Questions — CAIE (7646 questions)

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CAIE M2 2015 June Q6
9 marks Challenging +1.8
6 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a03ad6c1-b4a3-4007-8d3b-ce289a998a55-4_520_582_264_440} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a03ad6c1-b4a3-4007-8d3b-ce289a998a55-4_497_300_287_1411} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} A small ball \(B\) is projected with speed \(U \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(\theta ^ { \circ }\) above the horizontal from a point \(O\). At time 2 s after the instant of projection, \(B\) strikes a smooth wall which slopes at \(60 ^ { \circ }\) to the horizontal. The speed of \(B\) is \(18 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and its direction of motion is perpendicular to the wall at the instant of impact (see Fig. 1). \(B\) bounces off the wall with speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a direction perpendicular to the wall. At time 0.8 s after \(B\) bounces off the wall, \(B\) strikes the wall again at a lower point \(A\) (see Fig. 2).
  1. Find \(U\) and \(\theta\).
  2. By considering the motion of \(B\) after it bounces off the wall, calculate \(V\).
CAIE M2 2015 June Q7
12 marks Challenging +1.2
7 A force of magnitude \(0.4 t \mathrm {~N}\), applied at an angle of \(30 ^ { \circ }\) above the horizontal, acts on a particle \(P\), where \(t \mathrm {~s}\) is the time since the force starts to act. \(P\) is at rest on rough horizontal ground when \(t = 0\). The mass of \(P\) is 0.2 kg and the coefficient of friction between \(P\) and the ground is \(\mu\).
  1. Given that \(P\) is about to slip when \(t = 2\), find \(\mu\) and the value of \(t\) for the instant when \(P\) loses contact with the ground.
  2. While \(P\) is moving on the ground, it has velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t \mathrm {~s}\). Show that $$\frac { \mathrm { d } v } { \mathrm {~d} t } = 2.165 t - 4.330$$ where the coefficients are correct to 4 significant figures.
  3. Calculate the speed of \(P\) when it loses contact with the ground. {www.cie.org.uk} after the live examination series. }
CAIE M2 2015 June Q1
5 marks Standard +0.3
1 A uniform semicircular lamina has diameter \(A B\) of length 0.8 m .
  1. Find the distance of the centre of mass of the lamina from \(A B\). The lamina rests in a vertical plane, with the point \(B\) of the lamina in contact with a rough horizontal surface and with \(A\) vertically above \(B\). Equilibrium is maintained by a force of magnitude 6 N in the plane of the lamina, applied to the lamina at \(A\) and acting at an angle of \(20 ^ { \circ }\) below the horizontal.
  2. Calculate the mass of the lamina.
CAIE M2 2015 June Q2
5 marks Standard +0.3
2 A particle \(P\) is projected with speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(60 ^ { \circ }\) above the horizontal from a point \(O\) on horizontal ground. \(P\) is moving at an angle of \(45 ^ { \circ }\) above the horizontal at the instant 1.5 s after projection.
  1. Find \(V\).
  2. Hence calculate the horizontal and vertical displacements of \(P\) from \(O\) at the instant 1.5 s after projection.
CAIE M2 2015 June Q3
6 marks Standard +0.3
3 One end of a light elastic string of natural length 0.4 m and modulus of elasticity 20 N is attached to a fixed point \(A\) on a smooth plane inclined at \(30 ^ { \circ }\) to the horizontal. The other end of the string is attached to a particle \(P\) of mass 0.5 kg which rests in equilibrium on the plane.
  1. Calculate the extension of the string. \(P\) is projected down the plane from the equilibrium position with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The extension of the string is \(e \mathrm {~m}\) when the speed of the particle is \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) for the first time.
  2. Find \(e\).
CAIE M2 2015 June Q4
6 marks Moderate -0.3
4 A small ball \(B\) is projected from a point 1.5 m above horizontal ground with initial speed \(29 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(30 ^ { \circ }\) above the horizontal.
