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CAIE M2 2019 November Q2
5 marks Moderate -0.5
2 A small ball is projected from a point \(O\) on horizontal ground at an angle of \(30 ^ { \circ }\) above the horizontal. At time \(t \mathrm {~s}\) after projection the horizontal and vertically upwards displacements of the ball from \(O\) are \(x \mathrm {~m}\) and \(y \mathrm {~m}\) respectively. It is given that \(x = 40 t\).
  1. Calculate the initial speed of the ball, and express \(y\) in terms of \(t\).
  2. Hence find the equation of the trajectory of the ball.
CAIE M2 2019 November Q3
6 marks Standard +0.8
3 A particle \(P\) of mass 0.5 kg is attached to one end of a light elastic string of natural length 0.6 m and modulus of elasticity 12 N . The other end of the string is attached to a fixed point \(O\). The particle \(P\) is projected vertically downwards with speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from the point 0.5 m vertically below \(O\). For an instant when the acceleration of \(P\) is \(4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) downwards, find the extension of the string and the speed of \(P\).
CAIE M2 2019 November Q4
6 marks Standard +0.3
4 A particle is projected from a point \(O\) on horizontal ground with speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(60 ^ { \circ }\) above the horizontal. At the instant 3 s after projection the direction of motion of the particle is \(30 ^ { \circ }\) below the horizontal.
  1. Find \(V\).
    ..................................................................................................................................
  2. Calculate the distance of the particle from \(O\) at the instant 3 s after projection.
CAIE M2 2019 November Q5
10 marks Standard +0.8
5 \includegraphics[max width=\textwidth, alt={}, center]{9daebcbe-826e-4eda-afa7-c935c6ea2bfc-06_671_504_255_824} \(A\) and \(B\) are two fixed points on a vertical axis with \(A\) above \(B\). A particle \(P\) of mass 0.4 kg is attached to \(A\) by a light inextensible string of length 0.5 m . The particle \(P\) is attached to \(B\) by another light inextensible string. \(P\) moves with constant speed in a horizontal circle with centre \(O\) between \(A\) and \(B\). Angle \(B A P = 30 ^ { \circ }\) and angle \(A B P = 70 ^ { \circ }\) (see diagram).
  1. Given that the tensions in the two strings are equal, find the speed of \(P\).
  2. Given instead that the angular speed of \(P\) is \(12 \mathrm { rad } \mathrm { s } ^ { - 1 }\), find the tensions in the strings.
CAIE M2 2019 November Q6
9 marks Standard +0.8
6 A particle \(P\) of mass 0.2 kg is projected horizontally from a fixed point \(O\) on a smooth horizontal surface. When the displacement of \(P\) from \(O\) is \(x \mathrm {~m}\) the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). A horizontal force of variable magnitude \(0.09 \sqrt { } x \mathrm {~N}\) directed away from \(O\) acts on \(P\). An additional force of constant magnitude 0.3 N directed towards \(O\) acts on \(P\).
  1. Show that \(v \frac { \mathrm {~d} v } { \mathrm {~d} x } = 0.45 \sqrt { } x - 1.5\).
  2. Find the value of \(x\) for which the acceleration of \(P\) is zero.
  3. Given that the minimum value of \(v\) is positive, find the set of possible values for the speed of projection.
CAIE M2 2019 November Q7
11 marks Challenging +1.2
7 \includegraphics[max width=\textwidth, alt={}, center]{9daebcbe-826e-4eda-afa7-c935c6ea2bfc-10_451_574_258_781} \(A B C D\) is a uniform lamina in the shape of a trapezium which has centre of mass \(G\). The sides \(A D\) and \(B C\) are parallel and 1.8 m apart, with \(A D = 2.4 \mathrm {~m}\) and \(B C = 1.2 \mathrm {~m}\) (see diagram).
  1. Show that the distance of \(G\) from \(A D\) is 0.8 m .
    The lamina is freely suspended at \(A\) and hangs in equilibrium with \(A D\) making an angle of \(30 ^ { \circ }\) with the vertical.
  2. Calculate the distance \(A G\).
    With the lamina still freely suspended at \(A\) a horizontal force of magnitude 7 N acting in the plane of the lamina is applied at \(D\). The lamina is in equilibrium with \(A G\) making an angle of \(10 ^ { \circ }\) with the downward vertical.
  3. Find the two possible values for the weight of the lamina.
    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 M2 2019 November Q2
6 marks Moderate -0.3
2 A particle is projected from a point on horizontal ground with speed \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(\theta ^ { \circ }\) above the horizontal. The particle strikes the ground 2 s after projection.
