Questions M1 (1912 questions)

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CAIE M1 2014 November Q7
11 marks Standard +0.3
7
\includegraphics[max width=\textwidth, alt={}, center]{ffefbc81-402f-4048-8741-23c8bae30d5a-3_246_1006_1781_571} A block of mass 60 kg is pulled up a hill in the line of greatest slope by a force of magnitude 50 N acting at an angle \(\alpha ^ { \circ }\) above the hill. The block passes through points \(A\) and \(B\) with speeds \(8.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(3.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively (see diagram). The distance \(A B\) is 250 m and \(B\) is 17.5 m above the level of \(A\). The resistance to motion of the block is 6 N . Find the value of \(\alpha\).
[0pt] [11]
CAIE M1 2014 November Q1
Moderate -0.3
1 A particle \(P\) is projected vertically upwards with speed \(11 \mathrm {~ms} ^ { - 1 }\) from a point on horizontal ground. At the same instant a particle \(Q\) is released from rest at a point \(h \mathrm {~m}\) above the ground. \(P\) and \(Q\) hit the ground at the same instant, when \(Q\) has speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Find the time after projection at which \(P\) hits the ground.
  2. Hence find the values of \(h\) and \(V\).
CAIE M1 2014 November Q2
Moderate -0.8
2
\includegraphics[max width=\textwidth, alt={}, center]{c7133fc4-9a14-43fd-b5ed-788da72291cd-2_666_953_662_596} Three coplanar forces act at a point. The magnitudes of the forces are \(20 \mathrm {~N} , 25 \mathrm {~N}\) and 30 N , and the directions in which the forces act are as shown in the diagram, where \(\sin \alpha = 0.28\) and \(\cos \alpha = 0.96\), and \(\sin \beta = 0.6\) and \(\cos \beta = 0.8\).
  1. Show that the resultant of the three forces has a zero component in the \(x\)-direction.
  2. Find the magnitude and direction of the resultant of the three forces.
  3. The force of magnitude 20 N is replaced by another force. The effect is that the resultant force is unchanged in magnitude but reversed in direction. State the magnitude and direction of the replacement force.
CAIE M1 2014 November Q3
Standard +0.3
3 A train of mass 200000 kg moves on a horizontal straight track. It passes through a point \(A\) with speed \(28 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and later it passes through a point \(B\). The power of the train's engine at \(B\) is 1.2 times the power of the train's engine at \(A\). The driving force of the train's engine at \(B\) is 0.96 times the driving force of the train's engine at \(A\).
  1. Show that the speed of the train at \(B\) is \(35 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  2. For the motion from \(A\) to \(B\), find the work done by the train's engine given that the work done against the resistance to the train's motion is \(2.3 \times 10 ^ { 6 } \mathrm {~J}\).
CAIE M1 2014 November Q4
Moderate -0.3
4
\includegraphics[max width=\textwidth, alt={}, center]{c7133fc4-9a14-43fd-b5ed-788da72291cd-3_383_791_262_678} Forces of magnitude \(X \mathrm {~N}\) and 40 N act on a block \(B\) of mass 15 kg , which is in equilibrium in contact with a horizontal surface between points \(A\) and \(C\) on the surface. The forces act in the same vertical plane and in the directions shown in the diagram.
  1. Given that the surface is smooth, find the value of \(X\).
  2. It is given instead that the surface is rough and that the block is in limiting equilibrium. The frictional force acting on the block has magnitude 10 N in the direction towards \(A\). Find the coefficient of friction between the block and the surface.
CAIE M1 2014 November Q5
Standard +0.3
5
\includegraphics[max width=\textwidth, alt={}, center]{c7133fc4-9a14-43fd-b5ed-788da72291cd-3_289_567_1233_788} Particles \(A\) and \(B\), each of mass 0.3 kg , are connected by a light inextensible string. The string passes over a small smooth pulley fixed at the edge of a rough horizontal surface. Particle \(A\) hangs freely and particle \(B\) is held at rest in contact with the surface (see diagram). The coefficient of friction between \(B\) and the surface is 0.7 . Particle \(B\) is released and moves on the surface without reaching the pulley.
