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CAIE M1 2012 June Q2
6 marks Standard +0.3
2 \includegraphics[max width=\textwidth, alt={}, center]{918b65cc-617d-4942-8d96-b02eef21e417-2_471_621_870_762} A smooth ring \(R\) of mass 0.16 kg is threaded on a light inextensible string. The ends of the string are attached to fixed points \(A\) and \(B\). A horizontal force of magnitude 11.2 N acts on \(R\), in the same vertical plane as \(A\) and \(B\). The ring is in equilibrium. The string is taut with angle \(A R B = 90 ^ { \circ }\), and the part \(A R\) of the string makes an angle of \(\theta ^ { \circ }\) with the horizontal (see diagram). The tension in the string is \(T \mathrm {~N}\).
  1. Find two simultaneous equations involving \(T \sin \theta\) and \(T \cos \theta\).
  2. Hence find \(T\) and \(\theta\).
CAIE M1 2012 June Q3
7 marks Standard +0.3
3 A particle \(P\) travels from a point \(O\) along a straight line and comes to instantaneous rest at a point \(A\). The velocity of \(P\) at time \(t \mathrm {~s}\) after leaving \(O\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(v = 0.027 \left( 10 t ^ { 2 } - t ^ { 3 } \right)\). Find
  1. the distance \(O A\),
  2. the maximum velocity of \(P\) while moving from \(O\) to \(A\).
CAIE M1 2012 June Q4
7 marks Standard +0.3
4 A car of mass 1230 kg increases its speed from \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(21 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in 24.5 s . The table below shows corresponding values of time \(t \mathrm {~s}\) and speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
\(t\)00.516.324.5
\(v\)461921
  1. Using the values in the table, find the average acceleration of the car for \(0 < t < 0.5\) and for \(16.3 < t < 24.5\). While the car is increasing its speed the power output of its engine is constant and equal to \(P \mathrm {~W}\), and the resistance to the car's motion is constant and equal to \(R \mathrm {~N}\).
  2. Assuming that the values obtained in part (i) are approximately equal to the accelerations at \(v = 5\) and at \(v = 20\), find approximations for \(P\) and \(R\).
CAIE M1 2012 June Q5
8 marks Standard +0.3
5 A lorry of mass 16000 kg moves on a straight hill inclined at angle \(\alpha ^ { \circ }\) to the horizontal. The length of the hill is 500 m .
  1. While the lorry moves from the bottom to the top of the hill at constant speed, the resisting force acting on the lorry is 800 N and the work done by the driving force is 2800 kJ . Find the value of \(\alpha\).
  2. On the return journey the speed of the lorry is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the top of the hill. While the lorry travels down the hill, the work done by the driving force is 2400 kJ and the work done against the resistance to motion is 800 kJ . Find the speed of the lorry at the bottom of the hill.
    [0pt] [4]
CAIE M1 2012 June Q6
9 marks Standard +0.3
6 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{918b65cc-617d-4942-8d96-b02eef21e417-3_156_558_1592_450} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{918b65cc-617d-4942-8d96-b02eef21e417-3_138_559_1612_1137} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} A block of weight 6.1 N is at rest on a plane inclined at angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 11 } { 60 }\). The coefficient of friction between the block and the plane is \(\mu\). A force of magnitude 5.9 N acting parallel to a line of greatest slope is applied to the block.
  1. When the force acts up the plane (see Fig. 1) the block remains at rest. Show that \(\mu \geqslant \frac { 4 } { 5 }\).
  2. When the force acts down the plane (see Fig. 2) the block slides downwards. Show that \(\mu < \frac { 7 } { 6 }\).
  3. Given that the acceleration of the block is \(1.7 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) when the force acts down the plane, find the value of \(\mu\).
