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CAIE FP2 2014 June Q6
Easy -1.8
6 Employees at a particular company have been working seven hours each day, from 9 am to 4 pm . To try to reduce absence, the company decides to introduce 'flexi-time' and allow employees to work their seven hours each day at any time between 7 am and 9 pm . For a random sample of 10 employees, the numbers of hours of absence in the year before and the year after the introduction of flexi-time are given in the following table.
Employee\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)\(H\)\(I\)\(J\)
Before4235967420578451460
After34321007231261351400
Use a paired sample \(t\)-test to test, at the \(10 \%\) significance level, whether the population mean number of hours of absence has decreased, following the introduction of flexi-time.
CAIE FP2 2014 June Q7
Easy -1.8
7 James throws a discus repeatedly in an attempt to achieve a successful throw. A throw is counted as successful if the distance achieved is over 40 metres. For each throw, the probability that James is successful is \(\frac { 1 } { 4 }\), independently of all other throws. Find the probability that James takes
  1. exactly 5 throws to achieve the first successful throw,
  2. more than 8 throws to achieve the first successful throw. In order to qualify for a competition, a discus-thrower must throw over 40 metres within at most six attempts. When a successful throw is achieved, no further throws are taken. Find the probability that James qualifies for the competition. Colin is another discus-thrower. For each throw, the probability that he will achieve a throw over 40 metres is \(\frac { 1 } { 3 }\), independently of all other throws. Find the probability that exactly one of James and Colin qualifies for the competition.
CAIE FP2 2014 June Q8
Easy -4.0
8 A random sample of 200 is taken from the adult population of a town and classified by age-group and preferred type of car. The results are given in the following table.
HatchbackEstateConvertible
Under 25 years321117
Between 25 and 50 years45246
Over 50 years311618
Test, at the \(5 \%\) significance level, whether preferred type of car is independent of age-group.
CAIE FP2 2014 June Q9
Easy -3.0
9 The continuous random variable \(X\) has distribution function F given by $$\mathrm { F } ( x ) = \begin{cases} 0 & x < 2 , \\ \frac { 1 } { 8 } x - \frac { 1 } { 4 } & 2 \leqslant x \leqslant 10 , \\ 1 & x > 10 . \end{cases}$$ Find the value of \(k\) for which \(\mathrm { P } ( X \geqslant k ) = 0.6\). The random variable \(Y\) is defined by \(Y = 2 \ln X\). Find the distribution function of \(Y\). Find the probability density function of \(Y\) and sketch its graph.
CAIE FP2 2013 November Q8
Standard +0.3
8 The lifetime, in years, of an electrical component is the random variable \(T\), with probability density function f given by $$\mathrm { f } ( t ) = \begin{cases} A \mathrm { e } ^ { - \lambda t } & t \geqslant 0 \\ 0 & \text { otherwise } \end{cases}$$ where \(A\) and \(\lambda\) are positive constants.
  1. Show that \(A = \lambda\). It is known that out of 100 randomly chosen components, 16 failed within the first year.
  2. Find an estimate for the value of \(\lambda\), and hence find an estimate for the median value of \(T\).
CAIE FP2 2013 November Q10
Easy -2.0
10 Customers were asked which of three brands of coffee, \(A , B\) and \(C\), they prefer. For a random sample of 80 male customers and 60 female customers, the numbers preferring each brand are shown in the following table.
\(A\)\(B\)\(C\)
Male323612
Female183012
Test, at the \(5 \%\) significance level, whether there is a difference between coffee preferences of male and female customers. A larger random sample is now taken. It consists of \(80 n\) male customers and \(60 n\) female customers, where \(n\) is a positive integer. It is found that the proportions choosing each brand are identical to those in the smaller sample. Find the least value of \(n\) that would lead to a different conclusion for the 5\% significance level hypothesis test.
