Questions — CAIE (7279 questions)

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AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 Mechanics 1 PURE Pure 1 S1 S2 S3 S4 Stats 1 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 SPS SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
CAIE M2 2010 June Q5
  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 2017 March Q5
  1. Find the tension in the string \(A P\) and the value of \(\omega\).
  2. Find \(m\) and the tension in the string \(B Q\).
    \(6 O\) and \(A\) are fixed points on a rough horizontal surface, with \(O A = 1 \mathrm {~m}\). A particle \(P\) of mass 0.4 kg is projected horizontally with speed \(U \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from \(A\) in the direction \(O A\) and moves in a straight line. After projection, when the displacement of \(P\) from \(O\) is \(x \mathrm {~m}\), the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The coefficient of friction between the surface and \(P\) is 0.4 . A force of magnitude \(\frac { 0.8 } { x } \mathrm {~N}\) acts on \(P\) in the direction \(P O\).
  3. Show that, while the particle is in motion, \(v \frac { \mathrm {~d} v } { \mathrm {~d} x } = - 4 - \frac { 2 } { x }\).
    It is given that \(P\) comes to instantaneous rest between \(x = 2.0\) and \(x = 2.1\).
  4. Find the set of possible values of \(U\).
CAIE M2 2019 March Q6
  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
  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
  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
  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 2020 June Q4
  1. Show that \(\bar { x } = \frac { 400 - x ^ { 2 } } { 80 - 3 x }\) and find a corresponding expression for \(\bar { y }\).
    The shape \(A B E F D\) is in equilibrium in a vertical plane with the edge \(D F\) resting on a smooth horizontal surface.
  2. Find the greatest possible value of \(x\), giving your answer in the form \(\mathrm { a } + \mathrm { b } \sqrt { 2 }\), where \(a\) and \(b\) are constants to be determined.
CAIE Further Paper 3 2023 June Q4
  1. Show that \(\mathrm { x } = \frac { 32 \mathrm { a } ^ { 2 } + 3 \mathrm { ad } } { 16 \mathrm { a } + 3 \mathrm {~d} }\) and find an expression, in terms of \(a\) and \(d\), for \(\bar { y }\).
    The object is placed on a rough plane which is inclined to the horizontal at an angle \(\theta\) where \(\sin \theta = \frac { 1 } { 6 }\). The object is in equilibrium with \(C O\) horizontal, where \(C O\) lies in a vertical plane through a line of greatest slope.
  2. Find \(d\) in terms of \(a\).
CAIE Further Paper 3 2023 June Q3
  1. Find the value of \(e\).
  2. Find the loss in the total kinetic energy of the spheres as a result of the collision.
CAIE Further Paper 3 2021 November Q3
  1. Show that the distance of the centre of mass of the object from \(A B\) is \(\frac { 3 \mathrm { a } \left( 2 - \mathrm { k } ^ { 2 } \right) } { 2 ( 8 - 3 \mathrm { k } ) }\).
    When the object is freely suspended from the point \(A\), the line \(A B\) makes an angle \(\theta\) with the downward vertical, where \(\tan \theta = \frac { 7 } { 18 }\).
  2. Find the possible values of \(k\).
    \includegraphics[max width=\textwidth, alt={}, center]{e34abb4b-1c6c-4f39-836d-467ed18345eb-08_494_903_267_525} Two uniform smooth spheres \(A\) and \(B\) of equal radii have masses \(m\) and \(\frac { 3 } { 2 } m\) respectively. The two spheres are each moving with speed \(u\) on a horizontal surface when they collide. Immediately before the collision \(A\) 's direction of motion is along the line of centres, and \(B\) 's direction of motion makes an angle of \(60 ^ { \circ }\) with the line of centres (see diagram). The coefficient of restitution between the spheres is \(\frac { 2 } { 3 }\).
  3. Find the angle through which the direction of motion of \(B\) is deflected by the collision.
  4. Find the loss in the total kinetic energy of the system as a result of the collision.
CAIE Further Paper 3 2022 November Q3
  1. Show that \(\mathrm { N } = \frac { 8 } { 15 } \mathrm {~W} ( 1 + 2 \mathrm { k } )\).
  2. Find the value of \(k\).
CAIE Further Paper 3 2023 November Q3
  1. Find the value of \(\tan \theta\).
  2. Find the percentage loss in the total kinetic energy of the spheres as a result of this collision.
    \includegraphics[max width=\textwidth, alt={}, center]{0270d51a-e252-46d3-8c97-7f71ba91fa65-08_560_575_258_744} A bead of mass \(m\) moves on a smooth circular wire, with centre \(O\) and radius \(a\), in a vertical plane. The bead has speed \(\mathrm { v } _ { \mathrm { A } }\) when it is at the point \(A\) where \(O A\) makes an angle \(\alpha\) with the downward vertical through \(O\), and \(\cos \alpha = \frac { 3 } { 5 }\). Subsequently the bead has speed \(\mathrm { v } _ { \mathrm { B } }\) at the point \(B\), where \(O B\) makes an angle \(\theta\) with the upward vertical through \(O\). Angle \(A O B\) is a right angle (see diagram). The reaction of the wire on the bead at \(B\) is in the direction \(O B\) and has magnitude equal to \(\frac { 1 } { 6 }\) of the magnitude of the reaction when the bead is at \(A\).
  3. Find, in terms of \(m\) and \(g\), the magnitude of the reaction at \(B\).
  4. Given that \(\mathrm { V } _ { \mathrm { A } } = \sqrt { \mathrm { kag } }\), find the value of \(k\).
CAIE S1 2006 June Q6
  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.
