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OCR S4 2018 June Q1
5 marks Moderate -0.5
1 A Wilcoxon signed-rank test is carried out at the \(5 \%\) level of significance on a random sample of size 32 . The hypotheses are \(\mathrm { H } _ { 0 } : m = m _ { 0 } , \mathrm { H } _ { 1 } : m < m _ { 0 }\) where \(m\) is the population median and \(m _ { 0 }\) is a specific numerical value. The value obtained for the test statistic \(T\) is 162 . Find the outcome of the test.
OCR S4 2018 June Q2
8 marks Standard +0.3
2 The distances from home to work, in km , of 8 men and 5 women were recorded and are given below. The workers were chosen at random.
Men47101316172021
Women12141822
Carry out a Wilcoxon rank-sum test at the \(5 \%\) significance level to test whether there is a significant difference between the distances from home to work between men and women.
OCR S4 2018 June Q3
10 marks Standard +0.8
3 Events \(A\) and \(B\) are such that \(\mathrm { P } ( A ) = 0.6 , \mathrm { P } ( B ) = 0.4\) and \(\mathrm { P } ( A \cup B ) = 0.8\).
  1. Find \(\mathrm { P } ( A \cap B )\).
  2. Find \(\mathrm { P } \left( A \cap B ^ { \prime } \right)\).
  3. Find \(\mathrm { P } ( A \mid B )\). Events \(A\) and \(B\) are as above and a third event \(C\) is such that \(\mathrm { P } ( A \cup B \cup C ) = 1 , \mathrm { P } ( A \cap B \cap C ) = 0.05\), \(\mathrm { P } ( A \cap C ) = \mathrm { P } ( B \cap C )\) and \(\mathrm { P } \left( A \cap B ^ { \prime } \cap C ^ { \prime } \right) = 3 \mathrm { P } \left( A ^ { \prime } \cap B \cap C ^ { \prime } \right)\).
  4. Find \(\mathrm { P } ( C )\).
OCR S4 2018 June Q4
10 marks Standard +0.8
4 The random variable \(X\) has a \(\chi ^ { 2 }\) distribution with \(v\) degrees of freedom. The moment generating function of \(X\) is $$\mathrm { M } _ { X } ( t ) = ( 1 - 2 t ) ^ { - \frac { 1 } { 2 } v }$$
  1. Show that \(\mathrm { E } ( X ) = v\).
  2. Find \(\operatorname { Var } ( X )\).
  3. Obtain the moment generating function of the sum \(Y\) of two independent \(\chi ^ { 2 }\) random variables, one with 6 degrees of freedom and the other with 8 degrees of freedom.
  4. Identify the distribution of \(Y\).
OCR S4 2018 June Q5
11 marks Challenging +1.2
5 The independent discrete random variables \(U\) and \(V\) can each take the values 1, 2 and 3, all with probability \(\frac { 1 } { 3 }\). The random variables \(X\) and \(Y\) are defined as follows: $$X = | U - V | , Y = U + V .$$
  1. In the Printed Answer Book complete the table showing the joint probability distribution of \(X\) and \(Y\).
  2. Find \(\operatorname { Cov } ( X , Y )\).
  3. State with a reason whether \(X\) and \(Y\) are independent.
  4. Find \(\mathrm { P } ( Y = 3 \mid X = 1 )\).
OCR S4 2018 June Q6
13 marks Standard +0.8
6 In each round of a quiz a contestant can answer up to three questions. Each correct answer scores 1 point and allows the contestant to go on to the next question. A wrong answer scores 0 points and the contestant is allowed no further question in that round. If all 3 questions are answered correctly 1 bonus point is scored, making a total score of 4 for the round. For a certain contestant, \(A\), the probability of giving a correct answer is \(\frac { 3 } { 4 }\), independently of any other question. The random variable \(X _ { r }\) is the number of points scored by \(A\) during the \(r ^ { \text {th } }\) round.
  1. Find the probability generating function of \(X _ { r }\).
  2. Use the probability generating function found in part (i) to find the mean and variance of \(X _ { r }\).
  3. Write down an expression for the probability generating function of \(X _ { 1 } + X _ { 2 }\) and find the probability that \(A\) has a total score of 4 at the end of two rounds.
OCR S4 2018 June Q7
15 marks Challenging +1.2
7 Two independent observations \(X _ { 1 }\) and \(X _ { 2 }\) are made of a continuous random variable with probability density function $$f ( x ) = \begin{cases} \frac { 1 } { \theta } & 0 \leqslant x \leqslant \theta \\ 0 & \text { otherwise } \end{cases}$$ where \(\theta\) is a parameter whose value is to be estimated.
