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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.
Pre-U Pre-U 9794/2 2018 June Q6
12 marks Standard +0.8
6 \includegraphics[max width=\textwidth, alt={}, center]{f4b66aaa-16b9-4b15-b3f5-b9657fe98274-3_545_557_269_794} The diagram shows a sector \(A O B\) of a circle, centre \(O\) and radius \(r\). Angle \(A O B\) is \(\theta\) radians. The point \(C\) lies on \(O B\), and \(A C\) is perpendicular to \(O B\). The area of the triangle \(A O C\) is equal to the area of the segment bounded by the chord \(A B\) and the \(\operatorname { arc } A B\).
  1. Show that \(\theta = \sin \theta ( 1 + \cos \theta )\). The equation \(\theta = \sin \theta ( 1 + \cos \theta )\) has only one positive root.
  2. Use an iterative process based on this equation to find the value of the root correct to 3 significant figures. Use a starting value of 1 and show the result of each iteration. Use a change of sign to verify that the value you have found is correct to 3 significant figures.
Pre-U Pre-U 9794/2 2018 June Q7
10 marks Standard +0.8
7 A curve is given parametrically by \(x = t ^ { 2 } + 1 , y = t ^ { 3 } - 2 t\) where \(t\) is any real number.
  1. Show that the equation of the normal to the curve at the point where \(t = 2\) can be written in the form \(2 x + 5 y = 30\).
  2. Show that this normal does not meet the curve again.
Pre-U Pre-U 9794/2 2018 June Q8
8 marks Standard +0.3
8
  1. Use integration by parts twice to show that $$\int \mathrm { e } ^ { x } \sin x \mathrm {~d} x = \frac { 1 } { 2 } \mathrm { e } ^ { x } ( \sin x - \cos x ) + c .$$
  2. Hence find the equation of the curve which passes through the point \(( 0,2 )\) and for which \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { e } ^ { x } \sin x\).
Pre-U Pre-U 9794/2 2018 June Q9
13 marks Standard +0.8
9 In this question, \(x\) denotes an angle measured in degrees.
  1. Express \(4 \sin \left( 2 x + 30 ^ { \circ } \right) + 3 \cos 2 x\) in the form \(R \cos ( 2 x - \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\).
  2. Give full details of the sequence of transformations which maps the graph of \(y = \cos x\) onto the graph of \(y = 4 \sin \left( 2 x + 30 ^ { \circ } \right) + 3 \cos 2 x\).
  3. Find the smallest positive value of \(x\) that satisfies the equation \(4 \sin \left( 2 x + 30 ^ { \circ } \right) + 3 \cos 2 x = 6\).
Pre-U Pre-U 9794/2 2018 June Q10
10 marks Challenging +1.2
10
  1. By using the substitution \(u = 3 - 2 x\), or otherwise, show that \(\int _ { 0 } ^ { 1 } \left( \frac { 4 x } { 3 - 2 x } \right) ^ { 2 } \mathrm {~d} x = 16 - 12 \ln 3\).
  2. \includegraphics[max width=\textwidth, alt={}, center]{f4b66aaa-16b9-4b15-b3f5-b9657fe98274-4_595_588_927_817} The diagram shows the region \(R\), which is bounded by the curve \(y = \frac { 4 x } { 3 - 2 x }\), the \(y\)-axis and the line \(y = 4\). Find the exact volume generated when the region \(R\) is rotated completely around the \(x\)-axis. {www.cie.org.uk} after the live examination series. }
Pre-U Pre-U 9794/3 2018 June Q2
9 marks Moderate -0.3
2 A teacher is monitoring the progress of students. The length of time, \(x\) hours, spent revising in a given week is compared to the score, \(y\), achieved in an assessment at the end of the week. The scatter diagram for a random sample of 8 students is shown below. \includegraphics[max width=\textwidth, alt={}, center]{35d24778-1203-4d5d-be4b-bb375344fe09-2_866_967_715_589} The data are summarised as \(\Sigma x = 24.6 , \Sigma y = 404 , \Sigma x ^ { 2 } = 105.56 , \Sigma y ^ { 2 } = 20820\) and \(\Sigma x y = 1350.2\).
  1. Find the equation of the least squares regression line of \(y\) on \(x\).
  2. Calculate the product moment correlation coefficient for the data.
  3. A ninth student, Jane, revises for 1.5 hours.
    1. Estimate her score in the assessment.
    2. Comment on the reliability of this estimate.
Pre-U Pre-U 9794/3 2018 June Q3
5 marks Easy -1.2
3 John plays a game with two unbiased coins. John tosses the coins. If he gets two heads he wins \(\pounds 1\). If he gets two tails he wins 20 p. If he gets one head and one tail he wins nothing. Let \(X\) be the random variable for the amount of money, in pence, John wins per game.
  1. Construct a probability distribution table for \(X\).
  2. Calculate \(\mathrm { E } ( X )\).
  3. John pays \(s\) pence to play the game. State the values of \(s\) for which John should expect to make a loss.
Pre-U Pre-U 9794/3 2018 June Q4
6 marks Moderate -0.3
4 On a particular day at a busy international airport, 75\% of the scheduled flights depart on time. A random sample of 16 flights is chosen.
  1. Find the expected number of flights that depart on time.
  2. For these 16 flights, find the probability that fewer than 14 flights depart on time.
  3. For these 16 flights, the probability that at least \(k\) flights depart on time is greater than 0.9 . Find the largest possible value of \(k\).