  1. Find \(\theta\). \includegraphics[max width=\textwidth, alt={}, center]{42de91da-d65e-40e7-8de5-f40eda927850-03_67_1571_438_328}
  2. Calculate the time after projection at which the direction of motion of the particle is \(20 ^ { \circ }\) below the horizontal.
CAIE M2 2019 November Q5
9 marks Standard +0.3
5 A particle \(P\) of mass 0.3 kg is attached to one end of a light elastic string of natural length 0.6 m and modulus of elasticity 9 N . The other end of the string is attached to a fixed point \(O\) on a smooth plane inclined at \(30 ^ { \circ }\) to the horizontal. \(O A\) is a line of greatest slope of the plane with \(A\) below the level of \(O\) and \(O A = 0.8 \mathrm {~m}\). The particle \(P\) is released from rest at \(A\).
  1. Find the initial acceleration of \(P\).
  2. Find the greatest speed of \(P\). \(6 \quad A\) and \(B\) are two fixed points on a vertical axis with \(A 0.6 \mathrm {~m}\) above \(B\). A particle \(P\) of mass 0.3 kg is attached to \(A\) by a light inextensible string of length 0.5 m . The particle \(P\) is attached to \(B\) by a light elastic string with modulus of elasticity 46 N . The particle \(P\) moves with constant angular speed \(8 \mathrm { rad } \mathrm { s } ^ { - 1 }\) in a horizontal circle with centre at the mid-point of \(A B\).
  3. Find the speed of \(P\).
  4. Calculate the tension in the string \(B P\) and hence find the natural length of this string. \includegraphics[max width=\textwidth, alt={}, center]{42de91da-d65e-40e7-8de5-f40eda927850-10_540_574_260_781} \(A B C\) is the cross-section through the centre of mass of a uniform prism which rests with \(A B\) on a rough horizontal surface. \(A B = 0.4 \mathrm {~m}\) and \(C\) is 0.9 m above the surface (see diagram). The prism is on the point of toppling about its edge through \(B\).
  5. Show that angle \(B A C = 48.4 ^ { \circ }\), correct to 3 significant figures.
    A force of magnitude 18 N acting in the plane of the cross-section and perpendicular to \(A C\) is now applied to the prism at \(C\). The prism is on the point of rotating about its edge through \(A\).
  6. Calculate the weight of the prism.
  7. Given also that the prism is on the point of slipping, calculate the coefficient of friction between the prism and the surface.
    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 M2 Specimen Q1
4 marks Standard +0.3
1 A particle is projected with speed \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(50 ^ { \circ }\) above the horizontal. Calculate the time after projection when the particle has speed \(18 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is rising.
CAIE M2 Specimen Q2
5 marks Moderate -0.8
2 One end of a light inextensible string of length 0.5 m is attached to a fixed point \(A\). A particle \(P\) of mass 0.2 kg is attached to the other end of the string. \(P\) moves with constant speed in a horizontal circle with centre \(O\) which is 0.4 m vertically below \(A\).
  1. Show that the tension in the string is 2.5 N .
  2. Find the speed of \(P\).
CAIE M2 Specimen Q3
5 marks Standard +0.8
3 A particle \(P\) is projected with speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(\theta ^ { \circ }\) above the horizontal from a point \(O\) on horizontal ground. At the instant 4 s after projection the particle passes through the point \(A\), where \(O A = 40 \mathrm {~m}\) and the line \(O A\) makes an angle of \(30 ^ { \circ }\) with the horizontal. Calculate \(V\) and \(\theta\).
CAIE M2 Specimen Q4
7 marks Standard +0.3
4 \includegraphics[max width=\textwidth, alt={}, center]{add3948c-3b45-4e67-ac84-e2ca935afd64-05_392_621_255_762} A particle \(P\) of mass 0.4 kg moves with constant speed in a horizontal circle on the smooth inner surface of a fixed hollow hemisphere with centre \(O\) and radius 0.5 m (see diagram).
  1. Given that the speed of the particle is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and its angular speed is \(10 \mathrm { rad } \mathrm { s } ^ { - 1 }\), calculate the angle between \(O P\) and the vertical.
  2. Given instead that the magnitude of the force exerted on \(P\) by the hemisphere is 6 N , calculate
    (a) the angle between \(O P\) and the vertical,
    (b) the angular speed of \(P\).
CAIE M2 Specimen Q5
9 marks Challenging +1.2
5 A particle \(P\) of mass 0.5 kg is projected vertically upwards from a point on a horizontal surface. A resisting force of magnitude \(0.02 v ^ { 2 } \mathrm {~N}\) acts on \(P\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the upward velocity of \(P\) when it is a height of \(x \mathrm {~m}\) above the surface. The initial speed of \(P\) is \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Show that, while \(P\) is moving upwards, \(v \frac { \mathrm {~d} v } { \mathrm {~d} x } = - 10 - 0.04 v ^ { 2 }\).
