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

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Pre-U Pre-U 9795/2 2010 June Q1
7 marks Standard +0.8
1 A lorry moves along a straight horizontal road. The engine of the lorry produces a constant power of 80 kW . The mass of the lorry is 10 tonnes and the resistance to motion is constant at 4000 N .
  1. Express the driving force of the lorry in terms of its velocity and hence, using Newton's second law, write down a differential equation which connects the velocity of the lorry and the time for which it has been moving.
  2. Hence find the time taken, in seconds, for the lorry to accelerate from \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
Pre-U Pre-U 9795/2 2010 June Q2
9 marks Standard +0.8
2 At 1200 hours an aircraft, \(A\), sets out to intercept a second aircraft, \(B\), which is 200 km away on a bearing of \(300 ^ { \circ }\) and is flying due east at \(600 \mathrm {~km} \mathrm {~h} ^ { - 1 }\). Both aircraft are at the same altitude and continue to fly horizontally.
  1. (a) Find the bearing on which \(A\) should fly when travelling at \(800 \mathrm {~km} \mathrm {~h} ^ { - 1 }\).
    (b) Find the time at which \(A\) intercepts \(B\) in this case.
  2. Find the least steady speed at which \(A\) can fly to intercept \(B\).
Pre-U Pre-U 9795/2 2010 June Q3
11 marks Challenging +1.8
3 A particle is projected at an angle \(\theta\) above the horizontal from the foot of a plane which is inclined at \(45 ^ { \circ }\) to the horizontal. Subsequently the particle impacts on the plane at a higher point.
  1. Prove that the angle at which the particle strikes the plane is \(\phi\), where $$\tan \phi = \frac { \tan \theta - 1 } { 3 - \tan \theta }$$
  2. Find the angle to the horizontal at which the particle would have to be projected if it were to strike the plane horizontally.
Pre-U Pre-U 9795/2 2010 June Q4
11 marks Challenging +1.2
4 One end of a light elastic string of natural length 0.2 m and modulus of elasticity 100 N is attached to a fixed point \(A\). The other end is attached to a particle of mass 5 kg . The particle moves with angular speed \(\omega\) radians per second in a horizontal circle with the centre vertically below \(A\). The string makes an angle \(\theta\) with the vertical.
  1. By considering the horizontal component of the tension in the string, show that the tension in the string is \(( 1 + 5 x ) \omega ^ { 2 } \mathrm {~N}\), where \(x\) is the extension, in metres, of the string.
  2. (a) By considering vertical forces and also Hooke's law, deduce that \(\cos \theta = \frac { 1 } { 10 x }\).
    (b) Show that \(\omega > \frac { 10 \sqrt { 3 } } { 3 }\).
  3. When the value of \(\omega\) is \(5 \sqrt { 2 }\), find the radius of the circular motion.
Pre-U Pre-U 9795/2 2010 June Q5
11 marks Challenging +1.8
5 A particle of mass \(m\) is attached by a light elastic string of natural length \(l\) and modulus of elasticity \(\lambda\) to a fixed point \(A\), from which it is allowed to fall freely. The particle first comes to rest, instantaneously, at \(B\), where \(A B = 2 l\). Prove that
  1. \(\lambda = 4 m g\),
  2. while the string is taut, \(\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } = - \frac { 4 g } { l } x\), where \(x\) is the displacement from the equilibrium position at time \(t\),
  3. the time taken between the first occasion when the string becomes taut and the next occasion when it becomes slack is $$\left[ \frac { 1 } { 2 } \pi + \sin ^ { - 1 } \left( \frac { 1 } { 3 } \right) \right] \sqrt { \frac { l } { g } }$$
Pre-U Pre-U 9795/2 2010 June Q6
11 marks Challenging +1.8
6 Two smooth spheres, \(A\) and \(B\), have masses \(m\) and \(2 m\) respectively and equal radii. Sphere \(B\) is at rest on a smooth horizontal floor. Sphere \(A\) is projected with speed \(u\) along the floor in a direction parallel to a smooth vertical wall and strikes \(B\) obliquely. Subsequently \(B\) strikes the wall at an angle \(\alpha\) with the wall. The coefficient of restitution between \(A\) and \(B\) and between \(B\) and the wall is 0.5. After \(B\) has struck the wall, \(A\) and \(B\) are moving parallel to each other.
