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CAIE M2 2014 June Q5
Standard +0.3
\includegraphics{figure_5}
CAIE M2 2014 June Q6
Standard +0.8
\includegraphics{figure_6}
CAIE M2 2014 June Q7
Standard +0.3
\includegraphics{figure_7}
CAIE M2 2015 June Q1
3 marks Standard +0.3
One end of a light elastic string of natural length \(0.7\) m is attached to a fixed point \(A\) on a smooth horizontal surface. The other end of the string is attached to a particle \(P\) of mass \(0.3\) kg which is held at a point \(B\) on the horizontal surface, where \(AB = 1.2\) m. It is given that \(P\) is released from rest at \(B\) and that when \(AP = 0.9\) m, the particle has speed \(4\) m s\(^{-1}\). Calculate the modulus of elasticity of the string. [3]
CAIE M2 2015 June Q2
4 marks Moderate -0.8
A stone is projected from a point \(O\) on horizontal ground. The equation of the trajectory of the stone is $$y = 1.2x - 0.15x^2,$$ where \(x\) m and \(y\) m are respectively the horizontal and vertically upwards displacements of the stone from \(O\). Find
  1. the greatest height of the stone, [2]
  2. the distance from \(O\) of the point where the stone strikes the ground. [2]
CAIE M2 2015 June Q3
7 marks Standard +0.3
\includegraphics{figure_3} One end of a light inextensible string is attached to a fixed point \(A\) and the other end of the string is attached to a particle \(P\). The particle \(P\) moves with constant angular speed \(5\) rad s\(^{-1}\) in a horizontal circle which has its centre \(O\) vertically below \(A\). The string makes an angle \(\theta\) with the vertical (see diagram). The tension in the string is three times the weight of \(P\).
  1. Show that the length of the string is \(1.2\) m. [3]
  2. Find the speed of \(P\). [4]
CAIE M2 2015 June Q4
7 marks Moderate -0.3
\includegraphics{figure_4} A small ball \(B\) is projected from a point \(O\) above horizontal ground, with initial speed \(15\) m s\(^{-1}\) at an angle of projection of \(30°\) above the horizontal (see diagram). The ball strikes the ground \(3\) s after projection.
  1. Calculate the speed and direction of motion of the ball immediately before it strikes the ground. [5]
  2. Find the height of \(O\) above the ground. [2]
CAIE M2 2015 June Q5
8 marks Standard +0.3
A particle \(P\) of mass \(0.3\) kg is attached to one end of a light elastic string of natural length \(0.9\) m and modulus of elasticity \(18\) N. The other end of the string is attached to a fixed point \(O\) which is \(3\) m above the ground.
  1. Find the extension of the string when \(P\) is in the equilibrium position. [2]
\(P\) is projected vertically downwards from the equilibrium position with initial speed \(6\) m s\(^{-1}\). At the instant when the tension in the string is \(12\) N the string breaks. \(P\) continues to descend vertically.
    1. Calculate the height of \(P\) above the ground at the instant when the string breaks. [2]
    2. Find the speed of \(P\) immediately before it strikes the ground. [4]
CAIE M2 2015 June Q6
8 marks Standard +0.3
A particle \(P\) of mass \(0.1\) kg moves with decreasing speed in a straight line on a smooth horizontal surface. A horizontal resisting force of magnitude \(0.2e^{-x}\) N acts on \(P\), where \(x\) m is the displacement of \(P\) from a fixed point \(O\) on the line. The velocity of \(P\) is \(v\) m s\(^{-1}\) when its displacement from \(O\) is \(x\) m.
  1. Show that $$v\frac{dv}{dx} = ke^{-x},$$ where \(k\) is a constant to be found. [2]
\(P\) passes through \(O\) with velocity \(2.2\) m s\(^{-1}\).
