Questions — Pre-U (885 questions)

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Pre-U Pre-U 9795/2 2018 June Q3
Standard +0.3
3 The moment generating function of a random variable \(X\) is \(( 1 - 2 t ) ^ { - 3 }\).
  1. Find the mean and variance of \(X\).
  2. \(X _ { 1 }\) and \(X _ { 2 }\) are two independent observations of \(X\). Find \(\mathrm { E } \left[ \left( X _ { 1 } + X _ { 2 } \right) ^ { 3 } \right]\).
Pre-U Pre-U 9795/2 2018 June Q4
Challenging +1.2
4 The continuous random variable \(X\) has cumulative distribution function given by $$\mathrm { F } ( x ) = \begin{cases} 0 & x < 0 \\ \frac { 1 } { 8 } x ^ { 3 } & 0 \leqslant x \leqslant 2 \\ 1 & x > 2 \end{cases}$$
  1. Find \(\mathrm { E } ( X )\).
  2. Find the probability density function of \(Y\), where \(Y = \frac { 1 } { X ^ { 2 } }\).
Pre-U Pre-U 9795/2 2018 June Q5
Standard +0.3
5 A random sample of 12 seventeen-year-old boys and a random sample of 14 seventeen-year-old girls were given a certain task. The times, \(t\) minutes, taken to complete the task by the members of the two samples are summarised as follows.
\(n\)\(\Sigma t\)\(\Sigma t ^ { 2 }\)
Boys122044236
Girls143127126
  1. Stating any necessary assumption(s), find a \(95 \%\) symmetric confidence interval for the difference in the average times taken to complete the task by seventeen-year-old boys and seventeen-year-old girls.
  2. State with a reason whether the confidence interval calculated in part (i) suggests that there may in fact be no difference in the average times taken by seventeen-year-old boys and by seventeen-year-old girls.
Pre-U Pre-U 9795/2 2018 June Q6
Challenging +1.3
6 In a certain city there are \(N\) taxis. Each taxi displays a different licensing number which is an integer in the range 1 to \(N\). A visitor to the city attempts to estimate the value of \(N\), assuming that the licensing number of each taxi observed is equally likely to be any integer from 1 to \(N\) inclusive.
  1. The visitor observes one randomly chosen licensing number, \(X\). Using standard results for \(\sum _ { r = 1 } ^ { n } r\) and \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\), show that \(\mathrm { E } ( X ) = \frac { 1 } { 2 } ( N + 1 )\) and \(\operatorname { Var } ( X ) = \frac { 1 } { 12 } \left( N ^ { 2 } - 1 \right)\). The mean of 40 independent observations of \(X\) is denoted by \(A\).
  2. Find an unbiased estimator \(E _ { 1 }\) of \(N\) based on \(A\), and state the approximate distribution of \(E _ { 1 }\), giving the value(s) of any parameter(s). \(B\) is another random variable based on a random sample of 40 independent observations of \(X\). It is given that \(\mathrm { E } ( B ) = \frac { 40 } { 27 } N\) and that \(\operatorname { Var } ( B ) = \alpha N ^ { 2 }\) where \(\alpha\) is a constant.
  3. Find an unbiased estimator \(E _ { 2 }\) of \(N\) based on \(B\), and determine the set of values of \(\alpha\) for which \(\operatorname { Var } \left( E _ { 2 } \right) > \operatorname { Var } \left( E _ { 1 } \right)\) for all values of \(N\).
Pre-U Pre-U 9795/2 2018 June Q7
Moderate -0.3
7 A car has mass 800 kg .
  1. The car accelerates from \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in climbing a hill with a vertical height of 16 m . Ignoring resistive forces, find the work done by the engine.
  2. The engine produces a constant power output of 189 kW . The car now travels along horizontal ground. Modelling the resistive force as \(7 v ^ { 2 } \mathrm {~N}\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed, find the value of \(v\) for which the speed of the car is constant.
Pre-U Pre-U 9795/2 2018 June Q8
5 marks Standard +0.8
8 A light elastic string of natural length 0.2 m and modulus of elasticity 8 N has one end fixed to a point \(P\) on a horizontal ceiling. A particle of mass 0.4 kg is attached to the other end of the string.
  1. Find the extension of the string when the particle hangs in equilibrium vertically below \(P\).
  2. The particle is held at rest, with the string stretched, at a point \(x \mathrm {~m}\) vertically below \(P\) and is then released. Find the smallest value of \(x\) for which the particle will reach the ceiling.
Pre-U Pre-U 9795/2 2018 June Q9
9 marks Standard +0.3
9 \includegraphics[max width=\textwidth, alt={}, center]{09939c3a-7829-4784-8e6d-ee5356c22cd7-4_433_428_1219_863} A light inextensible string of length 1.4 m has its ends attached to two points \(A\) and \(C\), where \(A\) is 1 m vertically above \(C\). A smooth bead \(B\) of mass 0.2 kg is threaded on the string and rotates in a horizontal circle with the string taut. The distance \(B A\) is 0.8 m (see diagram). Find
  1. the tension in the string,
  2. the time taken for the bead to perform one complete circle.
