Questions Pre-U 9795/2 (184 questions)

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Pre-U Pre-U 9795/2 2013 November Q9
Challenging +1.2
9 A light string, of natural length 0.5 m and modulus of elasticity 4 N , has one end attached to the ceiling of a room. A particle of mass 0.2 kg is attached to the free end of the string and hangs in equilibrium.
  1. Find the extension of the string when the particle is in the equilibrium position. The particle is pulled down a further 0.5 m from the equilibrium position and released from rest. At time \(t\) seconds the displacement of the particle from the equilibrium position is \(x \mathrm {~m}\).
  2. Show that, while the string is taut, the equation of motion is \(\ddot { x } = - 40 x\).
  3. Find the time taken for the string to become slack for the first time.
  4. Show that the particle comes to instantaneous rest 0.125 m below the ceiling.
Pre-U Pre-U 9795/2 2013 November Q10
Standard +0.8
10 One end of a light inextensible string of length \(a\) is attached to a fixed point \(O\). A particle of mass \(m\) is attached to the free end of the string and the particle hangs at rest vertically below \(O\). The particle is projected horizontally with speed \(u\).
  1. Find the tension in the string when it makes an angle \(\theta\) with the downward vertical, whilst the string remains taut.
  2. Deduce that the particle will perform complete circles provided that \(u ^ { 2 } \geqslant 5 a g\).
  3. It is given that \(u ^ { 2 } = 4 a g\). Find
    1. the tension in the string when \(\theta = 60 ^ { \circ }\),
    2. the value of \(\theta\), to the nearest degree, at the instant when the string becomes slack.
Pre-U Pre-U 9795/2 2013 November Q11
Challenging +1.2
11 A smooth sphere of mass 2 kg has velocity \(( 24 \mathbf { i } - 7 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) and is travelling on a horizontal plane, where \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors in the horizontal plane. The sphere strikes a vertical wall. The line of intersection of the wall and the plane is in the direction \(( 4 \mathbf { i } + 3 \mathbf { j } )\).
  1. Show that the acute angle between the path of the sphere before the impact and the direction of the wall is \(\tan ^ { - 1 } \left( \frac { 4 } { 3 } \right)\).
  2. After the impact, the velocity of the sphere is \(( 7.2 \mathbf { i } + 15.4 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). Find
    1. the coefficient of restitution between the sphere and the wall,
    2. the magnitude of the impulse exerted by the sphere on the wall.
Pre-U Pre-U 9795/2 2013 November Q12
Challenging +1.2
12 A bullet of mass 0.0025 kg is fired vertically upwards from a point \(O\). At time \(t \mathrm {~s}\) after projection the speed of the bullet is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the resistance to motion has magnitude \(0.00001 v ^ { 2 } \mathrm {~N}\).
  1. Show that, while the bullet is rising, $$250 \frac { \mathrm {~d} v } { \mathrm {~d} t } = - 2500 - v ^ { 2 }$$
  2. It is given that the speed of projection is \(350 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find
    1. the time taken after projection for the bullet to reach its greatest height above \(O\),
    2. the greatest height above \(O\) reached by the bullet.
Pre-U Pre-U 9795/2 2015 June Q1
4 marks Standard +0.8
1 The independent random variables \(X\) and \(Y\) are such that $$X \sim \mathrm {~N} ( \mu , 11 ) , \quad Y \sim \mathrm {~N} \left( 10 , \sigma ^ { 2 } \right) \quad \text { and } \quad 2 X - 5 Y \sim \mathrm {~N} ( 0,144 ) .$$ Find
  1. the values of \(\mu\) and \(\sigma ^ { 2 }\),
  2. \(\mathrm { P } ( X - Y > 10 )\).
Pre-U Pre-U 9795/2 2015 June Q2
8 marks Standard +0.3
2 The pH value, \(X\), which is a measure of acidity, was measured for soil taken from a random sample of 20 villages in which rhododendrons grow well. The results are summarised below, where \(\bar { x }\) denotes the sample mean. You may assume that the sample is selected from a normal population. $$\Sigma x = 114 \quad \Sigma ( x - \bar { x } ) ^ { 2 } = 2.382$$
  1. Calculate a \(98 \%\) confidence interval for the mean pH value in villages where rhododendrons grow well, giving 3 decimal places in your answer.
  2. Comment, justifying your answer, on a suggestion that the average pH value in villages where rhododendrons grow well is 5.5.