  1. Calculate the value of \(x\) at the instant when the velocity of \(P\) is \(2\) m s\(^{-1}\). [4]
  2. Show that the speed of \(P\) does not fall below \(0.917\) m s\(^{-1}\), correct to \(3\) significant figures. [2]
CAIE M2 2015 June Q7
13 marks Challenging +1.2
\includegraphics{figure_7} The diagram shows the cross-section \(OABCDE\) through the centre of mass of a uniform prism on a rough inclined plane. The portion \(ADEO\) is a rectangle in which \(AD = OE = 0.6\) m and \(DE = AO = 0.8\) m; the portion \(BCD\) is an isosceles triangle in which angle \(BCD\) is a right angle, and \(A\) is the mid-point of \(BD\). The plane is inclined at \(45°\) to the horizontal, \(BC\) lies along a line of greatest slope of the plane and \(DE\) is horizontal.
  1. Calculate the distance of the centre of mass of the prism from \(BD\). [3]
The weight of the prism is \(21\) N, and it is held in equilibrium by a horizontal force of magnitude \(P\) N acting along \(ED\).
    1. Find the smallest value of \(P\) for which the prism does not topple. [2]
    2. It is given that the prism is about to slip for this smallest value of \(P\). Calculate the coefficient of friction between the prism and the plane. [3]
The value of \(P\) is gradually increased until the prism ceases to be in equilibrium.
  1. Show that the prism topples before it begins to slide, stating the value of \(P\) at which equilibrium is broken. [5]
CAIE M2 2016 June Q1
4 marks Standard +0.3
A small ball is projected with speed \(16 \text{ ms}^{-1}\) at an angle of \(45°\) above the horizontal from a point on horizontal ground. Calculate the period of time, before the ball lands, for which the speed of the ball is less than \(12 \text{ ms}^{-1}\). [4]
CAIE M2 2016 June Q2
5 marks Standard +0.3
\includegraphics{figure_2} A uniform wire has the shape of a semicircular arc, with diameter \(AB\) of length \(0.8 \text{ m}\). The wire is attached to a vertical wall by a smooth hinge at \(A\). The wire is held in equilibrium with \(AB\) inclined at \(70°\) to the upward vertical by a light string attached to \(B\). The other end of the string is attached to the point \(C\) on the wall \(0.8 \text{ m}\) vertically above \(A\). The tension in the string is \(15 \text{ N}\) (see diagram).
  1. Show that the horizontal distance of the centre of mass of the wire from the wall is \(0.463 \text{ m}\), correct to 3 significant figures. [3]
  2. Calculate the weight of the wire. [2]
CAIE M2 2016 June Q3
6 marks Standard +0.8
A particle \(P\) of mass \(0.4 \text{ kg}\) is released from rest at a point \(O\) on a smooth plane inclined at \(30°\) to the horizontal. When the displacement of \(P\) from \(O\) is \(x \text{ m}\) down the plane, the velocity of \(P\) is \(v \text{ ms}^{-1}\). A force of magnitude \(0.8e^{-x} \text{ N}\) acts on \(P\) up the plane along the line of greatest slope through \(O\).
  1. Show that \(v \frac{dv}{dx} = 5 - 2e^{-x}\). [2]
  2. Find \(v\) when \(x = 0.6\). [4]
CAIE M2 2016 June Q4
6 marks Challenging +1.2
\includegraphics{figure_4} A uniform solid cone has base radius \(0.4 \text{ m}\) and height \(4.4 \text{ m}\). A uniform solid cylinder has radius \(0.4 \text{ m}\) and weight equal to the weight of the cone. An object is formed by attaching the cylinder to the cone so that the base of the cone and a circular face of the cylinder are in contact and their circumferences coincide. The object rests in equilibrium with its circular base on a plane inclined at an angle of \(20°\) to the horizontal (see diagram).