  1. Show that \(B\) strikes the ground 3 s after projection.
  2. Find the speed and direction of motion of \(B\) immediately before it strikes the ground.
CAIE M2 2015 June Q5
8 marks Challenging +1.2
5 \includegraphics[max width=\textwidth, alt={}, center]{8f8492a7-8a83-4eb2-81ee-99b4a385b704-3_876_483_260_840} A uniform triangular prism of weight 20 N rests on a horizontal table. \(A B C\) is the cross-section through the centre of mass of the prism, where \(B C = 0.5 \mathrm {~m} , A B = 0.4 \mathrm {~m} , A C = 0.3 \mathrm {~m}\) and angle \(B A C = 90 ^ { \circ }\). The vertical plane \(A B C\) is perpendicular to the edge of the table. The point \(D\) on \(A C\) is at the edge of the table, and \(A D = 0.25 \mathrm {~m}\). One end of a light elastic string of natural length 0.6 m and modulus of elasticity 48 N is attached to \(C\) and a particle of mass 2.5 kg is attached to the other end of the string. The particle is released from rest at \(C\) and falls vertically (see diagram).
  1. Show that the tension in the string is 60 N at the instant when the prism topples.
  2. Calculate the speed of the particle at the instant when the prism topples.
CAIE M2 2015 June Q6
9 marks Standard +0.3
6 A cyclist and her bicycle have a total mass of 60 kg . The cyclist rides in a horizontal straight line, and exerts a constant force in the direction of motion of 150 N . The motion is opposed by a resistance of magnitude \(12 v \mathrm {~N}\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the cyclist's speed at time \(t \mathrm {~s}\) after passing through a fixed point \(A\).
  1. Show that \(5 \frac { \mathrm {~d} v } { \mathrm {~d} t } = 12.5 - v\).
  2. Given that the cyclist passes through \(A\) with speed \(11.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), solve this differential equation to show that \(v = 12.5 - \mathrm { e } ^ { - 0.2 t }\).
  3. Express the displacement of the cyclist from \(A\) in terms of \(t\).
CAIE M2 2015 June Q7
11 marks Standard +0.8
7 A particle \(P\) of mass 0.7 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 \(A\) which is \(h \mathrm {~m}\) above a smooth horizontal surface. \(P\) moves in contact with the surface with uniform circular motion about the point on the surface which is vertically below \(A\).
  1. Given that \(h = 0.14\), find an inequality for the angular speed of \(P\).
  2. Given instead that the magnitude of the force exerted by the surface on \(P\) is 1.4 N and that the speed of \(P\) is \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), calculate the tension in the string and the value of \(h\).
CAIE M2 2016 June Q1
5 marks Standard +0.3
1 A small ball \(B\) is projected with speed \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(30 ^ { \circ }\) above the horizontal from a point \(O\) on horizontal ground. At the instant 0.8 s after projection, \(B\) is 0.5 m vertically above the top of a vertical post.
  1. Calculate the height of the top of the post above the ground.
  2. Show that \(B\) is at its greatest height 0.2 s before passing over the post.
CAIE M2 2016 June Q2
8 marks Standard +0.3
2 One end of a light elastic string of natural length 0.4 m is attached to a fixed point \(O\). The other end of the string is attached to a particle of weight 5 N which hangs in equilibrium 0.6 m vertically below \(O\).
  1. Find the modulus of elasticity of the string. The particle is projected vertically upwards from the equilibrium position and comes to instantaneous rest after travelling 0.3 m upwards.
