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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 $14.5 per Subject. Attempt all the 7 questions Question 1 Code: 9709/42/M/J/13/3, Topic: - A particle$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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