Questions — CAIE M2 (519 questions)

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CAIE M2 2014 November Q1
4 marks Standard +0.8
A particle \(P\) is projected with speed \(V\) m s\(^{-1}\) at an angle of \(30°\) above the horizontal from a point \(O\) on horizontal ground. At the instant \(2\) s after projection, \(OP\) makes an angle of \(15°\) above the horizontal. Calculate \(V\). [4]
CAIE M2 2014 November Q2
4 marks Standard +0.8
\includegraphics{figure_2} A uniform solid cone with height \(0.8\) m and semi-vertical angle \(30°\) has weight \(20\) N. The cone rests in equilibrium with a single point \(P\) of its base in contact with a rough horizontal surface, and its vertex \(V\) vertically above \(P\). Equilibrium is maintained by a force of magnitude \(F\) N acting along the axis of symmetry of the cone and applied to \(V\) (see diagram).
  1. Show that the moment of the weight of the cone about \(P\) is \(6\) N m. [2]
  2. Hence find \(F\). [2]
CAIE M2 2014 November Q3
5 marks Standard +0.3
One end of a light elastic string of natural length \(1.6\) m and modulus of elasticity \(28\) N is attached to a fixed point \(O\). The other end of the string is attached to a particle \(P\) of mass \(0.35\) kg which hangs in equilibrium vertically below \(O\). The particle \(P\) is projected vertically upwards from the equilibrium position with speed \(1.8\) m s\(^{-1}\). Calculate the speed of \(P\) at the instant the string first becomes slack. [5]
CAIE M2 2014 November Q4
7 marks Standard +0.8
\includegraphics{figure_4} \(ABCDEF\) is the cross-section through the centre of mass of a uniform solid prism. \(ABCF\) is a rectangle in which \(AB = CF = 1.6\) m, and \(BC = AF = 0.4\) m. \(CDE\) is a triangle in which \(CD = 1.8\) m, \(CE = 0.4\) m, and angle \(DCE = 90°\). The prism stands on a rough horizontal surface. A horizontal force of magnitude \(T\) N acts at \(B\) in the direction \(CB\) (see diagram). The prism is in equilibrium.
  1. Show that the distance of the centre of mass of the prism from \(AB\) is \(0.488\) m. [4]
  2. Given that the weight of the prism is \(100\) N, find the greatest and least possible values of \(T\). [3]
CAIE M2 2014 November Q5
9 marks Standard +0.3
The equation of the trajectory of a small ball \(B\) projected from a fixed point \(O\) is $$y = -0.05x^2,$$ where \(x\) and \(y\) are, respectively, the displacements in metres of \(B\) from \(O\) in the horizontal and vertically upwards directions.
  1. Show that \(B\) is projected horizontally, and find its speed of projection. [3]
  2. Find the value of \(y\) when the direction of motion of \(B\) is \(60°\) below the horizontal, and find the corresponding speed of \(B\). [6]
CAIE M2 2014 November Q6
9 marks Challenging +1.2
\(O\), \(A\) and \(B\) are three points in a straight line on a smooth horizontal surface. A particle \(P\) of mass \(0.6\) kg moves along the line. At time \(t\) s the particle has displacement \(x\) m from \(O\) and speed \(v\) m s\(^{-1}\). The only horizontal force acting on \(P\) has magnitude \(0.4v^{\frac{1}{2}}\) N and acts in the direction \(OA\). Initially the particle is at \(A\), where \(x = 1\) and \(v = 1\).
  1. Show that \(3v^{\frac{1}{2}}\frac{dv}{dx} = 2\). [2]
  2. Express \(v\) in terms of \(x\). [4]
  3. Given that \(AB = 7\) m, find the value of \(t\) when \(P\) passes through \(B\). [3]
CAIE M2 2014 November Q7
12 marks Standard +0.8
\includegraphics{figure_7} One end of a light elastic string with modulus of elasticity \(15\) N is attached to a fixed point \(A\) which is \(2\) m vertically above a fixed small smooth ring \(R\). The string has natural length \(2\) m and it passes through \(R\). The other end of the string is attached to a particle \(P\) of mass \(m\) kg which moves with constant angular speed \(\omega\) rad s\(^{-1}\) in a horizontal circle which has its centre \(0.4\) m vertically below the ring. \(PR\) makes an acute angle \(\theta\) with the vertical (see diagram).
  1. Show that the tension in the string is \(\frac{3}{\cos\theta}\) N and hence find the value of \(m\). [4]
  2. Show that the value of \(\omega\) does not depend on \(\theta\). [4]
It is given that for one value of \(\theta\) the elastic potential energy stored in the string is twice the kinetic energy of \(P\).