  1. Find, for the first 0.9 m of \(B\) 's motion,
CAIE M1 2014 November Q7
Standard +0.3
7
\includegraphics[max width=\textwidth, alt={}, center]{c7133fc4-9a14-43fd-b5ed-788da72291cd-4_512_1351_998_397} The diagram shows the velocity-time graph for the motion of a particle \(P\) which moves on a straight line \(B A C\). It starts at \(A\) and travels to \(B\) taking 5 s. It then reverses direction and travels from \(B\) to \(C\) taking 10 s . For the first 3 s of \(P\) 's motion its acceleration is constant. For the remaining 12 s the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t \mathrm {~s}\) after leaving \(A\), where $$v = - 0.2 t ^ { 2 } + 4 t - 15 \text { for } 3 \leqslant t \leqslant 15$$
  1. Find the value of \(v\) when \(t = 3\) and the magnitude of the acceleration of \(P\) for the first 3 s of its motion.
  2. Find the maximum velocity of \(P\) while it is moving from \(B\) to \(C\).
  3. Find the average speed of \(P\),
    (a) while moving from \(A\) to \(B\),
    (b) for the whole journey. \footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }
CAIE M1 2014 November Q2
Easy -1.2
2
\includegraphics[max width=\textwidth, alt={}, center]{42463266-c145-4f8a-b6c3-15820cb15c3c-2_268_978_758_603}
\(h t h t\) th
\(t \quad \alpha t\) th ht
\(\beta - A\) th \(t\) the th hit \(t\)
\(v\) th \(t\)
Pt \(\quad\) htttt th tt th
Pt \(t\) th \(t t\) th
th \(t\) tt \(t v\) th th \(t\) tt th \(t\) th \(t\)
tt ht \(P\)
\(\beta t\) th ht \(h\)
\(h \quad k\) the the \(t h\)
\(h \quad k\) tht th \(t\) th
Pt
th
th
CAIE M1 2014 November Q3
Moderate -0.5
3
\includegraphics[max width=\textwidth, alt={}, center]{42463266-c145-4f8a-b6c3-15820cb15c3c-2_490_643_1530_717}
\(h\) th ht \(t t h\)
\(t\) t th \(t \quad t\)
\(h \quad h\) ht th \(t\) th \(t\) th \(t\)
th \(t\) th vt \(h\) th \(t\) vtht
\(t\) tht \(t\) hth th
th \(t v\)
th th \(t\)
\(\beta\)
\(\beta\)
  1. \(h\) tht \(\alpha\) thv \(\alpha\)
  2. thv \(\alpha\)
CAIE M1 2014 November Q4
Moderate -0.5
4 \(A t \quad\) tt \(t v\) tht \(t$$\quad\) h tt \(t\) \(t \quad\) th \(t \quad\) th
t \(t\)
\(t\) th \(t\) tt v
\(h\)
tt
  1. th \(v\)
    \(h\)
    \(h\) th \(v\)
  2. th \(v t\)
  3. th \(t\)
CAIE M1 2014 November Q5
Moderate -0.5
5 $$\begin{aligned} & A \quad k \quad h \quad t \\ & t \text { th } t \\ & \text { th } v \text { th th } v \text { th } t \text { th } t \\ & t \text { tt th th } \end{aligned}$$ $$\begin{gathered} \quad t \text { th ht } A \quad t \\ t \quad \text { tt } \quad \text { th } \quad W h \end{gathered}$$ $$\text { th v } \stackrel { h } { } \rightarrow \text { th }$$
CAIE M1 2014 November Q6
Easy -1.2
6
\includegraphics[max width=\textwidth, alt={}, center]{42463266-c145-4f8a-b6c3-15820cb15c3c-3_462_218_1343_287}
\includegraphics[max width=\textwidth, alt={}, center]{42463266-c145-4f8a-b6c3-15820cb15c3c-3_513_1018_1290_767}
\multirow{3}{*}{Pt tthhv th th tht \(t\) th \(t v\)}hv tt \(k h t\)\multirow[t]{4}{*}{tth \(t \mathrm { t } \quad\) ht htt}\multirow{5}{*}{\(h\) th \(v\) th}
\(t\) th \(t\) tt \(t v\) th th \(t\) tt \(h\) th
\(v t\)\(t\)\(t\) h th tv \(t v t\)
  1. th \(t\) th \(t\) th hh th \(t\) tt \(t v\) th \(h\)
    th \(t\)
  2. th \(v\)
  3. th tt hht \(v\) th \(h t\)
    [0pt] [ uestion 7 is printed on the next page.]
