Questions — Pre-U Pre-U 9795/2 (184 questions)

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Pre-U Pre-U 9795/2 2017 June Q2
13 marks Standard +0.8
2 A discrete random variable \(X\) has the following probability distribution.
\(x\)- 12
\(\mathrm { P } ( X = x )\)\(\frac { 1 } { 3 }\)\(\frac { 2 } { 3 }\)
  1. Write down the probability generating function of \(X\).
  2. \(T\) is the sum of ten independent observations of \(X\). Use the probability generating function of \(T\) to find
    1. \(\mathrm { E } ( T )\),
    2. \(\mathrm { P } ( T = 8 )\).
Pre-U Pre-U 9795/2 2017 June Q3
8 marks Standard +0.3
3 In a random sample of 100 voters from a constituency, 32 said that they would support the Cyan Party.
  1. Find an approximate \(99 \%\) confidence interval for the proportion of voters in the constituency who would support the Cyan Party.
  2. Using the given sample proportion, estimate the smallest size of sample needed for the width of a \(99 \%\) confidence interval to be less than 0.04 .
Pre-U Pre-U 9795/2 2017 June Q4
14 marks Standard +0.3
4 A continuous random variable \(X\) has probability density function $$\mathrm { f } ( x ) = \begin{cases} a & - 1 \leqslant x < 0 \\ a \left( 1 - x ^ { 2 } \right) & 0 \leqslant x \leqslant 1 \\ 0 & \text { otherwise } \end{cases}$$ where \(a\) is a constant.
  1. Find the value of \(a\).
  2. Find the cumulative distribution function of \(X\).
  3. Determine whether the upper quartile is greater than or less than 0.25 .
Pre-U Pre-U 9795/2 2017 June Q5
8 marks Standard +0.3
5 The number of calls to a car breakdown service during any one hour of the day is modelled by the distribution \(\operatorname { Po } ( 20 )\).
  1. Find the probability that in a randomly chosen 12 -minute period there are at least 7 calls to the service.
  2. Find the period of time, correct to the nearest second, for which the probability that no calls are made to the service is 0.6 .
  3. Use a suitable approximation to find the probability that, in a randomly chosen 3-hour period, there are no more than 65 calls to the service.
Pre-U Pre-U 9795/2 2017 June Q6
11 marks Standard +0.8
6 The random variable \(X\) has a uniform distribution on the interval \([ - 1,1 ]\), so that its probability density function is given by $$f ( x ) = \begin{cases} \frac { 1 } { 2 } & - 1 \leqslant x \leqslant 1 \\ 0 & \text { otherwise } \end{cases}$$
  1. Show from the definition of the moment generating function that the moment generating function of \(X\) is \(\frac { \sinh t } { t }\).
  2. By using the series expansion of \(\sinh t\), find the variance of \(X\) and the value of \(\mathrm { E } \left( X ^ { 4 } \right)\).
Pre-U Pre-U 9795/2 2017 June Q7
9 marks Challenging +1.2
7 The total mass of a can of pears is the sum of three independent random variables: the mass of pears, the mass of juice, and the mass of the container. The mass in grams of pears in a can has the distribution \(\mathrm { N } ( 300,400 )\). The mass in grams of juice has the distribution \(\mathrm { N } ( 200,60 )\). The mass in grams of the container has the distribution \(\mathrm { N } ( 70,10 )\).
  1. Find the probability that the total mass of a randomly chosen can is less than 530 g .
  2. Find the probability that the mass of the container of a randomly chosen can is more than one eighth of the total mass of the can.
Pre-U Pre-U 9795/2 2017 June Q8
5 marks Moderate -0.3
8 A horizontal turntable rotates about a vertical axis. Starting from rest, it accelerates uniformly to an angular velocity of \(8.4 \mathrm { rad } \mathrm { s } ^ { - 1 }\) in 2 s .
  1. Find the angular acceleration of the turntable.
  2. A particle rests on the turntable at a distance of 0.15 m from the axis. Find the radial and transverse components of the acceleration of the particle when the angular velocity is \(1.2 \mathrm { rad } \mathrm { s } ^ { - 1 }\). Find also the magnitude of the acceleration at this instant.
Pre-U Pre-U 9795/2 2017 June Q9
7 marks Standard +0.8
9 A particle is projected from a point \(O\) on horizontal ground with speed \(40 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\theta\) above the horizontal.
