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CAIE M1 2016 June Q1
4 marks Moderate -0.3
1 A particle of mass 8 kg is pulled at a constant speed a distance of 20 m up a rough plane inclined at an angle of \(30 ^ { \circ }\) to the horizontal by a force acting along a line of greatest slope.
  1. Find the change in gravitational potential energy of the particle.
  2. The total work done against gravity and friction is 1146 J . Find the frictional force acting on the particle.
CAIE M1 2016 June Q2
5 marks Moderate -0.3
2 Alan starts walking from a point \(O\), at a constant speed of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), along a horizontal path. Ben walks along the same path, also starting from \(O\). Ben starts from rest 5 s after Alan and accelerates at \(1.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) for 5 s . Ben then continues to walk at a constant speed until he is at the same point, \(P\), as Alan.
  1. Find how far Ben has travelled when he has been walking for 5 s and find his speed at this instant.
  2. Find the distance \(O P\).
CAIE M1 2016 June Q3
6 marks Moderate -0.3
3 \includegraphics[max width=\textwidth, alt={}, center]{aaf655c6-47f0-4f17-9a57-58aaf48728df-2_586_611_1171_767} The coplanar forces shown in the diagram are in equilibrium. Find the values of \(P\) and \(\theta\).
CAIE M1 2016 June Q4
7 marks Standard +0.3
4 A particle of mass 15 kg is stationary on a rough plane inclined at an angle of \(20 ^ { \circ }\) to the horizontal. The coefficient of friction between the particle and the plane is 0.2 . A force of magnitude \(X \mathrm {~N}\) acting parallel to a line of greatest slope of the plane is used to keep the particle in equilibrium. Show that the least possible value of \(X\) is 23.1 , correct to 3 significant figures, and find the greatest possible value of \(X\).
CAIE M1 2016 June Q5
8 marks Moderate -0.3
5 The motion of a car of mass 1400 kg is resisted by a constant force of magnitude 650 N .
  1. Find the constant speed of the car on a horizontal road, assuming that the engine works at a rate of 20 kW .
  2. The car is travelling at a constant speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) up a hill inclined at an angle of \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 7 }\). Find the power of the car's engine.
  3. The car descends the same hill with the engine working at \(80 \%\) of the power found in part (ii). Find the acceleration of the car at an instant when the speed is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
CAIE M1 2016 June Q6
10 marks Standard +0.8
6 Two particles of masses 1.3 kg and 0.7 kg are connected by a light inextensible string that passes over a fixed smooth pulley. The particles are held at the same vertical height with the string taut. The distance of each particle above a horizontal plane is 2 m , and the distance of each particle below the pulley is 4 m . The particles are released from rest.
  1. Find
    1. the tension in the string before the particle of mass 1.3 kg reaches the plane,
    2. the time taken for the particle of mass 1.3 kg to reach the plane.
    3. Find the greatest height of the particle of mass 0.7 kg above the plane.
CAIE M1 2016 June Q7
10 marks Standard +0.3
7 A particle \(P\) moves in a straight line. At time \(t \mathrm {~s}\), the displacement of \(P\) from \(O\) is \(s \mathrm {~m}\) and the acceleration of \(P\) is \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\), where \(a = 6 t - 2\). When \(t = 1 , s = 7\) and when \(t = 3 , s = 29\).
  1. Find the set of values of \(t\) for which the particle is decelerating.
  2. Find \(s\) in terms of \(t\).
  3. Find the time when the velocity of the particle is \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
CAIE M1 2017 June Q1
3 marks Moderate -0.8
1 A particle of mass 0.6 kg is dropped from a height of 8 m above the ground. The speed of the particle at the instant before hitting the ground is \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the work done against air resistance.
CAIE M1 2017 June Q2
6 marks Moderate -0.3
2 A particle of mass 0.8 kg is projected with a speed of \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) up a line of greatest slope of a rough plane inclined at an angle of \(10 ^ { \circ }\) to the horizontal. The coefficient of friction between the particle and the plane is 0.4 .