  1. Find this value of \(\theta\). [4]
CAIE M2 2015 November Q1
4 marks Moderate -0.3
A particle \(P\) moves in a straight line and passes through a point \(O\) of the line with velocity \(2\text{ m s}^{-1}\). At time \(t\) s after passing through \(O\), the velocity of \(P\) is \(v\text{ m s}^{-1}\) and the acceleration of \(P\) is given by \(\text{e}^{-0.5t}\text{ m s}^{-2}\). Calculate the velocity of \(P\) when \(t = 1.2\). [4]
CAIE M2 2015 November Q2
5 marks Challenging +1.2
\includegraphics{figure_2} A uniform rigid rod \(AB\) of length \(1.2\text{ m}\) and weight \(8\text{ N}\) has a particle of weight \(2\text{ N}\) attached at the end \(B\). The end \(A\) of the rod is freely hinged to a fixed point. One end of a light elastic string of natural length \(0.8\text{ m}\) and modulus of elasticity \(20\text{ N}\) is attached to the hinge. The string passes over a small smooth pulley \(P\) fixed \(0.8\text{ m}\) vertically above the hinge. The other end of the string is attached to a small light smooth ring \(R\) which can slide on the rod. The system is in equilibrium with the rod inclined at an angle \(\theta°\) to the vertical (see diagram).
  1. Show that the tension in the string is \(20\sin\theta\text{ N}\). [1]
  2. Explain why the part of the string attached to the ring is perpendicular to the rod. [1]
  3. Find \(\theta\). [3]
CAIE M2 2015 November Q3
5 marks Standard +0.3
A particle \(P\) of mass \(0.3\text{ kg}\) moves in a straight line on a smooth horizontal surface. \(P\) passes through a fixed point \(O\) of the line with velocity \(8\text{ m s}^{-1}\). A force of magnitude \(2x\text{ N}\) acts on \(P\) in the direction \(PO\), where \(x\text{ m}\) is the displacement of \(P\) from \(O\).
  1. Show that \(v\frac{\text{d}v}{\text{d}x} = kx\) and state the value of the constant \(k\). [2]
  2. Find the value of \(x\) at the instant when \(P\) comes to instantaneous rest. [3]
CAIE M2 2015 November Q4
8 marks Standard +0.3
\includegraphics{figure_4} One end of a light inextensible string is attached to a fixed point \(A\). The string passes through a smooth bead \(B\) of mass \(0.3\text{ kg}\) and the other end of the string is attached to a fixed point \(C\) vertically below \(A\). The bead \(B\) moves with constant speed in a horizontal circle of radius \(0.6\text{ m}\) which has its centre between \(A\) and \(C\). The string makes an angle of \(30°\) with the vertical at \(A\) and an angle of \(45°\) with the vertical at \(C\) (see diagram).
  1. Calculate the speed of \(B\). [5]
The lower end of the string is detached from \(C\), and \(B\) is now attached to this end of the string. The other end of the string remains attached to \(A\). The bead is set in motion so that it moves with angular speed \(3\text{ rad s}^{-1}\) in a horizontal circle which has its centre vertically below \(A\).
  1. Calculate the tension in the string. [3]
CAIE M2 2015 November Q5
8 marks Standard +0.8
A particle \(P\) of mass \(0.2\text{ kg}\) is attached to one end of a light elastic string of natural length \(0.75\text{ m}\) and modulus of elasticity \(21\text{ N}\). The other end of the string is attached to a fixed point \(A\) which is \(0.8\text{ m}\) vertically above a smooth horizontal surface. \(P\) rests in equilibrium on the surface.
  1. Find the magnitude of the force exerted on \(P\) by the surface. [2]
\(P\) is now projected horizontally along the surface with speed \(3\text{ m s}^{-1}\).
  1. Calculate the extension of the string at the instant when \(P\) leaves the surface. [3]
  2. Hence find the speed of \(P\) at the instant when it leaves the surface. [3]
CAIE M2 2015 November Q6
9 marks Standard +0.3
[diagram]
A uniform circular disc has centre \(O\) and radius \(1.2\text{ m}\). The centre of the disc is at the origin of \(x\)- and \(y\)-axes. Two circular holes with centres at \(A\) and \(B\) are made in the disc (see diagram). The point \(A\) is on the negative \(x\)-axis with \(OA = 0.5\text{ m}\). The point \(B\) is on the negative \(y\)-axis with \(OB = 0.7\text{ m}\). The hole with centre \(A\) has radius \(0.3\text{ m}\) and the hole with centre \(B\) has radius \(0.4\text{ m}\). Find the distance of the centre of mass of the object from
  1. the \(x\)-axis, [4]
  2. the \(y\)-axis. [3]
The object can rotate freely in a vertical plane about a horizontal axis through \(O\).