  2. Find the greatest height of \(P\) above the surface.
  3. Find the speed of \(P\) immediately before it strikes the surface after descending.
CAIE M2 Specimen Q6
9 marks Challenging +1.2
6 \includegraphics[max width=\textwidth, alt={}, center]{add3948c-3b45-4e67-ac84-e2ca935afd64-08_442_953_237_596} An object is formed by joining a hemispherical shell of radius 0.2 m and a solid cone with base radius 0.2 m and height \(h \mathrm {~m}\) along their circumferences. The centre of mass, \(G\), of the object is \(d \mathrm {~m}\) from the vertex of the cone on the axis of symmetry of the object. The object rests in equilibrium on a horizontal plane, with the curved surface of the cone in contact with the plane (see diagram). The object is on the point of toppling.
  1. Show that \(d = h + \frac { 0.04 } { h }\).
  2. It is given that the cone is uniform and of weight 4 N , and that the hemispherical shell is uniform and of weight \(W \mathrm {~N}\). Given also that \(h = 0.8\), find \(W\).
CAIE M2 Specimen Q7
11 marks Standard +0.8
7 A particle \(P\) of mass \(M \mathrm {~kg}\) is attached to one end of a light elastic string of natural length 0.8 m and modulus of elasticity 12.5 N . The other end of the string is attached to a fixed point \(A\). The particle is released from rest at \(A\) and falls vertically until it comes to instantaneous rest at the point \(B\). The greatest speed of \(P\) during its descent is \(4.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when the extension of the string is \(e \mathrm {~m}\).
  1. Show that \(e = 0.64 M\).
  2. Find a second equation in \(e\) and \(M\), and hence find \(M\).
  3. Calculate the distance \(A B\).
CAIE Further Paper 3 2020 November Q1
3 marks Challenging +1.2
1 A particle \(P\) of mass \(m\) is placed on a fixed smooth plane which is inclined at an angle \(\theta\) to the horizontal. A light spring, of natural length \(a\) and modulus of elasticity \(3 m g\), has one end attached to \(P\) and the other end attached to a fixed point \(O\) at the top of the plane. The spring lies along a line of greatest slope of the plane. The system is released from rest with the spring at its natural length. Find, in terms of \(a\) and \(\theta\), an expression for the greatest extension of the spring in the subsequent motion. \includegraphics[max width=\textwidth, alt={}, center]{1c53c407-25ea-43fc-a571-74ba1fffea8f-04_515_707_267_685} A particle \(P\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a fixed point \(O\). The particle \(P\) is held with the string taut and making an angle \(\theta\) with the downward vertical. The particle \(P\) is then projected with speed \(\frac { 4 } { 5 } \sqrt { 5 a g }\) perpendicular to the string and just completes a vertical circle (see diagram). Find the value of \(\cos \theta\).
CAIE Further Paper 3 2020 November Q3
6 marks Standard +0.8
3 One end of a light elastic string, of natural length \(a\) and modulus of elasticity \(4 m g\), is attached to a fixed point \(O\). The other end of the string is attached to a particle of mass \(m\). The particle moves in a horizontal circle with a constant angular speed \(\sqrt { \frac { \mathrm { g } } { \mathrm { a } } }\) with the string inclined at an angle \(\theta\) to the downward vertical through \(O\). The length of the string during this motion is \(( \mathrm { k } + 1 ) \mathrm { a }\).
  1. Find the value of \(k\).
  2. Find the value of \(\cos \theta\). \includegraphics[max width=\textwidth, alt={}, center]{1c53c407-25ea-43fc-a571-74ba1fffea8f-06_584_695_264_667} The diagram shows the cross-section \(A B C D\) of a uniform solid object which is formed by removing a cone with cross-section \(D C E\) from the top of a larger cone with cross-section \(A B E\). The perpendicular distance between \(A B\) and \(D C\) is \(h\), the diameter \(A B\) is \(6 r\) and the diameter \(D C\) is \(2 r\).
  3. Find an expression, in terms of \(h\), for the distance of the centre of mass of the solid object from \(A B\).
    The object is freely suspended from the point \(B\) and hangs in equilibrium. The angle between \(A B\) and the downward vertical through \(B\) is \(\theta\).
  4. Given that \(h = \frac { 13 } { 4 } r\), find the value of \(\tan \theta\).
CAIE S1 2020 June Q1
3 marks Easy -1.8
1 For \(n\) values of the variable \(x\), it is given that $$\Sigma ( x - 50 ) = 144 \quad \text { and } \quad \Sigma x = 944 .$$ Find the value of \(n\).