CAIE M1 2012 June Q7
10 marks Standard +0.8
7 \includegraphics[max width=\textwidth, alt={}, center]{918b65cc-617d-4942-8d96-b02eef21e417-4_506_471_255_836} Two particles \(A\) and \(B\) have masses 0.12 kg and 0.38 kg respectively. The particles are attached to the ends of a light inextensible string which passes over a fixed smooth pulley. \(A\) is held at rest with the string taut and both straight parts of the string vertical. \(A\) and \(B\) are each at a height of 0.65 m above horizontal ground (see diagram). \(A\) is released and \(B\) moves downwards. Find
  1. the acceleration of \(B\) while it is moving downwards,
  2. the speed with which \(B\) reaches the ground and the time taken for it to reach the ground. \(B\) remains on the ground while \(A\) continues to move with the string slack, without reaching the pulley. The string remains slack until \(A\) is at a height of 1.3 m above the ground for a second time. At this instant \(A\) has been in motion for a total time of \(T \mathrm {~s}\).
  3. Find the value of \(T\) and sketch the velocity-time graph for \(A\) for the first \(T \mathrm {~s}\) of its motion.
  4. Find the total distance travelled by \(A\) in the first \(T\) s of its motion.
CAIE M1 2013 June Q1
4 marks Moderate -0.3
1 A block is at rest on a rough horizontal plane. The coefficient of friction between the block and the plane is 1.25 .
  1. State, giving a reason for your answer, whether the minimum vertical force required to move the block is greater or less than the minimum horizontal force required to move the block. A horizontal force of continuously increasing magnitude \(P \mathrm {~N}\) and fixed direction is applied to the block.
  2. Given that the weight of the block is 60 N , find the value of \(P\) when the acceleration of the block is \(4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
CAIE M1 2013 June Q2
5 marks Moderate -0.3
2 A car of mass 1250 kg travels from the bottom to the top of a straight hill of length 600 m , which is inclined at an angle of \(2.5 ^ { \circ }\) to the horizontal. The resistance to motion of the car is constant and equal to 400 N . The work done by the driving force is 450 kJ . The speed of the car at the bottom of the hill is \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the speed of the car at the top of the hill.
CAIE M1 2013 June Q3
7 marks Standard +0.3
3 The top of a cliff is 40 metres above the level of the sea. A man in a boat, close to the bottom of the cliff, is in difficulty and fires a distress signal vertically upwards from sea level. Find
  1. the speed of projection of the signal given that it reaches a height of 5 m above the top of the cliff,
  2. the length of time for which the signal is above the level of the top of the cliff. The man fires another distress signal vertically upwards from sea level. This signal is above the level of the top of the cliff for \(\sqrt { } ( 17 ) \mathrm { s }\).
  3. Find the speed of projection of the second signal.
CAIE M1 2013 June Q4
6 marks Standard +0.3
4 A train of mass 400000 kg is moving on a straight horizontal track. The power of the engine is constant and equal to 1500 kW and the resistance to the train's motion is 30000 N . Find
  1. the acceleration of the train when its speed is \(37.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\),
  2. the steady speed at which the train can move.
CAIE M1 2013 June Q5
8 marks Standard +0.3
5 \includegraphics[max width=\textwidth, alt={}, center]{2c628138-0729-4160-a95c-d6ab0f199cc5-3_275_663_258_742} A light inextensible string has a particle \(A\) of mass 0.26 kg attached to one end and a particle \(B\) of mass 0.54 kg attached to the other end. The particle \(A\) is held at rest on a rough plane inclined at angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 5 } { 13 }\). The string is taut and parallel to a line of greatest slope of the plane. The string passes over a small smooth pulley at the top of the plane. Particle \(B\) hangs at rest vertically below the pulley (see diagram). The coefficient of friction between \(A\) and the plane is 0.2 . Particle \(A\) is released and the particles start to move.
  1. Find the magnitude of the acceleration of the particles and the tension in the string. Particle \(A\) reaches the pulley 0.4 s after starting to move.
  2. Find the distance moved by each of the particles.
CAIE M1 2013 June Q6
9 marks Standard +0.3
6 \includegraphics[max width=\textwidth, alt={}, center]{2c628138-0729-4160-a95c-d6ab0f199cc5-3_639_939_1260_603} A particle \(P\) of mass 0.5 kg lies on a smooth horizontal plane. Horizontal forces of magnitudes \(F \mathrm {~N}\), 2.5 N and 2.6 N act on \(P\). The directions of the forces are as shown in the diagram, where \(\tan \alpha = \frac { 12 } { 5 }\) and \(\tan \beta = \frac { 7 } { 24 }\).