  1. Write down a momentum equation and a restitution equation along the line of centres for the impact between \(A\) and \(B\). Hence find the components of velocity of \(A\) and \(B\) in this direction after this first impact.
  2. Find the value of \(\alpha\), giving your answer in degrees.
Pre-U Pre-U 9795/2 2010 June Q7
8 marks Standard +0.3
7 The number of goals scored by a hockey team in an interval of time of length \(t\) minutes follows a Poisson distribution with mean \(\frac { 1 } { 24 } t\). The random variable \(T\) is defined as the length of time, in minutes, between successive goals.
  1. (a) Show that \(\mathrm { P } ( T < t ) = 1 - \mathrm { e } ^ { - \frac { 1 } { 24 } t }\) for \(t \geqslant 0\).
    (b) Hence find the probability density function of \(T\).
  2. Find the exact value of the interquartile range of \(T\).
Pre-U Pre-U 9795/2 2010 June Q8
8 marks Standard +0.3
8 Two groups of Year 12 pupils, one at each of schools \(A\) and \(B\), are given the same mathematics test. The scores, \(x\) and \(y\), of pupils at schools \(A\) and \(B\) respectively are summarised as follows.
School \(A\)\(n _ { A } = 15\)\(\bar { x } = 53\)\(\Sigma ( x - \bar { x } ) ^ { 2 } = 925\)
School \(B\)\(n _ { B } = 12\)\(\bar { y } = 47\)\(\Sigma ( y - \bar { y } ) ^ { 2 } = 850\)
  1. Assuming that the two groups are random samples from independent normal populations with means \(\mu _ { A }\) and \(\mu _ { B }\) respectively and a common, but unknown, variance, construct a \(98 \%\) confidence interval for \(\mu _ { A } - \mu _ { B }\).
  2. Comment, with a reason, on any difference in ability between the two schools.
Pre-U Pre-U 9795/2 2010 June Q9
10 marks Challenging +1.2
9
  1. Two independent discrete random variables \(X\) and \(Y\) follow Poisson distributions with means \(\lambda\) and \(\mu\) respectively. Prove that the discrete random variable \(Z = X + Y\) follows a Poisson distribution with mean \(\lambda + \mu\). A garage has a white limousine and a green limousine for hire. Demands to hire the white limousine occur at a constant mean rate of 3 per week and demands to hire the green limousine occur at a constant mean rate of 2 per week. Demands for hire are received independently and randomly.
  2. Calculate the probability that in a period of two weeks
    1. no demands for hire are received, giving your answer to 3 significant figures,
    2. seven demands for hire are received.
    3. Find the least value of \(n\) such that the probability of at least \(n\) demands for hire in a period of three weeks is less than 0.005 .
Pre-U Pre-U 9795/2 2010 June Q10
11 marks Challenging +1.2
10 A box contains a large number, \(n\), of identical dice, which are thought to be biased. The probability that one of these dice will show a six in a single roll is \(p\). The \(n\) dice are rolled many times and the number of sixes obtained in each trial is recorded. In \(4.01 \%\) of these trials 56 or more dice showed a six. In \(10.56 \%\) of these trials 37 or fewer dice showed a six. Using a suitable normal approximation, find the values of \(n\) and \(p\).
Pre-U Pre-U 9795/2 2010 June Q11
12 marks Standard +0.8
11 The thickness of a randomly chosen paperback book is \(P \mathrm {~cm}\) and the thickness of a randomly chosen hardback is \(H \mathrm {~cm}\), where \(P\) and \(H\) have distributions \(\mathrm { N } ( 2.0,0.75 )\) and \(\mathrm { N } ( 5.0,2.25 )\) respectively. When more than one book is selected, any book is selected independently of all other books.
  1. Calculate the probability that a randomly chosen hardback is more than 1 cm thicker than a randomly chosen paperback.
  2. Calculate the probability that 2 paperbacks and 4 hardbacks, randomly chosen, have a combined thickness of less than 20 cm .
  3. Find the probability that a randomly chosen hardback is more than twice the thickness of a randomly chosen paperback.
Pre-U Pre-U 9795/2 2010 June Q12
11 marks Challenging +1.2
12 Two players, \(A\) and \(B\), are taking turns to shoot at a basket with a basketball. The winner of this game is the first player to score a basket. The probability that \(A\) scores a basket with any shot is \(\frac { 1 } { 4 }\) and the probability that \(B\) scores a basket with any shot is \(\frac { 1 } { 5 }\). Each shot is independent of all other shots. \(A\) shoots first.
