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

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Pre-U Pre-U 9795/2 2012 June Q8
8 marks Challenging +1.2
8 \includegraphics[max width=\textwidth, alt={}, center]{d8ca5464-435f-45e0-8e19-1830415a7c60-4_757_729_260_708} An aircraft carrier, \(A\), is heading due north at \(40 \mathrm {~km} \mathrm {~h} ^ { - 1 }\). A destroyer, \(D\), which is 8 km south-west of \(A\), is ordered to take up a position 3 km east of \(A\) as quickly as possible. The speed of \(D\) is \(60 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) (see diagram). Find the bearing, \(\theta\), of the course that \(D\) should take, giving your answer to the nearest degree.
Pre-U Pre-U 9795/2 2012 June Q9
9 marks Standard +0.3
9 \includegraphics[max width=\textwidth, alt={}, center]{d8ca5464-435f-45e0-8e19-1830415a7c60-4_666_816_1384_662} A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(l\). The other end of the string is attached to a fixed point \(A\). The particle moves with constant angular speed \(\omega\) in a horizontal circle whose centre is at a distance \(h\) vertically below \(A\) (see diagram).
  1. Show that however fast the particle travels \(A P\) will never become horizontal, and that the tension in the string is always greater than the weight of the particle.
  2. Find the tension in the string in terms of \(m , l\) and \(\omega\).
  3. Show that \(\omega ^ { 2 } h = g\) and calculate \(\omega\) when \(h\) is 0.5 m .
Pre-U Pre-U 9795/2 2012 June Q10
12 marks Challenging +1.8
10 \includegraphics[max width=\textwidth, alt={}, center]{d8ca5464-435f-45e0-8e19-1830415a7c60-5_432_949_258_598} A smooth sphere \(P\) of mass \(3 m\) is at rest on a smooth horizontal table. A second smooth sphere \(Q\) of mass \(m\) and the same radius as \(P\) is moving along the table towards \(P\) and strikes it obliquely (see diagram). After the collision, the directions of motion of the two spheres are perpendicular.
  1. Find the coefficient of restitution.
  2. Given that one sixth of the original kinetic energy is lost as a result of the collision, find the angle between the initial direction of motion of \(Q\) and the line of centres.
Pre-U Pre-U 9795/2 2012 June Q11
11 marks Challenging +1.2
11 Two light strings, each of natural length \(a\) and modulus of elasticity \(6 m g\), are attached at their ends to a particle \(P\) of mass \(m\). The other ends of the strings are attached to two fixed points \(A\) and \(B\), which are at a distance \(6 a\) apart on a smooth horizontal table. Initially \(P\) is at rest at the mid-point of \(A B\). The particle is now given a horizontal impulse in the direction perpendicular to \(A B\). At time \(t\) the displacement of \(P\) from the line \(A B\) is \(x\).
  1. Show that $$\ddot { x } = - \frac { 12 g x } { a } \left( 1 - \frac { a } { \sqrt { 9 a ^ { 2 } + x ^ { 2 } } } \right) .$$
  2. Given that \(\frac { x } { a }\) is small throughout the motion, show that the equation of motion is approximately $$\ddot { x } = - \frac { 8 g x } { a }$$ and state the period of the simple harmonic motion that this equation represents.
  3. Given that the initial speed of \(P\) is \(\sqrt { \frac { g a } { 200 } }\), show that the amplitude of the simple harmonic motion is \(\frac { 1 } { 40 } a\).
Pre-U Pre-U 9795/2 2012 June Q12
12 marks Challenging +1.8
12 A projectile is launched from the origin with speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\alpha\) above the horizontal.
  1. Prove that the equation of its trajectory is $$y = x \tan \alpha - \frac { x ^ { 2 } } { 80 } \left( 1 + \tan ^ { 2 } \alpha \right)$$
  2. Regarding the equation of the trajectory as a quadratic equation in \(\tan \alpha\), show that \(\tan \alpha\) has real values provided that $$y \leqslant 20 - \frac { x ^ { 2 } } { 80 }$$
  3. A plane is inclined at an angle \(\beta\) to the horizontal. The line \(l\), with equation \(y = x \tan \beta\), is a line of greatest slope in the plane. A particle is projected from a point on the plane, in the vertical plane containing \(l\). By considering the intersection of \(l\) with the bounding parabola \(y = 20 - \frac { x ^ { 2 } } { 80 }\), deduce that the maximum range up, or down, this inclined plane is \(\frac { 40 } { 1 + \sin \beta }\), or \(\frac { 40 } { 1 - \sin \beta }\), respectively.
Pre-U Pre-U 9795/2 2013 June Q1
3 marks Moderate -0.3
1 A company hires out narrowboats on a canal. It may be assumed that demands to hire a narrowboat occur independently and randomly at a constant mean rate of 25 per week. Using a suitable normal approximation, find
  1. the probability that 15 or fewer narrowboats are hired out during a certain week,
  2. the number of narrowboats that the company needs to have available for a week in order that the probability of running out of boats is 0.05 or less.
