}1123456
    22264106
    336312156
    4441212
    551015
    666612
    Two fair dice are thrown and the lowest common multiple of the two scores is found.
  2. Use the table to find the probabilities of the following events.
    (A) The lowest common multiple is greater than 6 .
    (B) The lowest common multiple is a multiple of 5 .
    (C) The lowest common multiple is both greater than 6 and a multiple of 5 .
  3. Use your answers to part (ii) to show that the events "the lowest common multiple is greater than 6 " and "the lowest common multiple is a multiple of 5 " are not independent.
Edexcel S1 Q2
9 marks Moderate -0.8
  1. Plot a scatter diagram showing these data. The student wanted to investigate further whether or not her data provided evidence of an increase in temperature in June each year. Using \(Y\) for the number of years since 1993 and \(T\) for the mean temperature, she calculated the following summary statistics. $$\Sigma Y = 28 , \quad \Sigma T = 182.5 , \quad \Sigma Y ^ { 2 } = 140 , \quad \Sigma T ^ { 2 } = 4173.93 , \quad \Sigma Y T = 644.7 .$$
  2. Calculate the product moment correlation coefficient for these data.
  3. Comment on your result in relation to the student's enquiry.
OCR MEI S1 Q3
6 marks Easy -1.8
  1. On die insert, complete the lable giving due lowest common multiples of all pairs of integers between 1 and 6 .
    Second integer
    \cline { 2 - 8 } \multicolumn{2}{|c|}{}123456
    \multirow{5}{*}{
    First
    integer
    }    		1123456
    \cline { 2 - 8 }22264106
    \cline { 2 - 8 }336312156
    \cline { 2 - 8 }4441212
    \cline { 2 - 8 }551015
    \cline { 2 - 8 }666612
    Two fair dice are thrown and the lowest common multiple of the two scores is found.
  2. Use the table to find the probabilities of the following events.
    (A) The lowest common multiple is greater than 6 .
    (B) The lowest common multiple is a multiple of 5 .
    (C) The lowest common multiple is both greater than 6 and a multiple of 5.
  3. Use your answers to part (ii) to show that the events "the lowest common multiple is greater than 6 " and "the lowest common multiple is a multiple of 5 " are not independent.
AQA S2 2011 June Q3
10 marks Standard +0.3
  1. State the null hypothesis that Emily used.
  2. Find the value of the test statistic, \(X ^ { 2 }\), giving your answer to one decimal place.
  3. State, in context, the conclusion that Emily should reach based on the results of her \(\chi ^ { 2 }\) test.
  4. Make one comment on the GCSE performances of 16-year-old students attending Bailey Language School.
  5. Emily's friend, Joanna, used the same data to correctly conduct a \(\chi ^ { 2 }\) test using the \(10 \%\) level of significance. State, with justification, the conclusion that Joanna should reach.
OCR S2 2007 June Q4
6 marks Moderate -0.3
  1. State two conditions needed for \(X\) to be well modelled by a normal distribution.
  2. It is given that \(X \sim \mathrm {~N} \left( 50.0,8 ^ { 2 } \right)\). The mean of 20 random observations of \(X\) is denoted by \(\bar { X }\). Find \(\mathrm { P } ( \bar { X } > 47.0 )\). 5 The number of system failures per month in a large network is a random variable with the distribution \(\operatorname { Po } ( \lambda )\). A significance test of the null hypothesis \(\mathrm { H } _ { 0 } : \lambda = 2.5\) is carried out by counting \(R\), the number of system failures in a period of 6 months. The result of the test is that \(\mathrm { H } _ { 0 }\) is rejected if \(R > 23\) but is not rejected if \(R \leqslant 23\).
  3. State the alternative hypothesis.
  4. Find the significance level of the test.
  5. Given that \(\mathrm { P } ( R > 23 ) < 0.1\), use tables to find the largest possible actual value of \(\lambda\). You should show the values of any relevant probabilities. 6 In a rearrangement code, the letters of a message are rearranged so that the frequency with which any particular letter appears is the same as in the original message. In ordinary German the letter \(e\) appears \(19 \%\) of the time. A certain encoded message of 20 letters contains one letter \(e\).
  6. Using an exact binomial distribution, test at the \(10 \%\) significance level whether there is evidence that the proportion of the letter \(e\) in the language from which this message is a sample is less than in German, i.e., less than \(19 \%\).