  1. Find the acceleration of the particle.
  2. Find the distance the particle moves up the plane before coming to rest.
CAIE M1 2017 June Q3
6 marks Moderate -0.3
3 \includegraphics[max width=\textwidth, alt={}, center]{4941e074-2f93-4a0c-80ba-0ca96a48389e-04_442_584_255_778} Two light inextensible strings are attached to a particle of weight 25 N . The strings pass over two smooth fixed pulleys and have particles of weights \(A \mathrm {~N}\) and \(B \mathrm {~N}\) hanging vertically at their ends. The sloping parts of the strings make angles of \(30 ^ { \circ }\) and \(40 ^ { \circ }\) respectively with the vertical (see diagram). The system is in equilibrium. Find the values of \(A\) and \(B\).
CAIE M1 2017 June Q4
6 marks Moderate -0.3
4 A car of mass 800 kg is moving up a hill inclined at \(\theta ^ { \circ }\) to the horizontal, where \(\sin \theta = 0.15\). The initial speed of the car is \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Twelve seconds later the car has travelled 120 m up the hill and has speed \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Find the change in the kinetic energy and the change in gravitational potential energy of the car.
  2. The engine of the car is working at a constant rate of 32 kW . Find the total work done against the resistive forces during the twelve seconds.
CAIE M1 2017 June Q5
7 marks Moderate -0.3
5 A particle \(P\) moves in a straight line \(A B C D\) with constant deceleration. The velocities of \(P\) at \(A , B\) and \(C\) are \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 } , 12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively.
  1. Find the ratio of distances \(A B : B C\).
  2. The particle comes to rest at \(D\). Given that the distance \(A D\) is 80 m , find the distance \(B C\).
CAIE M1 2017 June Q6
10 marks Standard +0.3
6 A particle \(P\) moves in a straight line passing through a point \(O\). At time \(t \mathrm {~s}\), the velocity of \(P , v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), is given by \(v = q t + r t ^ { 2 }\), where \(q\) and \(r\) are constants. The particle has velocity \(4 \mathrm {~ms} ^ { - 1 }\) when \(t = 1\) and when \(t = 2\).
  1. Show that, when \(t = 0.5\), the acceleration of \(P\) is \(4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
    ............................................................................................................................... .
  2. Find the values of \(t\) when \(P\) is at instantaneous rest.
  3. The particle is at \(O\) when \(t = 3\). Find the distance of \(P\) from \(O\) when \(t = 0\).
CAIE M1 2017 June Q7
12 marks Standard +0.3
7 \includegraphics[max width=\textwidth, alt={}, center]{4941e074-2f93-4a0c-80ba-0ca96a48389e-10_374_762_259_688} As shown in the diagram, a particle \(A\) of mass 0.8 kg lies on a plane inclined at an angle of \(30 ^ { \circ }\) to the horizontal and a particle \(B\) of mass 1.2 kg lies on a plane inclined at an angle of \(60 ^ { \circ }\) to the horizontal. The particles are connected by a light inextensible string which passes over a small smooth pulley \(P\) fixed at the top of the planes. The parts \(A P\) and \(B P\) of the string are parallel to lines of greatest slope of the respective planes. The particles are released from rest with both parts of the string taut.
  1. Given that both planes are smooth, find the acceleration of \(A\) and the tension in the string.
  2. It is given instead that both planes are rough, with the same coefficient of friction, \(\mu\), for both particles. Find the value of \(\mu\) for which the system is in limiting equilibrium.
CAIE Further Paper 2 2020 June Q1
6 marks Standard +0.3
1 Find the solution of the differential equation $$\frac { d y } { d x } + 5 y = e ^ { - 7 x }$$ for which \(y = 0\) when \(x = 0\). Give your answer in the form \(y = f ( x )\).
CAIE Further Paper 2 2020 June Q2
7 marks Standard +0.3
2 It is given that \(y = 2 ^ { x }\).
  1. By differentiating \(\ln y\) with respect to \(x\), show that \(\frac { \mathrm { dy } } { \mathrm { dx } } = 2 ^ { \mathrm { x } } \ln 2\).
  2. Write down \(\frac { d ^ { 2 } y } { d x ^ { 2 } }\).
  3. Hence find the first three terms in the Maclaurin's series for \(2 ^ { X }\).
CAIE Further Paper 2 2020 June Q3
8 marks Standard +0.8
3
  1. Find the roots of the equation \(z ^ { 3 } = - 1 - \mathrm { i }\), giving your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(0 \leqslant \theta < 2 \pi\).