Pre-U Pre-U 9795/2 2013 June Q2
9 marks Moderate -0.3
2
  1. The heights of boys in Year 9 are normally distributed with mean 156 cm and standard deviation 8 cm . The heights of girls in Year 10 are, independently, normally distributed with mean 160 cm and standard deviation 7 cm . Find the probability that the mean height of a random sample of 9 boys in Year 9 exceeds the mean height of a random sample of 16 girls in Year 10.
  2. State why the distributions of the sample means are normally distributed.
Pre-U Pre-U 9795/2 2013 June Q3
9 marks Standard +0.3
3
  1. Given that \(X \sim \operatorname { Po } ( 5 )\), find \(\mathrm { P } ( X > 6 \mid X > 3 )\).
  2. Given that \(Y \sim \operatorname { Po } ( \lambda )\) and \(\mathrm { P } ( Y \leqslant 1 ) = \frac { 1 } { 2 }\), show that \(\lambda\) satisfies the equation \(\lambda = \ln \{ 2 ( 1 + \lambda ) \}\).
  3. Starting with a suitable approximation from the table of cumulative Poisson probabilities, use iteration to find \(\lambda\) correct to 3 decimal places.
Pre-U Pre-U 9795/2 2013 June Q4
10 marks Standard +0.8
4 The broadband speed in village \(P\) was measured on 8 randomly selected occasions and the broadband speed in village \(Q\) was measured on 6 randomly selected occasions. The results, measured in megabits per second, are shown below.
Village \(P :\)4.83.52.93.74.24.65.13.3
Village \(Q :\)2.41.92.33.12.72.9
  1. Calculate a \(90 \%\) confidence interval for the difference in mean broadband speed in these two villages.
  2. State two assumptions that you have made in carrying out the calculation.
Pre-U Pre-U 9795/2 2013 June Q5
8 marks Standard +0.3
5 The discrete random variable \(X\) has probability generating function given by $$\mathrm { G } _ { X } ( t ) = k \left( 5 t ^ { - 1 } + 3 + 2 t ^ { 2 } \right) ,$$ where \(k\) is a constant.
  1. Find
    1. the value of \(k\),
    2. the modal value of \(X\).
    3. The random variables \(X _ { 1 }\) and \(X _ { 2 }\) are independent observations of \(X\).
      (a) Write down the probability generating function of \(Y\), where \(Y = X _ { 1 } + X _ { 2 }\).
      (b) Use your answer to part (ii)(a) to find \(\mathrm { E } ( Y )\) and \(\operatorname { Var } ( Y )\).
Pre-U Pre-U 9795/2 2013 June Q6
14 marks Challenging +1.2
6 A rectangle of area \(Y \mathrm {~m} ^ { 2 }\) has a perimeter of 16 m and a side of length \(X \mathrm {~m}\), where \(X\) is a random variable with probability density function, f, given by $$f ( x ) = \begin{cases} \frac { 1 } { 2 } & 0 \leqslant x \leqslant 2 \\ 0 & \text { otherwise } \end{cases}$$
  1. Obtain the cumulative distribution function, F , of \(X\).
  2. Show that $$16 - Y = ( 4 - X ) ^ { 2 }$$ and deduce that the probability density function of the random variable \(Y\) is $$g ( y ) = \begin{cases} \frac { 1 } { 4 \sqrt { 16 - y } } & 0 \leqslant y \leqslant 12 \\ 0 & \text { otherwise } \end{cases}$$
  3. Find the median of \(Y\).
  4. Find \(\mathrm { E } ( Y )\).
Pre-U Pre-U 9795/2 2013 June Q7
8 marks Standard +0.3
7 Find the power required to pump \(3 \mathrm {~m} ^ { 3 }\) of water per minute from a depth of 25 m and deliver it through a circular pipe of diameter 10 cm . Assume that friction may be neglected and that the density of water is \(1000 \mathrm {~kg} \mathrm {~m} ^ { - 3 }\).
Pre-U Pre-U 9795/2 2013 June Q8
9 marks Challenging +1.2
8 A particle, \(P\), is moving in a straight line with simple harmonic motion about a centre \(O\). When \(P\) is at the point \(A , 2 \mathrm {~m}\) from \(O\), it has speed \(4 \sqrt { 5 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\). When \(P\) is at the point \(B , \sqrt { 5 } \mathrm {~m}\) from \(O\), it has speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Find the amplitude and period of the motion.
  2. Given that \(A\) and \(B\) are on opposite sides of \(O\), find the time taken for \(P\) to travel directly from \(A\) to \(B\).