Pre-U Pre-U 9795/2 2015 June Q3
14 marks Challenging +1.2
3 The probability generating function of the random variable \(X\) is \(\frac { 1 } { 81 } \left( t + \frac { 2 } { t } \right) ^ { 4 }\).
  1. Use the probability generating function to find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
  2. The random variable \(Y\) is defined by \(Y = \frac { 1 } { 2 } ( X + 4 )\). By finding the probability distribution of \(X\), or otherwise, show that \(Y \sim \mathrm {~B} ( n , p )\), stating the values of \(n\) and \(p\).
Pre-U Pre-U 9795/2 2015 June Q4
11 marks Challenging +1.2
4
  1. (a) Derive the moment generating function for a Poisson distribution with mean \(\lambda\).
    (b) The independent random variables \(X\) and \(Y\) are such that \(X \sim \operatorname { Po } ( \mu )\) and \(Y \sim \operatorname { Po } ( v )\). Use moment generating functions to show that \(( X + Y ) \sim \operatorname { Po } ( \mu + v )\).
  2. The number of goals scored per match by Camford Academicals FC may be modelled by a Poisson distribution with mean 2. The number of goals scored against Camford during a match may be modelled by an independent Poisson distribution with mean \(k\). The probability that no goals are scored, by either side, in a match involving Camford is 0.045 . Find
    (a) the value of \(k\),
    (b) the probability that exactly 3 goals are scored against Camford in a match,
    (c) the probability that the total number of goals scored, in a match involving Camford, is between 2 and 5 inclusive.
Pre-U Pre-U 9795/2 2015 June Q5
11 marks Challenging +1.8
5 Each year a college has a large fixed number, \(n\), of places to fill. The probability, \(p\), that a randomly chosen student comes from abroad is constant. Using a suitable normal approximation and applying a continuity correction, it is calculated that the probability of more than 60 students coming from abroad is 0.0187 and the probability of fewer than 40 students coming from abroad is 0.0783 . Find the values of \(n\) and \(p\).
Pre-U Pre-U 9795/2 2015 June Q6
18 marks Challenging +1.8
6 The object distance, \(U \mathrm {~cm}\), and the image distance, \(V \mathrm {~cm}\), for a convex lens of focal length 40 cm are related by the lens law $$\frac { 1 } { U } + \frac { 1 } { V } = \frac { 1 } { 40 } .$$ The random variable \(U\) is uniformly distributed over the interval \(80 \leqslant u \leqslant 120\).
  1. Show that the probability density function of \(V\) is given by $$f ( v ) = \begin{cases} \frac { 40 } { ( v - 40 ) ^ { 2 } } & 60 \leqslant v \leqslant 80 \\ 0 & \text { otherwise } \end{cases}$$
  2. Find
    1. the median value of \(V\),
    2. the expected value of \(V\).
Pre-U Pre-U 9795/2 2015 June Q7
6 marks Challenging +1.2
7 \includegraphics[max width=\textwidth, alt={}, center]{86cc07e7-ea69-4480-96c8-82b818445199-3_599_499_1279_822} A light inextensible string of length \(4 a\) has one end fixed at a point \(P\) and the other end fixed at a point \(Q\), which is vertically below \(P\) and at a distance \(3 a\) from \(P\). A small smooth ring \(R\) of mass \(m\) is threaded on the string. \(R\) moves in a horizontal circle with centre \(Q\) and with the string taut (see diagram).
  1. Show that \(Q R = \frac { 7 } { 8 } a\).
  2. Find the speed of \(R\) in terms of \(a\) and \(g\).
Pre-U Pre-U 9795/2 2015 June Q8
4 marks Standard +0.3
8 \includegraphics[max width=\textwidth, alt={}, center]{86cc07e7-ea69-4480-96c8-82b818445199-4_182_803_264_671} A light spring of modulus of elasticity 8 N and natural length 0.4 m has one end fixed to a smooth horizontal table at a fixed point \(L\). A particle of mass 0.2 kg is attached to the other end of the spring and pulled out horizontally to a point \(M\) on the table, so that the spring is extended by 0.2 m . The particle is then released from rest. The mid-point of \(L M\) is \(N\) and the point \(O\) is on \(L M\) such that \(L O = 0.4 \mathrm {~m}\) (see diagram).
  1. Show that the particle moves in simple harmonic motion with centre \(O\) and state the exact period of its motion.
  2. Find the exact time taken for the particle to move directly from \(M\) to \(N\).
Pre-U Pre-U 9795/2 2015 June Q9
6 marks Standard +0.3
9 A car of mass 800 kg is descending a straight hill which is inclined at \(2 ^ { \circ }\) to the horizontal. The car passes through the points \(A\) and \(B\) with speeds \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. The distance \(A B\) is 400 m .
