Questions — CAIE (7279 questions)

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CAIE Further Paper 2 2020 June Q4
  1. By considering the sum of the areas of these rectangles, show that $$\int _ { 0 } ^ { 1 } x ^ { 2 } d x < \frac { 2 n ^ { 2 } + 3 n + 1 } { 6 n ^ { 2 } }$$
  2. Use a similar method to find, in terms of \(n\), a lower bound for \(\int _ { 0 } ^ { 1 } x ^ { 2 } \mathrm {~d} x\).
CAIE Further Paper 2 2020 November Q4
  1. By considering the sum of the areas of the rectangles, show that $$\int _ { 0 } ^ { 1 } \left( 1 - x ^ { 3 } \right) d x \leqslant \frac { 3 n ^ { 2 } + 2 n - 1 } { 4 n ^ { 2 } }$$
  2. Use a similar method to find, in terms of \(n\), a lower bound for \(\int _ { 0 } ^ { 1 } \left( 1 - x ^ { 3 } \right) \mathrm { dx }\).
CAIE Further Paper 2 2021 November Q4
  1. By considering the sum of the areas of these rectangles, show that $$\sum _ { r = 1 } ^ { N } \frac { \ln r } { r ^ { 2 } } < \frac { 2 + 3 \ln 2 } { 4 } - \frac { 1 + \ln N } { N }$$
  2. Use a similar method to find, in terms of \(N\), a lower bound for \(\sum _ { \mathrm { r } = 1 } ^ { \mathrm { N } } \frac { \ln \mathrm { r } } { \mathrm { r } ^ { 2 } }\).
CAIE FP2 2017 June Q4
  1. Find the moment of inertia of this object about an axis \(l\), which is perpendicular to the plane of the object and through the centre of \(\operatorname { disc } A\).
    The object is free to rotate about the horizontal axis \(l\). It is released from rest in the position shown, with the centre of disc \(B\) vertically above the centre of disc \(A\).
  2. Write down the change in the vertical position of the centre of mass of the object when the centre of disc \(B\) is vertically below the centre of disc \(A\). Hence find the angular velocity of the object when the centre of disc \(B\) is vertically below the centre of disc \(A\).
CAIE FP2 2017 June Q4
  1. Find the tension in the string in terms of \(W\).
  2. Find the modulus of elasticity of the string in terms of \(W\).
  3. Find the angle that the force acting on the rod at \(A\) makes with the horizontal.
    \includegraphics[max width=\textwidth, alt={}, center]{b10d2991-abff-4d2b-b470-1df844d1c7ee-10_442_442_260_849} A particle of mass \(m\) 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 is moving in complete vertical circles with the string taut. When the particle is at the point \(P\), where \(O P\) makes an angle \(\alpha\) with the upward vertical through \(O\), its speed is \(u\). When the particle is at the point \(Q\), where angle \(Q O P = 90 ^ { \circ }\), its speed is \(v\) (see diagram). It is given that \(\cos \alpha = \frac { 4 } { 5 }\).
  4. Show that \(v ^ { 2 } = u ^ { 2 } + \frac { 14 } { 5 } a g\).
    The tension in the string when the particle is at \(Q\) is twice the tension in the string when the particle is at \(P\).
  5. Obtain another equation relating \(u ^ { 2 } , v ^ { 2 } , a\) and \(g\), and hence find \(u\) in terms of \(a\) and \(g\).
  6. Find the least tension in the string during the motion.
CAIE FP2 2018 June Q5
  1. Show that the moment of inertia of the object about the axis \(l\) is \(180 M a ^ { 2 }\).
  2. Show that small oscillations of the object about the axis \(l\) are approximately simple harmonic, and state the period.
CAIE FP2 2019 June Q4
  1. Find the moment of inertia of the object, consisting of the rod and two spheres, about \(L\).
    The object is pivoted at \(A\) so that it can rotate freely about \(L\). The object is released from rest with the rod making an angle of \(60 ^ { \circ }\) to the downward vertical. The greatest angular speed attained by the object in the subsequent motion is \(\frac { 9 } { 20 } \sqrt { } \left( \frac { g } { a } \right)\).
  2. Find the value of \(k\).
CAIE FP2 2017 November Q5
  1. Show that the moment of inertia of the system, consisting of frame and small object, about an axis through \(O\) perpendicular to the plane of the frame, is \(\frac { 169 } { 3 } m a ^ { 2 }\).
  2. Show that small oscillations of the system about this axis are approximately simple harmonic and state their period.
CAIE FP2 2018 November Q3
  1. Show that \(u ^ { 2 } = 2 a g\).
  2. Find the maximum tension in the string as the particle moves from \(A\) to \(B\).
    \includegraphics[max width=\textwidth, alt={}, center]{a092cd45-dc19-476d-adf7-0198fbb2116e-06_543_807_255_669} A uniform rod \(A B\) of length \(2 a\) and weight \(W\) rests against a smooth horizontal peg at a point \(C\) on the rod, where \(A C = x\). The lower end \(A\) of the rod rests on rough horizontal ground. The rod is in equilibrium inclined at an angle of \(45 ^ { \circ }\) to the horizontal (see diagram). The coefficient of friction between the rod and the ground is \(\mu\). The rod is about to slip at \(A\).
