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CAIE P3 2024 November Q10
13 marks Challenging +1.2
10 A balloon in the shape of a sphere has volume \(V\) and radius \(r\). Air is pumped into the balloon at a constant rate of \(40 \pi\) starting when time \(t = 0\) and \(r = 0\). At the same time, air begins to flow out of the balloon at a rate of \(0.8 \pi r\). The balloon remains a sphere at all times.
  1. Show that \(r\) and \(t\) satisfy the differential equation $$\frac { \mathrm { d } r } { \mathrm {~d} t } = \frac { 50 - r } { 5 r ^ { 2 } }$$
  2. Find the quotient and remainder when \(5 r ^ { 2 }\) is divided by \(50 - r\).
  3. Solve the differential equation in part (a), obtaining an expression for \(t\) in terms of \(r\).
  4. Find the value of \(t\) when the radius of the balloon is 12 .
CAIE P3 2024 November Q11
14 marks Standard +0.8
11 Let \(\mathrm { f } ( x ) = \frac { 2 \mathrm { e } ^ { 2 x } } { \mathrm { e } ^ { 2 x } - 3 \mathrm { e } ^ { x } + 2 }\).
  1. Find \(\mathrm { f } ^ { \prime } ( x )\) and hence find the exact coordinates of the stationary point of the curve with equation \(y = \mathrm { f } ( x )\). \includegraphics[max width=\textwidth, alt={}, center]{9da6c2ac-31aa-4063-88b9-e15e38bedd8a-16_2718_35_107_2012}
  2. Use the substitution \(u = e ^ { x }\) and partial fractions to find the exact value of \(\int _ { \ln 3 } ^ { \ln 5 } \mathrm { f } ( x ) \mathrm { d } x\). Give your answer in the form \(\ln a\), where \(a\) is a rational number in its simplest form.
    If you use the following page to complete the answer to any question, the question number must be clearly shown. \includegraphics[max width=\textwidth, alt={}, center]{9da6c2ac-31aa-4063-88b9-e15e38bedd8a-18_2718_42_107_2007}
CAIE P3 2024 November Q1
4 marks Moderate -0.3
1 The complex number \(z\) satisfies \(| z | = 2\) and \(0 \leqslant \arg z \leqslant \frac { 1 } { 4 } \pi\).
  1. On the Argand diagram below, sketch the locus of the points representing \(z\).
  2. On the same diagram, sketch the locus of the points representing \(z ^ { 2 }\). \includegraphics[max width=\textwidth, alt={}, center]{6280ab81-0bdb-47b4-8651-bff1261a0adf-02_1074_1363_628_351} \includegraphics[max width=\textwidth, alt={}, center]{6280ab81-0bdb-47b4-8651-bff1261a0adf-02_1002_26_1820_2017}
CAIE P3 2024 November Q2
5 marks Standard +0.3
2 Let \(\mathrm { f } ( x ) = 2 x ^ { 3 } - 5 x ^ { 2 } + 4\).
  1. Show that if a sequence of values given by the iterative formula $$x _ { n + 1 } = \sqrt { \frac { 4 } { 5 - 2 x _ { n } } }$$ converges, then it converges to a root of the equation \(\mathrm { f } ( x ) = 0\).
  2. The equation has a root close to 1.2 . Use the iterative formula from part (a) and an initial value of 1.2 to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P3 2024 November Q3
5 marks Moderate -0.5
3 \includegraphics[max width=\textwidth, alt={}, center]{6280ab81-0bdb-47b4-8651-bff1261a0adf-04_527_634_255_717} The number of bacteria in a population, \(P\), at time \(t\) hours is modelled by the equation \(P = a \mathrm { e } ^ { k t }\), where \(a\) and \(k\) are constants. The graph of \(\ln P\) against \(t\), shown in the diagram, has gradient \(\frac { 1 } { 20 }\) and intersects the vertical axis at \(( 0,3 )\).
  1. State the value of \(k\) and find the value of \(a\) correct to 2 significant figures.
  2. Find the time taken for \(P\) to double. Give your answer correct to the nearest hour. \includegraphics[max width=\textwidth, alt={}, center]{6280ab81-0bdb-47b4-8651-bff1261a0adf-05_2723_33_99_22}
CAIE P3 2024 November Q4
5 marks Moderate -0.5
4 Find the complex number \(z\) satisfying the equation $$\frac { z - 3 \mathrm { i } } { z + 3 \mathrm { i } } = \frac { 2 - 9 \mathrm { i } } { 5 }$$ Give your answer in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
CAIE P3 2024 November Q5
6 marks Standard +0.3
5
  1. Show that \(\cos ^ { 4 } \theta - \sin ^ { 4 } \theta - 4 \sin ^ { 2 } \theta \cos ^ { 2 } \theta \equiv \cos ^ { 2 } 2 \theta + \cos 2 \theta - 1\).