Pre-U Pre-U 9795/2 2013 June Q9
10 marks Challenging +1.2
9 A particle of mass 2 kg is moving along the \(x\)-axis, which is horizontal, against a resistive force which is proportional to the cube of the speed of the particle at any instant. At time \(t\) seconds the particle's velocity is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and its displacement is \(x \mathrm {~m}\). When \(t = 0 , x = 0 , v = 4\) and the retardation is \(2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  1. Show that $$\frac { 1 } { v } = \frac { x + 8 } { 32 } .$$
  2. Find the time taken to cover the first 8 metres.
Pre-U Pre-U 9795/2 2013 June Q10
6 marks Standard +0.8
10 Ship \(A\) is 15 km due south of ship \(B\). Ship \(B\) is travelling at \(20 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) on a bearing of \(300 ^ { \circ }\). Ship \(A\) is travelling at \(16 \mathrm {~km} \mathrm {~h} ^ { - 1 }\). Find
  1. the bearing, to the nearest degree, that \(A\) must take in order to get as close as possible to \(B\), [4]
  2. the time, in minutes, that it takes for the ships to be as close as possible.
Pre-U Pre-U 9795/2 2013 June Q11
11 marks Standard +0.3
11 \includegraphics[max width=\textwidth, alt={}, center]{742ef62b-bd72-45b4-88e3-70399632e9d6-4_384_524_587_808} One end of a light inextensible string of length 10 m is attached to a fixed point \(O\). A particle of mass 5 kg is attached to the other end of the string. The particle rests in equilibrium below \(O\). The particle is pulled aside until the string makes an angle of \(60 ^ { \circ }\) with the downward vertical and released from rest (see diagram). At the instant when the string makes an angle \(\cos ^ { - 1 } \left( \frac { 4 } { 5 } \right)\) with the downward vertical,
  1. find the speed of the particle and the tension in the string,
  2. show that the magnitude of the acceleration of the particle is \(6 \sqrt { 2 } \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
Pre-U Pre-U 9795/2 2013 June Q12
6 marks Challenging +1.2
12 \includegraphics[max width=\textwidth, alt={}, center]{742ef62b-bd72-45b4-88e3-70399632e9d6-4_247_801_1535_671} A small smooth sphere is projected from a point \(A\) across a smooth horizontal surface. The sphere strikes a smooth vertical wall at the point \(P\). The acute angle between the direction of motion of the sphere and the wall is \(\theta\). After the impact, the sphere passes through the point \(B\), where angle \(A P B = \phi\) (see diagram). The coefficient of restitution between the sphere and the wall is \(e\).
  1. Given that \(\theta = \tan ^ { - 1 } 3\) and \(\phi = 90 ^ { \circ }\), find the exact value of \(e\).
  2. Given instead that \(e = \frac { 2 } { 3 }\) and \(\phi = 45 ^ { \circ }\), show that \(\theta = \tan ^ { - 1 } 3\).
Pre-U Pre-U 9795/2 2013 November Q1
Moderate -0.3
1 The lifetime, \(T\) years, of a mortgage may be modelled by the random variable \(T\) with probability density function \(\mathrm { f } ( t )\), where $$\mathrm { f } ( t ) = \begin{cases} k \sin \left( \frac { 3 } { 32 } t \right) & 0 \leqslant t \leqslant 8 \pi \\ 0 & \text { otherwise } \end{cases}$$
  1. Show that \(k = \frac { 3 } { 32 } ( 2 - \sqrt { 2 } )\).
  2. Sketch the graph of \(\mathrm { f } ( t )\) and state the mode.
Pre-U Pre-U 9795/2 2013 November Q2
Standard +0.8
2
  1. The statistic \(T\) is derived from a random sample taken from a population which has an unknown parameter \(\theta\). \(T\) is an unbiased estimator of \(\theta\). What does the statement ' \(T\) is an unbiased estimator of \(\theta ^ { \prime }\) imply?
  2. A random sample of size \(n\) is taken from each of two independent populations. The first population has a non-zero mean \(\mu\) and variance \(\sigma ^ { 2 }\) and \(\bar { X } _ { 1 }\) denotes the sample mean. The second population has mean \(\frac { 1 } { 2 } \mu\) and variance \(b \sigma ^ { 2 }\), where \(b\) is a positive constant, and \(\bar { X } _ { 2 }\) denotes the sample mean. Two unbiased estimators for \(\mu\) are defined by $$T _ { 1 } = 3 \bar { X } _ { 1 } - a \bar { X } _ { 2 } \quad \text { and } \quad T _ { 2 } = \frac { 1 } { 5 } \left( 4 \bar { X } _ { 1 } + 2 \bar { X } _ { 2 } \right) .$$
    1. Determine the value of \(a\).
    2. Show that \(\operatorname { Var } \left( T _ { 1 } \right) = \frac { \sigma ^ { 2 } } { n } ( 9 + 16 b )\) and find a similar expression for \(\operatorname { Var } \left( T _ { 2 } \right)\).
