Questions - CAIE M1 - A-Level Maths

Given that the particle is on the point of moving,find the value of $T$ .
Given instead that the particle is accelerating at $0.2 \mathrm {~ms} ^ { - 2 }$ ,find the value of $T$ .

CAIE M1 2024 November Q6

6 A particle moves in a straight line. It starts from rest, at time $t = 0$, and accelerates at $0.6 t \mathrm {~ms} ^ { - 2 }$ for 4 s , reaching a speed of $V \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The particle then travels at $V \mathrm {~m} \mathrm {~s} ^ { - 1 }$ for 11 s , and finally slows down, with constant deceleration, stopping after a further 5 s .

Show that $V = 4.8$.
Sketch a velocity-time graph for the motion.
\includegraphics[max width=\textwidth, alt={}, center]{3a6ecf05-127f-4ddf-959e-233f6bae9171-08_2722_40_107_2010}
Find an expression, in terms of $t$, for the velocity of the particle for $15 \leqslant t \leqslant 20$.
Find the total distance travelled by the particle.
\includegraphics[max width=\textwidth, alt={}, center]{3a6ecf05-127f-4ddf-959e-233f6bae9171-10_592_608_251_731} Two particles, $A$ and $B$, of masses 3 kg and 5 kg respectively, are connected by a light inextensible string that passes over a fixed smooth pulley. The particles are held with the string taut and its straight parts vertical. Particle $A$ is 1 m above a horizontal plane, and particle $B$ is 2 m above the plane (see diagram). The particles are released from rest. In the subsequent motion, $A$ does not reach the pulley, and after $B$ reaches the plane it remains in contact with the plane.
Find the tension in the string and the time taken for $B$ to reach the plane.
\includegraphics[max width=\textwidth, alt={}, center]{3a6ecf05-127f-4ddf-959e-233f6bae9171-10_2718_42_107_2007}
Find the time for which $A$ is at least 3.25 m above the plane.
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CAIE M1 2020 Specimen Q1

1 A rticle $P$ is p j ected rticallyw ard with $\mathbf { p }$ ed $\mathrm { ms } ^ { - 1 }$ frm a $\dot { \mathrm { p } } \mathrm { n } \mathbf { d }$ b gd

Fid b g eatest b in a th gd each dy $P$.
Fid $\mathbf { b }$ to al time frm $\mathrm { p } \dot { \mathrm { p } }$ ectim il $P$ retu $\mathbf { B }$ to $\mathbf { b } \mathbf { g d }$

CAIE M1 2020 Specimen Q2

2 ACB tan resistan e of mag itd ( ) N acts $\boldsymbol { \infty }$ car $\boldsymbol { 0 }$ mass $\boldsymbol { 0 }$ g

Th car is mi gr lor straig le lro de taco tan sp e $\varnothing \quad 3 \mathrm {~ms} ^ { - 1 }$. Fid $n \mathrm {~W} , \mathrm { t } \mathbf { b }$ rate at wh cht b eg $\mathbf { B }$ the car is work g
Th car tra ls at a co tan sp ed n a h ll in lie d at an ag e $\boldsymbol { 6 } \theta ^ { \circ }$ to to b izo al, wh re $\sin \theta ^ { \circ } = \frac { 1 } { 20 }$, w ittl b eg e wo kg t\\mathbb { N }$. Fid b sp e $\boldsymbol { \varnothing }$ th car.

CAIE M1 2020 Specimen Q3

3 Th ee small smo h se res $A , B$ ad $C 6$ eq l radi ad 6 masses $4 \mathrm {~g} \quad 2 \mathrm {~g}$ ad 3 g resp ctie ly, lie in th todr in a strait lie o a smo hb izt al p ae. In tially, $B$ ad $C$ are at rest ad $A$ is mi g ard $B$ with sp ed $6 \mathrm {~ms} ^ { - 1 }$. After th cb liso with $B$, se re $A$ co in s to mo in the same d rectim withs p ed $\mathrm { ms } ^ { - 1 }$.

