Questions — CAIE M2 (456 questions)

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CAIE M2 2018 November Q4
4 A small object is projected horizontally with speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point \(O\) above horizontal ground. At time \(t \mathrm {~s}\) after projection, the horizontal and vertically upwards displacements of the object from \(O\) are \(x \mathrm {~m}\) and \(y \mathrm {~m}\) respectively.
  1. Express \(x\) and \(y\) in terms of \(t\) and hence show that the equation of the path of the object is \(y = - \frac { 5 x ^ { 2 } } { V ^ { 2 } }\).
    The object passes through points with coordinates \(( a , - a )\) and \(\left( a ^ { 2 } , - 16 a \right)\), where \(a\) is a positive constant.
  2. Find the value of \(a\).
  3. Given that the object strikes the ground at the point where \(x = 5 a\), find the height of \(O\) above the ground .
CAIE M2 2018 November Q5
5 A particle \(P\) of mass 0.7 kg is attached to a fixed point \(O\) by a light elastic string of natural length 0.6 m and modulus of elasticity 15 N . The particle \(P\) is projected vertically downwards from the point \(A , 0.8 \mathrm {~m}\) vertically below \(O\). The initial speed of \(P\) is \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Find the distance below \(A\) of the point at which \(P\) comes to instantaneous rest.
  2. Find the greatest speed of \(P\) in the motion.
    \includegraphics[max width=\textwidth, alt={}, center]{f922bf53-94a0-4ccc-8c38-959d2f795629-10_478_652_260_751} The diagram shows a uniform lamina \(A B C D E F G H\). The lamina consists of a quarter-circle \(O A B\) of radius \(r \mathrm {~m}\), a rectangle \(D E F G\) and two isosceles right-angled triangles \(C O D\) and \(G O H\). The rectangle has \(D G = E F = r \mathrm {~m}\) and \(D E = F G = x \mathrm {~m}\).
  3. Given that the centre of mass of the lamina is at \(O\), express \(x\) in terms of \(r\).
  4. Given instead that the rectangle \(D E F G\) is a square with edges of length \(r \mathrm {~m}\), state with a reason whether the centre of mass of the lamina lies within the square or the quarter-circle.
    \includegraphics[max width=\textwidth, alt={}, center]{f922bf53-94a0-4ccc-8c38-959d2f795629-12_384_693_258_726} A rough horizontal rod \(A B\) of length 0.45 m rotates with constant angular velocity \(6 \mathrm { rad } \mathrm { s } ^ { - 1 }\) about a vertical axis through \(A\). A small ring \(R\) of mass 0.2 kg can slide on the rod. A particle \(P\) of mass 0.1 kg is attached to the mid-point of a light inextensible string of length 0.6 m . One end of the string is attached to \(R\) and the other end of the string is attached to \(B\), with angle \(R P B = 60 ^ { \circ }\) (see diagram). \(R\) and \(P\) move in horizontal circles as the system rotates. \(R\) is in limiting equilibrium.
  5. Show that the tension in the portion \(P R\) of the string is 1.66 N , correct to 3 significant figures.
  6. Find the coefficient of friction between the ring and the rod.
    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 2018 November Q1
4 marks
1 A small ball \(B\) is projected with speed \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(60 ^ { \circ }\) to the horizontal from a point on horizontal ground. Find the time after projection when the speed of \(B\) is \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) for the second time.
[0pt] [4]
\includegraphics[max width=\textwidth, alt={}, center]{5ab17f54-2408-4cf5-b852-611dd5aa112e-04_620_668_255_742} A uniform object is made by attaching the base of a solid hemisphere to the base of a solid cone so that the object has an axis of symmetry. The base of the cone has radius 0.3 m , and the hemisphere has radius 0.2 m . The object is placed on a horizontal plane with a point \(A\) on the curved surface of the hemisphere and a point \(B\) on the circumference of the cone in contact with the plane (see diagram).
  1. Given that the object is on the point of toppling about \(B\), find the distance of the centre of mass of the object from the base of the cone.
  2. Given instead that the object is on the point of toppling about \(A\), calculate the height of the cone.
    [0pt] [The volume of a cone is \(\frac { 1 } { 3 } \pi r ^ { 2 } h\). The volume of a hemisphere is \(\frac { 2 } { 3 } \pi r ^ { 3 }\).]
CAIE M2 2018 November Q4
4
\includegraphics[max width=\textwidth, alt={}, center]{5ab17f54-2408-4cf5-b852-611dd5aa112e-08_152_885_262_630} A particle \(P\) of mass 0.5 kg is projected along a smooth horizontal surface towards a fixed point \(A\). Initially \(P\) is at a point \(O\) on the surface, and after projection, \(P\) has a displacement from \(O\) of \(x \mathrm {~m}\) and velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The particle \(P\) is connected to \(A\) by a light elastic string of natural length 0.8 m and modulus of elasticity 16 N . The distance \(O A\) is 1.6 m (see diagram). The motion of \(P\) is resisted by a force of magnitude \(24 x ^ { 2 } \mathrm {~N}\).
