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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 5 6 6 7 7 6 11 11 59
Score

Get Mathematics 9709 Topical Questions (2010-2021) for $14.5 per Subject.
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Question 1 Code: 9709/42/O/N/11/1, Topic: -

A racing cyclist, whose mass with his cycle is $75 \mathrm{~kg}$, works at a rate of $720 \mathrm{~W}$ while moving on a straight horizontal road. The resistance to the cyclist's motion is constant and equal to $R \mathrm{~N}$.

$\text{(i)}$ Given that the cyclist is accelerating at $0.16 \mathrm{~m} \mathrm{~s}^{-2}$ at an instant when his speed is $12 \mathrm{~m} \mathrm{~s}^{-1}$, find the value of $R$. $[3]$

$\text{(ii)}$ Given that the cyclist's acceleration is positive, show that his speed is less than $15 \mathrm{~m} \mathrm{~s}^{-1}$. $[2]$

Question 2 Code: 9709/43/M/J/19/2, Topic: -

 

Coplanar forces of magnitudes $12 \mathrm{~N}, 24 \mathrm{~N}$ and $30 \mathrm{~N}$ act at a point in the directions shown in the diagram.

$\text{(i)}$ Find the components of the resultant of the three forces in the $x$-direction and in the $y$-direction. $[4]$

Component in $x$-direction.

Component in $y$-direction.

$\text{(ii)}$ Hence find the direction of the resultant. $[2]$

Question 3 Code: 9709/43/O/N/14/3, Topic: -

 

Each of three light inextensible strings has a particle attached to one of its ends. The other ends of the strings are tied together at a point $O$. Two of the strings pass over fixed smooth pegs and the particles hang freely in equilibrium. The weights of the particles and the angles between the sloping parts of the strings and the vertical are as shown in the diagram. It is given that $\sin \beta=0.8$ and $\cos \beta=0.6$.

$\text{(i)}$ Show that $W \cos \alpha=3.8$ and find the value of $W \sin \alpha$. $[3]$

$\text{(ii)}$ Hence find the values of $W$ and $\alpha$. $[3]$

Question 4 Code: 9709/42/M/J/14/5, Topic: -

 

A light inextensible rope has a block $A$ of mass $5 \mathrm{~kg}$ attached at one end, and a block $B$ of mass $16 \mathrm{~kg}$ attached at the other end. The rope passes over a smooth pulley which is fixed at the top of a rough plane inclined at an angle of $30^{\circ}$ to the horizontal. Block $A$ is held at rest at the bottom of the plane and block $B$ hangs below the pulley (see diagram). The coefficient of friction between $A$ and the plane is $\frac{1}{\sqrt{3}}.$ Block $A$ is released from rest and the system starts to move. When each of the blocks has moved a distance of $x \mathrm{~m}$ each has speed $v \mathrm{~m} \mathrm{~s}^{-1}$.

$\text{(i)}$ Write down the gain in kinetic energy of the system in terms of $v$. $[1]$

$\text{(ii)}$ Find, in terms of $x$,

$\text{(a)}$ the loss of gravitational potential energy of the system, $[2]$

$\text{(b)}$ the work done against the frictional force. $[3]$

$\text{(iii)}$ Show that $21 v^{2}=220 x$. $[2]$

Question 5 Code: 9709/41/M/J/16/5, Topic: -

 

Two particles of masses $5 \mathrm{~kg}$ and $10 \mathrm{~kg}$ are connected by a light inextensible string that passes over a fixed smooth pulley. The $5 \mathrm{~kg}$ particle is on a rough fixed slope which is at an angle of $\alpha$ to the horizontal, where $\tan \alpha=\frac{3}{4}$. The $10 \mathrm{~kg}$ particle hangs below the pulley (see diagram). The coefficient of friction between the slope and the $5 \mathrm{~kg}$ particle is $\frac{1}{2}$. The particles are released from rest. Find the acceleration of the particles and the tension in the string. $[7]$

Question 6 Code: 9709/42/O/N/16/5, Topic: -

A particle of mass $m \mathrm{~kg}$ is resting on a rough plane inclined at $30^{\circ}$ to the horizontal. A force of magnitude $10 \mathrm{~N}$ applied to the particle up a line of greatest slope of the plane is just sufficient to stop the particle sliding down the plane. When a force of $75 \mathrm{~N}$ is applied to the particle up a line of greatest slope of the plane, the particle is on the point of sliding up the plane. Find $m$ and the coefficient of friction between the particle and the plane. $[6]$

Question 7 Code: 9709/41/M/J/14/7, Topic: -

Two cyclists $P$ and $Q$ travel along a straight road $A B C$, starting simultaneously at $A$ and arriving simultaneously at $C$. Both cyclists pass through $B 400 \mathrm{~s}$ after leaving $A$. Cyclist $P$ starts with speed $3 \mathrm{~m} \mathrm{~s}^{-1}$ and increases this speed with constant acceleration $0.005 \mathrm{~m} \mathrm{~s}^{-2}$ until he reaches $B$.

$\text{(i)}$ Show that the distance $A B$ is $1600 \mathrm{~m}$ and find $P$ 's speed at $B$. $[3]$

Cyclist $Q$ travels from $A$ to $B$ with speed $v \mathrm{~m} \mathrm{~s}^{-1}$ at time $t$ seconds after leaving $A$, where

$$ v=0.04 t-0.0001 t^{2}+k \text {, } $$

and $k$ is a constant.

$\text{(ii)}$ Find the value of $k$ and the maximum speed of $Q$ before he has reached $B$. $[6]$

Cyclist $P$ travels from $B$ to $C$, a distance of $1400 \mathrm{~m}$, at the speed he had reached at $B$. Cyclist $Q$ travels from $B$ to $C$ with constant acceleration $a \mathrm{~m} \mathrm{~s}^{-2}$.

$\text{(iii)}$ Find the time taken for the cyclists to travel from $B$ to $C$ and find the value of $a$. $[4]$

Question 8 Code: 9709/43/O/N/17/7, Topic: -

 

A particle $P$ of mass $0.2 \mathrm{~kg}$ rests on a rough plane inclined at $30^{\circ}$ to the horizontal. The coefficient of friction between the particle and the plane is $0.3$. A force of magnitude $T \mathrm{~N}$ acts upwards on $P$ at $15^{\circ}$ above a line of greatest slope of the plane (see diagram).

$\text{(i)}$ Find the least value of $T$ for which the particle remains at rest. $[6]$

The force of magnitude $T \mathrm{~N}$ is now removed. A new force of magnitude $0.25 \mathrm{~N}$ acts on $P$ up the plane, parallel to a line of greatest slope of the plane. Starting from rest, $P$ slides down the plane. After moving a distance of $3 \mathrm{~m}, P$ passes through the point $A$.

$\text{(ii)}$ Use an energy method to find the speed of $P$ at $A$. $[5]$

Worked solutions: P1, P3 & P6 (S1)

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