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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 8 Total
Marks 4 7 8 7 11 10 11 11 69
Score

Get Mathematics 9709 Topical Questions (2010-2021) for $14.5 per Subject.
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Question 1 Code: 9709/43/M/J/15/1, Topic: -

A block is pulled along a horizontal floor by a horizontal rope. The tension in the rope is $500 \mathrm{~N}$ and the block moves at a constant speed of $2.75 \mathrm{~m} \mathrm{~s}^{-1}$. Find the work done by the tension in $40 \mathrm{~s}$ and find the power applied by the tension. $[4]$

Question 2 Code: 9709/42/M/J/12/3, Topic: -

A particle $P$ moves in a straight line, starting from the point $O$ with velocity $2 \mathrm{~m} \mathrm{~s}^{-1}$. The acceleration of $P$ at time $t \mathrm{~s}$ after leaving $O$ is $2 t^{\frac{2}{3}} \mathrm{~m} \mathrm{~s}^{-2}$.

$\text{(i)}$ Show that $t^{\frac{5}{3}}=\frac{5}{6}$ when the velocity of $P$ is $3 \mathrm{~m} \mathrm{~s}^{-1}$. $[4]$

$\text{(ii)}$ Find the distance of $P$ from $O$ when the velocity of $P$ is $3 \mathrm{~ms}^{-1}$. $[3]$

Question 3 Code: 9709/43/O/N/11/5, Topic: -

A particle $P$ moves in a straight line. It starts from rest at $A$ and comes to rest instantaneously at $B$. The velocity of $P$ at time $t$ seconds after leaving $A$ is $v \mathrm{~m} \mathrm{~s}^{-1}$, where $v=6 t^{2}-k t^{3}$ and $k$ is a constant.

$\text{(i)}$ Find an expression for the displacement of $P$ from $A$ in terms of $t$ and $k$. $[2]$

$\text{(ii)}$ Find an expression for $t$ in terms of $k$ when $P$ is at $B$. $[1]$

Given that the distance $A B$ is $108 \mathrm{~m}$, find

$\text{(iii)}$ the value of $k$, $[2]$

$\text{(iv)}$ the maximum value of $v$ when the particle is moving from $A$ towards $B$. $[3]$

Question 4 Code: 9709/43/M/J/15/5, Topic: -

 ![9709-43-M-J-15-5a.PNG]

Four coplanar forces of magnitudes $4 \mathrm{~N}, 8 \mathrm{~N}, 12 \mathrm{~N}$ and $16 \mathrm{~N}$ act at a point. The directions in which the forces act are shown in Fig. 1.

$\text{(i)}$ Find the magnitude and direction of the resultant of the four forces. $[5]$

 ![9709-43-M-J-15-5b.PNG]

The forces of magnitudes $4 \mathrm{~N}$ and $16 \mathrm{~N}$ exchange their directions and the forces of magnitudes $8 \mathrm{~N}$ and $12 \mathrm{~N}$ also exchange their directions (see Fig. 2).

$\text{(ii)}$ State the magnitude and direction of the resultant of the four forces in Fig. $2.$ $[5]$

Question 5 Code: 9709/42/O/N/19/6, Topic: -

A block of mass $3 \mathrm{~kg}$ is initially at rest on a rough horizontal plane. A force of magnitude $6 \mathrm{~N}$ is applied to the block at an angle of $\theta$ above the horizontal, where $\cos \theta=\frac{24}{25}$. The force is applied for a period of $5 \mathrm{~s}$, during which time the block moves a distance of $4.5 \mathrm{~m}$.

$\text{(i)}$ Find the magnitude of the frictional force on the block. $[4]$

$\text{(ii)}$ Show that the coefficient of friction between the block and the plane is $0.165$, correct to 3 significant figures. $[3]$

$\text{(iii)}$ When the block has moved a distance of $4.5 \mathrm{~m}$, the force of magnitude $6 \mathrm{~N}$ is removed and the block then decelerates to rest. Find the total time for which the block is in motion. $[4]$

Question 6 Code: 9709/41/O/N/11/7, Topic: -

A particle $P$ starts from a point $O$ and moves along a straight line. $P$ 's velocity $t$ s after leaving $O$ is $v \mathrm{~m} \mathrm{~s}^{-1}$, where

$$ v=0.16 t^{\frac{3}{2}}-0.016 t^{2} $$

$P$ comes to rest instantaneously at the point $A$.

$\text{(i)}$ Verify that the value of $t$ when $P$ is at $A$ is 100. $[1]$

$\text{(ii)}$ Find the maximum speed of $P$ in the interval $0< t <100$. $[4]$

$\text{(iii)}$ Find the distance $O A$. $[3]$

$\text{(iv)}$ Find the value of $t$ when $P$ passes through $O$ on returning from $A$. $[2]$

Question 7 Code: 9709/42/O/N/11/7, Topic: -

A tractor travels in a straight line from a point $A$ to a point $B$. The velocity of the tractor is $v \mathrm{~m} \mathrm{~s}^{-1}$ at time $t \mathrm{~s}$ after leaving $A$.

$\text{(i)}$

The diagram shows an approximate velocity-time graph for the motion of the tractor. The graph consists of two straight line segments. Use the graph to find an approximation for

$\text{(a)}$ the distance $A B$, $[2]$

$\text{(b)}$ the acceleration of the tractor for $0< t <400$ and for $400< t <800$. $[2]$

$\text{(ii)}$ The actual velocity of the tractor is given by $v=0.04 t-0.00005 t^{2}$ for $0 \leqslant t \leqslant 800$.

$\text{(a)}$ Find the values of $t$ for which the actual acceleration of the tractor is given correctly by the approximate velocity-time graph in part $\text{(i)}$. $[3]$

For the interval $0 \leqslant t \leqslant 400$, the approximate velocity of the tractor in part $\text{(i)}$ is denoted by $v_{1} \mathrm{~m} \mathrm{~s}^{-1}$

$\text{(b)}$ Express $v_{1}$ in terms of $t$ and hence show that $v_{1}-v=0.00005(t-200)^{2}-1$. $[2]$

$\text{(c)}$ Deduce that $-1 \leqslant v_{1}-v \leqslant 1$. $[2]$

Question 8 Code: 9709/41/M/J/13/7, Topic: -

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.

$\text{(i)}$ Find the distance $A B$. $[3]$

$\text{(ii)}$ Find the maximum speed of the car. $[4]$

$\text{(iii)}$ Find the acceleration of the car

$\text{(a)}$ as it starts from $A$,

$\text{(b)}$ as it arrives at $B$.

$[2]$

$\text{(iv)}$ Sketch the velocity-time graph for the journey. $[2]$

Worked solutions: P1, P3 & P6 (S1)

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