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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 2020 June Q4
7 marks Challenging +1.2
  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
8 marks Challenging +1.8
  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
7 marks Standard +0.3
  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
6 marks Challenging +1.2
  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
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 Further Paper 3 2023 November Q3
7 marks Challenging +1.8
  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
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.
CAIE S1 2015 June Q4
7 marks Standard +0.3
(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
8 marks Standard +0.3
  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
9 marks Standard +0.3
  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
9 marks Standard +0.3
  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
9 marks Standard +0.3
  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 \(X\).
Edexcel C12 2018 January Q8
6 marks Moderate -0.5
  1. \(y = \mathrm { f } ( - x )\)
  2. \(y = \mathrm { f } ( 2 x )\) On each diagram, show clearly the coordinates of any points of intersection of the curve with the two coordinate axes and the coordinates of the stationary points.
Edexcel C1 2006 January Q6
9 marks Moderate -0.8
  1. \(y = \mathrm { f } ( x + 1 )\),
  2. \(y = 2 \mathrm { f } ( x )\),
  3. \(y = \mathrm { f } \left( \frac { 1 } { 2 } x \right)\). On each diagram show clearly the coordinates of all the points where the curve meets the axes.
Edexcel C1 Specimen Q2
4 marks Moderate -0.8
  1. \(y = \mathrm { f } ( x + 1 )\),
  2. \(y = \mathrm { f } ( 2 x )\). On each diagram, show clearly the coordinates of the maximum point, and of each point at which the curve crosses the coordinate axes.
Edexcel C2 2013 June Q7
9 marks Moderate -0.3
  1. Find by calculation the \(x\)-coordinate of \(A\) and the \(x\)-coordinate of \(B\). The shaded region \(R\) is bounded by the line with equation \(y = 10\) and the curve as shown in Figure 1.
  2. Use calculus to find the exact area of \(R\).
Edexcel P3 2022 October Q8
9 marks Standard +0.3
  1. Express \(8 \sin x - 15 \cos x\) in the form \(R \sin ( x - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\) Give the exact value of \(R\), and give the value of \(\alpha\), in radians, to 4 significant figures. $$\mathrm { f } ( x ) = \frac { 15 } { 41 + 16 \sin x - 30 \cos x } \quad x > 0$$
  2. Find
    1. the minimum value of \(\mathrm { f } ( x )\)
    2. the smallest value of \(x\) at which this minimum value occurs.
  3. State the \(y\) coordinate of the minimum points on the curve with equation $$y = 2 \mathrm { f } ( x ) - 5 \quad x > 0$$
  4. State the smallest value of \(x\) at which a maximum point occurs for the curve with equation $$y = - \mathrm { f } ( 2 x ) \quad x > 0$$ \section*{8. In this question you must show all stages of your working.
    In this question you must show all stages of your working.}
Edexcel P4 2024 June Q6
10 marks
  1. show that \end{itemize} $$\frac { \mathrm { d } A } { \mathrm {~d} \theta } = K ( 1 - \cos \theta )$$ where \(K\) is a constant to be found.
  2. Find, in \(\mathrm { cm } ^ { 2 } \mathrm {~s} ^ { - 1 }\), the rate of increase of the area of the segment when \(\theta = \frac { \pi } { 3 }\) 5. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{e583bf92-d6a9-4f1a-b3c8-372afa6e0a0e-10_803_1086_248_493} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} Figure 2 shows a sketch of the curve defined by the parametric equations $$x = t ^ { 2 } + 2 t \quad y = \frac { 2 } { t ( 3 - t ) } \quad a \leqslant t \leqslant b$$ where \(a\) and \(b\) are constants.
    The ends of the curve lie on the line with equation \(y = 1\)
  3. Find the value of \(a\) and the value of \(b\) The region \(R\), shown shaded in Figure 2, is bounded by the curve and the line with equation \(y = 1\)
  4. Show that the area of region \(R\) is given by $$M - k \int _ { a } ^ { b } \frac { t + 1 } { t ( 3 - t ) } \mathrm { d } t$$ where \(M\) and \(k\) are constants to be found.
    1. Write \(\frac { t + 1 } { t ( 3 - t ) }\) in partial fractions.
    2. Use algebraic integration to find the exact area of \(R\), giving your answer in simplest form.
Edexcel P4 2021 October Q8
7 marks Standard +0.8
  1. Find \(\int x ^ { 2 } \ln x \mathrm {~d} x\) Figure 3 shows a sketch of part of the curve with equation $$y = x \ln x \quad x > 0$$ The region \(R\), shown shaded in Figure 3, lies entirely above the \(x\)-axis and is bounded by the curve, the \(x\)-axis and the line with equation \(x = \mathrm { e }\). This region is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid of revolution.
  2. Find the exact volume of the solid formed, giving your answer in simplest form. \section*{8. In this question you must show all stages of your working.
    In this question you must show all stages of your working.}
Edexcel C4 2013 January Q5
15 marks Moderate -0.3
  1. Show that \(A\) has coordinates \(( 0,3 )\).
  2. Find the \(x\) coordinate of the point \(B\).
  3. Find an equation of the normal to \(C\) at the point \(A\). The region \(R\), as shown shaded in Figure 2, is bounded by the curve \(C\), the line \(x = - 1\) and the \(x\)-axis.
  4. Use integration to find the exact area of \(R\).
Edexcel FP2 2008 June Q2
8 marks Challenging +1.2
  1. Find, in terms of \(a\), the polar coordinates of the points where the curve \(C _ { 1 }\) meets the circle \(C _ { 2 }\).(4) \end{enumerate} The regions enclosed by the curves \(C _ { 1 }\) and \(C _ { 2 }\) overlap and this common region \(R\) is shaded in the figure.
  2. Find, in terms of \(a\), an exact expression for the area of the \includegraphics[max width=\textwidth, alt={}, center]{863ef52d-ae75-450c-9eab-8102804868f5-1_523_707_1262_1255}
    region \(R\).(8)
  3. In a single diagram, copy the two curves in the diagram above and also sketch the curve \(C _ { 3 }\) with polar equation \(r = 2 a \cos \theta , 0 \leq \theta < 2 \pi\) Show clearly the coordinates of the points of intersection of \(\mathrm { C } _ { 1 } , \mathrm { C } _ { 2 }\) and \(\mathrm { C } _ { 3 }\) with the initial line, \(\theta = 0\).(3)(Total 15 marks)