CAIE M1 2017 June Q1
5 marks Moderate -0.8
1 A man pushes a wheelbarrow of mass 25 kg along a horizontal road with a constant force of magnitude 35 N at an angle of \(20 ^ { \circ }\) below the horizontal. There is a constant resistance to motion of 15 N . The wheelbarrow moves a distance of 12 m from rest.
  1. Find the work done by the man.
  2. Find the speed attained by the wheelbarrow after 12 m .
CAIE M1 2017 June Q2
5 marks Moderate -0.3
2 \includegraphics[max width=\textwidth, alt={}, center]{3d7f53af-dbf2-499b-9966-ae85514cef02-03_522_604_262_769} The four coplanar forces shown in the diagram are in equilibrium. Find the values of \(P\) and \(\theta\).
CAIE M1 2017 June Q3
6 marks Standard +0.3
3 A train travels between two stations, \(A\) and \(B\). The train starts from rest at \(A\) and accelerates at a constant rate for \(T\) s until it reaches a speed of \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It then travels at this constant speed before decelerating at a constant rate, coming to rest at \(B\). The magnitude of the train's deceleration is twice the magnitude of its acceleration. The total time taken for the journey is 180 s .
  1. Sketch the velocity-time graph for the train's journey from \(A\) to \(B\). \includegraphics[max width=\textwidth, alt={}, center]{3d7f53af-dbf2-499b-9966-ae85514cef02-04_496_857_516_685}
  2. Find an expression, in terms of \(T\), for the length of time for which the train is travelling with constant speed.
  3. The distance from \(A\) to \(B\) is 3300 m . Find how far the train travels while it is decelerating.
CAIE M1 2017 June Q4
6 marks Moderate -0.3
4 A particle \(P\) moves in a straight line starting from a point \(O\). At time \(t \mathrm {~s}\) after leaving \(O\), the velocity, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), of \(P\) is given by \(v = ( 2 t - 5 ) ^ { 3 }\).
  1. Find the values of \(t\) when the acceleration of \(P\) is \(54 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  2. Find an expression for the displacement of \(P\) from \(O\) at time \(t \mathrm {~s}\).
CAIE M1 2017 June Q5
6 marks Standard +0.3
5 A particle is projected vertically upwards from a point \(O\) with a speed of \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Two seconds later a second particle is projected vertically upwards from \(O\) with a speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At time \(t \mathrm {~s}\) after the second particle is projected, the two particles collide.
  1. Find \(t\). \includegraphics[max width=\textwidth, alt={}, center]{3d7f53af-dbf2-499b-9966-ae85514cef02-06_65_1569_488_328}
  2. Hence find the height above \(O\) at which the particles collide.
CAIE M1 2017 June Q6
8 marks Moderate -0.3
6 A car of mass 1200 kg is travelling along a horizontal road.
  1. It is given that there is a constant resistance to motion.
    1. The engine of the car is working at 16 kW while the car is travelling at a constant speed of \(40 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the resistance to motion.
    2. The power is now increased to 22.5 kW . Find the acceleration of the car at the instant it is travelling at a speed of \(45 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
    3. It is given instead that the resistance to motion of the car is \(( 590 + 2 v ) \mathrm { N }\) when the speed of the car is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The car travels at a constant speed with the engine working at 16 kW . Find this speed.
CAIE M1 2017 June Q7
14 marks Standard +0.3
7 \includegraphics[max width=\textwidth, alt={}, center]{3d7f53af-dbf2-499b-9966-ae85514cef02-10_336_803_258_671} Two particles \(A\) and \(B\) of masses \(m \mathrm {~kg}\) and 4 kg respectively are connected by a light inextensible string that passes over a fixed smooth pulley. Particle \(A\) is on a rough fixed slope which is at an angle of \(30 ^ { \circ }\) to the horizontal ground. Particle \(B\) hangs vertically below the pulley and is 0.5 m above the ground (see diagram). The coefficient of friction between the slope and particle \(A\) is 0.2 .