  1. Write down the equation of the trajectory, in terms of \(\tan \theta\).
  2. The particle passes through a point whose horizontal and vertical distances from \(O\) are 72 m and \(y \mathrm {~m}\) respectively. By considering the equation of the trajectory as a quadratic equation in \(\tan \theta\), or otherwise, find the greatest possible value of \(y\).
Pre-U Pre-U 9795/2 2017 June Q10
8 marks Moderate -0.3
10 The engine of a lorry of mass 4000 kg works at a constant rate of 75 kW . Resistance to motion is modelled by a constant resistive force. On a horizontal road the lorry travels at a constant speed of \(24 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Find the work done by the engine in travelling for 1 minute on the horizontal road.
  2. The lorry travels at a constant speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) up a slope of angle \(\sin ^ { - 1 } 0.05\) to the horizontal. Find the value of \(v\).
Pre-U Pre-U 9795/2 2017 June Q11
7 marks Standard +0.3
11 \includegraphics[max width=\textwidth, alt={}, center]{22640c3b-792f-4003-a4f8-78220efd73b0-4_280_1002_1722_568} A non-uniform \(\operatorname { rod } A B\) of mass 1.6 kg and length 1.25 m has its centre of mass at \(G\) where \(A G = 0.4 \mathrm {~m}\). The rod rests on a rough horizontal table. A force \(P \mathrm {~N}\) is applied at \(B\), acting at an angle \(\alpha\) above the horizontal, such that the rod is in equilibrium but about to rotate about \(A\) (see diagram).
  1. Assume that the rod is in contact with the table only at \(A\). By taking moments about \(A\), show that \(P \sin \alpha = 5.12\).
  2. The coefficient of friction between the rod and the table is \(\frac { 6 } { 17 }\). Show that \(P \leqslant 6.4\).
Pre-U Pre-U 9795/2 2017 June Q12
9 marks Challenging +1.2
12 A particle of mass 0.6 kg is projected vertically upwards from horizontal ground, with initial speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The upwards velocity at any instant is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and the displacement is \(x \mathrm {~m}\). Air resistance is modelled by a force \(0.024 v ^ { 2 } \mathrm {~N}\) acting downwards.
  1. Show that \(v\) and \(x\) satisfy the differential equation $$v \frac { \mathrm {~d} v } { \mathrm {~d} x } = - 10 - 0.04 v ^ { 2 } .$$
  2. Find the value of \(u\) if the maximum height reached is 50 m .
Pre-U Pre-U 9795/2 2017 June Q13
9 marks Challenging +1.2
13 A fairground game consists of a small ball fixed to one end of a light inextensible string of length 0.6 m . The other end of the string is fixed to a point \(O\). A small bell is fixed 0.6 m vertically above \(O\). Initially the ball hangs vertically in equilibrium. The object of the game is to project the ball with an initial horizontal velocity \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) so that it moves in a vertical circle and hits the bell.
  1. Find the smallest possible value of \(u\) for which the ball hits the bell.
  2. Given, instead, that the value of \(u\) is 5 , find the angle made by the string with the upward vertical at the moment when the string becomes slack.
Pre-U Pre-U 9795/2 2017 June Q14
9 marks Challenging +1.2
14 \includegraphics[max width=\textwidth, alt={}, center]{22640c3b-792f-4003-a4f8-78220efd73b0-5_86_1589_1297_278} A particle of mass 0.05 kg is attached to two identical light elastic strings, each of natural length 1.2 m and modulus of elasticity 0.6 N . The other ends of the strings are attached to points \(A\) and \(E\) on a smooth horizontal table. The distance \(A E\) is 2 m and points \(B , C\) and \(D\) lie between \(A\) and \(E\) so that \(A B = 0.7 \mathrm {~m} , B C = 0.1 \mathrm {~m} , C D = 0.4 \mathrm {~m}\) and \(D E = 0.8 \mathrm {~m}\) (see diagram). Initially the particle is held at \(B\) and it is then released. In the subsequent motion the displacement of the particle from \(C\), in the direction of \(A\), is denoted by \(x \mathrm {~m}\).
  1. Find the equation of motion for the particle when it is between \(B\) and \(C\).
  2. Find the velocity of the particle when it is at \(C\).
  3. Find the total time that elapses before the particle first returns to \(B\).
Pre-U Pre-U 9795/2 2018 June Q1
Moderate -0.3
1
  1. The random variable \(X\) has the distribution \(\mathrm { B } ( 200,0.2 )\). Use a suitable approximation to find \(\mathrm { P } ( X \leqslant 30 )\).
  2. The random variable \(Y\) has the distribution \(\mathrm { B } ( 200,0.02 )\). Use a suitable approximation to find \(\mathrm { P } ( Y \leqslant 3 )\).