  1. Find \(\mathrm { E } ( X )\).
  2. Show that \(S _ { 1 } = X _ { 1 } + X _ { 2 }\) is an unbiased estimator of \(\theta\). \(L\) is the larger of \(X _ { 1 }\) and \(X _ { 2 }\), or their common value if they are equal.
  3. Show that the probability density function of \(L\) is \(\frac { 2 l } { \theta ^ { 2 } }\) for \(0 \leqslant l \leqslant \theta\).
  4. Find \(\mathrm { E } ( L )\).
  5. Find an unbiased estimator \(S _ { 2 }\) of \(\theta\), based on \(L\).
  6. Determine which of the two estimators \(S _ { 1 }\) and \(S _ { 2 }\) is the more efficient.
OCR M1 2010 January Q1
6 marks Moderate -0.8
1 A particle \(P\) is projected vertically downwards from a fixed point \(O\) with initial speed \(4.2 \mathrm {~ms} ^ { - 1 }\), and takes 1.5 s to reach the ground. Calculate
  1. the speed of \(P\) when it reaches the ground,
  2. the height of \(O\) above the ground,
  3. the speed of \(P\) when it is 5 m above the ground.
OCR M1 2010 January Q2
8 marks Moderate -0.8
2 Two horizontal forces of magnitudes 12 N and 19 N act at a point. Given that the angle between the two forces is \(60 ^ { \circ }\), calculate
  1. the magnitude of the resultant force,
  2. the angle between the resultant and the 12 N force.
OCR M1 2010 January Q3
9 marks Standard +0.3
3 \includegraphics[max width=\textwidth, alt={}, center]{c9e725ad-561b-4e98-9b8f-7c9d3c8e67e6-2_153_1009_978_570} Three particles \(P , Q\) and \(R\), are travelling in the same direction in the same straight line on a smooth horizontal surface. \(P\) has mass \(m \mathrm {~kg}\) and speed \(9 \mathrm {~m} \mathrm {~s} ^ { - 1 } , Q\) has mass 0.8 kg and speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(R\) has mass 0.4 kg and speed \(2.75 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) (see diagram).
  1. A collision occurs between \(P\) and \(Q\), after which \(P\) and \(Q\) move in opposite directions, each with speed \(3.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Calculate
    (a) the value of \(m\),
    (b) the change in the momentum of \(P\).
  2. When \(Q\) collides with \(R\) the two particles coalesce. Find their subsequent common speed.
OCR M1 2010 January Q4
10 marks Standard +0.3
4 \includegraphics[max width=\textwidth, alt={}, center]{c9e725ad-561b-4e98-9b8f-7c9d3c8e67e6-3_494_255_258_945} Particles \(P\) and \(Q\), of masses 0.4 kg and 0.3 kg respectively, are attached to the ends of a light inextensible string. The string passes over a smooth fixed pulley and the sections of the string not in contact with the pulley are vertical. \(P\) rests in limiting equilibrium on a plane inclined at \(60 ^ { \circ }\) to the horizontal (see diagram).
  1. (a) Calculate the components, parallel and perpendicular to the plane, of the contact force exerted by the plane on \(P\).
    (b) Find the coefficient of friction between \(P\) and the plane. \(P\) is held stationary and a particle of mass 0.2 kg is attached to \(Q\). With the string taut, \(P\) is released from rest.
  2. Calculate the tension in the string and the acceleration of the particles. \includegraphics[max width=\textwidth, alt={}, center]{c9e725ad-561b-4e98-9b8f-7c9d3c8e67e6-3_579_1195_1553_475} The \(( t , v )\) diagram represents the motion of two cyclists \(A\) and \(B\) who are travelling along a horizontal straight road. At time \(t = 0 , A\), who cycles with constant speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), overtakes \(B\) who has initial speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). From time \(t = 0 B\) cycles with constant acceleration for 20 s . When \(t = 20\) her speed is \(11 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), which she subsequently maintains.
OCR M1 2010 January Q6
12 marks Standard +0.3
6 A swimmer \(C\) swims with velocity \(v \mathrm {~ms} ^ { - 1 }\) in a swimming pool. At time \(t \mathrm {~s}\) after starting, \(v = 0.006 t ^ { 2 } - 0.18 t + k\), where \(k\) is a constant. \(C\) swims from one end of the pool to the other in 28.4 s .