  1. In the case where the system is in equilibrium with particle \(A\) on the point of moving directly up the slope, show that \(m = 5.94\), correct to 3 significant figures.
  2. In the case where \(m = 3\), the system is released from rest with the string taut. Find the total distance travelled by \(A\) before coming to instantaneous rest. You may assume that \(A\) does not reach the pulley.
CAIE M1 2018 June Q1
4 marks Moderate -0.3
1 A man has mass 80 kg . He runs along a horizontal road against a constant resistance force of magnitude \(P \mathrm {~N}\). The total work done by the man in increasing his speed from \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(5.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) while running a distance of 60 metres is 1200 J . Find the value of \(P\).
CAIE M1 2018 June Q2
4 marks Standard +0.3
2 A train of mass 240000 kg travels up a slope inclined at an angle of \(4 ^ { \circ }\) to the horizontal. There is a constant resistance of magnitude 18000 N acting on the train. At an instant when the speed of the train is \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) its deceleration is \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Find the power of the engine of the train.
CAIE M1 2018 June Q3
4 marks Moderate -0.5
3 \includegraphics[max width=\textwidth, alt={}, center]{16640429-198d-4ea9-a2f6-6e2ef6ac1b4a-05_535_616_260_762} The three coplanar forces shown in the diagram have magnitudes \(3 \mathrm {~N} , 2 \mathrm {~N}\) and \(P \mathrm {~N}\). Given that the three forces are in equilibrium, find the values of \(\theta\) and \(P\).
CAIE M1 2018 June Q4
9 marks Moderate -0.3
4 A particle \(P\) moves in a straight line \(A B C D\) with constant acceleration. The distances \(A B\) and \(B C\) are 100 m and 148 m respectively. The particle takes 4 s to travel from \(A\) to \(B\) and also takes 4 s to travel from \(B\) to \(C\).
  1. Show that the acceleration of \(P\) is \(3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) and find the speed of \(P\) at \(A\).
  2. \(P\) reaches \(D\) with a speed of \(61 \mathrm {~ms} ^ { - 1 }\). Find the distance \(C D\).
CAIE M1 2018 June Q5
7 marks Standard +0.8
5 A particle of mass 20 kg is on a rough plane inclined at an angle of \(60 ^ { \circ }\) to the horizontal. Equilibrium is maintained by a force of magnitude \(P \mathrm {~N}\) acting on the particle, in a direction parallel to a line of greatest slope of the plane. The greatest possible value of \(P\) is twice the least possible value of \(P\). Find the value of the coefficient of friction between the particle and the plane.
CAIE M1 2018 June Q6
8 marks Standard +0.3
6 A particle \(P\) moves in a straight line passing through a point \(O\). At time \(t \mathrm {~s}\), the acceleration, \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\), of \(P\) is given by \(a = 6 - 0.24 t\). The particle comes to instantaneous rest at time \(t = 20\).
  1. Find the value of \(t\) at which the particle is again at instantaneous rest.
  2. Find the distance the particle travels between the times of instantaneous rest.
CAIE M1 2018 June Q7
14 marks Standard +0.3
7
[diagram]
As shown in the diagram, a particle \(A\) of mass 1.6 kg lies on a horizontal plane and a particle \(B\) of mass 2.4 kg lies on a plane inclined at an angle of \(30 ^ { \circ }\) to the horizontal. The particles are connected by a light inextensible string which passes over a small smooth pulley \(P\) fixed at the top of the inclined plane. The distance \(A P\) is 2.5 m and the distance of \(B\) from the bottom of the inclined plane is 1 m . There is a barrier at the bottom of the inclined plane preventing any further motion of \(B\). The part \(B P\) of the string is parallel to a line of greatest slope of the inclined plane. The particles are released from rest with both parts of the string taut.