  1. Calculate the least possible value of the coefficient of friction between the plane and the object. [2]
  2. Calculate the greatest possible height of the cylinder. [4]
CAIE M2 2016 June Q5
9 marks Standard +0.3
A particle is projected at an angle of \(θ°\) below the horizontal from a point at the top of a vertical cliff \(26 \text{ m}\) high. The particle strikes horizontal ground at a distance \(8 \text{ m}\) from the foot of the cliff \(2 \text{ s}\) after the instant of projection. Find
  1. the speed of projection of the particle and the value of \(θ\), [6]
  2. the direction of motion of the particle immediately before it strikes the ground. [3]
CAIE M2 2016 June Q6
9 marks Standard +0.3
\includegraphics{figure_6} A light inextensible string passes through a small smooth bead \(B\) of mass \(0.4 \text{ kg}\). One end of the string is attached to a fixed point \(A\) \(0.4 \text{ m}\) above a fixed point \(O\) on a smooth horizontal surface. The other end of the string is attached to a fixed point \(C\) which is vertically below \(A\) and \(0.3 \text{ m}\) above the surface. The bead moves with constant speed on the surface in a circle with centre \(O\) and radius \(0.3 \text{ m}\) (see diagram).
  1. Given that the tension in the string is \(2 \text{ N}\), calculate
    1. the angular speed of the bead, [3]
    2. the magnitude of the contact force exerted on the bead by the surface. [2]
  2. Given instead that the bead is about to lose contact with the surface, calculate the speed of the bead. [4]
CAIE M2 2016 June Q7
11 marks Standard +0.8
A particle \(P\) is attached to one end of a light elastic string of natural length \(1.2 \text{ m}\) and modulus of elasticity \(12 \text{ N}\). The other end of the string is attached to a fixed point \(O\) on a smooth plane inclined at an angle of \(30°\) to the horizontal. \(P\) rests in equilibrium on the plane, \(1.6 \text{ m}\) from \(O\).
  1. Calculate the mass of \(P\). [2]
A particle \(Q\), with mass equal to the mass of \(P\), is projected up the plane along a line of greatest slope. When \(Q\) strikes \(P\) the two particles coalesce. The combined particle remains attached to the string and moves up the plane, coming to instantaneous rest after moving \(0.2 \text{ m}\).
  1. Show that the initial kinetic energy of the combined particle is \(1 \text{ J}\). [4]
The combined particle subsequently moves down the plane.
  1. Calculate the greatest speed of the combined particle in the subsequent motion. [5]
CAIE M2 2016 June Q2
5 marks Standard +0.3
\includegraphics{figure_2} A uniform wire has the shape of a semicircular arc, with diameter \(AB\) of length \(0.8\) m. The wire is attached to a vertical wall by a smooth hinge at \(A\). The wire is held in equilibrium with \(AB\) inclined at \(70°\) to the upward vertical by a light string attached to \(B\). The other end of the string is attached to the point \(C\) on the wall \(0.8\) m vertically above \(A\). The tension in the string is \(15\) N (see diagram).
  1. Show that the horizontal distance of the centre of mass of the wire from the wall is \(0.463\) m, correct to 3 significant figures. [3]
  2. Calculate the weight of the wire. [2]
CAIE M2 2016 June Q3
6 marks Standard +0.8
A particle \(P\) of mass \(0.4\) kg is released from rest at a point \(O\) on a smooth plane inclined at \(30°\) to the horizontal. When the displacement of \(P\) from \(O\) is \(x\) m down the plane, the velocity of \(P\) is \(v \text{ ms}^{-1}\). A force of magnitude \(0.8e^{-5x}\) N acts on \(P\) up the plane along the line of greatest slope through \(O\).
  1. Show that \(v \frac{dv}{dx} = 5 - 2e^{-x}\). [2]
  2. Find \(v\) when \(x = 0.6\). [4]
CAIE M2 2016 June Q4
6 marks Challenging +1.2
\includegraphics{figure_4} A uniform solid cone has base radius \(0.4\) m and height \(4.4\) m. A uniform solid cylinder has radius \(0.4\) m and weight equal to the weight of the cone. An object is formed by attaching the cylinder to the cone so that the base of the cone and a circular face of the cylinder are in contact and their circumferences coincide. The object rests in equilibrium with its circular base on a plane inclined at an angle of \(20°\) to the horizontal (see diagram).