  1. Show that \(v \frac { \mathrm {~d} v } { \mathrm {~d} x } = 32 - 40 x - 48 x ^ { 2 }\) while \(P\) is in motion and the string is stretched.
    The maximum value of \(v\) is 4.5 .
  2. Find the initial value of \(v\).
CAIE M2 2018 November Q5
5 A particle \(P\) of mass 0.1 kg is attached to one end of a light inextensible string of length 0.5 m . The other end of the string is attached to a fixed point \(A\). The particle \(P\) moves in a circle which has its centre \(O\) on a smooth horizontal surface 0.3 m below \(A\). The tension in the string has magnitude \(T \mathrm {~N}\) and the magnitude of the force exerted on \(P\) by the surface is \(R \mathrm {~N}\).
  1. Given that the speed of \(P\) is \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), calculate \(T\) and \(R\).
  2. Given instead that \(T = R\), calculate the angular speed of \(P\).
    \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{5ab17f54-2408-4cf5-b852-611dd5aa112e-12_449_621_260_762} \captionsetup{labelformat=empty} \caption{Fig. 1}
    \end{figure} Fig. 1 shows the cross-section \(A B C D E\) through the centre of mass \(G\) of a uniform prism. The crosssection consists of a rectangle \(A B C F\) from which a triangle \(D E F\) has been removed; \(A B = 0.6 \mathrm {~m}\), \(B C = 0.7 \mathrm {~m}\) and \(D F = E F = 0.3 \mathrm {~m}\).
  3. Show that the distance of \(G\) from \(B C\) is 0.276 m , and find the distance of \(G\) from \(A B\).
    \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{5ab17f54-2408-4cf5-b852-611dd5aa112e-13_494_583_258_781} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure} The prism is placed with \(C D\) on a rough horizontal surface. A force of magnitude 2 N acting in the plane of the cross-section is applied to the prism. The line of action of the force passes through \(G\) and is perpendicular to \(D E\) (see Fig. 2). The prism is on the point of toppling about the edge through \(D\).
  4. Calculate the weight of the prism.
    \includegraphics[max width=\textwidth, alt={}, center]{5ab17f54-2408-4cf5-b852-611dd5aa112e-14_512_520_258_817} A small object is projected with speed \(24 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point \(O\) at the foot of a plane inclined at \(45 ^ { \circ }\) to the horizontal. The angle of projection of the object is \(15 ^ { \circ }\) above a line of greatest slope of the plane (see diagram). At time \(t \mathrm {~s}\) after projection, the horizontal and vertically upwards displacements of the object from \(O\) are \(x \mathrm {~m}\) and \(y \mathrm {~m}\) respectively.
  5. Express \(x\) and \(y\) in terms of \(t\), and hence find the value of \(t\) for the instant when the object strikes the plane.
  6. Express the vertical height of the object above the plane in terms of \(t\) and hence find the greatest vertical height of the object above the plane.
    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
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]{4cd525d5-d59b-4ab9-85a3-fc3d97fd09fc-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 Q3
3 A smooth horizontal surface has two fixed points \(O\) and \(A\) which are 0.8 m apart. A particle \(P\) of mass 0.25 kg is projected with velocity \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) horizontally from \(A\) in the direction away from \(O\). The velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when the displacement of \(P\) from \(O\) is \(x \mathrm {~m}\). A force of magnitude \(k v ^ { 2 } x ^ { - 2 } \mathrm {~N}\) opposes the motion of \(P\).
  1. Show that \(v \frac { \mathrm {~d} v } { \mathrm {~d} x } = - 4 k v ^ { 2 } x ^ { - 2 }\).
  2. Express \(v\) in terms of \(k\) and \(x\).
CAIE M2 2019 November Q4
4 A small ball \(B\) is projected with speed \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(60 ^ { \circ }\) above the horizontal from a point \(O\). At time \(t \mathrm {~s}\) after projection the horizontal and vertically upwards displacements of \(B\) from \(O\) are \(x \mathrm {~m}\) and \(y \mathrm {~m}\) respectively.
  1. Express \(x\) and \(y\) in terms of \(t\) and hence find the equation of the trajectory of the ball.
  2. Find the value of \(x\) for which \(O B\) makes an angle of \(45 ^ { \circ }\) above the horizontal.
CAIE M2 2019 November Q5
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]{4cd525d5-d59b-4ab9-85a3-fc3d97fd09fc-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 2019 November Q1
1 A particle 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 horizontal surface. The particle is projected horizontally from \(O\) with speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the greatest distance of the particle from \(O\).
CAIE M2 2019 November Q2
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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 \(A B\).