  1. Assuming that resistances to motion are negligible, calculate the work done by the car's engine over the distance from \(A\) to \(B\).
  2. Assuming also that the driving force produced by the car's engine remains constant, calculate the power of the car's engine at the mid-point of \(A B\).
Pre-U Pre-U 9795/2 2015 June Q10
5 marks Challenging +1.2
10 A small body of mass \(m\) is thrown vertically upwards with initial velocity \(u\). Resistance to motion is \(k v ^ { 2 }\) per unit mass, where the velocity is \(v\) and \(k\) is a positive constant. Find, in terms of \(u , g\) and \(k\),
  1. the time taken to reach the greatest height,
  2. the greatest height to which the body will rise.
Pre-U Pre-U 9795/2 2015 June Q11
11 marks Challenging +1.2
11 In a training exercise, a submarine is travelling due north at \(15 \mathrm {~km} \mathrm {~h} ^ { - 1 }\). The submarine commander sees his target 5 km away on a bearing of \(310 ^ { \circ }\). The target is travelling due east at \(20 \mathrm {~km} \mathrm {~h} ^ { - 1 }\).
  1. If each of the submarine and target maintains its present course and speed, find the shortest distance between them.
  2. In fact, as soon as he sees the target, the submarine commander changes course, without changing speed, so as to intercept the target as quickly as possible. Find
    1. the course, in degrees, set by the submarine commander,
    2. the time taken, in minutes, to intercept the target from the moment that the course changes.
Pre-U Pre-U 9795/2 2015 June Q12
14 marks Challenging +1.8
12 Points \(A\) and \(B\) lie on a line of greatest slope of a plane inclined at an angle \(\alpha\) to the horizontal, with \(B\) above \(A\). A particle is projected from \(A\) with speed \(u\) at an angle \(\theta\) to the plane and subsequently strikes the plane at right angles at \(B\).
  1. Show that \(2 \tan \alpha \tan \theta = 1\).
  2. In either order, show that
    1. the vertical height of \(B\) above \(A\) is \(\frac { 2 u ^ { 2 } \tan ^ { 2 } \alpha } { g \left( 1 + 4 \tan ^ { 2 } \alpha \right) }\),
    2. the time of flight from \(A\) to \(B\) is \(\frac { 2 u \sec \alpha } { g \sqrt { 1 + 4 \tan ^ { 2 } \alpha } }\).
Pre-U Pre-U 9795/2 2016 June Q1
5 marks Moderate -0.5
1 An investigation was carried out of the lengths of commuters' journeys. For a random sample of 500 commuters, the mean journey time was 75 minutes, and the standard deviation was 40 minutes.
  1. Calculate a 95\% confidence interval for the mean journey time.
  2. Explain whether you need to assume that journey times are normally distributed.
Pre-U Pre-U 9795/2 2016 June Q2
4 marks Standard +0.8
2 The mass in grams of a pre-cut piece of Brie cheese is a random variable with the distribution \(\mathrm { N } ( 150,1200 )\). Brie costs 80 p per 100 g .
  1. Find the probability that a randomly chosen piece of Brie costs more than \(\pounds 1.40\). The mass in grams of a pre-cut piece of Stilton cheese is an independent random variable with the distribution \(\mathrm { N } ( 180,1500 )\).
  2. Find the probability that the total mass of four randomly chosen pieces of Brie is less than the total mass of three randomly chosen pieces of Stilton.
Pre-U Pre-U 9795/2 2016 June Q3
5 marks Standard +0.8
3
  1. Show that the probability generating function of a random variable with the distribution \(\mathrm { B } ( n , p )\) is \(( 1 - p + p t ) ^ { n }\).
  2. \(R\) and \(S\) are independent random variables with the distributions \(\mathrm { B } \left( 8 , \frac { 1 } { 4 } \right)\) and \(\mathrm { B } \left( 8 , \frac { 3 } { 4 } \right)\) respectively. Show that the probability generating function of \(R + S\) can be expressed as $$\left( \frac { 3 } { 16 } + \frac { 1 } { 16 } t ( 10 + 3 t ) \right) ^ { 8 }$$ and use this result to find \(\mathrm { P } ( R + S = 1 )\).