  1. Calculate the least possible value of the coefficient of friction between the plane and the object. [2]
  2. Calculate the greatest possible height of the cylinder. [4]
CAIE M2 2016 June Q5
9 marks Standard +0.3
A particle is projected at an angle of \(θ°\) below the horizontal from a point at the top of a vertical cliff \(26\) m high. The particle strikes horizontal ground at a distance \(8\) m from the foot of the cliff \(2\) s after the instant of projection. Find
  1. the speed of projection of the particle and the value of \(θ\), [6]
  2. the direction of motion of the particle immediately before it strikes the ground. [3]
CAIE M2 2016 June Q6
9 marks Standard +0.3
\includegraphics{figure_6} A light inextensible string passes through a small smooth bead \(B\) of mass \(0.4\) kg. One end of the string is attached to a fixed point \(A\) \(0.4\) m above a fixed point \(O\) on a smooth horizontal surface. The other end of the string is attached to a fixed point \(C\) which is vertically below \(A\) and \(0.3\) m above the surface. The bead moves with constant speed on the surface in a circle with centre \(O\) and radius \(0.3\) m (see diagram).
  1. Given that the tension in the string is \(2\) N, calculate
    1. the angular speed of the bead, [3]
    2. the magnitude of the contact force exerted on the bead by the surface. [2]
  2. Given instead that the bead is about to lose contact with the surface, calculate the speed of the bead. [4]
CAIE M2 2016 June Q7
11 marks Standard +0.8
A particle \(P\) is attached to one end of a light elastic string of natural length \(1.2\) m and modulus of elasticity \(12\) N. The other end of the string is attached to a fixed point \(O\) on a smooth plane inclined at an angle of \(30°\) to the horizontal. \(P\) rests in equilibrium on the plane, \(1.6\) m from \(O\).
  1. Calculate the mass of \(P\). [2]
A particle \(Q\), with mass equal to the mass of \(P\), is projected up the plane along a line of greatest slope. When \(Q\) strikes \(P\) the two particles coalesce. The combined particle remains attached to the string and moves up the plane, coming to instantaneous rest after moving \(0.2\) m.
  1. Show that the initial kinetic energy of the combined particle is \(1\) J. [4]
The combined particle subsequently moves down the plane.
  1. Calculate the greatest speed of the combined particle in the subsequent motion. [5]
CAIE M2 2017 June Q1
4 marks Standard +0.3
A particle is projected with speed \(20\,\text{m}\,\text{s}^{-1}\) at an angle of \(60°\) above the horizontal. Calculate the time after projection when the particle is descending at an angle of \(40°\) below the horizontal. [4]
CAIE M2 2017 June Q2
6 marks Standard +0.3
\includegraphics{figure_1} One end of a light inextensible string is attached to a fixed point \(A\). The other end of the string is attached to a particle \(P\) of mass \(m\) kg which hangs vertically below \(A\). The particle is also attached to one end of a light elastic string of natural length \(0.25\) m. The other end of this string is attached to a point \(B\) which is \(0.6\) m from \(P\) and on the same horizontal level as \(P\). Equilibrium is maintained by a horizontal force of magnitude \(7\) N applied to \(P\) (see Fig. 1).
  1. Calculate the modulus of elasticity of the elastic string. [2]
  2. \(P\) is released from rest by removing the \(7\) N force. In its subsequent motion \(P\) first comes to instantaneous rest at a point where \(BP = 0.3\) m and the elastic string makes an angle of \(30°\) with the horizontal (see Fig. 2). \includegraphics{figure_2} Find the value of \(m\). [4]