  1. Find
    1. the probability that \(B\) wins with his first shot,
    2. the probability that \(A\) wins with his second shot,
    3. the probability that \(A\) wins the game.
    4. \(R\) is the total number of shots taken by \(A\) and \(B\) up to and including the shot that scores a basket.
      (a) Show that the probability generating function of \(R\) is given by $$\mathrm { G } ( t ) = \frac { 5 t + 3 t ^ { 2 } } { 4 \left( 5 - 3 t ^ { 2 } \right) }$$ (b) Hence find \(\mathrm { E } ( R )\).
Pre-U Pre-U 9795/2 2011 June Q1
3 marks Standard +0.3
1 The independent random variables \(X\) and \(Y\) have distributions \(\mathrm { N } ( 30,9 )\) and \(\mathrm { N } ( 20,4 )\) respectively.
  1. Give the distribution of $$\left( X _ { 1 } + X _ { 2 } + X _ { 3 } \right) - \left( Y _ { 1 } + Y _ { 2 } + Y _ { 3 } + Y _ { 4 } \right)$$ where \(X _ { i } , i = 1,2,3\), and \(Y _ { j } , j = 1,2,3,4\), are independent observations of \(X\) and \(Y\) respectively. The time for female cadets to complete an assault course is \(X\) minutes and the time for male cadets to complete the same assault course is \(Y\) minutes.
  2. Find the probability that the total time for three randomly chosen female cadets to complete the assault course is greater than the total time for four randomly chosen male cadets to complete the assault course.
Pre-U Pre-U 9795/2 2011 June Q2
8 marks Moderate -0.3
2 The discrete random variable \(X\) has a Poisson distribution with mean 12.25.
  1. Calculate \(\mathrm { P } ( X \leqslant 5 )\).
  2. Calculate an approximate value for \(\mathrm { P } ( X \leqslant 5 )\) using a normal approximation to the Poisson distribution.
  3. Comment, giving a reason, on the accuracy of using a normal approximation to the Poisson distribution in this case.
Pre-U Pre-U 9795/2 2011 June Q3
10 marks Moderate -0.8
3 The fuel economy of two similar cars produced by manufacturers \(A\) and \(B\) was compared. A random sample of 15 cars was selected from manufacturer \(A\) and a random sample of 10 cars was selected from manufacturer \(B\). All the selected cars were driven over the same distance and the petrol consumption in miles per gallon (mpg) was calculated for each car. The results, \(x _ { A } \operatorname { mpg }\) and \(x _ { B } \operatorname { mpg }\) for cars from manufacturers \(A\) and \(B\) respectively, are summarised below, where \(\bar { x }\) denotes the sample mean and \(n\) the sample size. $$\begin{array} { l l l } \Sigma x _ { A } = 460.5 & \Sigma \left( x _ { A } - \bar { x } _ { A } \right) ^ { 2 } = 156.88 & n _ { A } = 15 \\ \Sigma x _ { B } = 334 & \Sigma \left( x _ { B } - \bar { x } _ { B } \right) ^ { 2 } = 123.97 & n _ { B } = 10 \end{array}$$
  1. (a) Assuming that the populations are normally distributed with a common variance, show that the pooled estimate of this common variance is 12.21 , correct to 4 significant figures. [2]
    (b) Construct a 95\% confidence interval for \(\mu _ { B } - \mu _ { A }\), the difference in the population means for manufacturers \(A\) and \(B\).
  2. Comment on a claim that the fuel economy for manufacturer \(B\) 's cars is better than that for manufacturer \(A\) 's cars.
  3. A random variable \(X\) has probability density function given by $$\mathrm { f } ( x ) = \begin{cases} \frac { 1 } { \theta } \mathrm { e } ^ { - \frac { x } { \theta } } & x \geqslant 0 \\ 0 & x < 0 \end{cases}$$ where \(\theta\) is a positive constant. Find \(\mathrm { E } \left( X ^ { 2 } \right)\).