  1. Given that \(P\) is in equilibrium, find the values of \(F\) and \(\tan \theta\).
  2. The force of magnitude \(F \mathrm {~N}\) is removed. Find the magnitude and direction of the acceleration with which \(P\) starts to move.
CAIE M1 2013 June Q7
11 marks Standard +0.3
7 A car driver makes a journey in a straight line from \(A\) to \(B\), starting from rest. The speed of the car increases to a maximum, then decreases until the car is at rest at \(B\). The distance travelled by the car \(t\) seconds after leaving \(A\) is \(0.0000117 \left( 400 t ^ { 3 } - 3 t ^ { 4 } \right)\) metres.
  1. Find the distance \(A B\).
  2. Find the maximum speed of the car.
  3. Find the acceleration of the car
    (a) as it starts from \(A\),
    (b) as it arrives at \(B\).
  4. Sketch the velocity-time graph for the journey.
CAIE M1 2013 June Q3
Moderate -0.5
3 \includegraphics[max width=\textwidth, alt={}, center]{bc436b32-01f9-41dc-b2f7-ce49e18d3e6c-2_314_1193_1366_276} \({ } ^ { P A } { } _ { P A } ^ { P B }\) \(P\) \(P\) \begin{verbatim} " \end{verbatim}
CAIE M1 2013 June Q4
Standard +0.3
4 \(A\) B \(A \quad B\) \(\begin{array} { l l } B & \\ A & B \end{array}\) \(P B\) $$P \theta$$ \(\theta\) P \(\theta\) L \(P\) \(P\)
  1. (i)
    P \(\theta\) \(P \quad \underline { \theta }\)
  2. - \(\underline { \theta }\) \(5 \theta\) $$\begin{gathered} \\ \theta \end{gathered} \quad P$$
  3. \(P\)
CAIE M1 2013 June Q6
Standard +0.3
6 \(\begin{array} { c c c c c c c c } & P & & P & & & \theta & \\ \theta & P & & A & \theta & \theta & P & \\ \text { (i) } & & P & & & \theta & \\ \text { (ii) } & P & & & P & & & \\ \text { (iii) } & & & & & & \theta \\ & & & & & & \theta \end{array}\) 7 \includegraphics[max width=\textwidth, alt={}, center]{bc436b32-01f9-41dc-b2f7-ce49e18d3e6c-3_512_1095_1439_580}
  1. \(B\) \(\begin{array} { l l l } A & P & A \end{array}\) AN PA
CAIE M1 2013 June Q1
4 marks Moderate -0.3
1 A straight ice track of length 50 m is inclined at \(14 ^ { \circ }\) to the horizontal. A man starts at the top of the track, on a sledge, with speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). He travels on the sledge to the bottom of the track. The coefficient of friction between the sledge and the track is 0.02 . Find the speed of the sledge and the man when they reach the bottom of the track.
CAIE M1 2013 June Q2
5 marks Standard +0.3
2 \includegraphics[max width=\textwidth, alt={}, center]{ceb367ee-4e12-4cb2-9020-078ea5724d6e-2_529_691_529_726} Particle \(A\) of mass 1.6 kg and particle \(B\) of mass 2 kg are attached to opposite ends of a light inextensible string. The string passes over a small smooth pulley fixed at the top of a smooth plane, which is inclined at angle \(\theta\), where \(\sin \theta = 0.8\). Particle \(A\) is held at rest at the bottom of the plane and \(B\) hangs at a height of 3.24 m above the level of the bottom of the plane (see diagram). \(A\) is released from rest and the particles start to move.
  1. Show that the loss of potential energy of the system, when \(B\) reaches the level of the bottom of the plane, is 23.328 J .