Pre-U Pre-U 9795/2 2018 June Q10
6 marks Standard +0.8
10 A particle \(P\) is attached to one end of a light inextensible string of length 1.4 m . The other end of the string is fixed to the ceiling at \(C\). The angle between \(C P\) and the vertical is \(\theta\) radians. The particle is held with the string taut with \(\theta = 0.3\) and is then released.
  1. (a) Show that the motion of the system is approximately simple harmonic, and state its period.
    (b) Hence find an approximation for the speed of \(P\) when \(\theta = 0.2\).
  2. Find the speed of \(P\) when \(\theta = 0.2\) using an energy method, and hence find the percentage error in the answer to part (i) (b).
Pre-U Pre-U 9795/2 2018 June Q11
13 marks Challenging +1.8
11 A particle of mass 0.2 kg is projected so that it hits a smooth sloping plane \(\Pi\) that makes an angle of \(\sin ^ { - 1 } 0.6\) above the horizontal. The path of the particle is in a vertical plane containing a line of greatest slope of \(\Pi\). Immediately before the first impact between the particle and \(\Pi\), the particle is moving horizontally with speed \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The coefficient of restitution between the particle and \(\Pi\) is 0.5 .
  1. Find the magnitude of the impulse on the particle from \(\Pi\) at the first impact, and state the direction of this impulse.
  2. Find the distance between the points on \(\Pi\) where the first and second impacts occur.
  3. Find the time taken between the first and third impacts.
Pre-U Pre-U 9795/2 2018 June Q12
21 marks Standard +0.8
12 A uniform \(\operatorname { rod } A B\) has mass 5 kg and length 4 m .
  1. \includegraphics[max width=\textwidth, alt={}, center]{09939c3a-7829-4784-8e6d-ee5356c22cd7-5_529_540_995_840} The rod rests with \(A\) on a rough plane that makes an angle of \(60 ^ { \circ }\) to the horizontal. A string is attached to \(B\) and the rod is in equilibrium in the vertical plane containing the line of greatest slope of the plane, with the string vertical and \(A B\) perpendicular to the plane (see diagram). Find the magnitude of the frictional force at \(A\) and the tension in the string.
  2. \includegraphics[max width=\textwidth, alt={}, center]{09939c3a-7829-4784-8e6d-ee5356c22cd7-5_323_637_1850_794} The rod now rests horizontally with \(A\) in contact with a rough plane that makes an angle of \(60 ^ { \circ }\) with the horizontal and \(B\) in contact with a rough plane that makes an angle of \(30 ^ { \circ }\) with the horizontal (see diagram). The rod and the lines of greatest slope of the two planes are all in the same vertical plane. The coefficients of friction at \(A\) and \(B\) are \(\mu _ { A }\) and \(\mu _ { B }\) respectively. Friction is limiting at both \(A\) and \(B\), with \(A\) on the point of slipping downwards. Show that \(\mu _ { B } = \frac { 1 - \alpha \mu _ { A } } { \alpha + \mu _ { A } }\) where \(\alpha\) is an irrational number to be found.
Pre-U Pre-U 9794/1 2018 June Q1
4 marks Moderate -0.3
1 Solve \(5 x + 3 < | 3 x - 1 |\).
Pre-U Pre-U 9794/1 2018 June Q2
7 marks Moderate -0.3
2 It is given that \(\mathrm { f } ( x ) = 4 + 3 \sqrt { x }\), where \(x \geqslant 0\).
  1. State the range of f .
  2. State the value of \(\mathrm { ff } ( 16 )\).
  3. Find \(\mathrm { f } ^ { - 1 } ( x )\).
  4. On the same axes, sketch the graphs of \(y = \mathrm { f } ( x )\) and \(y = \mathrm { f } ^ { - 1 } ( x )\) and state how the graphs are related.
Pre-U Pre-U 9794/1 2018 June Q3
4 marks Easy -1.2
3 Given that \(z = 1\) is the real root of the equation \(z ^ { 3 } - 1 = 0\), find the two complex roots.
Pre-U Pre-U 9794/1 2018 June Q4
5 marks Moderate -0.3
4
  1. Sketch the graph of \(y = \sec \theta\) for \(0 \leqslant \theta \leqslant 2 \pi\).
  2. Solve \(\sec \theta = \operatorname { cosec } \theta\) for \(0 \leqslant \theta \leqslant 2 \pi\).
Pre-U Pre-U 9794/1 2018 June Q5
10 marks Standard +0.3
5 \includegraphics[max width=\textwidth, alt={}, center]{7895dcbc-2ae0-498f-8770-7b738feed7c9-2_746_1182_1304_479} The diagram shows the curve with equation \(y = x ^ { 3 } + 2 x ^ { 2 } - 13 x + 10\) and the tangent to the curve at the point ( 2,0 ).