  4. A random sample \(X _ { 1 } , X _ { 2 } , \ldots , X _ { n }\) is taken from a population with the distribution in part (i). The estimator \(T\) is defined by \(T = k \sum _ { i = 1 } ^ { n } X _ { i } ^ { 2 }\), where \(k\) is a constant. Find the value of \(k\) such that \(T\) is an unbiased estimator of \(\theta ^ { 2 }\).
  5. The discrete random variable \(X\) has distribution \(\operatorname { Geo } ( p )\). Show that the moment generating function of \(X\) is given by \(\mathrm { M } _ { X } ( t ) = \frac { p \mathrm { e } ^ { t } } { 1 - q \mathrm { e } ^ { t } }\), where \(q = 1 - p\).
  6. Use the moment generating function to find
    (a) \(\mathrm { E } ( X )\),
    (b) \(\operatorname { Var } ( X )\).
  7. An unbiased six-sided die is thrown repeatedly until a five is obtained, and \(Y\) denotes the number of throws up to and including the throw on which the five is obtained. Find \(\mathrm { P } ( | Y - \mu | < \sigma )\), where \(\mu\) and \(\sigma\) are the mean and standard deviation, respectively, of the distribution of \(Y\).
  8. The continuous random variable \(X\) has a uniform distribution over the interval \(0 < x < \frac { 1 } { 2 } \pi\). Show that the probability density function of \(Y\), where \(Y = \sin X\), is given by $$\mathrm { f } ( y ) = \begin{cases} \frac { 2 } { \pi \sqrt { 1 - y ^ { 2 } } } & 0 < y < 1 \\ 0 & \text { otherwise. } \end{cases}$$
  9. Deduce, using the probability density function, the exact values of
    (a) the median value of \(Y\),
    (b) \(\mathrm { E } ( Y )\).
Pre-U Pre-U 9795/2 2011 June Q7
3 marks Standard +0.3
7 \includegraphics[max width=\textwidth, alt={}, center]{963c0834-fe49-480b-9bb5-1ace4254641a-4_339_511_349_817} A particle of mass 0.3 kg is attached to one end \(A\) of a light inextensible string of length 1.5 m . The other end \(B\) of the string is attached to a ceiling, so that the particle may swing in a vertical plane. The particle is released from rest when the string is taut and makes an angle of \(75 ^ { \circ }\) with the vertical (see diagram). Air resistance may be regarded as being negligible.
  1. Show that, at an instant when the string makes an angle of \(40 ^ { \circ }\) with the vertical, the speed of the particle is \(3.90 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), correct to 3 significant figures.
  2. By considering Newton's second law, along and perpendicular to the string, find the radial and transverse components of acceleration, at this same instant, and hence the magnitude of the acceleration of the particle. \includegraphics[max width=\textwidth, alt={}, center]{963c0834-fe49-480b-9bb5-1ace4254641a-4_419_604_1370_772} A smooth sphere of mass 0.3 kg is moving in a straight line on a horizontal surface. It collides with a vertical wall when the velocity of the sphere is \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at \(60 ^ { \circ }\) to the wall (see diagram). The coefficient of restitution between the sphere and the wall is 0.4 .
  3. (a) Find the component of the velocity of the sphere perpendicular to the wall immediately after the collision.
    (b) Find the magnitude of the impulse exerted by the wall on the sphere.
  4. Determine the magnitude and direction of the velocity of the sphere immediately after the collision, giving the direction as an acute angle to the wall.
Pre-U Pre-U 9795/2 2011 June Q9
9 marks Standard +0.3
9 At noon a vessel, \(A\), leaves a port, \(O\), and travels at \(10 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) on a bearing of \(042 ^ { \circ }\). Also at noon a second vessel, \(B\), leaves another port, \(P , 13 \mathrm {~km}\) due north of \(O\), and travels at \(15 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) on a bearing of \(090 ^ { \circ }\). Take \(O\) as the origin and \(\mathbf { i }\) and \(\mathbf { j }\) as unit vectors east and north respectively.
  1. Express the velocity vector of \(A\) relative to \(B\) in the form \(a \mathbf { i } + b \mathbf { j }\), where \(a\) and \(b\) are constants to be determined.
  2. Express the position vector of \(A\) relative to \(B\), at time \(t\) hours after the vessels have left port, in terms of \(t , \mathbf { i }\) and \(\mathbf { j }\).
  3. Explain why the scalar product of the vectors in parts (i) and (ii) is zero when the two vessels are closest together.