  7. Give a reason why a binomial distribution might not be an appropriate model in this context. 7 Two continuous random variables \(S\) and \(T\) have probability density functions as follows. $$\begin{array} { l l } S : & f ( x ) = \begin{cases} \frac { 1 } { 2 } & - 1 \leqslant x \leqslant 1 \\ 0 & \text { otherwise } \end{cases} \\ T : & g ( x ) = \begin{cases} \frac { 3 } { 2 } x ^ { 2 } & - 1 \leqslant x \leqslant 1 \\ 0 & \text { otherwise } \end{cases} \end{array}$$
  8. Sketch on the same axes the graphs of \(y = \mathrm { f } ( x )\) and \(y = \mathrm { g } ( x )\). [You should not use graph paper or attempt to plot points exactly.]
  9. Explain in everyday terms the difference between the two random variables.
  10. Find the value of \(t\) such that \(\mathrm { P } ( T > t ) = 0.2\). 8 A random variable \(Y\) is normally distributed with mean \(\mu\) and variance 12.25. Two statisticians carry out significance tests of the hypotheses \(\mathrm { H } _ { 0 } : \mu = 63.0 , \mathrm { H } _ { 1 } : \mu > 63.0\).
  11. Statistician \(A\) uses the mean \(\bar { Y }\) of a sample of size 23, and the critical region for his test is \(\bar { Y } > 64.20\). Find the significance level for \(A\) 's test.
  12. Statistician \(B\) uses the mean of a sample of size 50 and a significance level of \(5 \%\).
    1. Find the critical region for \(B\) 's test.
    2. Given that \(\mu = 65.0\), find the probability that \(B\) 's test results in a Type II error.
    3. Given that, when \(\mu = 65.0\), the probability that \(A\) 's test results in a Type II error is 0.1365 , state with a reason which test is better. 9 (a) The random variable \(G\) has the distribution \(\mathrm { B } ( n , 0.75 )\). Find the set of values of \(n\) for which the distribution of \(G\) can be well approximated by a normal distribution.
      (b) The random variable \(H\) has the distribution \(\mathrm { B } ( n , p )\). It is given that, using a normal approximation, \(\mathrm { P } ( H \geqslant 71 ) = 0.0401\) and \(\mathrm { P } ( H \leqslant 46 ) = 0.0122\).
      1. Find the mean and standard deviation of the approximating normal distribution.
      2. Hence find the values of \(n\) and \(p\).
OCR S4 2010 June Q3
7 marks Challenging +1.8
  1. Assuming that all rankings are equally likely, show that \(\mathrm { P } ( R \leqslant 17 ) = \frac { 2 } { 231 }\). The marks of 5 randomly chosen students from School \(A\) and 6 randomly chosen students from School \(B\), who took the same examination, achieving different marks, were ranked. The rankings are shown in the table.
    Rank1234567891011
    School\(A\)\(A\)\(A\)\(B\)\(A\)\(A\)\(B\)\(B\)\(B\)\(B\)\(B\)
  2. For a Wilcoxon rank-sum test, obtain the exact smallest significance level for which there is evidence of a difference in performance at the two schools.
OCR M1 2010 January Q5
11 marks Moderate -0.3
  1. Find the value of \(t\) when \(A\) and \(B\) have the same speed.
  2. Calculate the value of \(t\) when \(B\) overtakes \(A\).
  3. On a single diagram, sketch the \(( t , x )\) graphs for the two cyclists for the time from \(t = 0\) until after \(B\) has overtaken \(A\).
OCR M1 2015 June Q3
8 marks Standard +0.3
  1. Calculate the distance \(A\) cycles, and hence find the period of time for which \(B\) walks before finding the bicycle.
  2. Find \(T\).
  3. Calculate the distance \(A\) and \(B\) each travel.
OCR MEI M1 2008 June Q7
17 marks Moderate -0.3
  1. What information in the question indicates that the tension in the string section CB is also 60 N ?
  2. Show that the string sections AC and CB are equally inclined to the horizontal (so that \(\alpha = \beta\) in Fig. 7.1).
  3. Calculate the angle of the string sections AC and CB to the horizontal. In a different situation the same box is supported by two separate light strings, PC and QC, that are tied to the box at C . There is also a horizontal force of 10 N acting at C . This force and the angles between these strings and the horizontal are shown in Fig. 7.2. The box is in equilibrium. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{170edb27-324e-44df-8dc1-7d8fbad680fe-4_323_503_1649_822} \captionsetup{labelformat=empty} \caption{Fig. 7.2}