Fid b sp e $\boldsymbol { \Phi } \quad B$ after th s cb liso Se re $B$ cb lid s with $C$.I it $h$ s cb lisd $\mathbf { b }$ se two se res co lesce tof $\mathbf { o }$ m am $\mathbf { b }$ ect $D$.
Fid b sp e $D$ after th s cb lisin
\includegraphics[max width=\textwidth, alt={}, center]{0a1cec7f-f9d1-4628-b979-443514c73eb9-05_65_1652_1146_242}

CAIE M1 2020 Specimen Q4

4 A $\boldsymbol { p }$ rticle of mass $\emptyset \mathrm { g }$ is $\mathbf { n }$ a rg p an in lin d at an an $\mathrm { e } \boldsymbol { 6 } \mathbf { B } ^ { \circ }$ to th $\mathbf { b }$ izn tal. A fo ce $\boldsymbol { 6 }$ mag te $\quad$ z N, actig at an ag e 6 O $^ { \circ }$ ab a lin 6 g eatest sle 6 th p aB , is s ed to p e n th $\mathbf { P }$ rticle frm slid g n th p as. Th co fficient

CAIE M1 2020 Specimen Q5

5 A car 6 mass $\mathbb { I } \quad$ g is p lig a trailer 6 mass $\theta \quad$ g ah ll in lin d at an ag e $6 \sin ^ { - 1 } ( \mathbb { I } )$ ) to th b izo al. Th car and to trailer are co cted b a lig rig d -b r wh ch is $\boldsymbol { \rho }$ rallel to th ro d Th d iv g fo ce $\varnothing$ th car's eg A is $\theta \mathrm { N }$ ad th resistan es to th car ad trailer are $\theta \mathrm { N }$ ad (1) N resp ctie ly.

Fid b acceleratio th sy tem ad b tensio it b tw -b r.
Wh it b car ad railer are tra llig tasp e $\boldsymbol { \Theta } \quad \mathbf { b } \mathrm { ms } ^ { - 1 } , \mathrm { t } \mathbf { b }$ divg $\mathbf { o }$ ce b cm es zero Fid th time, in sect , $\mathbf { b }$ fo e the sy tem cm es to rest ad th fo ce in th ro $\mathbf { r d }$ ig th s time.

CAIE M1 2020 Specimen Q6

6 A $\boldsymbol { p }$ rticle $P \mathrm {~m}$ s ira straitg lie . Tb ± lo ity $v \mathrm {~ms} ^ { - 1 }$ at time $t \mathrm {~s}$ is g ஓ ity $$\begin{array} { l l } v = 5 t ( t - 2 & \text { fo } 0 \leqslant t \leqslant 4
v = k & \text { fo } 4 \leqslant t \leqslant 4
v = 82 \quad t & \text { fo } 4 \leqslant t \leqslant \Omega \end{array}$$ wh re $k$ is a co tan.

Fid $k$.
Sk tcht b lo ity ime g aff $\mathbf { 0 } \quad 0 \leqslant t \leqslant 0$
Fid bet $\mathbf { 6 }$ le $\mathrm { s } \mathbf { 6 } t$ fo wh cht b acceleratio $P$ is $\mathbf { p }$ itie .
Fid $\mathbf { b }$ to ald stan e trac lledy $P$ irt $\mathbf { b }$ in era $10 \leqslant t \leqslant 0$

CAIE M1 2020 Specimen Q7

4 marks

7
\includegraphics[max width=\textwidth, alt={}, center]{0a1cec7f-f9d1-4628-b979-443514c73eb9-12_248_674_260_699} Two $\boldsymbol { \rho }$ rticles $A$ ad $B , 6$ masses 08 k ad 0 g resp ctie ly, are co cted $\varphi$ a lig inex en ib e strig Particle $A$ is p aced $\mathbf { n }$ ab izb al sn face. Th strig $\boldsymbol { p }$ sses rasmall smo hp ley $P$ fiæ d at th ed 6 th su face, and $B \mathbf { h }$ g freely. Th $\mathbf { b }$ izo alsectin 6 th strig $A P$, is $\mathbf { 6 }$ leg $\mathrm { h } \otimes \mathrm { m }$ (see id ag am). Th $\boldsymbol { p }$ rticles are released rm rest witb lo ectim of th strig atı.

