Questions Pre-U 9795/2 (184 questions)

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Pre-U Pre-U 9795/2 Specimen Q3
5 marks Standard +0.8
3 A light spring, of natural length 0.4 m and modulus of elasticity 6.4 N , has one end \(A\) attached to the ceiling of a room. A particle of mass \(m \mathrm {~kg}\) is attached to the free end of the spring and hangs in equilibrium. The particle is displaced vertically downwards and released from rest. In the subsequent motion the particle does not reach the ceiling and air resistance may be neglected.
  1. Show that the particle oscillates in simple harmonic motion.
  2. Given that the period of the motion is 1.12 s , find
    1. the value of \(m\), correct to 3 significant figures,
    2. the extension of the spring when the particle has a downwards acceleration of \(2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
Pre-U Pre-U 9795/2 Specimen Q4
7 marks Challenging +1.2
4 A particle is projected with velocity \(V\), at an angle of elevation of \(60 ^ { \circ }\) to the horizontal, from a point on a plane inclined at an angle of \(30 ^ { \circ }\) to the horizontal. The path of the particle is in a vertical plane through a line of greatest slope. If \(R _ { 1 }\) and \(R _ { 2 }\) are the respective ranges when the particle is projected up the plane and down the plane, show that $$R _ { 2 } = 2 R _ { 1 }$$
Pre-U Pre-U 9795/2 Specimen Q5
3 marks Standard +0.3
5 When a car of mass 990 kg moves at a constant speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) along a horizontal straight road, the power of its engine is 8.8 kW .
  1. Find the magnitude of the resistance to the motion of the car at this speed.
  2. Assuming that the resistance has magnitude \(k v ^ { 2 }\) newtons when the speed is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find the value of the constant \(k\). The power of the engine is now increased to 22 kW and remains constant at this value.
  3. Using the model in part (ii), show that $$\frac { \mathrm { d } v } { \mathrm {~d} x } = \frac { 20000 - v ^ { 3 } } { 900 v ^ { 2 } } .$$
  4. Hence show that the car moves about 300 m as its speed increases from \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
Pre-U Pre-U 9795/2 Specimen Q6
5 marks Challenging +1.2
6 A simple pendulum consists of a light inextensible string of length 1.5 m with a small bob of mass 0.2 kg at one end. When suspended from a fixed point and hanging at rest under gravity, the bob is given a horizontal speed of \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and it comes instantaneously to rest when the string makes an angle of 0.1 rad with the vertical. At time \(t\) seconds after projection the string makes an angle \(\theta\) with the vertical.
  1. Show that, neglecting air resistance, $$\left( \frac { \mathrm { d } \theta } { \mathrm {~d} t } \right) ^ { 2 } = \frac { 40 } { 3 } \{ \cos \theta - \cos ( 0.1 ) \}$$
  2. Find, correct to 2 significant figures,
    1. the value of \(u\),
    2. the tension in the string when \(\theta = 0.05 \mathrm { rad }\).
    3. By differentiating the above equation for \(\left( \frac { \mathrm { d } \theta } { \mathrm { d } t } \right) ^ { 2 }\), or otherwise, show that the motion of the bob can be modelled approximately by simple harmonic motion.
    4. Hence find the value of \(t\) at which the bob first comes instantaneously to rest.
Pre-U Pre-U 9795/2 Specimen Q7
1 marks Standard +0.8
7 The time taken for me to walk from my house to the bus stop has a normal distribution with mean 10 minutes and standard deviation 1.5 minutes. The arrival time of the bus is normally distributed with mean 0900 and standard deviation 1 minute. If the bus arrives early it does not wait. I leave home at 0845 . Find, correct to 3 decimal places, the probability that I catch the bus.
Pre-U Pre-U 9795/2 Specimen Q8
5 marks Standard +0.3
8 Specimens of rain were collected at random from the north and south sides of an island and analysed for sulphur content. The results (in suitable units) are given below.
North side0.120.610.790.160.08
South side1.120.270.060.120.240.78
Assume that the sulphur contents have normal distributions with population means \(\mu _ { N }\) and \(\mu _ { S }\) and a common, but unknown, variance.
  1. Calculate a symmetric \(95 \%\) confidence interval for the difference in population mean sulphur contents of the rain on the north and south sides of the island, \(\mu _ { S } - \mu _ { N }\).
  2. Comment on a claim that the mean sulphur content is the same on both sides of the island.
Pre-U Pre-U 9795/2 Specimen Q9
4 marks Standard +0.8
9 The service time, \(X\) minutes, for each customer in a post office is modelled by the probability density function given by $$\mathrm { f } ( x ) = \begin{cases} 0.2 \mathrm { e } ^ { - 0.2 x } & x > 0 , \\ 0 & \text { otherwise } . \end{cases}$$ Two customers begin to be served independently at the same instant. The larger of the two service times is \(T\) minutes.