  1. Find the acceleration of \(C\) in terms of \(t\).
  2. Given that the minimum speed of \(C\) is \(0.65 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), show that \(k = 2\).
  3. Express the distance travelled by \(C\) in terms of \(t\), and calculate the length of the pool.
OCR M1 2010 January Q7
16 marks Standard +0.3
7 \includegraphics[max width=\textwidth, alt={}, center]{c9e725ad-561b-4e98-9b8f-7c9d3c8e67e6-4_129_798_756_676} A winch drags a \(\log\) of mass 600 kg up a slope inclined at \(10 ^ { \circ }\) to the horizontal by means of an inextensible cable of negligible mass parallel to the slope (see diagram). The coefficient of friction between the \(\log\) and the slope is 0.15 , and the \(\log\) is initially at rest at the foot of the slope. The acceleration of the \(\log\) is \(0.11 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  1. Calculate the tension in the cable. The cable suddenly breaks after dragging the log a distance of 10 m .
  2. (a) Show that the deceleration of the log while continuing to move up the slope is \(3.15 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), correct to 3 significant figures.
    (b) Calculate the time taken, after the cable breaks, for the log to return to its original position at the foot of the slope. {www.ocr.org.uk}) after the live examination series.
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OCR M1 2011 January Q1
6 marks Moderate -0.8
1 Two particles \(P\) and \(Q\) are projected directly towards each other on a smooth horizontal surface. \(P\) has mass 0.5 kg and initial speed \(2.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and \(Q\) has mass 0.8 kg and initial speed \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). After a collision between \(P\) and \(Q\), the speed of \(P\) is \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the direction of its motion is reversed. Calculate
  1. the change in the momentum of \(P\),
  2. the speed of \(Q\) after the collision.
OCR M1 2011 January Q2
6 marks Moderate -0.5
2 \includegraphics[max width=\textwidth, alt={}, center]{4c6c9323-8238-4ec2-94a1-6e8188a34521-02_597_885_676_630} Three horizontal forces of magnitudes \(F \mathrm {~N} , 8 \mathrm {~N}\) and 10 N act at a point and are in equilibrium. The \(F \mathrm {~N}\) and 8 N forces are perpendicular to each other, and the 10 N force acts at an obtuse angle \(( 90 + \alpha ) ^ { \circ }\) to the \(F \mathrm {~N}\) force (see diagram). Calculate
  1. \(\alpha\),
  2. \(F\).
OCR M1 2011 January Q3
10 marks Moderate -0.8
3 A particle is projected vertically upwards with velocity \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point 2.5 m above the ground.
  1. Calculate the speed of the particle when it strikes the ground.
  2. Calculate the time after projection when the particle reaches the ground.
  3. Sketch on separate diagrams
    (a) the \(( t , v )\) graph,
    (b) the \(( t , x )\) graph,
    representing the motion of the particle.
OCR M1 2011 January Q4
10 marks Standard +0.3
4 \includegraphics[max width=\textwidth, alt={}, center]{4c6c9323-8238-4ec2-94a1-6e8188a34521-03_156_1141_258_502} A block \(B\) of mass 0.8 kg and a particle \(P\) of mass 0.3 kg are connected by a light inextensible string inclined at \(10 ^ { \circ }\) to the horizontal. They are pulled across a horizontal surface with acceleration \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), by a horizontal force of 2 N applied to \(B\) (see diagram).
  1. Given that contact between \(B\) and the surface is smooth, calculate the tension in the string.
  2. Calculate the coefficient of friction between \(P\) and the surface.
OCR M1 2011 January Q5
11 marks Standard +0.8
5 \includegraphics[max width=\textwidth, alt={}, center]{4c6c9323-8238-4ec2-94a1-6e8188a34521-03_538_917_918_614} \(X\) is a point on a smooth plane inclined at \(\theta ^ { \circ }\) to the horizontal. \(Y\) is a point directly above the line of greatest slope passing through \(X\), and \(X Y\) is horizontal. A particle \(P\) is projected from \(X\) with initial speed \(4.9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) down the line of greatest slope, and simultaneously a particle \(Q\) is released from rest at \(Y\). \(P\) moves with acceleration \(4.9 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), and \(Q\) descends freely under gravity (see diagram). The two particles collide at the point on the plane directly below \(Y\) at time \(T\) s after being set in motion.