Gie it $\mathbf { h }$ th sn face is smo lf id $\mathbf { b }$ time tak if $\mathbf { D }$ A tor eacht $\mathbf { b }$ p ley. [
It is $\dot { \mathbf { g } }$ ven in tead that th sn face is rg ad th t th sp ed $6 A$ immed ately $\mathbf { b }$ fo e it reach s th p leỳ $\mathrm { s } v \mathrm {~ms} ^ { - 1 }$. Th wo ld ag in t frictim $\mathrm { s } A \mathrm {~m}$ s frm rest to b p leyi s 2 J . Use an ee rgn eth $\quad$ f id $v$. [4] If B e th follw ig lin dpg to cm p ete th an wer(s) to ay q stin (s), th q stin $\mathrm { m } \quad \mathbf { b } \quad \mathrm { r } ( \mathrm { s } )$ ms tb clearlys n n

CAIE M1 2002 June Q1

1
\includegraphics[max width=\textwidth, alt={}, center]{430f1f9a-7a3a-47a0-b742-daf74e68adfd-2_300_748_274_708} One end of a light inextensible string is attached to a ring which is threaded on a fixed horizontal bar. The string is used to pull the ring along the bar at a constant speed of $0.4 \mathrm {~ms} ^ { - 1 }$. The string makes a constant angle of $30 ^ { \circ }$ with the bar and the tension in the string is 5 N (see diagram). Find the work done by the tension in 10 s .

CAIE M1 2002 June Q2

2 A basket of mass 5 kg slides down a slope inclined at $12 ^ { \circ }$ to the horizontal. The coefficient of friction between the basket and the slope is 0.2 .

Find the frictional force acting on the basket.
Determine whether the speed of the basket is increasing or decreasing.

CAIE M1 2002 June Q3

3
\includegraphics[max width=\textwidth, alt={}, center]{430f1f9a-7a3a-47a0-b742-daf74e68adfd-2_368_584_1302_794} Two forces, each of magnitude 10 N , act at a point $O$ in the directions of $O A$ and $O B$, as shown in the diagram. The angle between the forces is $\theta$. The resultant of these two forces has magnitude 12 N .

Find $\theta$.
Find the component of the resultant force in the direction of $O A$.

CAIE M1 2002 June Q4

4 A box of mass 4.5 kg is pulled at a constant speed of $2 \mathrm {~ms} ^ { - 1 }$ along a rough horizontal floor by a horizontal force of magnitude 15 N .

Find the coefficient of friction between the box and the floor. The horizontal pulling force is now removed. Find
the deceleration of the box in the subsequent motion,
the distance travelled by the box from the instant the horizontal force is removed until the box comes to rest.
A cyclist travels in a straight line from $A$ to $B$ with constant acceleration $0.06 \mathrm {~m} \mathrm {~s} ^ { - 2 }$. His speed at $A$ is $3 \mathrm {~ms} ^ { - 1 }$ and his speed at $B$ is $6 \mathrm {~ms} ^ { - 1 }$. Find
(a) the time taken by the cyclist to travel from $A$ to $B$,
(b) the distance $A B$.
A car leaves $A$ at the same instant as the cyclist. The car starts from rest and travels in a straight line to $B$. The car reaches $B$ at the same instant as the cyclist. At time $t \mathrm {~s}$ after leaving $A$ the speed of the car is $k t ^ { 2 } \mathrm {~ms} ^ { - 1 }$, where $k$ is a constant. Find
(a) the value of $k$,
(b) the speed of the car at $B$.
A lorry $P$ of mass 15000 kg climbs a straight hill of length 800 m at a steady speed. The hill is inclined at $2 ^ { \circ }$ to the horizontal. For $P$ 's journey from the bottom of the hill to the top, find
(a) the gain in gravitational potential energy,
(b) the work done by the driving force, which has magnitude 7000 N ,
(c) the work done against the force resisting the motion.
A second lorry, $Q$, also has mass 15000 kg and climbs the same hill as $P$. The motion of $Q$ is subject to a constant resisting force of magnitude 900 N , and $Q$ s speed falls from $20 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at the bottom of the hill to $10 \mathrm {~ms} ^ { - 1 }$ at the top. Find the work done by the driving force as $Q$ climbs from the bottom of the hill to the top.
\includegraphics[max width=\textwidth, alt={}, center]{430f1f9a-7a3a-47a0-b742-daf74e68adfd-3_483_231_1537_973} Particles $A$ and $B$, of masses 0.15 kg and 0.25 kg respectively, are attached to the ends of a light inextensible string which passes over a smooth fixed pulley. The system is held at rest with the string taut and with $A$ and $B$ at the same horizontal level, as shown in the diagram. The system is then released.
Find the downward acceleration of $B$. After $2 \mathrm {~s} B$ hits the floor and comes to rest without rebounding. The string becomes slack and $A$ moves freely under gravity.
Find the time that elapses until the string becomes taut again.
Sketch on a single diagram the velocity-time graphs for both particles, for the period from their release until the instant that $B$ starts to move upwards.