    Let \(\mathbf { w } = \mathbf { z } _ { 1 } ^ { 3 \mathrm { k } } + \mathbf { z } _ { 2 } ^ { 3 \mathrm { k } } + \mathbf { z } _ { 3 } ^ { 3 \mathrm { k } }\), where \(k\) is a positive integer and \(\mathrm { z } _ { 1 } , \mathrm { z } _ { 2 } , \mathrm { z } _ { 3 }\) are the roots of \(\mathrm { z } ^ { 3 } = - 1 - \mathrm { i }\).
  2. Express \(w\) in the form \(R \mathrm { e } ^ { \mathrm { i } \alpha }\), where \(R > 0\), giving \(R\) and \(\alpha\) in terms of \(k\). \includegraphics[max width=\textwidth, alt={}, center]{20e14db3-0eb0-4954-91cf-027e16f8bf14-06_889_824_267_616} The diagram shows the curve with equation \(\mathrm { y } = \mathrm { x } ^ { 2 }\) for \(0 \leqslant x \leqslant 1\), together with a set of \(n\) rectangles of width \(\frac { 1 } { n }\).
CAIE Further Paper 2 2020 June Q5
11 marks Challenging +1.2
5 The curves \(C _ { 1 } : y = \cosh x\) and \(C _ { 2 } : y = \sinh 2 x\) intersect at the point where \(x = a\).
  1. Find the exact value of \(a\), giving your answer in logarithmic form.
  2. Sketch \(C _ { 1 }\) and \(C _ { 2 }\) on the same diagram.
  3. Find the exact value of the length of the arc of \(C _ { 1 }\) from \(x = 0\) to \(\mathrm { x } = \mathrm { a }\).
CAIE Further Paper 2 2020 June Q6
10 marks Challenging +1.8
6 The integral \(\mathrm { I } _ { \mathrm { n } }\), where \(n\) is an integer, is defined by \(\mathrm { I } _ { \mathrm { n } } = \int _ { 0 } ^ { \frac { 1 } { 2 } } \left( 1 - \mathrm { x } ^ { 2 } \right) ^ { - \frac { 1 } { 2 } \mathrm { n } } \mathrm { dx }\).
  1. Find the exact value of \(I _ { 1 }\).
  2. By considering \(\frac { \mathrm { d } } { \mathrm { dx } } \left( \mathrm { x } \left( 1 - \mathrm { x } ^ { 2 } \right) ^ { - \frac { 1 } { 2 } \mathrm { n } } \right)\), or otherwise, show that $$\mathrm { nl } _ { \mathrm { n } + 2 } = 2 ^ { \mathrm { n } - 1 } 3 ^ { - \frac { 1 } { 2 } \mathrm { n } } + ( \mathrm { n } - 1 ) \mathrm { I } _ { \mathrm { n } } .$$
  3. Find the exact value of \(I _ { 5 }\) giving the answer in the form \(k \sqrt { 3 }\), where \(k\) is a rational number to be determined. \includegraphics[max width=\textwidth, alt={}, center]{20e14db3-0eb0-4954-91cf-027e16f8bf14-11_78_1576_336_321}
CAIE Further Paper 2 2020 June Q7
11 marks Challenging +1.8
7 It is given that \(x = t ^ { 3 } y\) and $$t ^ { 3 } \frac { d ^ { 2 } y } { d t ^ { 2 } } + \left( 4 t ^ { 3 } + 6 t ^ { 2 } \right) \frac { d y } { d t } + \left( 13 t ^ { 3 } + 12 t ^ { 2 } + 6 t \right) y = 61 e ^ { \frac { 1 } { 2 } t }$$
  1. Show that $$\frac { d ^ { 2 } x } { d t ^ { 2 } } + 4 \frac { d x } { d t } + 13 x = 61 e ^ { \frac { 1 } { 2 } t }$$
  2. Find the general solution for \(y\) in terms of \(t\).
CAIE Further Paper 2 2020 June Q8
14 marks Standard +0.8
8
  1. Find the values of \(a\) for which the system of equations $$\begin{aligned} 3 x + y + z & = 0 \\ a x + 6 y - z & = 0 \\ a y - 2 z & = 0 \end{aligned}$$ does not have a unique solution.
