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Pre-U Pre-U 9795/2 2016 Specimen Q7
Moderate -0.3
7 A child of mass 20 kg slides down a rough slope of length 16 m against a constant frictional force \(F \mathrm {~N}\). Starting with an initial speed of \(2 \mathrm {~ms} ^ { - 1 }\) at a point 8 m above the ground, she reaches the ground with a speed of \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the value of \(F\).
Pre-U Pre-U 9795/2 2016 Specimen Q8
Standard +0.3
8 \includegraphics[max width=\textwidth, alt={}, center]{c4bbba86-2968-4247-b300-357217cf213b-4_670_819_548_621} A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(l\). The other end of the string is attached to a fixed point \(A\). The particle moves with constant angular speed \(\omega\) in a horizontal circle whose centre is at a distance \(h\) vertically below \(A\) (see diagram).
  1. Find the tension in the string in terms of \(m , l\) and \(\omega\).
  2. Show that \(\omega ^ { 2 } h = g\).
  3. Deduce an expression in terms of \(g\) and \(h\) for the time taken for \(P\) to complete one full circle during its motion.
Pre-U Pre-U 9795/2 2016 Specimen Q9
8 marks Standard +0.3
9 The diagram shows a uniform rod \(A B\) of length 40 cm and mass 2 kg placed with the end \(A\) resting against a smooth vertical wall and the end \(B\) on rough horizontal ground. The angle between \(A B\) and the horizontal is \(60 ^ { \circ }\). \includegraphics[max width=\textwidth, alt={}, center]{c4bbba86-2968-4247-b300-357217cf213b-4_657_647_1923_708} Given that the value of the coefficient of friction between the rod and the ground is 0.2 , determine whether the rod slips.
Pre-U Pre-U 9795/2 2016 Specimen Q10
9 marks Standard +0.3
10 A cyclist and her bicycle have a combined mass of 90 kg and she is riding along a straight horizontal road. She is working at a constant power of 75 W . At time \(t\) seconds her speed is \(v \mathrm {~ms} ^ { - 1 }\) and the resistance to motion is \(k v \mathrm {~N}\), where \(k\) is a constant.
  1. Given that the steady speed at which the cyclist can move is \(10 \mathrm {~ms} ^ { - 1 }\), show that \(k = \frac { 3 } { 4 }\).
  2. Show that $$\frac { 25 } { v } - \frac { v } { 4 } = 30 \frac { \mathrm {~d} v } { \mathrm {~d} t } .$$
  3. Find the time taken for the cyclist to accelerate from a speed of \(3 \mathrm {~ms} ^ { - 1 }\) to a speed of \(7 \mathrm {~ms} ^ { - 1 }\).
Pre-U Pre-U 9795/2 2016 Specimen Q11
12 marks Challenging +1.8
11 \includegraphics[max width=\textwidth, alt={}, center]{c4bbba86-2968-4247-b300-357217cf213b-5_432_949_909_557} A smooth sphere \(P\) of mass \(3 m\) is at rest on a smooth horizontal table. A second smooth sphere \(Q\) of mass \(m\) and the same radius as \(P\) is moving along the table towards \(P\) and strikes it obliquely (see diagram). After the collision, the directions of motion of the two spheres are perpendicular.
  1. Find the coefficient of restitution.
  2. Given that one-sixth of the original kinetic energy is lost as a result of the collision, find the angle between the initial direction of motion of \(Q\) and the line of centres.
Pre-U Pre-U 9795/2 2016 Specimen Q12
12 marks Challenging +1.2
12 A particle is projected from the origin with speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\alpha\) above the horizontal.
  1. Prove that the equation of its trajectory is $$y = x \tan \alpha - \frac { x ^ { 2 } } { 80 } \left( 1 + \tan ^ { 2 } \alpha \right) .$$
  2. Regarding the equation of the trajectory as a quadratic equation in \(\tan \alpha\), show that \(\tan \alpha\) has real values provided that $$y \leqslant 20 - \frac { x ^ { 2 } } { 80 } .$$
  3. A plane is inclined at an angle of \(30 ^ { \circ }\) to the horizontal. The line \(l\), with equation \(y = x \tan 30 ^ { \circ }\), is a line of greatest slope in the plane. The particle is projected from the origin with speed \(20 \mathrm {~ms} ^ { - 1 }\) from a point on the plane, in the vertical plane containing \(l\). By considering the intersection of \(l\) with the curve \(y = 20 - \frac { x ^ { 2 } } { 80 }\), find the maximum range up this inclined plane.
