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MATHEMATICS 9709

Cambridge International AS and A Level

Name of student Date
Adm. number Year/grade Stream
Subject Mechanics 1 (M1) Variant(s) P41, P42, P43
Start time Duration Stop time

Qtn No. 1 2 3 4 5 6 7 Total
Marks 5 5 7 8 8 7 9 49
Score

Get Mathematics 9709 Topical Questions (2010-2021) for $14.5 per Subject.
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Question 1 Code: 9709/42/M/J/13/3, Topic: -

 

A particle $P$ of mass $2.1 \mathrm{~kg}$ is attached to one end of each of two light inextensible strings. The other ends of the strings are attached to points $A$ and $B$ which are at the same horizontal level. $P$ hangs in equilibrium at a point $40 \mathrm{~cm}$ below the level of $A$ and $B$, and the strings $P A$ and $P B$ have lengths $50 \mathrm{~cm}$ and $104 \mathrm{~cm}$ respectively (see diagram). Show that the tension in the string $P A$ is $20 \mathrm{~N}$, and find the tension in the string $P B$. $[5]$

Question 2 Code: 9709/42/M/J/15/3, Topic: -

A plane is inclined at an angle of $\sin ^{-1}\left(\frac{1}{8}\right)$ to the horizontal. $A$ and $B$ are two points on the same line of greatest slope with $A$ higher than $B$. The distance $A B$ is $12 \mathrm{~m}$. A small object $P$ of mass $8 \mathrm{~kg}$ is released from rest at $A$ and slides down the plane, passing through $B$ with speed $4.5 \mathrm{~m} \mathrm{~s}^{-1}$. For the motion of $P$ from $A$ to $B$, find

$\text{(i)}$ the increase in kinetic energy of $P$ and the decrease in potential energy of $P$, $[3]$

$\text{(ii)}$ the magnitude of the constant resisting force that opposes the motion of $P$. $[2]$

Question 3 Code: 9709/41/M/J/19/3, Topic: -

A lorry has mass $12000 \mathrm{~kg}$.

$\text{(i)}$ The lorry moves at a constant speed of $5 \mathrm{~m} \mathrm{~s}^{-1}$ up a hill inclined at an angle of $\theta$ to the horizontal, where $\sin \theta=0.08$. At this speed, the magnitude of the resistance to motion on the lorry is $1500 \mathrm{~N}$. Show that the power of the lorry's engine is $55.5 \mathrm{~kW}$. $[3]$

When the speed of the lorry is $v \mathrm{~m} \mathrm{~s}^{-1}$ the magnitude of the resistance to motion is $k v^{2} \mathrm{~N}$, where $k$ is a constant.

$\text{(ii)}$ Show that $k=60$. $[1]$

$\text{(iii)}$ The lorry now moves at a constant speed on a straight level road. Given that its engine is still working at $55.5 \mathrm{~kW}$, find the lorry's speed. $[3]$

Question 4 Code: 9709/43/M/J/12/5, Topic: -

A lorry of mass $16000 \mathrm{~kg}$ moves on a straight hill inclined at angle $\alpha^{\circ}$ to the horizontal. The length of the hill is $500 \mathrm{~m}$.

$\text{(i)}$ While the lorry moves from the bottom to the top of the hill at constant speed, the resisting force acting on the lorry is $800 \mathrm{~N}$ and the work done by the driving force is $2800 \mathrm{~kJ}$. Find the value of $\alpha$. $[4]$

$\text{(ii)}$ 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 $2400 \mathrm{~kJ}$ and the work done against the resistance to motion is $800 \mathrm{~kJ}$. Find the speed of the lorry at the bottom of the hill. $[4]$

Question 5 Code: 9709/43/M/J/14/5, Topic: -

A lorry of mass $16000 \mathrm{~kg}$ travels at constant speed from the bottom, $O$, to the top, $A$, of a straight hill. The distance $O A$ is $1200 \mathrm{~m}$ and $A$ is $18 \mathrm{~m}$ above the level of $O$. The driving force of the lorry is constant and equal to $4500 \mathrm{~N}$.

$\text{(i)}$ Find the work done against the resistance to the motion of the lorry. $[3]$

On reaching $A$ the lorry continues along a straight horizontal road against a constant resistance of $2000 \mathrm{~N}$. The driving force of the lorry is not now constant, and the speed of the lorry increases from $9 \mathrm{~ms}^{-1}$ at $A$ to $21 \mathrm{~m} \mathrm{~s}^{-1}$ at the point $B$ on the road. The distance $A B$ is $2400 \mathrm{~m}$.

$\text{(ii)}$ Use an energy method to find $F$, where $F \mathrm{~N}$ is the average value of the driving force of the lorry while moving from $A$ to $B$. $[3]$

$\text{(iii)}$ Given that the driving force at $A$ is $1280 \mathrm{~N}$ greater than $F \mathrm{~N}$ and that the driving force at $B$ is $1280 \mathrm{~N}$ less than $F \mathrm{~N}$, show that the power developed by the lorry's engine is the same at $B$ as it is at $A$. $[2]$

Question 6 Code: 9709/42/M/J/16/5, Topic: -

 

A block of mass $2.5 \mathrm{~kg}$ is placed on a plane which is inclined at an angle of $30^{\circ}$ to the horizontal. The block is kept in equilibrium by a light string making an angle of $20^{\circ}$ above a line of greatest slope. The tension in the string is $T \mathrm{~N}$, as shown in the diagram. The coefficient of friction between the block and plane is $\frac{1}{4}$. The block is in limiting equilibrium and is about to move up the plane. Find the value of $T$. $[7]$

Question 7 Code: 9709/43/M/J/14/6, Topic: -

A particle starts from rest at a point $O$ and moves in a horizontal straight line. The velocity of the particle is $v \mathrm{~m} \mathrm{~s}^{-1}$ at time $t \mathrm{~s}$ after leaving $O$. For $0 \leqslant t<60$, the velocity is given by

$$ v=0.05 t-0.0005 t^{2} . $$

The particle hits a wall at the instant when $t=60$, and reverses the direction of its motion. The particle subsequently comes to rest at the point $A$ when $t=100$, and for $60 $$ v=0.025 t-2.5 \text {. } $$

$\text{(i)}$ Find the velocity of the particle immediately before it hits the wall, and its velocity immediately after its hits the wall. $[2]$

$\text{(ii)}$ Find the total distance travelled by the particle. $[4]$

$\text{(iii)}$ Find the maximum speed of the particle and sketch the particle's velocity-time graph for $0 \leqslant t \leqslant 100$, showing the value of $t$ for which the speed is greatest. $[4]$

Worked solutions: P1, P3 & P6 (S1)

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