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Pre-U Pre-U 9794/2 2017 June Q6
7 marks Moderate -0.3
6 Find the solution of the differential equation $$x y ^ { 2 } \frac { \mathrm {~d} y } { \mathrm {~d} x } = x + 1$$ given that \(y = 3\) when \(x = 1\). Give your answer in the form \(y = \mathrm { f } ( x )\).
Pre-U Pre-U 9794/2 2017 June Q7
10 marks Standard +0.3
7 A curve, \(C\), is given parametrically by \(x = 2 \cos \theta , y = 3 \sin \theta , 0 < \theta < \frac { 1 } { 2 } \pi\).
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { 3 } { 2 } \cot \theta\). A tangent to \(C\) intersects the \(x\)-axis and \(y\)-axis at \(P\) and \(Q\) respectively.
  2. Show that the midpoint of \(P Q\) has coordinates \(\left( \sec \theta , \frac { 3 } { 2 } \operatorname { cosec } \theta \right)\).
  3. Hence show that the midpoint of \(P Q\) lies on the curve \(\frac { 4 } { x ^ { 2 } } + \frac { 9 } { y ^ { 2 } } = 4\).
Pre-U Pre-U 9794/2 2017 June Q8
10 marks Standard +0.3
8
  1. Express \(\frac { 7 x ^ { 2 } - 12 x + 1 } { \left( x ^ { 2 } + 1 \right) ( x - 2 ) }\) in the form \(\frac { A x + B } { x ^ { 2 } + 1 } + \frac { C } { x - 2 }\) where \(A , B\) and \(C\) are constants to be found.
  2. Hence find the exact value of \(\int _ { 0 } ^ { 1 } \frac { 7 x ^ { 2 } - 12 x + 1 } { \left( x ^ { 2 } + 1 \right) ( x - 2 ) } \mathrm { d } x\).
Pre-U Pre-U 9794/2 2017 June Q9
12 marks Standard +0.8
9
  1. Show that \(\int x ( x - 2 ) ^ { \frac { 3 } { 2 } } \mathrm {~d} x = \frac { 2 } { 35 } ( 5 x + 4 ) ( x - 2 ) ^ { \frac { 5 } { 2 } } + c\).
  2. Hence find the coordinates of the stationary points of the curve $$y = \frac { 2 } { 35 } ( 5 x + 4 ) ( x - 2 ) ^ { \frac { 5 } { 2 } } + x ^ { 2 } - \frac { 1 } { 3 } x ^ { 3 }$$
Pre-U Pre-U 9794/2 2017 June Q10
11 marks Challenging +1.2
10 An arithmetic sequence and a geometric sequence have \(n\)th terms \(a _ { n }\) and \(g _ { n }\) respectively, where \(n = 1,2,3 , \ldots\). It is given that \(a _ { 1 } = g _ { 1 } , a _ { 2 } = g _ { 2 } , a _ { 5 } = g _ { 3 } , a _ { 1 } \neq a _ { 2 }\) and \(a _ { 1 } \neq 0\).
  1. Show that the common ratio of the geometric sequence is 3 .
  2. Find the common difference of the arithmetic sequence in terms of \(a _ { 1 }\).
  3. Let \(a _ { 1 } = g _ { 1 } = 5\).
    1. Find the first three terms of both sequences.
    2. Show that every term of the geometric sequence is also a term of the arithmetic sequence.
Pre-U Pre-U 9794/3 2017 June Q1
5 marks Moderate -0.8
1 Levels of nitrogen dioxide in the atmosphere are being monitored at the side of a road in a busy city centre. A sample of 18 measurements taken (in suitable units) is as follows. $$\begin{array} { l l l l l l l l l l l l l l l l l l } 83 & 44 & 95 & 92 & 98 & 63 & 69 & 76 & 19 & 91 & 70 & 91 & 74 & 65 & 62 & 70 & 95 & 108 \end{array}$$
  1. Find the mean and standard deviation of the sample.
  2. Hence identify, with justification, any possible outliers.
Pre-U Pre-U 9794/3 2017 June Q2
9 marks Moderate -0.8
2 The table shows the turnover, in millions of pounds, of a small company at 3-year intervals over a period of 15 years, starting in 2000.
Year since 200003691215
Turnover ( \(\pounds\) millions)2.302.943.373.974.936.13
  1. For the information in the table find the equation of the least squares regression line of \(y\) on \(x\), where \(x\) is the year since 2000 and \(y\) is the turnover in millions of pounds.
  2. Use the equation of the regression line to calculate the residual for 2009.
  3. Use the equation of the regression line to estimate the turnover in 2024, and explain why it is inadvisable to rely on this estimate.