Pre-U Pre-U 9795/2 2016 Specimen Q13
11 marks Challenging +1.8
13 Two light strings, each of natural length \(l\) and modulus of elasticity \(6 m g\), are attached at their ends to a particle \(P\) of mass \(m\). The other ends of the strings are attached to two fixed points \(A\) and \(B\), which are at a distance \(6 l\) apart on a smooth horizontal table. Initially \(P\) is at rest at the mid-point of \(A B\). The particle is now given a horizontal impulse in the direction perpendicular to \(A B\). At time \(t\) the displacement of \(P\) from the line \(A B\) is \(x\).
  1. Show that the tension in each string is \(\frac { 6 m g } { l } \left( \sqrt { 9 l ^ { 2 } + x ^ { 2 } } - l \right)\).
  2. Show that $$\ddot { x } = - \frac { 12 g x } { l } \left( 1 - \frac { l } { \sqrt { 9 l ^ { 2 } + x ^ { 2 } } } \right) .$$
  3. Given that throughout the motion \(\frac { x ^ { 2 } } { l ^ { 2 } }\) is small enough to be negligible, show that the equation of motion is approximately $$\ddot { x } = - \frac { 8 g x } { l } .$$
  4. Given that the initial speed of \(P\) is \(\sqrt { \frac { g l } { 200 } }\), find the time taken for the particle to travel a distance of \(\frac { 1 } { 80 } l\).
Pre-U Pre-U 9795/2 2017 June Q1
6 marks Standard +0.3
1
  1. Explain the meaning of the term ' \(95 \%\) confidence interval'.
  2. The values of five independent observations of a normally distributed random variable are as follows. $$\begin{array} { l l l l l } 35.2 & 38.2 & 39.7 & 41.6 & 43.9 \end{array}$$ Obtain a 95\% confidence interval for the population mean.
Pre-U Pre-U 9795/2 2017 June Q2
13 marks Standard +0.8
2 A discrete random variable \(X\) has the following probability distribution.
\(x\)- 12
\(\mathrm { P } ( X = x )\)\(\frac { 1 } { 3 }\)\(\frac { 2 } { 3 }\)
  1. Write down the probability generating function of \(X\).
  2. \(T\) is the sum of ten independent observations of \(X\). Use the probability generating function of \(T\) to find
    1. \(\mathrm { E } ( T )\),
    2. \(\mathrm { P } ( T = 8 )\).
Pre-U Pre-U 9795/2 2017 June Q3
8 marks Standard +0.3
3 In a random sample of 100 voters from a constituency, 32 said that they would support the Cyan Party.
  1. Find an approximate \(99 \%\) confidence interval for the proportion of voters in the constituency who would support the Cyan Party.
  2. Using the given sample proportion, estimate the smallest size of sample needed for the width of a \(99 \%\) confidence interval to be less than 0.04 .
Pre-U Pre-U 9795/2 2017 June Q4
14 marks Standard +0.3
4 A continuous random variable \(X\) has probability density function $$\mathrm { f } ( x ) = \begin{cases} a & - 1 \leqslant x < 0 \\ a \left( 1 - x ^ { 2 } \right) & 0 \leqslant x \leqslant 1 \\ 0 & \text { otherwise } \end{cases}$$ where \(a\) is a constant.
  1. Find the value of \(a\).
  2. Find the cumulative distribution function of \(X\).
  3. Determine whether the upper quartile is greater than or less than 0.25 .
Pre-U Pre-U 9795/2 2017 June Q5
8 marks Standard +0.3
5 The number of calls to a car breakdown service during any one hour of the day is modelled by the distribution \(\operatorname { Po } ( 20 )\).