  2. Hence find the speed of the particles when \(B\) reaches the level of the bottom of the plane.
CAIE M1 2013 June Q3
6 marks Standard +0.3
3 A car has mass 800 kg . The engine of the car generates constant power \(P \mathrm {~kW}\) as the car moves along a straight horizontal road. The resistance to motion is constant and equal to \(R \mathrm {~N}\). When the car's speed is \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) its acceleration is \(1.4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), and when the car's speed is \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) its acceleration is \(0.33 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Find the values of \(P\) and \(R\).
CAIE M1 2013 June Q4
7 marks Standard +0.3
4 An aeroplane moves along a straight horizontal runway before taking off. It starts from rest at \(O\) and has speed \(90 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the instant it takes off. While the aeroplane is on the runway at time \(t\) seconds after leaving \(O\), its acceleration is \(( 1.5 + 0.012 t ) \mathrm { m } \mathrm { s } ^ { - 2 }\). Find
  1. the value of \(t\) at the instant the aeroplane takes off,
  2. the distance travelled by the aeroplane on the runway.
CAIE M1 2013 June Q5
8 marks Standard +0.3
5 A particle \(P\) is projected vertically upwards from a point on the ground with speed \(17 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Another particle \(Q\) is projected vertically upwards from the same point with speed \(7 \mathrm {~ms} ^ { - 1 }\). Particle \(Q\) is projected \(T\) seconds later than particle \(P\).
  1. Given that the particles reach the ground at the same instant, find the value of \(T\).
  2. At a certain instant when both \(P\) and \(Q\) are in motion, \(P\) is 5 m higher than \(Q\). Find the magnitude and direction of the velocity of each of the particles at this instant.
CAIE M1 2013 June Q6
9 marks Moderate -0.3
6 \includegraphics[max width=\textwidth, alt={}, center]{ceb367ee-4e12-4cb2-9020-078ea5724d6e-3_703_700_255_721} A small box of mass 40 kg is moved along a rough horizontal floor by three men. Two of the men apply horizontal forces of magnitudes 100 N and 120 N , making angles of \(30 ^ { \circ }\) and \(60 ^ { \circ }\) respectively with the positive \(x\)-direction. The third man applies a horizontal force of magnitude \(F \mathrm {~N}\) making an angle of \(\alpha ^ { \circ }\) with the negative \(x\)-direction (see diagram). The resultant of the three horizontal forces acting on the box is in the positive \(x\)-direction and has magnitude 136 N .
  1. Find the values of \(F\) and \(\alpha\).
  2. Given that the box is moving with constant speed, state the magnitude of the frictional force acting on the box and hence find the coefficient of friction between the box and the floor.
CAIE M1 2013 June Q7
11 marks Standard +0.3
7 \includegraphics[max width=\textwidth, alt={}, center]{ceb367ee-4e12-4cb2-9020-078ea5724d6e-3_430_860_1585_641} Particle \(A\) of mass 1.26 kg and particle \(B\) of mass 0.9 kg are attached to the ends of a light inextensible string. The string passes over a small smooth pulley \(P\) which is fixed at the edge of a rough horizontal table. \(A\) is held at rest at a point 0.48 m from \(P\), and \(B\) hangs vertically below \(P\), at a height of 0.45 m above the floor (see diagram). The coefficient of friction between \(A\) and the table is \(\frac { 2 } { 7 } . A\) is released and the particles start to move.
  1. Show that the magnitude of the acceleration of the particles is \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) and find the tension in the string.
  2. Find the speed with which \(B\) reaches the floor.
  3. Find the speed with which \(A\) reaches the pulley.
CAIE M1 2014 June Q1
Easy -1.2
1 A ta v at tat th ta \(k a\) th tta ta
a a hta taht tak \(v\) that th \(t\) th tat \(k\) th va \(2 A h a\) at a a th \(a\) Th \(a\)
  1. th va
  2. th at va th \(t\) th hta \(A\) a at at th \(a t t\)
CAIE M1 2014 June Q3
Moderate -0.5
3 \includegraphics[max width=\textwidth, alt={}, center]{9950eb77-25f9-4275-88ab-8dce9fb07cec-2_524_572_806_749}
aa at at a \(t\) Th at th \(a \quad a\) th \(t\) hh th at \(a h\) th aa that a \(t\) th
tat th