  1. Given that both planes are smooth, find the acceleration of \(A\) and the tension in the string.
  2. It is given instead that the horizontal plane is rough and that the coefficient of friction between \(A\) and the horizontal plane is 0.2 . The inclined plane is smooth. Find the total distance travelled by \(A\).
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE M1 2019 June Q1
3 marks Moderate -0.5
1 \includegraphics[max width=\textwidth, alt={}, center]{555678d3-f37d-4822-a005-de8c6094dc50-03_563_503_262_820} Given that \(\tan \alpha = \frac { 12 } { 5 }\) and \(\tan \theta = \frac { 4 } { 3 }\), show that the coplanar forces shown in the diagram are in equilibrium.
CAIE M1 2019 June Q2
7 marks Moderate -0.5
2 A particle \(P\) is projected vertically upwards with speed \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point 3 m above horizontal ground.
  1. Find the time taken for \(P\) to reach its greatest height.
  2. Find the length of time for which \(P\) is higher than 23 m above the ground.
  3. \(P\) is higher than \(h \mathrm {~m}\) above the ground for 1 second. Find \(h\).
CAIE M1 2019 June Q3
7 marks Standard +0.3
3 A lorry has mass 12000 kg .
  1. The lorry moves at a constant speed of \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) up a hill inclined at an angle of \(\theta\) to the horizontal, where \(\sin \theta = 0.08\). At this speed, the magnitude of the resistance to motion on the lorry is 1500 N . Show that the power of the lorry's engine is 55.5 kW .
    When the speed of the lorry is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) the magnitude of the resistance to motion is \(k v ^ { 2 } \mathrm {~N}\), where \(k\) is a constant.
  2. Show that \(k = 60\).
  3. The lorry now moves at a constant speed on a straight level road. Given that its engine is still working at 55.5 kW , find the lorry's speed.
CAIE M1 2019 June Q4
9 marks Standard +0.3
4 A particle of mass 1.3 kg rests on a rough plane inclined at an angle \(\theta\) to the horizontal, where \(\tan \theta = \frac { 12 } { 5 }\). The coefficient of friction between the particle and the plane is \(\mu\).
  1. A force of magnitude 20 N parallel to a line of greatest slope of the plane is applied to the particle and the particle is on the point of moving up the plane. Show that \(\mu = 1.6\).
    The force of magnitude 20 N is now removed.
  2. Find the acceleration of the particle.
  3. Find the work done against friction during the first 2 s of motion.
CAIE M1 2019 June Q5
10 marks Standard +0.3
5 A particle \(P\) moves in a straight line from a fixed point \(O\). The velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) of \(P\) at time \(t \mathrm {~s}\) is given by $$v = t ^ { 2 } - 8 t + 12 \quad \text { for } 0 \leqslant t \leqslant 8$$
  1. Find the minimum velocity of \(P\).
  2. Find the total distance travelled by \(P\) in the interval \(0 \leqslant t \leqslant 8\). \includegraphics[max width=\textwidth, alt={}, center]{555678d3-f37d-4822-a005-de8c6094dc50-12_401_1102_260_520} Two particles \(A\) and \(B\), of masses 0.4 kg and 0.2 kg respectively, are connected by a light inextensible string. Particle \(A\) is held on a smooth plane inclined at an angle of \(\theta ^ { \circ }\) to the horizontal. The string passes over a small smooth pulley \(P\) fixed at the top of the plane, and \(B\) hangs freely 0.5 m above horizontal ground (see diagram). The particles are released from rest with both sections of the string taut.
  1. Given that the system is in equilibrium, find \(\theta\).
  2. It is given instead that \(\theta = 20\). In the subsequent motion particle \(A\) does not reach \(P\) and \(B\) remains at rest after reaching the ground.
    1. Find the tension in the string and the acceleration of the system.
    2. Find the speed of \(A\) at the instant \(B\) reaches the ground.
    3. Use an energy method to find the total distance \(A\) moves up the plane before coming to instantaneous rest.
      If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.