CAIE M1 2003 June Q1

1 A crate of mass 800 kg is lifted vertically, at constant speed, by the cable of a crane. Find

the tension in the cable,
the power applied to the crate in increasing the height by 20 m in 50 s .

CAIE M1 2003 June Q2

2
\includegraphics[max width=\textwidth, alt={}, center]{cb04a09c-af23-4e9d-b3da-da9e351fe879-2_405_384_550_884} Three coplanar forces of magnitudes $10 \mathrm {~N} , 10 \mathrm {~N}$ and 6 N act at a point $P$ in the directions shown in the diagram. $P Q$ is the bisector of the angle between the two forces of magnitude 10 N .

Find the component of the resultant of the three forces
(a) in the direction of $P Q$,
(b) in the direction perpendicular to $P Q$.
Find the magnitude of the resultant of the three forces.

CAIE M1 2003 June Q3

3
\includegraphics[max width=\textwidth, alt={}, center]{cb04a09c-af23-4e9d-b3da-da9e351fe879-2_556_974_1548_587} The diagram shows the velocity-time graphs for the motion of two cyclists $P$ and $Q$, who travel in the same direction along a straight path. Both cyclists start from rest at the same point $O$ and both accelerate at $2 \mathrm {~m} \mathrm {~s} ^ { - 2 }$ up to a speed of $10 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. Both then continue at a constant speed of $10 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. $Q$ starts his journey $T$ seconds after $P$.

Show in a sketch of the diagram the region whose area represents the displacement of $P$, from $O$, at the instant when $Q$ starts. Given that $P$ has travelled 16 m at the instant when $Q$ starts, find
the value of $T$,
the distance between $P$ and $Q$ when $Q$ 's speed reaches $10 \mathrm {~ms} ^ { - 1 }$.

CAIE M1 2003 June Q4

4 A particle moves in a straight line. Its displacement $t$ seconds after leaving the fixed point $O$ is $x$ metres, where $x = \frac { 1 } { 2 } t ^ { 2 } + \frac { 1 } { 30 } t ^ { 3 }$. Find

the speed of the particle when $t = 10$,
the value of $t$ for which the acceleration of the particle is twice its initial acceleration.

CAIE M1 2003 June Q5

5
\includegraphics[max width=\textwidth, alt={}, center]{cb04a09c-af23-4e9d-b3da-da9e351fe879-3_504_387_598_881}
$S _ { 1 }$ and $S _ { 2 }$ are light inextensible strings, and $A$ and $B$ are particles each of mass 0.2 kg . Particle $A$ is suspended from a fixed point $O$ by the string $S _ { 1 }$, and particle $B$ is suspended from $A$ by the string $S _ { 2 }$. The particles hang in equilibrium as shown in the diagram.

Find the tensions in $S _ { 1 }$ and $S _ { 2 }$. The string $S _ { 1 }$ is cut and the particles fall. The air resistance acting on $A$ is 0.4 N and the air resistance acting on $B$ is 0.2 N .
Find the acceleration of the particles and the tension in $S _ { 2 }$.

CAIE M1 2003 June Q6

6 A small block of mass 0.15 kg moves on a horizontal surface. The coefficient of friction between the block and the surface is 0.025 .

Find the frictional force acting on the block.
Show that the deceleration of the block is $0.25 \mathrm {~m} \mathrm {~s} ^ { - 2 }$. The block is struck from a point $A$ on the surface and, 4 s later, it hits a boundary board at a point $B$. The initial speed of the block is $5.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.
Find the distance $A B$. The block rebounds from the board with a speed of $3.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and moves along the line $B A$. Find
the speed with which the block passes through $A$,
the total distance moved by the block, from the instant when it was struck at $A$ until the instant when it comes to rest.