Pre-U Pre-U 9795/2 2018 June Q2
Moderate -0.8
2 Secret radio messages received under difficult conditions are subject to errors caused by random instantaneous breaks in transmission. The number of errors caused by breaks in transmission in a 10-minute period is denoted by \(B\).
  1. State two conditions, other than randomness, needed for a Poisson distribution to be a suitable model for \(B\). Assume now that \(B \sim \mathrm { Po } ( 5 )\).
  2. Calculate the probability that in a 15-minute period there are between 6 and 10 errors, inclusive, caused by random breaks in transmission. Secret radio messages are also subject to errors caused by mistakes made by the sender. The number of errors caused by mistakes made by the sender in a 10 -minute period, \(M\), has the independent distribution \(\operatorname { Po } ( 8 )\).
  3. Calculate the period of time, in seconds, for which the probability that a message contains no errors of either sort is 0.6 .
Pre-U Pre-U 9795/2 2018 June Q3
Standard +0.3
3 The moment generating function of a random variable \(X\) is \(( 1 - 2 t ) ^ { - 3 }\).
  1. Find the mean and variance of \(X\).
  2. \(X _ { 1 }\) and \(X _ { 2 }\) are two independent observations of \(X\). Find \(\mathrm { E } \left[ \left( X _ { 1 } + X _ { 2 } \right) ^ { 3 } \right]\).
Pre-U Pre-U 9795/2 2018 June Q4
Challenging +1.2
4 The continuous random variable \(X\) has cumulative distribution function given by $$\mathrm { F } ( x ) = \begin{cases} 0 & x < 0 \\ \frac { 1 } { 8 } x ^ { 3 } & 0 \leqslant x \leqslant 2 \\ 1 & x > 2 \end{cases}$$
  1. Find \(\mathrm { E } ( X )\).
  2. Find the probability density function of \(Y\), where \(Y = \frac { 1 } { X ^ { 2 } }\).
Pre-U Pre-U 9795/2 2018 June Q5
Standard +0.3
5 A random sample of 12 seventeen-year-old boys and a random sample of 14 seventeen-year-old girls were given a certain task. The times, \(t\) minutes, taken to complete the task by the members of the two samples are summarised as follows.
\(n\)\(\Sigma t\)\(\Sigma t ^ { 2 }\)
Boys122044236
Girls143127126
  1. Stating any necessary assumption(s), find a \(95 \%\) symmetric confidence interval for the difference in the average times taken to complete the task by seventeen-year-old boys and seventeen-year-old girls.
  2. State with a reason whether the confidence interval calculated in part (i) suggests that there may in fact be no difference in the average times taken by seventeen-year-old boys and by seventeen-year-old girls.
Pre-U Pre-U 9795/2 2018 June Q6
Challenging +1.3
6 In a certain city there are \(N\) taxis. Each taxi displays a different licensing number which is an integer in the range 1 to \(N\). A visitor to the city attempts to estimate the value of \(N\), assuming that the licensing number of each taxi observed is equally likely to be any integer from 1 to \(N\) inclusive.
  1. The visitor observes one randomly chosen licensing number, \(X\). Using standard results for \(\sum _ { r = 1 } ^ { n } r\) and \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\), show that \(\mathrm { E } ( X ) = \frac { 1 } { 2 } ( N + 1 )\) and \(\operatorname { Var } ( X ) = \frac { 1 } { 12 } \left( N ^ { 2 } - 1 \right)\). The mean of 40 independent observations of \(X\) is denoted by \(A\).
  2. Find an unbiased estimator \(E _ { 1 }\) of \(N\) based on \(A\), and state the approximate distribution of \(E _ { 1 }\), giving the value(s) of any parameter(s). \(B\) is another random variable based on a random sample of 40 independent observations of \(X\). It is given that \(\mathrm { E } ( B ) = \frac { 40 } { 27 } N\) and that \(\operatorname { Var } ( B ) = \alpha N ^ { 2 }\) where \(\alpha\) is a constant.
  3. Find an unbiased estimator \(E _ { 2 }\) of \(N\) based on \(B\), and determine the set of values of \(\alpha\) for which \(\operatorname { Var } \left( E _ { 2 } \right) > \operatorname { Var } \left( E _ { 1 } \right)\) for all values of \(N\).
Pre-U Pre-U 9795/2 2018 June Q7
Moderate -0.3
7 A car has mass 800 kg .
  1. The car accelerates from \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in climbing a hill with a vertical height of 16 m . Ignoring resistive forces, find the work done by the engine.