  2. Solve the equation \(\cos ^ { 4 } \alpha - \sin ^ { 4 } \alpha = 4 \sin ^ { 2 } \alpha \cos ^ { 2 } \alpha\) for \(0 ^ { \circ } \leqslant \alpha \leqslant 180 ^ { \circ }\).
CAIE P3 2024 November Q6
7 marks Standard +0.3
6 The lines \(l\) and \(m\) have vector equations $$l : \quad \mathbf { r } = 2 \mathbf { i } + \mathbf { j } - 3 \mathbf { k } + \lambda ( - \mathbf { i } + 2 \mathbf { k } ) \quad \text { and } \quad m : \quad \mathbf { r } = 2 \mathbf { i } + \mathbf { j } - 3 \mathbf { k } + \mu ( 2 \mathbf { i } - \mathbf { j } + 5 \mathbf { k } ) .$$ Lines \(l\) and \(m\) intersect at the point \(P\).
  1. State the coordinates of \(P\).
  2. Find the exact value of the cosine of the acute angle between \(l\) and \(m\).
  3. The point \(A\) on line \(I\) has coordinates ( \(0,1,1\) ). The point \(B\) on line \(m\) has coordinates ( \(0,2 , - 8\) ). Find the exact area of triangle \(A P B\).
CAIE P3 2024 November Q7
8 marks Standard +0.3
7 The parametric equations of a curve are $$x = 3 \sin 2 t , \quad y = \tan t + \cot t$$ for \(0 < t < \frac { 1 } { 2 } \pi\).
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { - 2 } { 3 \sin ^ { 2 } 2 t }\). \includegraphics[max width=\textwidth, alt={}, center]{6280ab81-0bdb-47b4-8651-bff1261a0adf-10_2716_40_109_2009} \includegraphics[max width=\textwidth, alt={}, center]{6280ab81-0bdb-47b4-8651-bff1261a0adf-11_2723_33_99_22}
  2. Find the equation of the normal to the curve at the point where \(t = \frac { 1 } { 4 } \pi\). Give your answer in the form \(p y + q x + r = 0\), where \(p , q\) and \(r\) are integers.
CAIE P3 2024 November Q8
8 marks Standard +0.3
8 Let \(\mathrm { f } ( x ) = \frac { 7 a ^ { 2 } } { ( a - 2 x ) ( 3 a + x ) }\), where \(a\) is a positive constant.
  1. Express \(\mathrm { f } ( x )\) in partial fractions. \includegraphics[max width=\textwidth, alt={}, center]{6280ab81-0bdb-47b4-8651-bff1261a0adf-12_2718_40_107_2009} \includegraphics[max width=\textwidth, alt={}, center]{6280ab81-0bdb-47b4-8651-bff1261a0adf-13_2726_33_97_22}
  2. Hence obtain the expansion of \(\mathrm { f } ( x )\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\). [4]
  3. State the set of values of \(x\) for which the expansion in part (b) is valid.
CAIE P3 2024 November Q9
8 marks Standard +0.3
9
  1. Find the quotient and remainder when \(x ^ { 4 } + 16\) is divided by \(x ^ { 2 } + 4\).
  2. Hence show that \(\int _ { 2 } ^ { 2 \sqrt { 3 } } \frac { x ^ { 4 } + 16 } { x ^ { 2 } + 4 } \mathrm {~d} x = \frac { 4 } { 3 } ( \pi + 4 )\).
CAIE P3 2024 November Q10
8 marks Moderate -0.3
10 A water tank is in the shape of a cuboid with base area \(40000 \mathrm {~cm} ^ { 2 }\). At time \(t\) minutes the depth of water in the tank is \(h \mathrm {~cm}\). Water is pumped into the tank at a rate of \(50000 \mathrm {~cm} ^ { 3 }\) per minute. Water is leaking out of the tank through a hole in the bottom at a rate of \(600 \mathrm {~cm} ^ { 3 }\) per minute.
  1. Show that \(200 \frac { \mathrm {~d} h } { \mathrm {~d} t } = 250 - 3 h\).
    \includegraphics[max width=\textwidth, alt={}, center]{6280ab81-0bdb-47b4-8651-bff1261a0adf-17_2723_33_99_22}
  2. It is given that when \(t = 0 , h = 50\). Find the time taken for the depth of water in the tank to reach 80 cm . Give your answer correct to 2 significant figures.