  1. Find the probability that in a randomly chosen 12 -minute period there are at least 7 calls to the service.
  2. Find the period of time, correct to the nearest second, for which the probability that no calls are made to the service is 0.6 .
  3. Use a suitable approximation to find the probability that, in a randomly chosen 3-hour period, there are no more than 65 calls to the service.
Pre-U Pre-U 9795/2 2017 June Q6
11 marks Standard +0.8
6 The random variable \(X\) has a uniform distribution on the interval \([ - 1,1 ]\), so that its probability density function is given by $$f ( x ) = \begin{cases} \frac { 1 } { 2 } & - 1 \leqslant x \leqslant 1 \\ 0 & \text { otherwise } \end{cases}$$
  1. Show from the definition of the moment generating function that the moment generating function of \(X\) is \(\frac { \sinh t } { t }\).
  2. By using the series expansion of \(\sinh t\), find the variance of \(X\) and the value of \(\mathrm { E } \left( X ^ { 4 } \right)\).
Pre-U Pre-U 9795/2 2017 June Q7
9 marks Challenging +1.2
7 The total mass of a can of pears is the sum of three independent random variables: the mass of pears, the mass of juice, and the mass of the container. The mass in grams of pears in a can has the distribution \(\mathrm { N } ( 300,400 )\). The mass in grams of juice has the distribution \(\mathrm { N } ( 200,60 )\). The mass in grams of the container has the distribution \(\mathrm { N } ( 70,10 )\).
  1. Find the probability that the total mass of a randomly chosen can is less than 530 g .
  2. Find the probability that the mass of the container of a randomly chosen can is more than one eighth of the total mass of the can.
Pre-U Pre-U 9795/2 2017 June Q8
5 marks Moderate -0.3
8 A horizontal turntable rotates about a vertical axis. Starting from rest, it accelerates uniformly to an angular velocity of \(8.4 \mathrm { rad } \mathrm { s } ^ { - 1 }\) in 2 s .
  1. Find the angular acceleration of the turntable.
  2. A particle rests on the turntable at a distance of 0.15 m from the axis. Find the radial and transverse components of the acceleration of the particle when the angular velocity is \(1.2 \mathrm { rad } \mathrm { s } ^ { - 1 }\). Find also the magnitude of the acceleration at this instant.
Pre-U Pre-U 9795/2 2017 June Q9
7 marks Standard +0.8
9 A particle is projected from a point \(O\) on horizontal ground with speed \(40 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\theta\) above the horizontal.
  1. Write down the equation of the trajectory, in terms of \(\tan \theta\).
  2. The particle passes through a point whose horizontal and vertical distances from \(O\) are 72 m and \(y \mathrm {~m}\) respectively. By considering the equation of the trajectory as a quadratic equation in \(\tan \theta\), or otherwise, find the greatest possible value of \(y\).
Pre-U Pre-U 9795/2 2017 June Q10
8 marks Moderate -0.3
10 The engine of a lorry of mass 4000 kg works at a constant rate of 75 kW . Resistance to motion is modelled by a constant resistive force. On a horizontal road the lorry travels at a constant speed of \(24 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Find the work done by the engine in travelling for 1 minute on the horizontal road.
  2. The lorry travels at a constant speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) up a slope of angle \(\sin ^ { - 1 } 0.05\) to the horizontal. Find the value of \(v\).
Pre-U Pre-U 9795/2 2017 June Q11
7 marks Standard +0.3
11 \includegraphics[max width=\textwidth, alt={}, center]{22640c3b-792f-4003-a4f8-78220efd73b0-4_280_1002_1722_568} A non-uniform \(\operatorname { rod } A B\) of mass 1.6 kg and length 1.25 m has its centre of mass at \(G\) where \(A G = 0.4 \mathrm {~m}\). The rod rests on a rough horizontal table. A force \(P \mathrm {~N}\) is applied at \(B\), acting at an angle \(\alpha\) above the horizontal, such that the rod is in equilibrium but about to rotate about \(A\) (see diagram).