CAIE M1 2003 June Q7

7
\includegraphics[max width=\textwidth, alt={}, center]{cb04a09c-af23-4e9d-b3da-da9e351fe879-4_257_988_267_580} The diagram shows a vertical cross-section $A B C D$ of a surface. The parts $A B$ and $C D$ are straight and have lengths 2.5 m and 5.2 m respectively. $A D$ is horizontal, and $A B$ is inclined at $60 ^ { \circ }$ to the horizontal. The points $B$ and $C$ are at the same height above $A D$. The parts of the surface containing $A B$ and $B C$ are smooth. A particle $P$ is given a velocity of $8 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at $A$, in the direction $A B$, and it subsequently reaches $D$. The particle does not lose contact with the surface during this motion.

Find the speed of $P$ at $B$.
Show that the maximum height of the cross-section, above $A D$, is less than 3.2 m .
State briefly why $P$ 's speed at $C$ is the same as its speed at $B$.
The frictional force acting on the particle as it travels from $C$ to $D$ is 1.4 N . Given that the mass of $P$ is 0.4 kg , find the speed with which $P$ reaches $D$.

CAIE M1 2004 June Q1

1
\includegraphics[max width=\textwidth, alt={}, center]{e060fc3b-ae93-46b5-b476-dcecb14d6d06-2_200_588_267_781} A ring of mass 1.1 kg is threaded on a fixed rough horizontal rod. A light string is attached to the ring and the string is pulled with a force of magnitude 13 N at an angle $\alpha$ below the horizontal, where $\tan \alpha = \frac { 5 } { 12 }$ (see diagram). The ring is in equilibrium.

Find the frictional component of the contact force on the ring.
Find the normal component of the contact force on the ring.
Given that the equilibrium of the ring is limiting, find the coefficient of friction between the ring and the rod.

CAIE M1 2004 June Q2

6 marks

2
\includegraphics[max width=\textwidth, alt={}, center]{e060fc3b-ae93-46b5-b476-dcecb14d6d06-2_684_257_1114_945} Coplanar forces of magnitudes $250 \mathrm {~N} , 100 \mathrm {~N}$ and 300 N act at a point in the directions shown in the diagram. The resultant of the three forces has magnitude $R \mathrm {~N}$, and acts at an angle $\alpha ^ { \circ }$ anticlockwise from the force of magnitude 100 N . Find $R$ and $\alpha$.
[0pt] [6]

CAIE M1 2004 June Q3

3 marks

3
\includegraphics[max width=\textwidth, alt={}, center]{e060fc3b-ae93-46b5-b476-dcecb14d6d06-3_727_899_267_625} A boy runs from a point $A$ to a point $C$. He pauses at $C$ and then walks back towards $A$ until reaching the point $B$, where he stops. The diagram shows the graph of $v$ against $t$, where $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ is the boy's velocity at time $t$ seconds after leaving $A$. The boy runs and walks in the same straight line throughout.

Find the distances $A C$ and $A B$.
Sketch the graph of $x$ against $t$, where $x$ metres is the boy's displacement from $A$. Show clearly the values of $t$ and $x$ when the boy arrives at $C$, when he leaves $C$, and when he arrives at $B$. [3]

CAIE M1 2004 June Q4

4 The top of an inclined plane is at a height of 0.7 m above the bottom. A block of mass 0.2 kg is released from rest at the top of the plane and slides a distance of 2.5 m to the bottom. Find the kinetic energy of the block when it reaches the bottom of the plane in each of the following cases:

the plane is smooth,
the coefficient of friction between the plane and the block is 0.15 .

CAIE M1 2004 June Q5

5 A particle $P$ moves in a straight line that passes through the origin $O$. The velocity of $P$ at time $t$ seconds is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$, where $v = 20 t - t ^ { 3 }$. At time $t = 0$ the particle is at rest at a point whose displacement from $O$ is - 36 m .

Find an expression for the displacement of $P$ from $O$ in terms of $t$.
Find the displacement of $P$ from $O$ when $t = 4$.
Find the values of $t$ for which the particle is at $O$.

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