  3. Find an expression for \(x\) in terms of \(a\) and \(\mu\).
  4. Hence show that \(\mu \geqslant \frac { 1 } { 3 }\).
  5. Given that \(x = \frac { 3 } { 2 } a\), find the value of \(\mu\) and the magnitude of the resultant force on the rod at \(A\).
CAIE FP2 2019 November Q5
  1. Show that the moment of inertia of the object about \(L\) is \(\left( \frac { 408 + 7 \lambda } { 12 } \right) M a ^ { 2 }\).
    The period of small oscillations of the object about \(L\) is \(5 \pi \sqrt { } \left( \frac { 2 a } { g } \right)\).
  2. Find the value of \(\lambda\).
CAIE FP2 2017 Specimen Q3
  1. Find the value of \(k\).
  2. The particle \(P\) is released from rest at a point between \(A\) and \(B\) where both strings are taut. Show that \(P\) performs simple harmonic motion and state the period of the motion.
  3. In the case where \(P\) is released from rest at a distance \(0.2 a \mathrm {~m}\) from \(M\), the speed of \(P\) is \(0.7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when \(P\) is \(0.05 a \mathrm {~m}\) from \(M\). Find the value of \(a\).
CAIE M1 2022 June Q4
  1. In the case where \(F = 20\), find the tensions in each of the strings.
  2. Find the greatest value of \(F\) for which the block remains in equilibrium in the position shown.
CAIE M1 2022 June Q6
  1. It is given that the plane \(B C\) is smooth and that the particles are released from rest. Find the tension in the string and the magnitude of the acceleration of the particles.
  2. It is given instead that the plane \(B C\) is rough. A force of magnitude 3 N is applied to \(Q\) directly up the plane along a line of greatest slope of the plane. Find the least value of the coefficient of friction between \(Q\) and the plane \(B C\) for which the particles remain at rest.
CAIE M1 2023 June Q4
  1. Given that the forces are in equilibrium, find the value of \(F\) and the value of \(\theta\).
  2. Given instead that \(F = 10 \sqrt { 2 }\) and \(\theta = 45\), find the direction and the exact magnitude the resultant force.
    \includegraphics[max width=\textwidth, alt={}, center]{f9e3d562-ae3c-49cc-bc92-56956d939252-10_518_627_264_756} Two particles \(P\) and \(Q\), of masses 0.2 kg and 0.1 kg respectively, are attached to the ends of a light inextensible string. The string passes over a fixed smooth pulley at \(B\) which is attached to two inclined planes. Particle \(P\) lies on a smooth plane \(A B\) which is inclined at \(60 ^ { \circ }\) to the horizontal. Particle \(Q\) lies on a plane \(B C\) which is inclined at an angle of \(\theta ^ { \circ }\) to the horizontal. The string is taut and the particles can move on lines of greatest slope of the two planes (see diagram).
  3. It is given that \(\theta = 60\), the plane \(B C\) is rough and the coefficient of friction between \(Q\) and the plane \(B C\) is 0.7 . The particles are released from rest. Determine whether the particles move.
  4. It is given instead that the plane \(B C\) is smooth. The particles are released from rest and in the subsequent motion the tension in the string is \(( \sqrt { 3 } - 1 ) \mathrm { N }\). Find the magnitude of the acceleration of \(P\) as it moves on the plane, and find the value of \(\theta\).
CAIE M1 2023 March Q5
  1. Find the magnitude of the force in each of the struts \(A D\) and \(B D\).
    A horizontal force of magnitude \(F \mathrm {~N}\) is applied to the block in a direction parallel to \(A B\).
  2. Find the value of \(F\) for which the magnitude of the force in the strut \(A D\) is zero.
    \includegraphics[max width=\textwidth, alt={}, center]{b2cd1b68-523f-40c3-8a51-acb2b55ae8c0-08_456_782_260_687} A block \(B\), of mass 2 kg , lies on a rough inclined plane sloping at \(30 ^ { \circ }\) to the horizontal. A light rope, inclined at an angle of \(20 ^ { \circ }\) above a line of greatest slope, is attached to \(B\). The tension in the rope is \(T \mathrm {~N}\). There is a friction force of \(F \mathrm {~N}\) acting on \(B\) (see diagram). The coefficient of friction between \(B\) and the plane is \(\mu\).
  3. It is given that \(F = 5\) and that the acceleration of \(B\) up the plane is \(1.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
    1. Find the value of \(T\).
    2. Find the value of \(\mu\).
  4. It is given instead that \(\mu = 0.8\) and \(T = 15\). Determine whether \(B\) will move up the plane.
CAIE M1 2011 June Q4
  1. Make a rough copy of the diagram and shade the region whose area represents the displacement of \(P\) from \(X\) at the instant when \(Q\) starts. It is given that \(P\) has travelled 70 m at the instant when \(Q\) starts.