  2. Calculate the speed of projection of the particle.
  3. Calculate the greatest extension of the string in the subsequent motion.
CAIE M2 2016 June Q3
7 marks Standard +0.3
3 The point \(O\) is 8 m above a horizontal plane. A particle \(P\) is projected from \(O\). After projection, the horizontal and vertically upwards displacements of \(P\) from \(O\) are \(x \mathrm {~m}\) and \(y \mathrm {~m}\) respectively. The equation of the trajectory of \(P\) is $$y = 2 x - x ^ { 2 }$$
  1. Find the value of \(x\) for the point where \(P\) strikes the plane.
  2. Find the angle and speed of projection of \(P\).
  3. Calculate the speed of \(P\) immediately before it strikes the plane.
CAIE M2 2016 June Q4
8 marks Standard +0.3
4 \includegraphics[max width=\textwidth, alt={}, center]{f8633b64-b20c-4471-9641-ccc3e6854f2c-3_784_556_260_790} A uniform object is made by drilling a cylindrical hole through a rectangular block. The axis of the cylindrical hole is perpendicular to the cross-section \(A B C D\) through the centre of mass of the object. \(A B = C D = 0.7 \mathrm {~m} , B C = A D = 0.4 \mathrm {~m}\), and the centre of the hole is 0.1 m from \(A B\) and 0.2 m from \(A D\) (see diagram). The hole has a cross-section of area \(0.03 \mathrm {~m} ^ { 2 }\).
  1. Show that the distance of the centre of mass of the object from \(A B\) is 0.212 m , and calculate the distance of the centre of mass from \(A D\). The object has weight 70 N and is placed on a rough horizontal surface, with \(A D\) in contact with the surface. A vertically upwards force of magnitude \(F \mathrm {~N}\) acts on the object at \(C\). The object is on the point of toppling.
  2. Find the value of \(F\). The force acting at \(C\) is removed, and the object is placed on a rough plane inclined at an angle \(\theta ^ { \circ }\) to the horizontal. \(A D\) lies along a line of greatest slope, with \(A\) higher than \(D\). The plane is sufficiently rough to prevent sliding, and the object does not topple.
  3. Find the greatest possible value of \(\theta\).
CAIE M2 2016 June Q5
10 marks Standard +0.3
5 A particle \(P\) of mass 0.4 kg is placed at rest at a point \(A\) on a rough horizontal surface. A horizontal force, directed away from \(A\) and with magnitude \(0.6 t \mathrm {~N}\), acts on \(P\), where \(t \mathrm {~s}\) is the time after \(P\) is placed at \(A\). The coefficient of friction between \(P\) and the surface is 0.3 , and \(P\) has displacement from \(A\) of \(x \mathrm {~m}\) at time \(t \mathrm {~s}\).
  1. Show that \(P\) starts to move when \(t = 2\). Show also that when \(P\) is in motion it has acceleration \(( 1.5 t - 3 ) \mathrm { m } \mathrm { s } ^ { - 2 }\).
  2. Express the velocity of \(P\) in terms of \(t\), for \(t \geqslant 2\).
  3. Express \(x\) in terms of \(t\), for \(t \geqslant 2\).
CAIE M2 2016 June Q6
12 marks Standard +0.8
6 \includegraphics[max width=\textwidth, alt={}, center]{f8633b64-b20c-4471-9641-ccc3e6854f2c-4_479_499_255_824} \(O A\) is a rod which rotates in a horizontal circle about a vertical axis through \(O\). A particle \(P\) of mass 0.2 kg is attached to the mid-point of a light inextensible string. One end of the string is attached to the \(\operatorname { rod }\) at \(A\) and the other end of the string is attached to a point \(B\) on the axis. It is given that \(O A = O B\), angle \(O A P =\) angle \(O B P = 30 ^ { \circ }\), and \(P\) is 0.4 m from the axis. The rod and the particle rotate together about the axis with \(P\) in the plane \(O A B\) (see diagram).
  1. Calculate the tensions in the two parts of the string when the speed of \(P\) is \(1.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The angular speed of the rod is increased to \(5 \mathrm { rad } \mathrm { s } ^ { - 1 }\), and it is given that the system now rotates with angle \(O A P =\) angle \(O B P = 60 ^ { \circ }\).