CAIE M1 2014 November Q7
Moderate -0.5
7
\includegraphics[max width=\textwidth, alt={}, center]{42463266-c145-4f8a-b6c3-15820cb15c3c-4_659_798_260_699}
\(A k k t t t t\) th tt
\(t\) th ht \(h\)
tt th \(\begin{aligned} & \beta v \text { th } h \\ & \text { thh } t \end{aligned}\)
  1. th \(t\) th \(k\) th \(k\)
  2. th \(t\) t th \(k\) th At th tt tht th \(k\) thh
  3. th \(t\) th \(k\) tv th \(t\)
    \(h t\)
    \(\alpha\)
    \(\alpha \quad \alpha \quad A t\) thk \(t\)
    \(\beta \quad \beta \quad h k t t v\)
    th ht
    \(t \quad\) th \(k\) tht \(t\)
    th \(t\) tt
    thh
CAIE M1 2015 November Q1
Easy -1.2
1 A weightlifter performs an exercise in which he raises a mass of 200 kg from rest vertically through a distance of 0.7 m and holds it at that height.
  1. Find the work done by the weightlifter.
  2. Given that the time taken to raise the mass is 1.2 s , find the average power developed by the weightlifter.
CAIE M1 2015 November Q2
Moderate -0.3
2 A particle of mass 0.5 kg starts from rest and slides down a line of greatest slope of a smooth plane. The plane is inclined at an angle of \(30 ^ { \circ }\) to the horizontal.
  1. Find the time taken for the particle to reach a speed of \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). When the particle has travelled 3 m down the slope from its starting point, it reaches rough horizontal ground at the bottom of the slope. The frictional force acting on the particle is 1 N .
  2. Find the distance that the particle travels along the ground before it comes to rest.
CAIE M1 2015 November Q3
Moderate -0.3
3 A lorry of mass 24000 kg is travelling up a hill which is inclined at \(3 ^ { \circ }\) to the horizontal. The power developed by the lorry's engine is constant, and there is a constant resistance to motion of 3200 N .
  1. When the speed of the lorry is \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), its acceleration is \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Find the power developed by the lorry's engine.
  2. Find the steady speed at which the lorry moves up the hill if the power is 500 kW and the resistance remains 3200 N .
CAIE M1 2015 November Q4
Standard +0.3
4
\includegraphics[max width=\textwidth, alt={}, center]{2a91fb7a-0eaf-4c50-8a2c-4755c0b44c17-2_499_784_1617_685} Blocks \(P\) and \(Q\), of mass \(m \mathrm {~kg}\) and 5 kg respectively, are attached to the ends of a light inextensible string. The string passes over a small smooth pulley which is fixed at the top of a rough plane inclined at \(35 ^ { \circ }\) to the horizontal. Block \(P\) is at rest on the plane and block \(Q\) hangs vertically below the pulley (see diagram). The coefficient of friction between block \(P\) and the plane is 0.2 . Find the set of values of \(m\) for which the two blocks remain at rest.
CAIE M1 2015 November Q5
Moderate -0.3
5
\includegraphics[max width=\textwidth, alt={}, center]{2a91fb7a-0eaf-4c50-8a2c-4755c0b44c17-3_355_1048_255_552} A small bead \(Q\) can move freely along a smooth horizontal straight wire \(A B\) of length 3 m . Three horizontal forces of magnitudes \(F \mathrm {~N} , 10 \mathrm {~N}\) and 20 N act on the bead in the directions shown in the diagram. The magnitude of the resultant of the three forces is \(R \mathrm {~N}\) in the direction shown in the diagram.
  1. Find the values of \(F\) and \(R\).
  2. Initially the bead is at rest at \(A\). It reaches \(B\) with a speed of \(11.7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the mass of the bead.