Pre-U Pre-U 9794/3 2017 June Q3
8 marks Standard +0.3
3 The probability distribution of the discrete random variable \(X\) is defined as follows. $$\mathrm { P } ( X = x ) = k ( 2 + x ) ( 5 - x ) \quad \text { for } x = 0,1,2,3,4$$
  1. Show that \(k = \frac { 1 } { 50 }\).
  2. Find the variance of \(X\).
  3. Find \(\mathrm { P } ( X = 4 \mid X > 0 )\).
Pre-U Pre-U 9794/3 2017 June Q4
9 marks Moderate -0.3
4 The letters of the word 'STATISTICS' are to be rearranged.
  1. How many distinct arrangements are there?
  2. How many of the arrangements start and end with the letter S ?
  3. What is the probability that, in a randomly chosen arrangement, the S's are all together?
Pre-U Pre-U 9794/3 2017 June Q5
9 marks Standard +0.3
5 The random variable \(X\) has a geometric distribution: \(X \sim \operatorname { Geo } ( p )\).
  1. Show that \(\mathrm { P } ( X > n ) = q ^ { n }\), where \(q = 1 - p\). You are given that \(\mathrm { P } ( X \geqslant 4 ) = 0.216\).
  2. Use the result given in part (i) to find the value of \(p\) and \(\mathrm { P } ( X \leqslant 8 )\).
  3. Write down \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
Pre-U Pre-U 9794/3 2017 June Q6
11 marks Moderate -0.3
6 A crate, which has a mass of 220 kg , is being lowered on the end of a cable onto the back of a lorry.
  1. Draw a diagram to show the forces acting on the crate. The crate is lowered in three stages.
    Stage 1 It starts from rest and accelerates at \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) until it reaches a speed of \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
    Stage 2 It descends at a constant speed of \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
    Stage 3 It decelerates at \(0.75 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) and eventually comes to rest.
  2. Find the tension in the cable in each of the three stages.
  3. Sketch the velocity-time graph for the complete downward motion of the crate.
  4. The crate is lowered 15 m altogether. By considering your velocity-time graph, find the total time taken.
Pre-U Pre-U 9794/3 2017 June Q7
9 marks Moderate -0.8
7 A building 33.8 m high stands on horizontal ground. A particle is projected horizontally from the top of the building and hits the ground 31.2 m away.
  1. Find the initial speed of the particle.
  2. Find the magnitude and direction of the velocity of the particle when it hits the ground.
Pre-U Pre-U 9794/3 2017 June Q8
6 marks Standard +0.3
8 An object of weight 16 N is supported in equilibrium by a force of \(P \mathrm {~N}\) at \(30 ^ { \circ }\) to the vertical and by another of 10 N at \(\theta ^ { \circ }\) to the vertical as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{85c5c346-8eb5-47ea-b94e-80b1a0038ce1-4_549_483_397_831}
  1. Draw a triangle to show that the forces acting on the object are in equilibrium.
  2. Find the two possible values of \(\theta\) and the corresponding values of \(P\).
Pre-U Pre-U 9794/3 2017 June Q9
8 marks Moderate -0.8
9 A particle moves along a straight line such that its displacement from \(O\), a fixed point on the line, is \(x\). The particle travels from rest from the point \(P\), where \(x = 2\), to the point \(Q\), where \(x = 5.6\). All distances are in metres. Two models for the motion of the particle are proposed.
  1. In Model 1, the acceleration of the particle is assumed to be constant and the particle takes 18 seconds to travel from \(P\) to \(Q\). Find the velocity of the particle when it reaches \(Q\).
  2. In Model 2, the velocity after \(t\) seconds is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(v = \frac { 1 } { 270 } \left( 18 t - t ^ { 2 } \right)\).
    1. Write down the values of \(t\) when \(v = 0\).
    2. Show that \(x = 5.6\) when \(t = 18\).
    3. The particle represents a fragile instrument that is being moved from \(P\) to \(Q\) across a laboratory. Explain why Model 2 might be more appropriate than Model 1.
Pre-U Pre-U 9794/3 2017 June Q10
5 marks Moderate -0.8
10 A cyclist travelling at a steady speed of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) passes a bus which is at rest at a bus stop. 5 seconds later the bus sets off following the cyclist and accelerating at \(\frac { 1 } { 2 } \mathrm {~m} \mathrm {~s} ^ { - 2 }\). How soon after setting off does the bus catch up with the cyclist? How fast is the bus going at this time? {www.cie.org.uk} after the live examination series. }
Pre-U Pre-U 9795/2 2018 June Q1
Moderate -0.3
1
  1. The random variable \(X\) has the distribution \(\mathrm { B } ( 200,0.2 )\). Use a suitable approximation to find \(\mathrm { P } ( X \leqslant 30 )\).
  2. The random variable \(Y\) has the distribution \(\mathrm { B } ( 200,0.02 )\). Use a suitable approximation to find \(\mathrm { P } ( Y \leqslant 3 )\).
Pre-U Pre-U 9795/2 2018 June Q2
Moderate -0.8
2 Secret radio messages received under difficult conditions are subject to errors caused by random instantaneous breaks in transmission. The number of errors caused by breaks in transmission in a 10-minute period is denoted by \(B\).
  1. State two conditions, other than randomness, needed for a Poisson distribution to be a suitable model for \(B\). Assume now that \(B \sim \mathrm { Po } ( 5 )\).
  2. Calculate the probability that in a 15-minute period there are between 6 and 10 errors, inclusive, caused by random breaks in transmission. Secret radio messages are also subject to errors caused by mistakes made by the sender. The number of errors caused by mistakes made by the sender in a 10 -minute period, \(M\), has the independent distribution \(\operatorname { Po } ( 8 )\).
  3. Calculate the period of time, in seconds, for which the probability that a message contains no errors of either sort is 0.6 .
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).