  2. Show that the tension in the part \(A P\) of the string is zero. {www.cie.org.uk} after the live examination series. }
CAIE M2 2017 June Q1
5 marks Moderate -0.5
1 A particle \(P\) of mass 0.2 kg moves with speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and angular speed \(5 \mathrm { rad } \mathrm { s } ^ { - 1 }\) in a horizontal circle on a smooth surface. \(P\) is attached to one end of a light elastic string of natural length 0.6 m . The other end of the string is attached to the point on the surface which is the centre of the circular motion of \(P\).
  1. Find the radius of this circle.
  2. Find the modulus of elasticity of the string.
CAIE M2 2017 June Q2
5 marks Standard +0.3
2
[diagram]
The ends of two light inextensible strings of length 0.7 m are attached to a particle \(P\). The other ends of the strings are attached to two fixed points \(A\) and \(B\) which lie in the same vertical line with \(A\) above \(B\). The particle \(P\) moves in a horizontal circle which has its centre at the mid-point of \(A B\). Both strings are inclined at \(60 ^ { \circ }\) to the vertical. The tension in the string attached to \(A\) is 6 N and the tension in the string attached to \(B\) is 4 N (see diagram).
  1. Find the mass of \(P\).
  2. Calculate the speed of \(P\).
CAIE M2 2017 June Q3
7 marks Challenging +1.2
3 An open box in the shape of a cube with edges of length 0.2 m is placed with its base horizontal and its four sides vertical. The four sides and base are uniform laminas, each with weight 3 N .
  1. Calculate the height of the centre of mass of the box above its base.
    The box is now fitted with a thin uniform square lid of weight 3 N and with edges of length 0.2 m . The lid is attached to the box by a hinge of length 0.2 m and weight 2 N . The lid of the box is held partly open.
  2. Find the angle which the lid makes with the horizontal when the centre of mass of the box (including the lid and hinge) is 0.12 m above the base of the box.
CAIE M2 2017 June Q4
8 marks Standard +0.8
4 A small object of mass 0.4 kg is released from rest at a point 8 m above the ground. The object descends vertically and when its downwards displacement from its initial position is \(x \mathrm {~m}\) the object has velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). While the object is moving, a force of magnitude \(0.2 v ^ { 2 } \mathrm {~N}\) opposes the motion.
  1. Show that \(v \frac { \mathrm {~d} v } { \mathrm {~d} x } = 10 - 0.5 v ^ { 2 }\).
  2. Express \(v\) in terms of \(x\).
  3. Find the increase in the value of \(v\) during the final 4 m of the descent of the object.
CAIE M2 2017 June Q5
8 marks Standard +0.3
5 A particle of mass 0.3 kg is attached to one end of a light elastic string of natural length 0.8 m and modulus of elasticity 6 N . The other end of the string is attached to a fixed point \(O\). The particle is projected vertically downwards from \(O\) with initial speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Calculate the greatest speed of the particle during its descent.
  2. Find the greatest distance of the particle below \(O\). \includegraphics[max width=\textwidth, alt={}, center]{2b0425b2-2f8f-491a-996c-3d3b589bd7df-12_558_554_260_794} The end \(A\) of a non-uniform rod \(A B\) of length 0.6 m and weight 8 N rests on a rough horizontal plane, with \(A B\) inclined at \(60 ^ { \circ }\) to the horizontal. Equilibrium is maintained by a force of magnitude 3 N applied to the rod at \(B\). This force acts at \(30 ^ { \circ }\) above the horizontal in the vertical plane containing the rod (see diagram).
  1. Find the distance of the centre of mass of the rod from \(A\).
    The 3 N force is removed, and the rod is held in equilibrium by a force of magnitude \(P \mathrm {~N}\) applied at \(B\), acting in the vertical plane containing the rod, at an angle of \(30 ^ { \circ }\) below the horizontal.
  2. Calculate \(P\).
    In one of the two situations described, the \(\operatorname { rod } A B\) is in limiting equilibrium.