CAIE M1 2015 November Q6
Standard +0.3
6 A particle \(P\) moves in a straight line, starting from a point \(O\). The velocity of \(P\), measured in \(\mathrm { m } \mathrm { s } ^ { - 1 }\), at time \(t \mathrm {~s}\) after leaving \(O\) is given by $$v = 0.6 t - 0.03 t ^ { 2 }$$
  1. Verify that, when \(t = 5\), the particle is 6.25 m from \(O\). Find the acceleration of the particle at this time.
  2. Find the values of \(t\) at which the particle is travelling at half of its maximum velocity.
CAIE M1 2015 November Q7
Standard +0.3
7 A cyclist starts from rest at point \(A\) and moves in a straight line with acceleration \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) for a distance of 36 m . The cyclist then travels at constant speed for 25 s before slowing down, with constant deceleration, to come to rest at point \(B\). The distance \(A B\) is 210 m .
  1. Find the total time that the cyclist takes to travel from \(A\) to \(B\). 24 s after the cyclist leaves point \(A\), a car starts from rest from point \(A\), with constant acceleration \(4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), towards \(B\). It is given that the car overtakes the cyclist while the cyclist is moving with constant speed.
  2. Find the time that it takes from when the cyclist starts until the car overtakes her.
CAIE M1 2015 November Q1
Moderate -0.3
1
\includegraphics[max width=\textwidth, alt={}, center]{48f66bd5-33c1-4ce9-85f9-69faf10e871c-2_558_529_258_808} Four horizontal forces act at a point \(O\) and are in equilibrium. The magnitudes of the forces are \(F \mathrm {~N}\), \(G \mathrm {~N} , 15 \mathrm {~N}\) and \(F \mathrm {~N}\), and the forces act in directions as shown in the diagram.
  1. Show that \(F = 41.0\), correct to 3 significant figures.
  2. Find the value of \(G\).
CAIE M1 2015 November Q2
Moderate -0.8
2 A particle is released from rest at a point \(H \mathrm {~m}\) above horizontal ground and falls vertically. The particle passes through a point 35 m above the ground with a speed of \(( V - 10 ) \mathrm { m } \mathrm { s } ^ { - 1 }\) and reaches the ground with a speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find
  1. the value of \(V\),
  2. the value of \(H\).
CAIE M1 2015 November Q3
Standard +0.3
3 A particle \(P\) moves along a straight line for 100 s . It starts at a point \(O\) and at time \(t\) seconds after leaving \(O\) the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where $$v = 0.00004 t ^ { 3 } - 0.006 t ^ { 2 } + 0.288 t$$
  1. Find the values of \(t\) at which the acceleration of \(P\) is zero.
  2. Find the displacement of \(P\) from \(O\) when \(t = 100\).
CAIE M1 2015 November Q4
Standard +0.3
4
\includegraphics[max width=\textwidth, alt={}, center]{48f66bd5-33c1-4ce9-85f9-69faf10e871c-3_574_483_260_829} The diagram shows a vertical cross-section \(A B C\) of a surface. The part of the surface containing \(A B\) is smooth and \(A\) is 2.5 m above the level of \(B\). The part of the surface containing \(B C\) is rough and is at \(45 ^ { \circ }\) to the horizontal. The distance \(B C\) is 4 m (see diagram). A particle \(P\) of mass 0.2 kg is released from rest at \(A\) and moves in contact with the curve \(A B\) and then with the straight line \(B C\). The coefficient of friction between \(P\) and the part of the surface containing \(B C\) is 0.4 . Find the speed with which \(P\) reaches \(C\).
CAIE M1 2015 November Q5
Standard +0.3
5
\includegraphics[max width=\textwidth, alt={}, center]{48f66bd5-33c1-4ce9-85f9-69faf10e871c-3_259_828_1288_660} A smooth inclined plane of length 2.5 m is fixed with one end on the horizontal floor and the other end at a height of 0.7 m above the floor. Particles \(P\) and \(Q\), of masses 0.5 kg and 0.1 kg respectively, are attached to the ends of a light inextensible string which passes over a small smooth pulley fixed at the top of the plane. Particle \(Q\) is held at rest on the floor vertically below the pulley. The string is taut and \(P\) is at rest on the plane (see diagram). \(Q\) is released and starts to move vertically upwards towards the pulley and \(P\) moves down the plane.
  1. Find the tension in the string and the magnitude of the acceleration of the particles before \(Q\) reaches the pulley. At the instant just before \(Q\) reaches the pulley the string breaks; \(P\) continues to move down the plane and reaches the floor with a speed of \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  2. Find the length of the string.