    \end{figure}
  4. Calculate the tensions in the two strings.
OCR MEI M1 2013 June Q7
18 marks Moderate -0.3
  1. Represent the forces acting on the object as a fully labelled triangle of forces.
  2. Find \(F\) and \(\theta\). Show that the distance between the object and the vertical section of rope A is 3 m . Abi then pulls harder and the object moves upwards. Bob adjusts the tension in rope B so that the object moves along a vertical line. Fig. 7.2 shows the situation when the object is part of the way up. The tension in rope A is \(S \mathrm {~N}\) and the tension in rope B is \(T \mathrm {~N}\). The ropes make angles \(\alpha\) and \(\beta\) with the vertical as shown in the diagram. Abi and Bob are taking a rest and holding the object stationary and in equilibrium. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{83e69140-4abf-4713-85da-922ce7530e47-5_383_360_534_854} \captionsetup{labelformat=empty} \caption{Fig. 7.2}
    \end{figure}
  3. Give the equations, involving \(S , T , \alpha\) and \(\beta\), for equilibrium in the vertical and horizontal directions.
  4. Find the values of \(S\) and \(T\) when \(\alpha = 8.5 ^ { \circ }\) and \(\beta = 35 ^ { \circ }\).
  5. Abi's mass is 40 kg . Explain why it is not possible for her to raise the object to a position in which \(\alpha = 60 ^ { \circ }\).
AQA M2 2012 January Q7
11 marks Standard +0.3
  1. Show that \(v ^ { 2 } = u ^ { 2 } - 4 a g\).
  2. The ratio of the tensions in the string when the bead is at the two points \(A\) and \(B\) is \(2 : 5\).
    1. Find \(u\) in terms of \(g\) and \(a\).
    2. Find the ratio \(u : v\).
AQA M2 2010 June Q7
12 marks Standard +0.8
  1. Draw a diagram to show the forces acting on the rod.
  2. Find the magnitude of the normal reaction force between the rod and the ground.
    1. Find the normal reaction acting on the rod at \(C\).
    2. Find the friction force acting on the rod at \(C\).
  4. In this position, the rod is on the point of slipping. Calculate the coefficient of friction between the rod and the peg.
    \includegraphics[max width=\textwidth, alt={}]{3ffa0a2b-aa7d-46eb-b92b-3e3ee59f235c-15_2484_1709_223_153}
OCR M2 2008 January Q6
11 marks Standard +0.3
  1. Show that the tension in the string is 4.16 N , correct to 3 significant figures.
  2. Calculate \(\omega\).
    (ii) \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{982647bd-8514-40cf-b4ee-674f51df32c5-3_510_417_1238_904} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure} The lower part of the string is now attached to a point \(R\), vertically below \(P\). \(P B\) makes an angle \(30 ^ { \circ }\) with the vertical and \(R B\) makes an angle \(60 ^ { \circ }\) with the vertical. The bead \(B\) now moves in a horizontal circle of radius 1.5 m with constant speed \(v _ { \mathrm { m } } \mathrm { m } ^ { - 1 }\) (see Fig. 2).
    1. Calculate the tension in the string.
    2. Calculate \(v\).
OCR M2 2006 June Q6
11 marks Standard +0.3
  1. Calculate the tension in the string and hence find the angular speed of \(Q\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{d6d87705-be4b-407d-b699-69fb441d88a7-4_489_1358_1286_392} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure} The particle \(Q\) on the plane is now fixed to a point 0.2 m from the hole at \(A\) and the particle \(P\) rotates in a horizontal circle of radius 0.2 m (see Fig. 2).
  2. Calculate the tension in the string.
  3. Calculate the speed of \(P\).
OCR M2 2008 June Q5
8 marks Standard +0.3
  1. Show that the distance from the ball to the centre of mass of the toy is 10.7 cm , correct to 1 decimal place.
  2. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{6ae57fe9-3b6f-46c2-95b8-d48903ed796b-3_312_1051_1509_587} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure} The toy lies on horizontal ground in a position such that the ball is touching the ground (see Fig. 2). Determine whether the toy is lying in equilibrium or whether it will move to a position where the rod is vertical.
OCR M2 2009 June Q5
11 marks Standard +0.3
  1. Fig. 1 Fig. 1 shows a uniform lamina \(B C D\) in the shape of a quarter circle of radius 6 cm . Show that the distance of the centre of mass of the lamina from \(B\) is 3.60 cm , correct to 3 significant figures. A uniform rectangular lamina \(A B D E\) has dimensions \(A B = 12 \mathrm {~cm}\) and \(A E = 6 \mathrm {~cm}\). A single plane object is formed by attaching the rectangular lamina to the lamina \(B C D\) along \(B D\) (see Fig. 2). The mass of \(A B D E\) is 3 kg and the mass of \(B C D\) is 2 kg . \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{e85c2bf4-21a8-4d9a-93c5-d5679b2a8233-3_959_447_1123_849} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure}
  2. Taking \(x\) - and \(y\)-axes along \(A E\) and \(A B\) respectively, find the coordinates of the centre of mass of the object. The object is freely suspended at \(C\) and rests in equilibrium.