  1. Find the equation of this tangent and verify that the tangent intersects the curve when \(x = - 6\).
  2. Calculate the exact area of the region bounded by the curve and the tangent.
Pre-U Pre-U 9794/1 2018 June Q6
8 marks Standard +0.3
6 Two straight lines have equations $$\mathbf { r } = - 3 \mathbf { i } + 11 \mathbf { j } - 9 \mathbf { k } + \lambda ( 4 \mathbf { i } + 7 \mathbf { j } + 8 \mathbf { k } )$$ and $$\mathbf { r } = 21 \mathbf { i } + 2 \mathbf { j } + 15 \mathbf { k } + \mu ( 3 \mathbf { i } + \mathbf { j } + 4 \mathbf { k } )$$
  1. Show that the lines intersect and find the coordinates of their point of intersection.
  2. Find the acute angle between the two lines.
Pre-U Pre-U 9794/1 2018 June Q7
8 marks Standard +0.3
7 Find the coordinates of the two stationary points of the curve $$9 x ^ { 2 } + 4 y ^ { 2 } - 6 x - 4 y = 34$$ showing that one is a maximum and one is a minimum.
Pre-U Pre-U 9794/1 2018 June Q8
7 marks Standard +0.3
8
  1. Using the quotient rule, show that \(\frac { \mathrm { d } } { \mathrm { d } \theta } ( \tan 3 \theta ) = 3 + 3 \tan ^ { 2 } 3 \theta\) for \(- \frac { 1 } { 6 } \pi < \theta < \frac { 1 } { 6 } \pi\).
  2. Hence find the value of \(\int _ { \frac { 1 } { 12 } \pi } ^ { \frac { 1 } { 9 } \pi } \tan ^ { 2 } 3 \theta \mathrm {~d} \theta\), giving your answer in the simplest exact form.
Pre-U Pre-U 9794/1 2018 June Q9
12 marks Standard +0.8
9
  1. Find the coordinates of the stationary point of the curve with equation $$y = \ln x - k x , \text { where } k > 0 \text { and } x > 0$$ and determine its nature.
  2. Hence show that the equation \(\ln x - k x = 0\) has real roots if \(0 < k \leqslant \frac { 1 } { \mathrm { e } }\).
  3. In the particular case that \(k = \frac { 1 } { 3 }\), the equation \(\ln x - k x = 0\) has two roots, one of which is near \(x = 5\). Use the Newton-Raphson process to find, correct to 3 significant figures, the root of the equation \(\ln x - \frac { 1 } { 3 } x = 0\) which is near \(x = 5\).
  4. Show that the equation \(\ln x - k x = 0\) has one real root if \(k \leqslant 0\).
  5. Explain why the equation \(\ln x - k x = 0\) has two distinct real roots if \(0 < k < \frac { 1 } { \mathrm { e } }\).
Pre-U Pre-U 9794/1 2018 June Q10
12 marks Standard +0.8
10
  1. Using partial fractions, find the general solution of the differential equation $$2 \frac { \mathrm {~d} y } { \mathrm {~d} x } = y - y ^ { 3 } \text { for } 0 < y < 1$$ giving your solution in the form \(y = \mathrm { f } ( x )\).
  2. Determine \(\lim _ { x \rightarrow - \infty } \mathrm { f } ( x )\) and \(\lim _ { x \rightarrow + \infty } \mathrm { f } ( x )\).
Pre-U Pre-U 9794/2 2018 June Q1
4 marks Easy -1.2
1 A geometric progression \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by \(u _ { 1 } = 32\) and \(u _ { n + 1 } = 0.75 u _ { n }\) for \(n \geqslant 1\).
  1. Find \(u _ { 5 }\).
  2. Find \(\sum _ { n = 1 } ^ { \infty } u _ { n }\).
Pre-U Pre-U 9794/2 2018 June Q2
11 marks Standard +0.3
2
  1. Express \(2 x ^ { 2 } + 6 x + 5\) in the form \(p ( x + q ) ^ { 2 } + r\).
  2. State the equation of the line of symmetry of the curve \(y = 2 x ^ { 2 } + 6 x + 5\).
  3. Find the value of the constant \(k\) for which the line \(y = k - 2 x\) is a tangent to the curve \(y = 2 x ^ { 2 } + 6 x + 5\).
Pre-U Pre-U 9794/2 2018 June Q3
11 marks Moderate -0.8
3 Solve the equation \(6 ^ { 2 x - 1 } = 3 ^ { x + 2 }\), giving your answer in the form \(x = \frac { \ln a } { \ln b }\) where \(a\) and \(b\) are integers.
Pre-U Pre-U 9794/2 2018 June Q4
12 marks Moderate -0.3
4 Solve the equation \(x + 2 \sqrt { x } - 6 = 0\), giving your answer in the form \(x = c + d \sqrt { 7 }\) where \(c\) and \(d\) are integers.
Pre-U Pre-U 9794/2 2018 June Q5
10 marks Standard +0.3
5 The complex numbers \(u\) and \(v\) are given by \(u = 3 + 2 \mathrm { i }\) and \(v = 1 + 4 \mathrm { i }\).
  1. Given that \(a u ^ { 2 } + b v ^ { * } = 7 + 36 \mathrm { i }\) find the values of the real constants \(a\) and \(b\).
  2. Show the points representing \(u\) and \(v\) on an Argand diagram and hence sketch the locus given by \(| z - u | = | z - v |\). Find the point of intersection of this locus with the imaginary axis.