Questions — Pre-U (885 questions)

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Pre-U Pre-U 9794/2 2013 November Q7
Moderate -0.3
7
  1. Differentiate \(3 \ln \left( x ^ { 2 } + 1 \right)\).
  2. Find \(\int \frac { x ^ { 2 } } { 3 - 4 x ^ { 3 } } \mathrm {~d} x\).
Pre-U Pre-U 9794/2 2013 November Q8
Moderate -0.3
8 Find the exact volume of the solid of revolution generated by rotating the graph of \(y = 3 \mathrm { e } ^ { x }\) between \(x = 0\) and \(x = 2\) through \(360 ^ { \circ }\) about the \(x\)-axis.
Pre-U Pre-U 9794/2 2013 November Q9
Moderate -0.3
9 Two straight lines have equations $$\mathbf { r } = \left( \begin{array} { r } 16 \\ 2 \\ 3 \end{array} \right) + \lambda \left( \begin{array} { r } 3 \\ 2 \\ - 1 \end{array} \right) \quad \text { and } \quad \mathbf { r } = \left( \begin{array} { r } - 3 \\ 8 \\ 12 \end{array} \right) + \mu \left( \begin{array} { r } 5 \\ - 6 \\ - 3 \end{array} \right) .$$ Show that the two lines intersect and find the coordinates of their point of intersection.
Pre-U Pre-U 9794/2 2013 November Q10
Standard +0.3
10
  1. Given that \(10 + 4 x - x ^ { 2 } \equiv p - ( x - q ) ^ { 2 }\), show that \(q = 2\) and find the value of \(p\).
  2. Hence find the coordinates of all the points of intersection of the curve \(y = 10 + 4 x - x ^ { 2 }\) and the circle \(( x - 2 ) ^ { 2 } + ( y - 1 ) ^ { 2 } = 25\).
Pre-U Pre-U 9794/2 2013 November Q11
Standard +0.3
11
  1. Expand \(( 1 + x ) ^ { - 1 }\) up to and including the term in \(x ^ { 2 }\).
  2. (a) Expand \(\sqrt { 2 + 3 x ^ { 2 } }\) up to and including the term in \(x ^ { 4 }\).
    (b) For what range of values of \(x\) is this expansion valid?
  3. Find the first three terms of the expansion of \(\frac { \sqrt { 2 + 3 x ^ { 2 } } } { 1 + x }\) in ascending powers of \(x\) and hence show that \(\int _ { 0 } ^ { 0.1 } \frac { \sqrt { 2 + 3 x ^ { 2 } } } { 1 + x } \mathrm {~d} x \approx 0.135\).
Pre-U Pre-U 9794/2 2013 November Q12
Standard +0.3
12 A curve \(C\) is given by the parametric equations \(x = 2 \tan \theta , y = 1 + \operatorname { cosec } \theta\) for \(0 < \theta < 2 \pi , \theta \neq \frac { 1 } { 2 } \pi , \pi , \frac { 3 } { 2 } \pi\).
  1. Show that the cartesian equation for \(C\) is \(\frac { 4 } { x ^ { 2 } } = y ^ { 2 } - 2 y\).
  2. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and hence show that \(C\) has no stationary points.
  3. \(P\) is the point on \(C\) where \(\theta = \frac { 1 } { 4 } \pi\). The tangent to \(C\) at \(P\) intersects the \(y\)-axis at \(Q\) and the \(x\)-axis at \(R\). Find the exact area of triangle \(O Q R\).
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.
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 9794/1 2014 June Q1
5 marks Easy -1.2
1
  1. Express \(x ^ { 2 } - 8 x + 10\) in the form \(( x - a ) ^ { 2 } + b\) where \(a\) and \(b\) are integers to be found.
  2. Hence write down the minimum value of \(x ^ { 2 } - 8 x + 10\) and the corresponding value of \(x\).
Pre-U Pre-U 9794/1 2014 June Q2
3 marks Moderate -0.8
2 Sketch the curve with equation \(y = \tan x\) for \(- \frac { 1 } { 2 } \pi < x < \frac { 1 } { 2 } \pi\).
On the same diagram, sketch the curve with equation \(y = \tan ^ { - 1 } x\) for all \(x\).
State the geometrical relationship between the curves.
Pre-U Pre-U 9794/1 2014 June Q3
3 marks Easy -1.2
3 Solve the inequality \(| 2 x - 1 | < 3\).
Pre-U Pre-U 9794/1 2014 June Q4
2 marks Moderate -0.8
4 The graph of \(\mathrm { f } ( x )\) is shown below. \includegraphics[max width=\textwidth, alt={}, center]{69792771-6de6-4886-9c71-e794fcb7aaba-2_949_1127_1041_507} Draw the graphs of
  1. \(\mathrm { f } ( x + 2 ) + 1\),
  2. \(- \frac { 1 } { 2 } \mathrm { f } ( x )\).
Pre-U Pre-U 9794/1 2014 June Q5
4 marks Easy -1.2
5 A root of the equation \(z ^ { 2 } + p z + q = 0\) is \(3 + \mathrm { i }\), where \(p\) and \(q\) are real. Write down the other root of the equation and hence calculate the values of \(p\) and \(q\).
Pre-U Pre-U 9794/1 2014 June Q6
7 marks Standard +0.3
6 The diagram shows the curve with equation \(y = 7 x - 10 - x ^ { 2 }\) and the tangent to the curve at the point where \(x = 3\). \includegraphics[max width=\textwidth, alt={}, center]{69792771-6de6-4886-9c71-e794fcb7aaba-3_648_679_342_733}
  1. Show that the curve crosses the \(x\)-axis at \(x = 2\).
  2. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and hence find the equation of the tangent to the curve at \(x = 3\). Show that the tangent crosses the \(x\)-axis at \(x = 1\).
  3. Evaluate \(\int _ { 2 } ^ { 3 } \left( 7 x - 10 - x ^ { 2 } \right) \mathrm { d } x\) and hence find the exact area of the shaded region bounded by the curve, the tangent and the \(x\)-axis.
Pre-U Pre-U 9794/1 2014 June Q7
4 marks Standard +0.3
7 Taking \(x = 2\) as a first approximation, use the Newton-Raphson process to find a root of the equation \(\frac { 1 } { x ^ { 2 } } - 0.119 - 0.018 x = 0\). Give your answer correct to 3 significant figures.