CAIE P3 2024 November Q11
11 marks Standard +0.8
11 \includegraphics[max width=\textwidth, alt={}, center]{6280ab81-0bdb-47b4-8651-bff1261a0adf-18_565_634_260_717} The diagram shows the curve \(y = 2 \sin x \sqrt { 2 + \cos x }\), for \(0 \leqslant x \leqslant 2 \pi\), and its minimum point \(M\), where \(x = a\).
  1. Find the value of \(a\) correct to 2 decimal places. \includegraphics[max width=\textwidth, alt={}, center]{6280ab81-0bdb-47b4-8651-bff1261a0adf-18_2716_38_109_2012} \includegraphics[max width=\textwidth, alt={}, center]{6280ab81-0bdb-47b4-8651-bff1261a0adf-19_2726_33_97_22}
  2. Use the substitution \(u = 2 + \cos x\) to find the exact area of the shaded region \(R\).
    If you use the following page to complete the answer to any question, the question number must be clearly shown.
CAIE M1 2020 June Q2
5 marks Moderate -0.8
2 A car of mass 1800 kg is towing a trailer of mass 400 kg along a straight horizontal road. The car and trailer are connected by a light rigid tow-bar. The car is accelerating at \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). There are constant resistance forces of 250 N on the car and 100 N on the trailer.
  1. Find the tension in the tow-bar.
  2. Find the power of the engine of the car at the instant when the speed is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
CAIE M1 2020 June Q3
7 marks Moderate -0.8
3 A particle \(P\) is projected vertically upwards with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point \(A\) which is 2.8 m above horizontal ground.
  1. Find the greatest height above the ground reached by \(P\).
  2. Find the length of time for which \(P\) is at a height of more than 3.6 m above the ground.
    The diagram shows a ring of mass 0.1 kg threaded on a fixed horizontal rod. The rod is rough and the coefficient of friction between the ring and the rod is 0.8 . A force of magnitude \(T \mathrm {~N}\) acts on the ring in a direction at \(30 ^ { \circ }\) to the rod, downwards in the vertical plane containing the rod. Initially the ring is at rest.
  3. Find the greatest value of \(T\) for which the ring remains at rest.
  4. Find the acceleration of the ring when \(T = 3\).
CAIE M1 2020 June Q5
7 marks Standard +0.3
5 \includegraphics[max width=\textwidth, alt={}, center]{c175972b-e298-4c86-bbef-3b05c6aca76f-08_412_339_260_897} A child of mass 35 kg is swinging on a rope. The child is modelled as a particle \(P\) and the rope is modelled as a light inextensible string of length 4 m . Initially \(P\) is held at an angle of \(45 ^ { \circ }\) to the vertical (see diagram).
  1. Given that there is no resistance force, find the speed of \(P\) when it has travelled half way along the circular arc from its initial position to its lowest point.
  2. It is given instead that there is a resistance force. The work done against the resistance force as \(P\) travels from its initial position to its lowest point is \(X \mathrm {~J}\). The speed of \(P\) at its lowest point is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find \(X\).
CAIE M1 2020 June Q6
11 marks Standard +0.3
6 A particle moves in a straight line \(A B\). The velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) of the particle \(t \mathrm {~s}\) after leaving \(A\) is given by \(v = k \left( t ^ { 2 } - 10 t + 21 \right)\), where \(k\) is a constant. The displacement of the particle from \(A\), in the direction towards \(B\), is 2.85 m when \(t = 3\) and is 2.4 m when \(t = 6\).
  1. Find the value of \(k\). Hence find an expression, in terms of \(t\), for the displacement of the particle from \(A\).
  2. Find the displacement of the particle from \(A\) when its velocity is a minimum.
CAIE M1 2020 June Q7
10 marks Standard +0.3
7 \includegraphics[max width=\textwidth, alt={}, center]{c175972b-e298-4c86-bbef-3b05c6aca76f-12_399_1121_262_511} A particle \(P\) of mass 0.3 kg , lying on a smooth plane inclined at \(30 ^ { \circ }\) to the horizontal, is released from rest. \(P\) slides down the plane for a distance of 2.5 m and then reaches a horizontal plane. There is no change in speed when \(P\) reaches the horizontal plane. A particle \(Q\) of mass 0.2 kg lies at rest on the horizontal plane 1.5 m from the end of the inclined plane (see diagram). \(P\) collides directly with \(Q\).
  1. It is given that the horizontal plane is smooth and that, after the collision, \(P\) continues moving in the same direction, with speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the speed of \(Q\) after the collision.
  2. It is given instead that the horizontal plane is rough and that when \(P\) and \(Q\) collide, they coalesce and move with speed \(1.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the coefficient of friction between \(P\) and the horizontal plane.