  1. Calculate the angle which \(OA\) makes with the vertical when the object rests in equilibrium. [2]
CAIE M2 2015 November Q7
11 marks Challenging +1.2
A particle \(P\) is projected with speed \(V\text{ m s}^{-1}\) at an angle of \(60°\) above the horizontal from a point \(O\). At the instant \(1\text{ s}\) later a particle \(Q\) is projected from \(O\) with the same initial speed at an angle of \(45°\) above the horizontal. The two particles collide when \(Q\) has been in motion for \(t\text{ s}\).
  1. Show that \(t = 2.414\), correct to 3 decimal places. [3]
  2. Find the value of \(V\). [4]
The collision occurs after \(P\) has passed through the highest point of its trajectory.
  1. Calculate the vertical distance of \(P\) below its greatest height when \(P\) and \(Q\) collide. [4]
CAIE M2 2015 November Q1
4 marks Moderate -0.3
A particle \(P\) moves in a straight line and passes through a point \(O\) of the line with velocity \(2\,\text{m s}^{-1}\). At time \(t\) s after passing through \(O\), the velocity of \(P\) is \(v\,\text{m s}^{-1}\) and the acceleration of \(P\) is given by \(e^{-0.5t}\,\text{m s}^{-2}\). Calculate the velocity of \(P\) when \(t = 1.2\). [4]
CAIE M2 2015 November Q2
5 marks Standard +0.8
\includegraphics{figure_2} A uniform rigid rod \(AB\) of length \(1.2\,\text{m}\) and weight \(8\,\text{N}\) has a particle of weight \(2\,\text{N}\) attached at the end \(B\). The end \(A\) of the rod is freely hinged to a fixed point. One end of a light elastic string of natural length \(0.8\,\text{m}\) and modulus of elasticity \(20\,\text{N}\) is attached to the hinge. The string passes over a small smooth pulley \(P\) fixed \(0.8\,\text{m}\) vertically above the hinge. The other end of the string is attached to a small light smooth ring \(R\) which can slide on the rod. The system is in equilibrium with the rod inclined at an angle \(\theta°\) to the vertical (see diagram).
  1. Show that the tension in the string is \(20\sin\theta\,\text{N}\). [1]
  2. Explain why the part of the string attached to the ring is perpendicular to the rod. [1]
  3. Find \(\theta\). [3]
CAIE M2 2015 November Q3
5 marks Standard +0.3
A particle \(P\) of mass \(0.3\,\text{kg}\) moves in a straight line on a smooth horizontal surface. \(P\) passes through a fixed point \(O\) of the line with velocity \(8\,\text{m s}^{-1}\). A force of magnitude \(2x\,\text{N}\) acts on \(P\) in the direction \(PO\), where \(x\,\text{m}\) is the displacement of \(P\) from \(O\).
  1. Show that \(v\frac{dv}{dx} = kx\) and state the value of the constant \(k\). [2]
  2. Find the value of \(x\) at the instant when \(P\) comes to instantaneous rest. [3]
CAIE M2 2015 November Q4
8 marks Standard +0.3
\includegraphics{figure_4} One end of a light inextensible string is attached to a fixed point \(A\). The string passes through a smooth bead \(B\) of mass \(0.3\,\text{kg}\) and the other end of the string is attached to a fixed point \(C\) vertically below \(A\). The bead \(B\) moves with constant speed in a horizontal circle of radius \(0.6\,\text{m}\) which has its centre between \(A\) and \(C\). The string makes an angle of \(30°\) with the vertical at \(A\) and an angle of \(45°\) with the vertical at \(C\) (see diagram).
  1. Calculate the speed of \(B\). [5]
The lower end of the string is detached from \(C\), and \(B\) is now attached to this end of the string. The other end of the string remains attached to \(A\). The bead is set in motion so that it moves with angular speed \(3\,\text{rad s}^{-1}\) in a horizontal circle which has its centre vertically below \(A\).
  1. Calculate the tension in the string. [3]
CAIE M2 2015 November Q5
8 marks Standard +0.3
A particle \(P\) of mass \(0.2\,\text{kg}\) is attached to one end of a light elastic string of natural length \(0.75\,\text{m}\) and modulus of elasticity \(21\,\text{N}\). The other end of the string is attached to a fixed point \(A\) which is \(0.8\,\text{m}\) vertically above a smooth horizontal surface. \(P\) rests in equilibrium on the surface.
  1. Find the magnitude of the force exerted on \(P\) by the surface. [2]
\(P\) is now projected horizontally along the surface with speed \(3\,\text{m s}^{-1}\).