    The matrix \(\mathbf { A }\) is given by $$\mathbf { A } = \left( \begin{array} { r r r } 3 & 1 & 1 \\ 0 & 6 & - 1 \\ 0 & 0 & - 2 \end{array} \right) .$$
  2. Use the characteristic equation of \(\mathbf { A }\) to find the inverse of \(\mathbf { A } ^ { 2 }\).
  3. Find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { A } ^ { 5 } = \mathbf { P D P } ^ { - 1 }\).
    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 Further Paper 2 2020 June Q3
8 marks Standard +0.8
3
  1. Find the roots of the equation \(z ^ { 3 } = - 1 - \mathrm { i }\), giving your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(0 \leqslant \theta < 2 \pi\).
    Let \(\mathbf { w } = \mathbf { z } _ { 1 } ^ { 3 \mathrm { k } } + \mathbf { z } _ { 2 } ^ { 3 \mathrm { k } } + \mathbf { z } _ { 3 } ^ { 3 \mathrm { k } }\), where \(k\) is a positive integer and \(\mathrm { z } _ { 1 } , \mathrm { z } _ { 2 } , \mathrm { z } _ { 3 }\) are the roots of \(\mathrm { z } ^ { 3 } = - 1 - \mathrm { i }\).
  2. Express \(w\) in the form \(R \mathrm { e } ^ { \mathrm { i } \alpha }\), where \(R > 0\), giving \(R\) and \(\alpha\) in terms of \(k\). \includegraphics[max width=\textwidth, alt={}, center]{1de67949-6262-4ade-b986-02b6563ae404-06_889_824_267_616} The diagram shows the curve with equation \(\mathrm { y } = \mathrm { x } ^ { 2 }\) for \(0 \leqslant x \leqslant 1\), together with a set of \(n\) rectangles of width \(\frac { 1 } { n }\).
CAIE Further Paper 2 2020 June Q6
10 marks Challenging +1.8
6 The integral \(\mathrm { I } _ { \mathrm { n } }\), where \(n\) is an integer, is defined by \(\mathrm { I } _ { \mathrm { n } } = \int _ { 0 } ^ { \frac { 1 } { 2 } } \left( 1 - \mathrm { x } ^ { 2 } \right) ^ { - \frac { 1 } { 2 } \mathrm { n } } \mathrm { dx }\).
  1. Find the exact value of \(I _ { 1 }\).
  2. By considering \(\frac { \mathrm { d } } { \mathrm { dx } } \left( \mathrm { x } \left( 1 - \mathrm { x } ^ { 2 } \right) ^ { - \frac { 1 } { 2 } \mathrm { n } } \right)\), or otherwise, show that $$\mathrm { nl } _ { \mathrm { n } + 2 } = 2 ^ { \mathrm { n } - 1 } 3 ^ { - \frac { 1 } { 2 } \mathrm { n } } + ( \mathrm { n } - 1 ) \mathrm { I } _ { \mathrm { n } } .$$
  3. Find the exact value of \(I _ { 5 }\) giving the answer in the form \(k \sqrt { 3 }\), where \(k\) is a rational number to be determined. \includegraphics[max width=\textwidth, alt={}, center]{1de67949-6262-4ade-b986-02b6563ae404-11_78_1576_336_321}
CAIE Further Paper 2 2021 June Q1
5 marks Standard +0.3
1
  1. Given that \(a\) is an integer, show that the system of equations $$\begin{aligned} a x + 3 y + z & = 14 \\ 2 x + y + 3 z & = 0 \\ - x + 2 y - 5 z & = 17 \end{aligned}$$ has a unique solution and interpret this situation geometrically.
  2. Find the value of \(a\) for which \(x = 1 , y = 4 , z = - 2\) is the solution to the system of equations in part (a).
CAIE Further Paper 2 2021 June Q2
7 marks Standard +0.3
2 The variables \(x\) and \(y\) are related by the differential equation $$\frac { d ^ { 2 } y } { d x ^ { 2 } } + 3 \frac { d y } { d x } + 2 y = 2 x + 1$$
  1. Find the general solution for \(y\) in terms of \(x\).
  2. State an approximate solution for large positive values of \(x\).