CAIE S1 2015 June Q4
(ii) Given that Nikita's mother does not like her present, find the probability that the present is a scarf.
CAIE S1 2014 November Q4
  1. Draw a fully labelled tree diagram to illustrate the various choices that Sharik can make until the computer indicates that he has answered the question correctly.
  2. The random variable \(X\) is the number of attempts that Sharik makes up to and including the one that the computer indicates is correct. Draw up the probability distribution table for \(X\) and find \(\mathrm { E } ( X )\).
    (a) The time, \(X\) hours, for which people sleep in one night has a normal distribution with mean 7.15 hours and standard deviation 0.88 hours.
  3. Find the probability that a randomly chosen person sleeps for less than 8 hours in a night.
  4. Find the value of \(q\) such that \(\mathrm { P } ( X < q ) = 0.75\).
    (b) The random variable \(Y\) has the distribution \(\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)\), where \(2 \sigma = 3 \mu\) and \(\mu \neq 0\). Find \(\mathrm { P } ( Y > 4 \mu )\).
CAIE S2 2016 June Q6
  1. Find \(\mathrm { P } ( X + Y = 4 )\). A random sample of 75 values of \(X\) is taken.
  2. State the approximate distribution of the sample mean, \(\bar { X }\), including the values of the parameters.
  3. Hence find the probability that the sample mean is more than 1.7.
  4. Explain whether the Central Limit theorem was needed to answer part (ii).
CAIE S2 2019 June Q6
  1. Show that \(b = \frac { a } { a - 1 }\).
  2. Given that the median of \(X\) is \(\frac { 3 } { 2 }\), find the values of \(a\) and \(b\).
  3. Use your values of \(a\) and \(b\) from part (ii) to find \(\mathrm { E } ( X )\).
CAIE Further Paper 4 2023 November Q4
  1. Given that \(\mathrm { P } ( X \leqslant 2 ) = \frac { 1 } { 3 }\), show that \(m = \frac { 1 } { 6 }\) and find the values of \(k\) and \(c\).
  2. Find the exact numerical value of the interquartile range of \(X\).
CAIE P3 2017 June Q6
6 Throughout this question the use of a calculator is not permitted. The complex number \(2 - \mathrm { i }\) is denoted by \(u\).
  1. It is given that \(u\) is a root of the equation \(x ^ { 3 } + a x ^ { 2 } - 3 x + b = 0\), where the constants \(a\) and \(b\) are real. Find the values of \(a\) and \(b\).
  2. On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying both the inequalities \(| z - u | < 1\) and \(| z | < | z + \mathrm { i } |\).
CAIE P3 2019 June Q5
5 Throughout this question the use of a calculator is not permitted. It is given that the complex number \(- 1 + ( \sqrt { } 3 ) \mathrm { i }\) is a root of the equation $$k x ^ { 3 } + 5 x ^ { 2 } + 10 x + 4 = 0$$ where \(k\) is a real constant.
  1. Write down another root of the equation.
  2. Find the value of \(k\) and the third root of the equation.
CAIE P3 2013 November Q8
8 Throughout this question the use of a calculator is not permitted.
  1. The complex numbers \(u\) and \(v\) satisfy the equations $$u + 2 v = 2 \mathrm { i } \quad \text { and } \quad \mathrm { i } u + v = 3$$ Solve the equations for \(u\) and \(v\), giving both answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
  2. On an Argand diagram, sketch the locus representing complex numbers \(z\) satisfying \(| z + \mathrm { i } | = 1\) and the locus representing complex numbers \(w\) satisfying \(\arg ( w - 2 ) = \frac { 3 } { 4 } \pi\). Find the least value of \(| z - w |\) for points on these loci.
CAIE P3 2014 November Q5
5 Throughout this question the use of a calculator is not permitted. The complex numbers \(w\) and \(z\) satisfy the relation $$w = \frac { z + \mathrm { i } } { \mathrm { i } z + 2 }$$
  1. Given that \(z = 1 + \mathrm { i }\), find \(w\), giving your answer in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
  2. Given instead that \(w = z\) and the real part of \(z\) is negative, find \(z\), giving your answer in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
CAIE S1 2021 November Q1
1 Each of the 180 students at a college plays exactly one of the piano, the guitar and the drums. The numbers of male and female students who play the piano, the guitar and the drums are given in the following table.
PianoGuitarDrums
Male254411
Female423820
A student at the college is chosen at random.
  1. Find the probability that the student plays the guitar.
  2. Find the probability that the student is male given that the student plays the drums.
  3. Determine whether the events 'the student plays the guitar' and 'the student is female' are independent, justifying your answer.
CAIE S1 2021 November Q2
2 A group of 6 people is to be chosen from 4 men and 11 women.
  1. In how many different ways can a group of 6 be chosen if it must contain exactly 1 man?
    Two of the 11 women are sisters Jane and Kate.
  2. In how many different ways can a group of 6 be chosen if Jane and Kate cannot both be in the group?
CAIE S1 2021 November Q3
3 A bag contains 5 yellow and 4 green marbles. Three marbles are selected at random from the bag, without replacement.
  1. Show that the probability that exactly one of the marbles is yellow is \(\frac { 5 } { 14 }\).
    The random variable \(X\) is the number of yellow marbles selected.
  2. Draw up the probability distribution table for \(X\).
  3. Find \(\mathrm { E } ( X )\).