  1. Assume that the rod is in contact with the table only at \(A\). By taking moments about \(A\), show that \(P \sin \alpha = 5.12\).
  2. The coefficient of friction between the rod and the table is \(\frac { 6 } { 17 }\). Show that \(P \leqslant 6.4\).
Pre-U Pre-U 9795/2 2017 June Q12
9 marks Challenging +1.2
12 A particle of mass 0.6 kg is projected vertically upwards from horizontal ground, with initial speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The upwards velocity at any instant is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and the displacement is \(x \mathrm {~m}\). Air resistance is modelled by a force \(0.024 v ^ { 2 } \mathrm {~N}\) acting downwards.
  1. Show that \(v\) and \(x\) satisfy the differential equation $$v \frac { \mathrm {~d} v } { \mathrm {~d} x } = - 10 - 0.04 v ^ { 2 } .$$
  2. Find the value of \(u\) if the maximum height reached is 50 m .
Pre-U Pre-U 9795/2 2017 June Q13
9 marks Challenging +1.2
13 A fairground game consists of a small ball fixed to one end of a light inextensible string of length 0.6 m . The other end of the string is fixed to a point \(O\). A small bell is fixed 0.6 m vertically above \(O\). Initially the ball hangs vertically in equilibrium. The object of the game is to project the ball with an initial horizontal velocity \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) so that it moves in a vertical circle and hits the bell.
  1. Find the smallest possible value of \(u\) for which the ball hits the bell.
  2. Given, instead, that the value of \(u\) is 5 , find the angle made by the string with the upward vertical at the moment when the string becomes slack.
Pre-U Pre-U 9795/2 2017 June Q14
9 marks Challenging +1.2
14 \includegraphics[max width=\textwidth, alt={}, center]{22640c3b-792f-4003-a4f8-78220efd73b0-5_86_1589_1297_278} A particle of mass 0.05 kg is attached to two identical light elastic strings, each of natural length 1.2 m and modulus of elasticity 0.6 N . The other ends of the strings are attached to points \(A\) and \(E\) on a smooth horizontal table. The distance \(A E\) is 2 m and points \(B , C\) and \(D\) lie between \(A\) and \(E\) so that \(A B = 0.7 \mathrm {~m} , B C = 0.1 \mathrm {~m} , C D = 0.4 \mathrm {~m}\) and \(D E = 0.8 \mathrm {~m}\) (see diagram). Initially the particle is held at \(B\) and it is then released. In the subsequent motion the displacement of the particle from \(C\), in the direction of \(A\), is denoted by \(x \mathrm {~m}\).
  1. Find the equation of motion for the particle when it is between \(B\) and \(C\).
  2. Find the velocity of the particle when it is at \(C\).
  3. Find the total time that elapses before the particle first returns to \(B\).
Pre-U Pre-U 9795/1 2017 June Q1
4 marks Moderate -0.8
1 Without using a calculator, determine the possible values of \(a\) and \(b\) for which \(( a + \mathrm { i } b ) ^ { 2 } = 21 - 20 \mathrm { i }\).
Pre-U Pre-U 9795/1 2017 June Q2
4 marks Standard +0.3
2 The equation \(x ^ { 3 } + 2 x ^ { 2 } + 3 x + 7 = 0\) has roots \(\alpha , \beta\) and \(\gamma\). Evaluate \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 }\) and use your answer to comment on the nature of these roots.
Pre-U Pre-U 9795/1 2017 June Q3
6 marks Standard +0.8
3
  1. Sketch the curve with polar equation \(r = \frac { 1 } { 1 + \theta } , 0 \leqslant \theta \leqslant 2 \pi\).
  2. Find, in terms of \(\pi\), the area of the region enclosed by the curve and the part of the initial line between the endpoints of the curve.
Pre-U Pre-U 9795/1 2017 June Q4
7 marks Challenging +1.8
4 The curve \(C\) has parametric equations \(x = \frac { 1 } { 2 } t ^ { 2 } - \ln t , y = 2 t\), for \(1 \leqslant t \leqslant 4\). When \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis, a surface of revolution is formed of surface area \(S\). Determine the exact value of \(S\).