    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 M1 2020 June Q1
6 marks Standard +0.3
1 A tram starts from rest and moves with uniform acceleration for 20 s . The tram then travels at a constant speed, \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\), for 170 s before being brought to rest with a uniform deceleration of magnitude twice that of the acceleration. The total distance travelled by the tram is 2.775 km .
  1. Sketch a velocity-time graph for the motion, stating the total time for which the tram is moving.
    [0pt] [2]
  2. Find \(V\).
  3. Find the magnitude of the acceleration. \includegraphics[max width=\textwidth, alt={}, center]{e11bebff-1c09-4576-9b9b-a1678ea2b226-03_625_627_260_758} Coplanar forces of magnitudes \(20 \mathrm {~N} , P \mathrm {~N} , 3 P \mathrm {~N}\) and \(4 P \mathrm {~N}\) act at a point in the directions shown in the diagram. The system is in equilibrium. Find \(P\) and \(\theta\).
CAIE M1 2020 June Q3
8 marks Standard +0.3
3 \includegraphics[max width=\textwidth, alt={}, center]{e11bebff-1c09-4576-9b9b-a1678ea2b226-04_397_759_264_694} A particle of mass 2.5 kg is held in equilibrium on a rough plane inclined at \(20 ^ { \circ }\) to the horizontal by a force of magnitude \(T \mathrm {~N}\) making an angle of \(60 ^ { \circ }\) with a line of greatest slope of the plane (see diagram). The coefficient of friction between the particle and the plane is 0.3 . Find the greatest and least possible values of \(T\).
CAIE M1 2020 June Q4
10 marks Standard +0.3
4 Small smooth spheres \(A\) and \(B\), of equal radii and of masses 4 kg and 2 kg respectively, lie on a smooth horizontal plane. Initially \(B\) is at rest and \(A\) is moving towards \(B\) with speed \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). After the spheres collide \(A\) continues to move in the same direction but with half the speed of \(B\).
  1. Find the speed of \(B\) after the collision.
    A third small smooth sphere \(C\), of mass 1 kg and with the same radius as \(A\) and \(B\), is at rest on the plane. \(B\) now collides directly with \(C\). After this collision \(B\) continues to move in the same direction but with one third the speed of \(C\).
  2. Show that there is another collision between \(A\) and \(B\).
  3. \(\quad A\) and \(B\) coalesce during this collision. Find the total loss of kinetic energy in the system due to the three collisions.
CAIE M1 2020 June Q5
10 marks Moderate -0.3
5 A car of mass 1250 kg is moving on a straight road.
  1. On a horizontal section of the road, the car has a constant speed of \(32 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and there is a constant force of 750 N resisting the motion.
    1. Calculate, in kW , the power developed by the engine of the car.
    2. Given that this power is suddenly decreased by 8 kW , find the instantaneous deceleration of the car.
  2. On a section of the road inclined at \(\sin ^ { - 1 } 0.096\) to the horizontal, the resistance to the motion of the car is \(( 1000 + 8 v ) \mathrm { N }\) when the speed of the car is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The car travels up this section of the road at constant speed with the engine working at 60 kW . Find this constant speed.
CAIE M1 2020 June Q6
10 marks Standard +0.3
6 A particle \(P\) moves in a straight line. The velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t \mathrm {~s}\) is given by $$\begin{array} { l l } v = 2 t + 1 & \text { for } 0 \leqslant t \leqslant 5 , \\ v = 36 - t ^ { 2 } & \text { for } 5 \leqslant t \leqslant 7 , \\ v = 2 t - 27 & \text { for } 7 \leqslant t \leqslant 13.5 . \end{array}$$
  1. Sketch the velocity-time graph for \(0 \leqslant t \leqslant 13.5\).
  2. Find the acceleration at the instant when \(t = 6\).
  3. Find the total distance travelled by \(P\) in the interval \(0 \leqslant t \leqslant 13.5\).
    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 M1 2020 June Q1
3 marks Easy -1.2
1 Particles \(P\) of mass \(m \mathrm {~kg}\) and \(Q\) of mass 0.2 kg are free to move on a smooth horizontal plane. \(P\) is projected at a speed of \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) towards \(Q\) which is stationary. After the collision \(P\) and \(Q\) move in opposite directions with speeds of \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. Find \(m\).
CAIE M1 2020 June Q2
4 marks Easy -1.2
2 A minibus of mass 4000 kg is travelling along a straight horizontal road. The resistance to motion is 900 N .
  1. Find the driving force when the acceleration of the minibus is \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  2. Find the power required for the minibus to maintain a constant speed of \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).