  3. Find the coefficient of friction at \(A\). \(7 \quad\) A particle \(P\) is projected from a point \(O\) with speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At time \(t \mathrm {~s}\) after projection the horizontal and vertically upwards displacements of \(P\) from \(O\) are \(x \mathrm {~m}\) and \(y \mathrm {~m}\) respectively. The equation of the trajectory of \(P\) is \(y = 2 x - \frac { 25 x ^ { 2 } } { V ^ { 2 } }\).
  1. Write down the value of \(\tan \theta\), where \(\theta\) is the angle of projection of \(P\).
    When \(t = 4 , P\) passes through the point \(A\) where \(x = y = a\).
  2. Calculate \(V\) and \(a\).
  3. Find the direction of motion of \(P\) when it passes through \(A\).
CAIE M2 2018 June Q6
9 marks Standard +0.8
6 \includegraphics[max width=\textwidth, alt={}, center]{8dda6c21-7cb5-43b6-9a34-485bdf4042c4-10_262_732_264_705} A particle \(P\) of mass 0.2 kg is attached to one end of a light inextensible string of length 0.6 m . The other end of the string is attached to a particle \(Q\) of mass 0.3 kg . The string passes through a small hole \(H\) in a smooth horizontal surface. A light elastic string of natural length 0.3 m and modulus of elasticity 15 N joins \(Q\) to a fixed point \(A\) which is 0.4 m vertically below \(H\). The particle \(P\) moves on the surface in a horizontal circle with centre \(H\) (see diagram).
  1. Calculate the greatest possible speed of \(P\) for which the elastic string is not extended.
  2. Find the distance \(H P\) given that the angular speed of \(P\) is \(8 \mathrm { rad } \mathrm { s } ^ { - 1 }\).
CAIE M2 2018 June Q7
10 marks Standard +0.8
7 \includegraphics[max width=\textwidth, alt={}, center]{8dda6c21-7cb5-43b6-9a34-485bdf4042c4-12_732_581_260_774} A uniform solid cone has height 1.2 m and base radius 0.5 m . A uniform object is made by drilling a cylindrical hole of radius 0.2 m through the cone along the axis of symmetry (see diagram).
  1. Show that the height of the object is 0.72 m and that the volume of the cone removed by the drilling is \(0.0352 \pi \mathrm {~m} ^ { 3 }\).
    [0pt] [The volume of a cone is \(\frac { 1 } { 3 } \pi r ^ { 2 } h\).]
  2. Find the distance of the centre of mass of the object from its base.
    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 June Q1
5 marks Moderate -0.3
1 \includegraphics[max width=\textwidth, alt={}, center]{bba68fb2-88c6-4883-931b-f738cda2dce3-03_231_970_258_591} A particle \(P\) of mass 0.3 kg is attached to a fixed point \(A\) by a light inextensible string of length 0.8 m . The fixed point \(O\) is 0.15 m vertically below \(A\). The particle \(P\) moves with constant speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a horizontal circle with centre \(O\) (see diagram).
  1. Show that the tension in the string is 16 N .
  2. Find the value of \(v\).
CAIE M2 2019 June Q2
5 marks Standard +0.3
2 A particle is projected with speed \(\mathrm { V } \mathrm { m } \mathrm { s } ^ { - 1 }\) at an angle of \(\theta ^ { \circ }\) above the horizontal. At the instant 4 s after projection the speed of the particle is \(16 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and its direction of motion is \(30 ^ { \circ }\) above the horizontal. Find \(V\) and \(\theta\).
CAIE M2 2019 June Q3
5 marks Standard +0.3
3 \includegraphics[max width=\textwidth, alt={}, center]{bba68fb2-88c6-4883-931b-f738cda2dce3-05_448_802_258_676} The diagram shows the cross-section through the centre of mass of a uniform solid object. The object is a cylinder of radius 0.2 m and length 0.7 m , from which a hemisphere of radius 0.2 m has been removed at one end. The point \(A\) is the centre of the plane face at the other end of the object. Find the distance of the centre of mass of the object from \(A\).
[0pt] [The volume of a hemisphere is \(\frac { 2 } { 3 } \pi r ^ { 3 }\).]