  4. Find the time at which the two vessels are closest together. \(10 A\) and \(B\) are two points 6 m apart on a smooth horizontal surface. A particle, \(P\), of mass 0.5 kg is attached to \(A\) by a light elastic string of natural length 2 m and modulus of elasticity 20 N , and to \(B\) by a light elastic string of natural length 1 m and modulus of elasticity 10 N , such that \(P\) is between \(A\) and \(B\).
  5. Find the length \(A P\) when \(P\) is in equilibrium. \(P\) is held at the point \(C\), where \(C\) is between \(A\) and \(B\) and \(A C = 4.5 \mathrm {~m} . P\) is then released from rest. At time \(t\) seconds after being released, the displacement of \(P\) from the equilibrium position is \(y\) metres.
  6. Show that $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } = - 40 y$$
  7. Find the time taken for \(P\) to reach the mid-point of \(A B\) for the first time. \includegraphics[max width=\textwidth, alt={}, center]{963c0834-fe49-480b-9bb5-1ace4254641a-6_750_1187_258_479} Two particles, \(P\) and \(Q\), are projected simultaneously from the same point on a plane inclined at \(\alpha\) to the horizontal. \(P\) is projected up the plane and \(Q\) down the plane. Each particle is projected with speed \(V\) at an angle \(\theta\) to the plane. Both particles move in a vertical plane containing a line of greatest slope of the inclined plane and you are given that \(\alpha + \theta < \frac { 1 } { 2 } \pi\) (see diagram).
  8. Show that the range of \(P\), up the plane, is given by $$\frac { 2 V ^ { 2 } \sin \theta } { g \cos ^ { 2 } \alpha } ( \cos \theta \cos \alpha - \sin \theta \sin \alpha ) .$$
  9. Write down a similar expression for the range of \(Q\), down the plane.
  10. Given that the range up the plane is a quarter of the range down the plane and that \(\alpha = \tan ^ { - 1 } \left( \frac { 1 } { 2 } \right)\), find \(\theta\).
Pre-U Pre-U 9795/2 2011 June Q12
9 marks Standard +0.3
12 A train of mass 250 tonnes is ascending an incline of \(\sin ^ { - 1 } \left( \frac { 1 } { 500 } \right)\) and working at 400 kW against resistance to motion which may be regarded as a constant force of 20000 N .
  1. Find the constant speed, \(V\), with which the train can ascend the incline working at this power.
  2. The train begins to ascend the incline at \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the same power and against the same resistance. Find the distance covered in reaching a speed of \(\frac { 3 } { 4 } V\).
Pre-U Pre-U 9795/2 2012 June Q1
5 marks Standard +0.8
1 The random variable \(X\) has probability density function \(\mathrm { f } ( x )\), where $$\mathrm { f } ( x ) = \begin{cases} k \mathrm { e } ^ { - k x } & x \geqslant 0 , \\ 0 & x < 0 , \end{cases}$$ and \(k\) is a positive constant.
  1. Show that the moment generating function of \(X\) is \(\mathrm { M } _ { X } ( t ) = k ( k - t ) ^ { - 1 } , t < k\).
  2. Use the moment generating function to find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
Pre-U Pre-U 9795/2 2012 June Q2
9 marks Standard +0.3
2 The independent random variables \(X\) and \(Y\) have normal distributions where \(X \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)\) and \(Y \sim \mathrm {~N} \left( 3 \mu , 4 \sigma ^ { 2 } \right)\). Two random samples each of size \(n\) are taken, one from each of these normal populations.
  1. Show that \(a \bar { X } + b \bar { Y }\) is an unbiased estimator of \(\mu\) provided that \(a + 3 b = 1\), where \(a\) and \(b\) are constants and \(\bar { X }\) and \(\bar { Y }\) are the respective sample means. In the remainder of the question assume that \(a \bar { X } + b \bar { Y }\) is an unbiased estimator of \(\mu\).
  2. Show that \(\operatorname { Var } ( a \bar { X } + b \bar { Y } )\) can be written as \(\frac { \sigma ^ { 2 } } { n } \left( 1 - 6 b + 13 b ^ { 2 } \right)\).
  3. The value of the constant \(b\) can be varied. Find the value of \(b\) that gives the minimum of \(\operatorname { Var } ( a \bar { X } + b \bar { Y } )\), and hence find the minimum of \(\operatorname { Var } ( a \bar { X } + b \bar { Y } )\) in terms of \(\sigma\) and \(n\).
