Questions — CAIE (7646 questions)

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CAIE P3 2019 November Q6
7 marks Standard +0.3
  1. Express \(w\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real and exact.
    The complex number \(1 + 2 \mathrm { i }\) is denoted by \(u\). The complex number \(v\) is such that \(| v | = 2 | u |\) and \(\arg v = \arg u + \frac { 1 } { 3 } \pi\).
  2. Sketch an Argand diagram showing the points representing \(u\) and \(v\).
  3. Explain why \(v\) can be expressed as \(2 u w\). Hence find \(v\), giving your answer in the form \(a + \mathrm { i } b\), where \(a\) and \(b\) are real and exact.
CAIE FP1 2017 November Q8
11 marks Challenging +1.8
  1. Find the value of \(I _ { 2 }\).
  2. Show that, for \(n > 2\), $$( n - 1 ) I _ { n } = 2 ^ { \frac { 1 } { 2 } n - 1 } + ( n - 2 ) I _ { n - 2 }$$
  3. The curve \(C\) has equation \(y = \sec ^ { 3 } x\) for \(0 \leqslant x \leqslant \frac { 1 } { 4 } \pi\). The region \(R\) is bounded by \(C\), the \(x\)-axis, the \(y\)-axis and the line \(x = \frac { 1 } { 4 } \pi\). Find the volume of revolution generated when \(R\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
CAIE FP1 2019 November Q5
9 marks Challenging +1.8
  1. Use standard results from the List of Formulae (MF10) to show that $$S _ { N } = \frac { 1 } { 3 } N \left( 25 N ^ { 2 } + 90 N + 83 \right)$$
  2. Use the method of differences to express \(T _ { N }\) in terms of \(N\).
  3. Find \(\lim _ { N \rightarrow \infty } \left( N ^ { - 3 } S _ { N } T _ { N } \right)\).
CAIE Further Paper 2 2020 June Q4
8 marks Challenging +1.2
  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
8 marks Challenging +1.2
  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
10 marks Challenging +1.8
  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 2018 June Q5
12 marks Challenging +1.2
  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
11 marks Challenging +1.3
  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
12 marks Challenging +1.3
  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 2017 Specimen Q3
11 marks Standard +0.8
  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
8 marks Standard +0.3
  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
10 marks Standard +0.3
  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 2011 June Q4
7 marks Standard +0.3
  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
11 marks Challenging +1.2
  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
10 marks Standard +0.8
  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 2011 November Q5
8 marks Standard +0.8
  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
8 marks Standard +0.3
  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
9 marks Standard +0.3
  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
    1. the acceleration of \(P\) while moving
      1. from \(A\) to \(B\),
      2. from \(B\) to \(C\),
      3. the distance \(A B\),
      4. the time taken for \(P\) to move from \(A\) to \(C\).
CAIE M2 2010 June Q5
8 marks Standard +0.8
  1. It is given that when the ball moves with speed \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) the tension in the string \(Q B\) is three times the tension in the string \(P B\). Calculate the radius of the circle. The ball now moves along this circular path with the minimum possible speed.
  2. State the tension in the string \(P B\) in this case, and find the speed of the ball.
CAIE M2 2019 March Q6
8 marks Challenging +1.2
  1. Find, in terms of \(r\), the distance of the centre of mass of the prism from the centre of the cylinder.
    \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{b8e52188-f9a6-46fc-90bf-97965c6dd324-11_633_729_258_708} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure} The prism has weight \(W \mathrm {~N}\) and is placed with its curved surface on a rough horizontal plane. The axis of symmetry of the cross-section makes an angle of \(30 ^ { \circ }\) with the vertical. A horizontal force of magnitude \(P \mathrm {~N}\) acting in the plane of the cross-section through the centre of mass is applied to the cylinder at the highest point of this cross-section (see Fig. 2). The prism rests in limiting equilibrium.
  2. Find the coefficient of friction between the prism and the plane.
CAIE M2 2003 November Q4
10 marks Standard +0.3
  1. Show that the distance of the centre of mass of the lamina from the side \(B C\) is 6.37 cm . \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{be83d46f-bf5b-4382-b424-bb5067626adc-3_671_608_1050_772} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure} The lamina is smoothly hinged to a wall at \(A\) and is supported, with \(A B\) horizontal, by a light wire attached at \(B\). The other end of the wire is attached to a point on the wall, vertically above \(A\), such that the wire makes an angle of \(30 ^ { \circ }\) with \(A B\) (see Fig. 2). The mass of the lamina is 8 kg . Find
  2. the tension in the wire,
  3. the magnitude of the vertical component of the force acting on the lamina at \(A\).
CAIE M2 2008 November Q4
7 marks Standard +0.3
  1. the base of the cylinder,
  2. the curved surface of the cylinder.
    (ii) \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{5109244c-3062-4f5f-9277-fc6b5b28f2d4-3_348_745_1183_740} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure} Sphere \(A\) is now attached to one end of a light inextensible string. The string passes through a small smooth hole in the middle of the base of the cylinder. Another small sphere \(B\), of mass 0.25 kg , is attached to the other end of the string. \(B\) hangs in equilibrium below the hole while \(A\) is moving in a horizontal circle of radius 0.2 m (see Fig. 2). Find the angular speed of \(A\).
CAIE M2 2012 November Q4
8 marks Challenging +1.2
  1. Find \(r\). The upper cylinder is now fixed to the lower cylinder to create a uniform object.
  2. Show that the centre of mass of the object is $$\frac { 25 h ^ { 2 } + 180 h + 81 } { 50 h + 180 } \mathrm {~m}$$ from \(A\). The object is placed with the plane face containing \(A\) in contact with a rough plane inclined at \(\alpha ^ { \circ }\) to the horizontal, where \(\tan \alpha = 0.5\). The object is on the point of toppling without sliding.
  3. Calculate \(h\).
CAIE Further Paper 3 2022 November Q3
7 marks Challenging +1.2
  1. Show that \(\mathrm { N } = \frac { 8 } { 15 } \mathrm {~W} ( 1 + 2 \mathrm { k } )\).
  2. Find the value of \(k\).
CAIE S1 2006 June Q6
9 marks Easy -1.2
  1. How many teams play in only 1 match?
  2. How many teams play in exactly 2 matches?
  3. Draw up a frequency table for the numbers of matches which the teams play.
  4. Calculate the mean and variance of the numbers of matches which the teams play.