  3. Calculate the angle that \(A C\) makes with the vertical.
OCR M2 2015 June Q7
11 marks Standard +0.8
  1. Show that \(\mu = \frac { 2 } { 3 }\). A small object of weight \(a W \mathrm {~N}\) is placed on the ladder at its mid-point and the object \(S\) of weight \(2 W \mathrm {~N}\) is placed on the ladder at its lowest point \(A\).
  2. Given that the system is in equilibrium, find the set of possible values of \(a\).
OCR MEI M2 2010 June Q2
18 marks Standard +0.3
  1. Calculate the coordinates of the centre of mass of the stand. A small object of mass 5 kg is fixed to the rod AB at a distance of 40 cm from A .
  2. Show that the coordinates of the centre of mass of the stand with the object are ( 22,68 ). The stand is tilted about the edge PQ until it is on the point of toppling. The angle through which the stand is tilted is called 'the angle of tilt'. This procedure is repeated about the edges QR and RS.
  3. Making your method clear, determine which edge requires the smallest angle of tilt for the stand to topple. The small object is removed. A light string is attached to the stand at A and pulled at an angle of \(50 ^ { \circ }\) to the downward vertical in the plane \(\mathrm { O } x y\) in an attempt to tip the stand about the edge RS.
  4. Assuming that the stand does not slide, find the tension in the string when the stand is about to turn about the edge RS.
OCR MEI M2 2016 June Q3
18 marks Standard +0.3
  1. Use an energy method to find the magnitude of the frictional force acting on the block. Calculate the coefficient of friction between the block and the plane.
  2. Calculate the power of the tension in the string when the block has a speed of \(7 \mathrm {~ms} ^ { - 1 }\). Fig. 3.1 shows a thin planar uniform rigid rectangular sheet of metal, OPQR, of width 1.65 m and height 1.2 m . The mass of the sheet is \(M \mathrm {~kg}\). The sides OP and PQ have thin rigid uniform reinforcements attached with masses \(0.6 M \mathrm {~kg}\) and \(0.4 M \mathrm {~kg}\), respectively. Fig. 3.1 also shows coordinate axes with origin at O . The sheet with its reinforcements is to be used as an inn sign.
  1. Calculate the coordinates of the centre of mass of the inn sign. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{8fb49c8b-92e5-49e5-9a3a-e8391c82d9a1-4_421_492_210_1334} \captionsetup{labelformat=empty} \caption{Fig. 3.1}
    \end{figure} The inn sign has a weight of 300 N . It hangs in equilibrium with QR horizontal when vertical forces \(Y _ { \mathrm { Q } } \mathrm { N }\) and \(Y _ { \mathrm { R } } \mathrm { N }\) act at Q and R respectively.
  2. Calculate the value of \(Y _ { \mathrm { Q } }\) and show that \(Y _ { \mathrm { R } } = 120\). The inn sign is hung from a framework, ABCD , by means of two light vertical inextensible wires attached to the sign at Q and R and the framework at B and C , as shown in Fig. 3.2. QR and BC are horizontal. The framework is made from light rigid rods \(\mathrm { AB } , \mathrm { BC } , \mathrm { CA }\) and CD freely pin-jointed together at \(\mathrm { A } , \mathrm { B }\) and C and to a vertical wall at A and D . Fig. 3.3 shows the dimensions of the framework in metres as well as the external forces \(X _ { \mathrm { A } } \mathrm { N } , Y _ { \mathrm { A } } \mathrm { N }\) acting at A and \(X _ { \mathrm { D } } \mathrm { N } , Y _ { \mathrm { D } } \mathrm { N }\) acting at D . You are given that \(\sin \alpha = \frac { 5 } { 13 } , \cos \alpha = \frac { 12 } { 13 } , \sin \beta = \frac { 4 } { 5 }\) and \(\cos \beta = \frac { 3 } { 5 }\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{8fb49c8b-92e5-49e5-9a3a-e8391c82d9a1-4_543_526_1420_253} \captionsetup{labelformat=empty} \caption{Fig. 3.2}
    \end{figure} \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{8fb49c8b-92e5-49e5-9a3a-e8391c82d9a1-4_629_793_1343_964} \captionsetup{labelformat=empty} \caption{Fig. 3.3}
    \end{figure}
  3. Mark on the diagram in your Printed Answer Book all the forces acting on the pin-joints at \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) and D , including those internal to the rods, when the inn sign is hanging from the framework.