  2. Find the value of \(T\).
  3. Find the distance between \(P\) and \(Q\) when \(Q\) 's speed reaches \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  4. Sketch a single diagram showing the displacement-time graphs for both \(P\) and \(Q\), with values shown on the \(t\)-axis at which the speed of either particle changes.
CAIE M1 2015 June Q6
  1. Find the value of \(h\).
  2. Find the value of \(m\), and find also the tension in the string while \(Q\) is moving.
  3. The string is slack while \(Q\) is at rest on the ground. Find the total time from the instant that \(P\) is released until the string becomes taut again.
CAIE M1 2019 June Q4
  1. Show that, before the string breaks, the magnitude of the acceleration of each particle is \(3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) and find the tension in the string.
  2. Find the difference in the times that it takes the particles to hit the ground.
CAIE M1 2007 November Q5
  1. The normal and frictional components of the contact force exerted on the ring by the rod are \(R \mathrm {~N}\) and \(F\) N respectively. Find \(R\) and \(F\) in terms of \(T\).
  2. The coefficient of friction between the rod and the ring is 0.7 . Find the value of \(T\) for which the ring is about to slip.
  3. A man walks in a straight line from \(A\) to \(B\) with constant acceleration \(0.004 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). His speed at \(A\) is \(1.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and his speed at \(B\) is \(2.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the time taken for the man to walk from \(A\) to \(B\), and find the distance \(A B\).
  4. A woman cyclist leaves \(A\) at the same instant as the man. She starts from rest and travels in a straight line to \(B\), reaching \(B\) at the same instant as the man. At time \(t \mathrm {~s}\) after leaving \(A\) the cyclist's speed is \(k \left( 200 t - t ^ { 2 } \right) \mathrm { m } \mathrm { s } ^ { - 1 }\), where \(k\) is a constant. Find
    (a) the value of \(k\),
    (b) the cyclist's speed at \(B\).
  5. Sketch, using the same axes, the velocity-time graphs for the man's motion and the woman's motion from \(A\) to \(B\).
CAIE M1 2011 November Q5
  1. Show that \(\mu \geqslant \frac { 6 } { 17 }\). When the applied force acts upwards as in Fig. 2 the block slides along the floor.
  2. Find another inequality for \(\mu\).
CAIE M1 2012 November Q5
  1. Find the value of \(\theta\). At time 4.8 s after leaving \(A\), the particle comes to rest at \(C\).
  2. Find the coefficient of friction between \(P\) and the rough part of the plane.
CAIE M1 2014 November Q6
  1. the work done against the frictional force acting on \(B\),
  2. the loss of potential energy of the system,
  3. the gain in kinetic energy of the system. At the instant when \(B\) has moved 0.9 m the string breaks. \(A\) is at a height of 0.54 m above a horizontal floor at this instant.
    (ii) Find the speed with which \(A\) reaches the floor.
    \(6 \quad A B C\) is a line of greatest slope of a plane inclined at angle \(\alpha\) to the horizontal, where \(\sin \alpha = 0.28\) and \(\cos \alpha = 0.96\). The point \(A\) is at the top of the plane, the point \(C\) is at the bottom of the plane and the length of \(A C\) is 5 m . The part of the plane above the level of \(B\) is smooth and the part below the level of \(B\) is rough. A particle \(P\) is released from rest at \(A\) and reaches \(C\) with a speed of \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The coefficient of friction between \(P\) and the part of the plane below \(B\) is 0.5 . Find
    (i) the acceleration of \(P\) while moving
  4. from \(A\) to \(B\),
  5. from \(B\) to \(C\),
    (ii) the distance \(A B\),
    (iii) the time taken for \(P\) to move from \(A\) to \(C\).
CAIE M1 2017 November Q6
  1. Show that the coefficient of friction between \(P\) and the plane is \(\frac { 4 } { 3 }\).
    A force of magnitude 10 N is applied to \(P\), acting up a line of greatest slope of the plane, and \(P\) accelerates at \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  2. Find the value of \(m\).
CAIE M1 2019 November Q4
  1. Find the acceleration of the blocks and the tension in the string.
  2. At a particular instant, the speed of the blocks is \(1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the time, after this instant, that it takes for the blocks to travel 0.65 m .
    \includegraphics[max width=\textwidth, alt={}, center]{dd1828e1-5b90-4584-92de-f00f9c4f9657-08_574_895_260_625} A small ring \(P\) is threaded on a fixed smooth horizontal \(\operatorname { rod } A B\). Three horizontal forces of magnitudes \(4.5 \mathrm {~N} , 7.5 \mathrm {~N}\) and \(F \mathrm {~N}\) act on \(P\) (see diagram).
  3. Given that these three forces are in equilibrium, find the values of \(F\) and \(\theta\).
  4. It is given instead that the values of \(F\) and \(\theta\) are 9.5 and 30 respectively, and the acceleration of the ring is \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Find the mass of the ring.
CAIE M1 Specimen Q4
  1. Find the values of \(F\) and \(R\).
  2. Initially the bead is at rest at \(A\). It reaches \(B\) with a speed of \(11.7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the mass of the bead.