  1. By considering the probability that both customers have been served in less than \(t\) minutes, show that the cumulative distribution function of \(T\) is given by $$\mathrm { G } ( t ) = \begin{cases} 1 - 2 \mathrm { e } ^ { - 0.2 t } + \mathrm { e } ^ { - 0.4 t } & t > 0 \\ 0 & \text { otherwise } \end{cases}$$
  2. Find the probability that both customers are served within 10 minutes.
  3. Find the value of the interquartile range of \(T\).
Pre-U Pre-U 9795/2 Specimen Q10
4 marks Standard +0.3
10
  1. \(X , Y\) and \(Z\) are independent random variables having Poisson distributions with means \(\lambda , \mu\) and \(\lambda + \mu\) respectively. Find \(\mathrm { P } ( X = 0\) and \(Y = 2 ) , \mathrm { P } ( X = 1\) and \(Y = 1 )\) and \(\mathrm { P } ( X = 2\) and \(Y = 0 )\). Hence verify that \(\mathrm { P } ( X + Y = 2 ) = \mathrm { P } ( Z = 2 )\).
  2. In an office the male absence rate, i.e. the number of working days lost each month due to the absence of male employees, has a Poisson distribution with mean 4.5. In the same office the female absence rate has an independent Poisson distribution with mean 4.1. Calculate the probability that
    1. during a particular month both the male absence rate and the female absence rate are equal to 3,
    2. during a particular month the total of the male and female absence rates is equal to 6,
    3. during a particular month the male and female absence rates were each equal to 3 , given that the total of the male and female absence rates was equal to 6 .
Pre-U Pre-U 9795/2 Specimen Q11
11 marks Standard +0.3
11 The time, \(T\) years, before a particular type of washing machine breaks down may be taken to have probability density function f given by $$\mathrm { f } ( t ) = \begin{cases} a t \mathrm { e } ^ { - b t } & t > 0 \\ 0 & \text { otherwise } \end{cases}$$ where \(a\) and \(b\) are positive constants. It may be assumed that, if \(n\) is a positive integer, $$\int _ { 0 } ^ { \infty } t ^ { n } \mathrm { e } ^ { - b t } \mathrm {~d} t = \frac { n ! } { b ^ { n + 1 } }$$
  1. Records show that the mean of \(T\) is 1.5 . Show that \(b = \frac { 4 } { 3 }\) and find the value of \(a\).
  2. Find \(\operatorname { Var } ( T )\).
  3. Calculate \(\mathrm { P } ( T < 1.5 )\). State, giving a reason, whether this value indicates that the median of \(T\) is smaller than the mean of \(T\) or greater than the mean of \(T\).
Pre-U Pre-U 9795/2 Specimen Q12
8 marks Standard +0.8
12 A game is played in which the number of points scored, \(X\), has the probability distribution given in the following table.
\(x\)- 113
\(\mathrm { P } ( X = x )\)\(\frac { 16 } { 25 }\)\(\frac { 8 } { 25 }\)\(\frac { 1 } { 25 }\)
  1. Write down the probability generating function of \(X\).
  2. Use this generating function to find the mean and variance of \(X\).
  3. The game is played 4 times (independently) and the total number of points scored is denoted by \(Y\). Show that the probability generating function of \(Y\) can be written in the form $$\frac { \left( a + t ^ { 2 } \right) ^ { 8 } } { b t ^ { 4 } }$$ where \(a\) and \(b\) are constants.
  4. Hence find \(\mathrm { P } ( Y < 0 )\).
Pre-U Pre-U 9795/2 2014 June Q1
8 marks Standard +0.3
A machine is selecting independently and at random long rods and short rods. The length of the long rods, \(X\) cm, is normally distributed with mean 25 cm and variance 3 cm\(^2\) and the length of the short rods, \(Y\) cm, is normally distributed with mean 15 cm and variance 2 cm\(^2\). Assume that \(X\) and \(Y\) are independent random variables.
  1. One long rod and one short rod are chosen at random. Find the probability that the difference in the lengths, \(X - Y\), is between 8 cm and 11 cm. [4]
  2. Two long rods and two short rods are chosen at random and are assembled into an approximately rectangular frame. Find the probability that the perimeter of the resulting frame is more than 75 cm. [4]
Pre-U Pre-U 9795/2 2014 June Q2
8 marks Challenging +1.2
The mean of a random sample of \(n\) observations drawn from a normal distribution with mean \(\mu\) and variance \(\sigma^2\) is denoted by \(\bar{X}\). It is given that P(\(\mu - 0.5\sigma < \bar{X} < \mu + 0.5\sigma\)) > 0.95.