  2. The engine produces a constant power output of 189 kW . The car now travels along horizontal ground. Modelling the resistive force as \(7 v ^ { 2 } \mathrm {~N}\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed, find the value of \(v\) for which the speed of the car is constant.
Pre-U Pre-U 9795/2 2018 June Q8
5 marks Standard +0.8
8 A light elastic string of natural length 0.2 m and modulus of elasticity 8 N has one end fixed to a point \(P\) on a horizontal ceiling. A particle of mass 0.4 kg is attached to the other end of the string.
  1. Find the extension of the string when the particle hangs in equilibrium vertically below \(P\).
  2. The particle is held at rest, with the string stretched, at a point \(x \mathrm {~m}\) vertically below \(P\) and is then released. Find the smallest value of \(x\) for which the particle will reach the ceiling.
Pre-U Pre-U 9795/2 2018 June Q9
9 marks Standard +0.3
9 \includegraphics[max width=\textwidth, alt={}, center]{09939c3a-7829-4784-8e6d-ee5356c22cd7-4_433_428_1219_863} A light inextensible string of length 1.4 m has its ends attached to two points \(A\) and \(C\), where \(A\) is 1 m vertically above \(C\). A smooth bead \(B\) of mass 0.2 kg is threaded on the string and rotates in a horizontal circle with the string taut. The distance \(B A\) is 0.8 m (see diagram). Find
  1. the tension in the string,
  2. the time taken for the bead to perform one complete circle.
Pre-U Pre-U 9795/2 2018 June Q10
6 marks Standard +0.8
10 A particle \(P\) is attached to one end of a light inextensible string of length 1.4 m . The other end of the string is fixed to the ceiling at \(C\). The angle between \(C P\) and the vertical is \(\theta\) radians. The particle is held with the string taut with \(\theta = 0.3\) and is then released.
  1. (a) Show that the motion of the system is approximately simple harmonic, and state its period.
    (b) Hence find an approximation for the speed of \(P\) when \(\theta = 0.2\).
  2. Find the speed of \(P\) when \(\theta = 0.2\) using an energy method, and hence find the percentage error in the answer to part (i) (b).
Pre-U Pre-U 9795/2 2018 June Q11
13 marks Challenging +1.8
11 A particle of mass 0.2 kg is projected so that it hits a smooth sloping plane \(\Pi\) that makes an angle of \(\sin ^ { - 1 } 0.6\) above the horizontal. The path of the particle is in a vertical plane containing a line of greatest slope of \(\Pi\). Immediately before the first impact between the particle and \(\Pi\), the particle is moving horizontally with speed \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The coefficient of restitution between the particle and \(\Pi\) is 0.5 .
  1. Find the magnitude of the impulse on the particle from \(\Pi\) at the first impact, and state the direction of this impulse.
  2. Find the distance between the points on \(\Pi\) where the first and second impacts occur.
  3. Find the time taken between the first and third impacts.
Pre-U Pre-U 9795/2 2018 June Q12
21 marks Standard +0.8
12 A uniform \(\operatorname { rod } A B\) has mass 5 kg and length 4 m .
  1. \includegraphics[max width=\textwidth, alt={}, center]{09939c3a-7829-4784-8e6d-ee5356c22cd7-5_529_540_995_840} The rod rests with \(A\) on a rough plane that makes an angle of \(60 ^ { \circ }\) to the horizontal. A string is attached to \(B\) and the rod is in equilibrium in the vertical plane containing the line of greatest slope of the plane, with the string vertical and \(A B\) perpendicular to the plane (see diagram). Find the magnitude of the frictional force at \(A\) and the tension in the string.
  2. \includegraphics[max width=\textwidth, alt={}, center]{09939c3a-7829-4784-8e6d-ee5356c22cd7-5_323_637_1850_794} The rod now rests horizontally with \(A\) in contact with a rough plane that makes an angle of \(60 ^ { \circ }\) with the horizontal and \(B\) in contact with a rough plane that makes an angle of \(30 ^ { \circ }\) with the horizontal (see diagram). The rod and the lines of greatest slope of the two planes are all in the same vertical plane. The coefficients of friction at \(A\) and \(B\) are \(\mu _ { A }\) and \(\mu _ { B }\) respectively. Friction is limiting at both \(A\) and \(B\), with \(A\) on the point of slipping downwards. Show that \(\mu _ { B } = \frac { 1 - \alpha \mu _ { A } } { \alpha + \mu _ { A } }\) where \(\alpha\) is an irrational number to be found.