Pre-U Pre-U 9795/2 2012 June Q3
10 marks Standard +0.3
3 Small amounts of a potentially hazardous chemical are discharged into a river from a nearby industrial site. A random sample of size 6 was taken from the river and the concentration of the chemical present in each item was measured in grams per litre. The results are shown below. $$\begin{array} { l l l l l l } 1.64 & 1.53 & 1.78 & 1.60 & 1.73 & 1.77 \end{array}$$
  1. Assuming that the sample was taken from a normal distribution with known variance 0.01 , construct a \(99 \%\) confidence interval for the mean concentration of the chemical present in the river.
  2. If instead the sample was taken from a normal distribution, but with unknown variance, construct a revised \(99 \%\) confidence interval for the mean concentration of the chemical present in the river.
  3. If the mean concentration of the chemical in the river exceeds 1.8 grams per litre, then remedial action needs to be taken. Comment briefly on the need for remedial action in the light of the results in parts (i) and (ii).
Pre-U Pre-U 9795/2 2012 June Q4
10 marks Challenging +1.3
4
  1. The random variable \(X\) has the distribution \(\operatorname { Po } ( \lambda )\). Prove that the probability generating function, \(\mathrm { G } _ { X } ( t )\), is given by $$\mathrm { G } _ { X } ( t ) = \mathrm { e } ^ { \lambda ( t - 1 ) } .$$
  2. The independent random variables \(X\) and \(Y\) have distributions \(\operatorname { Po } ( \lambda )\) and \(\operatorname { Po } ( \mu )\) respectively. Use probability generating functions to show that the distribution of \(X + Y\) is \(\operatorname { Po } ( \lambda + \mu )\).
  3. Given that \(X \sim \operatorname { Po } ( 1.5 )\) and \(Y \sim \operatorname { Po } ( 2.5 )\), find \(\mathrm { P } ( X \leqslant 2 \mid X + Y = 4 )\).
Pre-U Pre-U 9795/2 2012 June Q5
11 marks Standard +0.3
5
  1. The probability that a shopper obtains a parking space on the river embankment on any given Saturday morning is 0.2 . Using a suitable normal approximation, find the probability that, over a period of 100 Saturday mornings, the shopper finds a parking space
    1. at least 15 times,
    2. no more than 12 times.
    3. The number of parking tickets that a traffic warden issues on the river embankment during the course of a week has a Poisson distribution with mean 36 . The probability that the traffic warden issues more than \(N\) parking tickets is less than 0.05 . Using a suitable normal approximation, find the least possible value of \(N\).
Pre-U Pre-U 9795/2 2012 June Q6
12 marks Standard +0.3
6 The lengths of time, in years, that sales representatives for a certain company keep their company cars may be modelled by the distribution with probability density function \(\mathrm { f } ( x )\), where $$f ( x ) = \begin{cases} \frac { 4 } { 27 } x ^ { 2 } ( 3 - x ) & 0 \leqslant x \leqslant 3 \\ 0 & \text { otherwise } \end{cases}$$
  1. Draw a sketch of this probability density function.
  2. Calculate the mean and the mode of \(X\).
  3. Comment briefly on the values obtained in part (ii) in relation to the sketch in part (i).
  4. Given that \(\sigma ^ { 2 } = 0.36\), find \(\mathrm { P } ( | X - \mu | < \sigma )\), where \(\mu\) and \(\sigma ^ { 2 }\) denote the mean and the variance of \(X\) respectively.
Pre-U Pre-U 9795/2 2012 June Q7
8 marks Standard +0.3
7 A cyclist and her machine have a combined mass of 90 kg and she is riding along a straight horizontal road. She is working at a constant power of 75 W . At time \(t\) seconds her speed is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the resistance to motion is \(k v \mathrm {~N}\), where \(k\) is a constant.
  1. If the cyclist's maximum steady speed is \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), show that \(k = \frac { 3 } { 4 }\).
  2. Use Newton's second law to show that $$\frac { 25 } { v } - \frac { v } { 4 } = 30 \frac { \mathrm {~d} v } { \mathrm {~d} t } .$$
  3. Find the time taken for the cyclist to accelerate from a speed of \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to a speed of \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).