  1. Find the smallest possible value of \(n\). [5]
  2. With this value of \(n\), find P(\(\bar{X} > \mu - 0.1\sigma\)). [3]
Pre-U Pre-U 9795/2 2014 June Q3
8 marks Standard +0.8
A random sample of 400 seabirds is taken from a colony, ringed, and returned, unharmed, to the colony. After a suitable period of time has elapsed, a second random sample of 400 seabirds is taken, and 20 of this second sample are found to be ringed. You may assume that the probability that a seabird is captured is independent of whether or not it has been ringed and that the colony remains unchanged at the time of the second sampling.
  1. Estimate the number of seabirds in the colony. [1]
  2. Find a 98% confidence interval for the proportion of seabirds in the colony which are ringed. [5]
  3. Deduce a 98% confidence interval for the number of seabirds in the colony. [2]
Pre-U Pre-U 9795/2 2014 June Q4
10 marks Challenging +1.2
The continuous random variable \(X\) has probability density function given by $$f(x) = \begin{cases} 3e^{-x} & 0 \leq x \leq k, \\ 0 & \text{otherwise,} \end{cases}$$ where \(k\) is a constant.
  1. Show that \(e^{-k} = \frac{2}{3}\). [2]
  2. Show that the moment generating function of \(X\) is given by \(M_X(t) = \frac{3}{1-t}\left(1 - \frac{2}{3}e^{kt}\right)\). [4]
  3. By expanding \(M_X(t)\) as a power series in \(t\), up to and including the term in \(t^2\), show that $$M_X(t) = 1 + (1 - 2k)t + (1 - 2k - k^2)t^2 + \ldots.$$ [3] [You may use the standard series for \((1-t)^{-1}\) and \(e^{kt}\) without proof.]
  4. Deduce that the exact value of E\((X)\) is \(1 - 2\ln\left(\frac{2}{3}\right)\). [1]
Pre-U Pre-U 9795/2 2014 June Q5
13 marks Standard +0.3
  1. The discrete random variable \(X\) has a Poisson distribution with mean \(\lambda\). Use the probability generating function for \(X\) to show that both the mean and the variance have the value \(\lambda\). [5]
  2. The number of eggs laid by a certain insect has a Poisson distribution with variance 250. Find, using a suitable approximation, the probability that between 230 and 260 (inclusive) eggs are laid. [5]
  3. An insect lays 250 eggs. The probability that any egg that is laid survives to maturity is 0.1. Use a suitable approximation to find the probability that more than 30 eggs survive to maturity. [3]
Pre-U Pre-U 9795/2 2014 June Q6
13 marks Challenging +1.2
The continuous random variable \(X\) has probability density function given by $$f(x) = \begin{cases} \frac{4}{\pi(1+x^2)} & 0 \leq x \leq 1, \\ 0 & \text{otherwise.} \end{cases}$$
  1. Verify that the median value of \(X\) lies between 0.41 and 0.42. [3]
  2. Show that E\((X) = \frac{2}{\pi}\ln 2\). [2]
  3. Find Var\((X)\). [5]
  4. Given that \(\tan\frac{1}{8}\pi = \sqrt{2} - 1\), find the exact value of P(\(X > \frac{1}{4}\sqrt{3}|X > \sqrt{2} - 1\)). [3]
Pre-U Pre-U 9795/2 2014 June Q7
8 marks Challenging +1.2
\includegraphics{figure_7} A light inextensible string of length 8 m is threaded through a smooth fixed ring, \(R\), and carries a particle at each end. One particle, \(P\), of mass 0.5 kg is at rest at a distance 3 m below \(R\). The other particle, \(Q\), is rotating in a horizontal circle whose centre coincides with the position of \(P\) (see diagram). Find the angular speed and the mass of \(Q\). [8]
Pre-U Pre-U 9795/2 2014 June Q8
9 marks Challenging +1.8
\includegraphics{figure_8} A smooth sphere with centre \(A\) and of mass 2 kg, moving at 13 m s\(^{-1}\) on a smooth horizontal plane, strikes a smooth sphere with centre \(B\) and of mass 3 kg moving at 5 m s\(^{-1}\) on the same smooth horizontal plane. The spheres have equal radii. The directions of motion immediately before impact are at angles \(\tan^{-1}\left(\frac{2}{13}\right)\) to \(\overrightarrow{AB}\) and \(\tan^{-1}\left(\frac{4}{3}\right)\) to \(\overrightarrow{BA}\) respectively (see diagram). Given that the coefficient of restitution is \(\frac{2}{3}\), find the speeds of the spheres after impact. [9]
Pre-U Pre-U 9795/2 2014 June Q9
11 marks Challenging +1.2
An engine is travelling along a straight horizontal track against negligible resistances. In travelling a distance of 750 m its speed increases from 5 m s\(^{-1}\) to 15 m s\(^{-1}\). Find the time taken if the engine was
  1. exerting a constant tractive force, [2]
  2. working at constant power. [9]
Pre-U Pre-U 9795/2 2014 June Q10
12 marks Standard +0.3
One end of a light spring of length 0.5 m is attached to a fixed point \(F\). A particle \(P\) of mass 2.5 kg is attached to the other end of the spring and hangs in equilibrium 0.55 m below \(F\). Another particle \(Q\), of mass 1.5 kg, is attached to \(P\), without moving it, and both particles are then released.