  1. (a) Express in terms of \(T\) the distances travelled by the particles before the collision.
    (b) Calculate \(\theta\).
    (c) Using the answers to parts (a) and (b), show that \(T = \frac { 2 } { 3 }\).
  2. Calculate the speeds of the particles immediately before they collide.
OCR M1 2011 January Q6
15 marks Moderate -0.3
6 The velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) of a particle at time \(t \mathrm {~s}\) is given by \(v = t ^ { 2 } - 9\). The particle travels in a straight line and passes through a fixed point \(O\) when \(t = 2\).
  1. Find the displacement of the particle from \(O\) when \(t = 0\).
  2. Calculate the distance the particle travels from its position at \(t = 0\) until it changes its direction of motion.
  3. Calculate the distance of the particle from \(O\) when the acceleration of the particle is \(10 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
OCR M1 2011 January Q7
14 marks Standard +0.3
7 A particle \(P\) of mass 0.6 kg is projected up a line of greatest slope of a plane inclined at \(30 ^ { \circ }\) to the horizontal. \(P\) moves with deceleration \(10 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) and comes to rest before reaching the top of the plane.
  1. Calculate the frictional force acting on \(P\), and the coefficient of friction between \(P\) and the plane.
  2. Find the magnitude of the contact force exerted on \(P\) by the plane and the angle between the contact force and the upward direction of the line of greatest slope,
    (a) when \(P\) is in motion,
    (b) when \(P\) is at rest.
OCR M1 2011 January Q8
Moderate -0.8
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    \section*{OCR} RECOGNISING ACHIEVEMENT
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    OCR M1 2012 January Q1
    6 marks Moderate -0.8
    1 Particles \(P\) and \(Q\), of masses 0.3 kg and 0.5 kg respectively, are moving in the same direction along the same straight line on a smooth horizontal surface. \(P\) is moving with speed \(2.2 \mathrm {~ms} ^ { - 1 }\) and \(Q\) is moving with speed \(0.8 \mathrm {~ms} ^ { - 1 }\) immediately before they collide. In the collision, the speed of \(P\) is reduced by \(50 \%\) and its direction of motion is unchanged.
    1. Calculate the speed of \(Q\) immediately after the collision.
    2. Find the distance \(P Q\) at the instant 3 seconds after the collision.
    OCR M1 2012 January Q2
    6 marks Moderate -0.3
    2 In the sport of curling, a heavy stone is projected across a horizontal ice surface. One player projects a stone of weight 180 N , which moves 36 m in a straight line and comes to rest 24 s after the instant of projection. The only horizontal force acting on the stone after its projection is a constant frictional force between the stone and the ice.
    1. Calculate the deceleration of the stone.
    2. Find the magnitude of the frictional force acting on the stone, and calculate the coefficient of friction between the stone and the ice.
    OCR M1 2012 January Q3
    9 marks Standard +0.3
    3 A car is travelling along a straight horizontal road with velocity \(32.5 \mathrm {~ms} ^ { - 1 }\). The driver applies the brakes and the car decelerates at \(( 8 - 0.6 t ) \mathrm { ms } ^ { - 2 }\), where \(t \mathrm {~s}\) is the time which has elapsed since the brakes were first applied.
    1. Show that, while the car is decelerating, its velocity is \(\left( 32.5 - 8 t + 0.3 t ^ { 2 } \right) \mathrm { m } \mathrm { s } ^ { - 1 }\).
    2. Find the time taken to bring the car to rest.
    3. Show that the distance travelled while the car is decelerating is 75 m .
    OCR M1 2012 January Q4
    9 marks Moderate -0.3
    4 \includegraphics[max width=\textwidth, alt={}, center]{2b3457b6-1fe9-4e67-91d4-a8bc4a5b1709-2_325_481_1699_792} Three horizontal forces of magnitudes \(8 \mathrm {~N} , 15 \mathrm {~N}\) and 20 N act at a point. The 8 N and 15 N forces are at right angles. The 20 N force makes an angle of \(150 ^ { \circ }\) with the 8 N force and an angle of \(120 ^ { \circ }\) with the 15 N force (see diagram).
    1. Calculate the components of the resultant force in the directions of the 8 N and 15 N forces.
    2. Calculate the magnitude of the resultant force, and the angle it makes with the direction of the 8 N force. The directions in which the three horizontal forces act can be altered.
    3. State the greatest and least possible magnitudes of the resultant force.