CAIE S1 2020 June Q2
5 marks Moderate -0.8
2 A total of 500 students were asked which one of four colleges they attended and whether they preferred soccer or hockey. The numbers of students in each category are shown in the following table.
\cline { 2 - 4 } \multicolumn{1}{c|}{}SoccerHockeyTotal
Amos543286
Benn8472156
Canton225678
Devar12060180
Total280220500
  1. Find the probability that a randomly chosen student is at Canton college and prefers hockey.
  2. Find the probability that a randomly chosen student is at Devar college given that he prefers soccer.
  3. One of the students is chosen at random. Determine whether the events 'the student prefers hockey' and 'the student is at Amos college or Benn college' are independent, justifying your answer.
CAIE S1 2020 June Q3
8 marks Easy -1.2
3 Two machines, \(A\) and \(B\), produce metal rods of a certain type. The lengths, in metres, of 19 rods produced by machine \(A\) and 19 rods produced by machine \(B\) are shown in the following back-to-back stem-and-leaf diagram. \begin{table}[h]
\(A\)\(B\)
21124
76302224556
8743112302689
55532243346
4310256
\captionsetup{labelformat=empty} \caption{Key: 7 | 22 | 4 means 0.227 m for machine \(A\) and 0.224 m for machine \(B\).}
\end{table}
  1. Find the median and the interquartile range for machine \(A\).
    It is given that for machine \(B\) the median is 0.232 m , the lower quartile is 0.224 m and the upper quartile is 0.243 m .
  2. Draw box-and-whisker plots for \(A\) and \(B\). \includegraphics[max width=\textwidth, alt={}, center]{a3b3ebd1-db9e-4552-9abe-bfdeba786d02-05_812_1205_616_511}
  3. Hence make two comparisons between the lengths of the rods produced by machine \(A\) and those produced by machine \(B\).
CAIE S1 2020 June Q4
8 marks Moderate -0.8
4 Trees in the Redian forest are classified as tall, medium or short, according to their height. The heights can be modelled by a normal distribution with mean 40 m and standard deviation 12 m . Trees with a height of less than 25 m are classified as short.
  1. Find the probability that a randomly chosen tree is classified as short.
    Of the trees that are classified as tall or medium, one third are tall and two thirds are medium.
  2. Show that the probability that a randomly chosen tree is classified as tall is 0.298 , correct to 3 decimal places.
  3. Find the height above which trees are classified as tall.
CAIE S1 2020 June Q5
8 marks Moderate -0.8
5 A fair three-sided spinner has sides numbered 1, 2, 3. A fair five-sided spinner has sides numbered \(1,1,2,2,3\). Both spinners are spun once. For each spinner, the number on the side on which it lands is noted. The random variable \(X\) is the larger of the two numbers if they are different, and their common value if they are the same.
  1. Show that \(\mathrm { P } ( X = 3 ) = \frac { 7 } { 15 }\). \includegraphics[max width=\textwidth, alt={}, center]{a3b3ebd1-db9e-4552-9abe-bfdeba786d02-08_69_1569_541_328}
  2. Draw up the probability distribution table for \(X\).
  3. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
CAIE S1 2020 June Q6
9 marks Moderate -0.3
6
  1. Find the number of different ways in which the 10 letters of the word SUMMERTIME can be arranged so that there is an E at the beginning and an E at the end.
  2. Find the number of different ways in which the 10 letters of the word SUMMERTIME can be arranged so that the Es are not together.
  3. Four letters are selected from the 10 letters of the word SUMMERTIME. Find the number of different selections if the four letters include at least one M and exactly one E .
CAIE S1 2020 June Q7
9 marks Moderate -0.3
7 On any given day, the probability that Moena messages her friend Pasha is 0.72 .
  1. Find the probability that for a random sample of 12 days Moena messages Pasha on no more than 9 days.
  2. Moena messages Pasha on 1 January. Find the probability that the next day on which she messages Pasha is 5 January.
  3. Use an approximation to find the probability that in any period of 100 days Moena messages Pasha on fewer than 64 days.
    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 S1 2020 June Q1
6 marks Easy -1.2
1 Juan goes to college each day by any one of car or bus or walking. The probability that he goes by car is 0.2 , the probability that he goes by bus is 0.45 and the probability that he walks is 0.35 . When Juan goes by car, the probability that he arrives early is 0.6 . When he goes by bus, the probability that he arrives early is 0.1 . When he walks he always arrives early.
  1. Draw a fully labelled tree diagram to represent this information.
  2. Find the probability that Juan goes to college by car given that he arrives early.