  1. Calculate the extension of the string at the instant when \(P\) leaves the surface. [3]
  2. Hence find the speed of \(P\) at the instant when it leaves the surface. [3]
CAIE M2 2015 November Q6
9 marks Standard +0.3
\includegraphics{figure_6} A uniform circular disc has centre \(O\) and radius \(1.2\,\text{m}\). The centre of the disc is at the origin of \(x\)- and \(y\)-axes. Two circular holes with centres at \(A\) and \(B\) are made in the disc (see diagram). The point \(A\) is on the negative \(x\)-axis with \(OA = 0.5\,\text{m}\). The point \(B\) is on the negative \(y\)-axis with \(OB = 0.7\,\text{m}\). The hole with centre \(A\) has radius \(0.3\,\text{m}\) and the hole with centre \(B\) has radius \(0.4\,\text{m}\). Find the distance of the centre of mass of the object from
  1. the \(x\)-axis, [4]
  2. the \(y\)-axis. [3]
The object can rotate freely in a vertical plane about a horizontal axis through \(O\).
  1. Calculate the angle which \(OA\) makes with the vertical when the object rests in equilibrium. [2]
CAIE M2 2015 November Q7
11 marks Challenging +1.2
A particle \(P\) is projected with speed \(V\,\text{m s}^{-1}\) at an angle of \(60°\) above the horizontal from a point \(O\). At the instant \(1\,\text{s}\) later a particle \(Q\) is projected from \(O\) with the same initial speed at an angle of \(45°\) above the horizontal. The two particles collide when \(Q\) has been in motion for \(t\,\text{s}\).
  1. Show that \(t = 2.414\), correct to \(3\) decimal places. [3]
  2. Find the value of \(V\). [4]
The collision occurs after \(P\) has passed through the highest point of its trajectory.
  1. Calculate the vertical distance of \(P\) below its greatest height when \(P\) and \(Q\) collide. [4]
CAIE M2 2016 November Q1
4 marks Standard +0.3
A particle \(P\) of mass 0.3 kg moves in a circle with centre \(O\) on a smooth horizontal surface. \(P\) is attached to \(O\) by a light elastic string of modulus of elasticity 12 N and natural length \(l\) m. The speed of \(P\) is 4 m s\(^{-1}\), and the radius of the circle in which it moves is 2l m. Calculate \(l\). [4]
CAIE M2 2016 November Q2
7 marks Standard +0.8
\includegraphics{figure_2} A uniform wire is bent to form an object which has a semicircular arc with diameter \(AB\) of length 1.2 m, with a smaller semicircular arc with diameter \(BC\) of length 0.6 m. The end \(C\) of the smaller arc is at the centre of the larger arc (see diagram). The two semicircular arcs of the wire are in the same plane.
  1. Show that the distance of the centre of mass of the object from the line \(ACB\) is 0.191 m, correct to 3 significant figures. [3]
The object is freely suspended at \(A\) and hangs in equilibrium.
  1. Find the angle between \(ACB\) and the vertical. [4]
CAIE M2 2016 November Q3
7 marks Standard +0.3
A small block \(B\) of mass 0.25 kg is released from rest at a point \(O\) on a smooth horizontal surface. After its release the velocity of \(B\) is \(v\) m s\(^{-1}\) when its displacement is \(x\) m from \(O\). The force acting on \(B\) has magnitude \((2 + 0.3x^2)\) N and is directed horizontally away from \(O\).
  1. Show that \(v\frac{dv}{dx} = 1.2x^2 + 8\). [2]
  2. Find the velocity of \(B\) when \(x = 1.5\). [3]
An extra force acts on \(B\) after \(x = 1.5\). It is given that, when \(x > 1.5\), $$v\frac{dv}{dx} = 1.2x^2 + 6 - 3x.$$
  1. Find the magnitude of this extra force and state the direction in which it acts. [2]
CAIE M2 2016 November Q4
7 marks Standard +0.3
\includegraphics{figure_4} The diagram shows the cross-section \(ABCD\) through the centre of mass of a uniform solid prism. \(AB = 0.9\) m, \(BC = 2a\) m, \(AD = a\) m and angle \(ABC =\) angle \(BAD = 90°\).
  1. Calculate the distance of the centre of mass of the prism from \(AD\). [2]
  2. Express the distance of the centre of mass of the prism from \(AB\) in terms of \(a\). [2]
The prism has weight 18 N and rests in equilibrium on a rough horizontal surface, with \(AD\) in contact with the surface. A horizontal force of magnitude 6 N is applied to the prism. This force acts through the centre of mass in the direction \(BC\).
  1. Given that the prism is on the point of toppling, calculate \(a\). [3]