  1. Show that the modulus of elasticity of the spring is 250 N. [2]
  2. Prove that the motion is simple harmonic. [4]
  3. Find
    1. the amplitude of the motion, [1]
    2. the greatest speed of the particles, [1]
    3. the period of the motion, [1]
    4. the time taken for the spring to be extended by 0.1 m for the first time. [3]
Pre-U Pre-U 9795/2 2014 June Q11
10 marks Challenging +1.8
It is given that the trajectory of a projectile which is launched with speed \(V\) at an angle \(\alpha\) above the horizontal has equation $$y = x\tan\alpha - \frac{gx^2}{2V^2}(1 + \tan^2\alpha),$$ where the point of projection is the origin, and the \(x\)- and \(y\)-axes are horizontal and vertically upwards respectively.
  1. Express the above equation as a quadratic equation in \(\tan\alpha\) and deduce that the boundary of all accessible points for this projectile has equation $$y = \frac{1}{2gV^2}(V^4 - g^2x^2).$$ [4]
  2. A stone is thrown with speed \(\sqrt{gh}\) from the top of a vertical tower, of height \(h\), which stands on horizontal ground. Find
    1. the maximum distance, from the foot of the tower, at which the stone can land, [3]
    2. the angle of elevation at which the stone must be thrown to achieve this maximum distance. [3]
Pre-U Pre-U 9795/2 2014 June Q12
10 marks Challenging +1.2
A cyclist, when travelling due west at 15 km h\(^{-1}\), finds that the wind appears to be blowing from a bearing of 150°. When the cyclist is travelling due west at 10 km h\(^{-1}\), the wind appears to be blowing from a bearing of 135°. Find the velocity of the wind. [10]
Pre-U Pre-U 9795/2 Specimen Q1
8 marks Challenging +1.8
A smooth sphere \(A\) of mass \(m\) is projected with speed \(u\) along a smooth horizontal surface and strikes a stationary smooth sphere \(B\) of equal radius but of mass \(M\). The direction of motion of \(A\) before the impact makes an acute angle \(\theta\) with the line of centres at the moment of contact. After the impact, the direction of motion of \(A\) is perpendicular to the initial direction of motion of \(A\). The coefficient of restitution between the two spheres is \(e\). Given that \(Me \geq m\), prove that $$\tan^2 \theta = \frac{Me - m}{m + M}.$$ [8]
Pre-U Pre-U 9795/2 Specimen Q2
9 marks Standard +0.8
One end of a light inextensible string of length \(l\) is attached to a fixed point \(O\). The other end is attached to a particle \(P\) of mass \(m\). The particle hangs at rest vertically below \(O\). The particle is then given a horizontal speed \(u\).
    1. Show that when \(OP\) has turned through an angle \(\theta\) the tension in the string is given by $$T = mg(3\cos \theta - 2) + \frac{mu^2}{l}$$ as long as the string remains taut. [5]
    2. Deduce that \(u^2 \geq 5gl\) in order for the particle to perform complete circles. [1]
    1. In the case \(u^2 = 3gl\), find the angle that \(OP\) makes with the downward vertical at \(O\) at the instant when the string becomes slack. [2]
    2. Describe the nature of the motion while the string is slack. [1]
Pre-U Pre-U 9795/2 Specimen Q3
11 marks Standard +0.8
A stone of mass \(m\) is projected vertically upwards with initial velocity \(u\). At time \(t\), the height risen above the point of projection is \(x\) and the resistance to motion is \(kv\) when the velocity of the stone is \(v\).
  1. Write down a first-order differential equation relating \(v\) and \(t\) and hence find \(t\) in terms of \(v\). [5]
  2. Write down a first-order differential equation relating \(v\) and \(x\) and hence find \(x\) in terms of \(v\). [6]