  4. Show that \(X _ { \mathrm { D } } = 261\).
  5. Calculate the forces internal to the rods \(\mathrm { AB } , \mathrm { BC }\) and CD , stating whether each rod is in tension or thrust (compression). Calculate also the values of \(Y _ { \mathrm { D } }\) and \(Y _ { \mathrm { A } }\). [Your working in this part should correspond to your diagram in part (iii).]
OCR M3 2009 January Q4
10 marks Standard +0.8
  1. Show that \(v ^ { 2 } = 9 + 9.8 \sin \theta\).
  2. Find, in terms of \(\theta\), the radial and tangential components of the acceleration of \(P\).
  3. Show that the tension in the string is \(( 3.6 + 5.88 \sin \theta ) \mathrm { N }\) and hence find the value of \(\theta\) at the instant when the string becomes slack, giving your answer correct to 1 decimal place.
OCR M3 2010 January Q6
13 marks Challenging +1.2
  1. By considering the total energy of the system, obtain an expression for \(v ^ { 2 }\) in terms of \(\theta\).
  2. Show that the magnitude of the force exerted on \(P\) by the cylinder is \(( 7.12 \sin \theta - 4.64 \theta ) \mathrm { N }\).
  3. Given that \(P\) leaves the surface of the cylinder when \(\theta = \alpha\), show that \(1.53 < \alpha < 1.54\).
OCR M3 2007 June Q6
13 marks Challenging +1.8
  1. Show that, when \(P\) is in equilibrium, \(O P = 7.25 \mathrm {~m}\).
  2. Verify that \(P\) and \(Q\) together just reach the safety net.
  3. At the lowest point of their motion \(P\) releases \(Q\). Prove that \(P\) subsequently just reaches \(O\).
  4. State two additional modelling assumptions made when answering this question.
OCR M4 2008 June Q6
15 marks Challenging +1.3
  1. Show that the moment of inertia of the lamina about the axis through \(X\) is \(\frac { 4 } { 3 } m a ^ { 2 }\).
  2. At an instant when \(\cos \theta = \frac { 3 } { 5 }\), show that \(\omega ^ { 2 } = \frac { 6 g } { 5 a }\).
  3. At an instant when \(\cos \theta = \frac { 3 } { 5 }\), show that \(R = 0\), and given also that \(\sin \theta = \frac { 4 } { 5 }\) find \(S\) in terms of \(m\) and \(g\).
OCR M4 2010 June Q6
12 marks Challenging +1.2
  1. Show that \(\theta = 0\) is a position of stable equilibrium.
  2. Show that the kinetic energy of the system is \(4 m a ^ { 2 } \dot { \theta } ^ { 2 }\).
  3. By differentiating the energy equation, then making suitable approximations for \(\sin \theta\) and \(\cos \theta\), find the approximate period of small oscillations about the equilibrium position \(\theta = 0\). \section*{[Question 7 is printed overleaf.]}
OCR M4 2013 June Q5
14 marks Standard +0.8
  1. Find the magnitude and bearing of the velocity of \(U\) relative to \(P\).
  2. Find the shortest distance between \(P\) and \(U\) in the subsequent motion.
    (ii) Plane \(Q\) is flying with constant velocity \(160 \mathrm {~ms} ^ { - 1 }\) in the direction which brings it as close as possible to \(U\).
    1. Find the bearing of the direction in which \(Q\) is flying.
    2. Find the shortest distance between \(Q\) and \(U\) in the subsequent motion. \includegraphics[max width=\textwidth, alt={}, center]{6e3d5f5e-7ffa-4111-903d-468fb4d20192-3_771_769_262_646} A square frame \(A B C D\) consists of four uniform rods \(A B , B C , C D , D A\), rigidly joined at \(A , B , C , D\). Each rod has mass 0.6 kg and length 1.5 m . The frame rotates freely in a vertical plane about a fixed horizontal axis passing through the mid-point \(O\) of \(A D\). At time \(t\) seconds the angle between \(A D\) and the horizontal, measured anticlockwise, is \(\theta\) radians (see diagram).
      1. Show that the moment of inertia of the frame about the axis through \(O\) is \(3.15 \mathrm {~kg} \mathrm {~m} ^ { 2 }\).
      2. Show that \(\frac { \mathrm { d } ^ { 2 } \theta } { \mathrm {~d} t ^ { 2 } } = - 5.6 \sin \theta\).
      3. Deduce that the frame can make small oscillations which are approximately simple harmonic, and find the period of these oscillations. The frame is at rest with \(A D\) horizontal. A couple of constant moment 25 Nm about the axis is then applied to the frame.
      4. Find the angular speed of the frame when it has rotated through 1.2 radians.