    3. The estimator with the smaller variance is preferred. State which of \(T _ { 1 }\) and \(T _ { 2 }\) is the preferred estimator of \(\mu\).
Pre-U Pre-U 9795/2 2013 November Q3
Standard +0.3
3 The number of signal failures in a certain region of the railway network averages 10 every 3 weeks. Assume that signal failures occur independently, randomly and at constant mean rate.
  1. Find the probability that
    1. there are between 7 and 12 (inclusive) signal failures in a three-week period,
    2. there are more than 4 signal failures in a one-week period.
    3. It has been calculated, using a suitable distributional approximation, that the probability of more than 62 signal failures in a period of \(n\) weeks is 0.0385 . Find the value of \(n\).
Pre-U Pre-U 9795/2 2013 November Q4
Challenging +1.2
4 It is given that \(X\) and \(Y\) are independent random variables with distributions \(\mathrm { N } \left( \mu _ { x } , \sigma _ { x } ^ { 2 } \right)\) and \(\mathrm { N } \left( \mu _ { y } , \sigma _ { y } ^ { 2 } \right)\) respectively, and that \(W\) is a random variable such that \(W = X + Y\).
  1. Use moment generating functions to show that the distribution of \(W\) is \(\mathrm { N } \left( \mu _ { x } + \mu _ { y } , \sigma _ { x } ^ { 2 } + \sigma _ { y } ^ { 2 } \right)\).
  2. State the distribution of \(X - Y\). The diameters of the central poles of one brand of rotary clothes lines are normally distributed with mean 3.75 cm and variance \(0.000125 \mathrm {~cm} ^ { 2 }\). The diameters of the cylindrical tubes, into which the central poles fit, are normally distributed with mean 3.85 cm and variance \(0.0001 \mathrm {~cm} ^ { 2 }\). Poles and tubes are chosen at random. The 'clearance' between a tube and a pole is the diameter of the tube minus the diameter of the pole.
  3. Find the probability that a pole and tube have a clearance between 0.08 cm and 0.13 cm .
  4. Given that a pole and tube have a clearance between 0.08 cm and 0.13 cm , find the probability that the clearance is between 0.11 cm and 0.125 cm .
Pre-U Pre-U 9795/2 2013 November Q5
Standard +0.3
5 The random variable \(X\) has a binomial distribution with parameters \(n\) and \(p\), where \(p > 0.5\). A random sample of \(4 n\) observations of \(X\) is taken and \(\bar { X }\) denotes the sample mean. It is given that \(\mathrm { E } ( \bar { X } ) = 180\) and \(\operatorname { Var } ( \bar { X } ) = 0.0225\).
  1. Find
    1. the values of \(p\) and \(n\),
    2. \(\mathrm { P } ( \bar { X } < 179.8 )\),
    3. the value of \(a\) for which \(\mathrm { P } ( 180 - a < \bar { X } < 180 + a ) = 0.99\), giving your answer correct to 2 decimal places.
    4. State how you have used the Central Limit Theorem in part (i).
Pre-U Pre-U 9795/2 2013 November Q6
Challenging +1.2
6
  1. Verify that \(\left( 1 - t ^ { 6 } \right) = ( 1 - t ) \left( 1 + t + t ^ { 2 } + t ^ { 3 } + t ^ { 4 } + t ^ { 5 } \right)\).
  2. An unbiased six-faced die is rolled \(r\) times. Show that the probability generating function for the total score is $$\left[ \frac { t \left( 1 - t ^ { 6 } \right) } { 6 ( 1 - t ) } \right] ^ { r }$$
  3. Hence show that the probability of the total score being ( \(r + 3\) ) is $$\left( \frac { 1 } { 6 } \right) ^ { r + 1 } r ( r + 1 ) ( r + 2 )$$
Pre-U Pre-U 9795/2 2013 November Q7
Standard +0.8
7 At a given instant two stunt cars, \(X\) and \(Y\), are at distances 500 m and 800 m respectively from the point of intersection, \(O\), of two straight roads crossing at right angles. The stunt cars are approaching \(O\) at uniform speeds of \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively, one on each road. Find, in either order,
  1. the time taken to reach the point of closest approach,
  2. the shortest distance between the stunt cars.
Pre-U Pre-U 9795/2 2013 November Q8
Standard +0.3
8 A car of mass 1 tonne reaches the foot of an incline travelling at \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It reaches the top of the incline 50 seconds later travelling at \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The length of the incline is 1200 m and the angle made with the horizontal is \(\sin ^ { - 1 } \left( \frac { 1 } { 8 } \right)\). The constant resistance to motion is 400 N . Find the average power developed by the engine of the car.