Pre-U Pre-U 9795/2 2016 June Q4
7 marks Standard +0.3
4 In a Football League match, the number of goals scored by the home team can be modelled by the distribution \(\mathrm { Po } ( 2.4 )\). The number of goals scored by the away team can be modelled by the distribution Po(1.8).
  1. State a necessary assumption for the total number of goals scored in one match to be modelled by the distribution \(\operatorname { Po } ( 4.2 )\).
  2. Assume now that this assumption holds.
    1. Write down an expression for the probability that the total number of goals scored in \(n\) randomly chosen games is less than 4 .
    2. Find the probability that the result of a randomly chosen game is either 0-0 or 1-1.
Pre-U Pre-U 9795/2 2016 June Q5
10 marks Challenging +1.2
5 The random variable \(R\) has the distribution \(\mathrm { B } ( n , p )\).
  1. State two conditions that \(n\) and \(p\) must satisfy if the distribution of \(R\) can be well approximated by a normal distribution. Assume now that these conditions hold. Using the normal approximation, it is given that \(\mathrm { P } ( R < 25 ) = 0.8282\) and \(\mathrm { P } ( R \geqslant 28 ) = 0.0393\), correct to 4 decimal places.
  2. Find the mean and standard deviation of the approximating normal distribution.
  3. Hence find the value of \(p\) and the value of \(n\).
Pre-U Pre-U 9795/2 2016 June Q6
6 marks Standard +0.3
6 A continuous random variable \(X\) has probability density function $$\mathrm { f } ( x ) = \begin{cases} 4 x \mathrm { e } ^ { - 2 x } & x \geqslant 0 \\ 0 & \text { otherwise } . \end{cases}$$
  1. Show that the moment generating function \(\mathrm { M } _ { X } ( t )\) of \(X\) is \(\frac { 4 } { ( 2 - t ) ^ { 2 } }\). You may assume that \(x \mathrm { e } ^ { - k x } \rightarrow 0\) as \(x \rightarrow + \infty\).
  2. What condition on \(t\) is needed in finding \(\mathrm { M } _ { X } ( t )\) ?
  3. \(Y\) is the sum of three independent observations of \(X\). Find the moment generating function of \(Y\), and use your answer to find \(\operatorname { Var } ( Y )\).
Pre-U Pre-U 9795/2 2016 June Q7
10 marks Challenging +1.8
7 A continuous random variable \(X\) has probability density function $$f ( x ) = \begin{cases} \frac { 3 x ^ { 2 } } { k ^ { 3 } } & 0 \leqslant x \leqslant k \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a parameter.
  1. Find \(\mathrm { E } ( X )\). Hence show that \(\frac { 4 } { 3 } X\) is an unbiased estimator of \(k\). Three independent observations of \(X\) are denoted by \(X _ { 1 } , X _ { 2 }\) and \(X _ { 3 }\), and the largest value of \(X _ { 1 } , X _ { 2 }\) and \(X _ { 3 }\) is denoted by \(M\).
  2. Write down an expression for \(\mathrm { P } ( M \leqslant x )\), and hence show that the probability density function of \(M\) is $$f _ { M } ( x ) = \begin{cases} \frac { 9 x ^ { 8 } } { k ^ { 9 } } & 0 \leqslant x \leqslant k \\ 0 & \text { otherwise } . \end{cases}$$
  3. Find \(\mathrm { E } ( M )\) and use your answer to construct an unbiased estimator of \(k\) based on \(M\).
Pre-U Pre-U 9795/2 2016 June Q8
5 marks Moderate -0.3
8 A rough plane is inclined at \(20 ^ { \circ }\) to the horizontal. A particle of mass 0.4 kg is projected down the plane, along a line of greatest slope, at \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). After it has travelled 3 m down the plane its speed is \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). By considering the change in energy, find the magnitude of the frictional force, assumed constant.
Pre-U Pre-U 9795/2 2016 June Q9
4 marks Standard +0.3
9 \includegraphics[max width=\textwidth, alt={}, center]{1a89caec-6da8-4b83-9ffa-efc209ecbc8d-4_506_730_625_712} Particles \(P\) and \(Q\), of masses 1.2 kg and 1.5 kg respectively, are attached to the ends of a light inextensible string. The string passes through a small smooth ring which is attached to the ceiling but which is free to rotate. \(P\) rotates at \(\omega \mathrm { rad } \mathrm { s } ^ { - 1 }\) in a horizontal circle of radius 0.12 m , and \(Q\) hangs vertically in equilibrium (see diagram). Determine
  1. the vertical distance below the ring at which \(P\) rotates,
  2. the value of \(\omega\).