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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 9 10 11 12 Total
Marks 5 5 7 9 8 6 10 9 8 8 11 14 100
Score

Get Mathematics 9709 Topical Questions (2010-2021) for $14.5 per Subject. Attempt all the 12 questions Question 1 Code: 9709/41/M/J/16/1, Topic: - A lift moves upwards from rest and accelerates at$0.9 \mathrm{~m} \mathrm{~s}^{-2}$for$3 \mathrm{~s}$. The lift then travels for$6 \mathrm{~s}$at constant speed and finally slows down, with a constant deceleration, stopping in a further$4 \mathrm{~s}$.$\text{(i)}$Sketch a velocity-time graph for the motion.$[3]\text{(ii)}$Find the total distance travelled by the lift.$[2]$Question 2 Code: 9709/42/M/J/16/1, Topic: - Coplanar forces of magnitudes$7 \mathrm{~N}, 6 \mathrm{~N}$and$8 \mathrm{~N}$act at a point in the directions shown in the diagram. Given that$\sin \alpha=\frac{3}{5}$, find the magnitude and direction of the resultant of the three forces.$[5]$Question 3 Code: 9709/43/M/J/12/4, Topic: - A car of mass$1230 \mathrm{~kg}$increases its speed from$4 \mathrm{~m} \mathrm{~s}^{-1}$to$21 \mathrm{~m} \mathrm{~s}^{-1}$in$24.5 \mathrm{~s}$. The table below shows corresponding values of time$t \mathrm{~s}$and speed$v \mathrm{~m} \mathrm{~s}^{-1}$. $$\begin{array}{|c|c|c|c|c|} \hline t & 0 & 0.5 & 16.3 & 24.5 \\ \hline v & 4 & 6 & 19 & 21 \\ \hline \end{array}$$$\text{(i)}$Using the values in the table, find the average acceleration of the car for$0< t <0.5$and for$16.3< t <24.5.[2]$While the car is increasing its speed the power output of its engine is constant and equal to$P \mathrm{~W}$, and the resistance to the car's motion is constant and equal to$R \mathrm{~N}$.$\text{(ii)}$Assuming that the values obtained in part$\text{(i)}$are approximately equal to the accelerations at$v=5$and at$v=20$, find approximations for$P$and$R$.$[5]$Question 4 Code: 9709/43/M/J/11/5, Topic: - A small block of mass$1.25 \mathrm{~kg}$is on a horizontal surface. Three horizontal forces, with magnitudes and directions as shown in the diagram, are applied to the block. The angle$\theta$is such that$\cos \theta=0.28$and$\sin \theta=0.96$. A horizontal frictional force also acts on the block, and the block is in equilibrium.$\text{(i)}$Show that the magnitude of the frictional force is$7.5 \mathrm{~N}$and state the direction of this force.$[4]\text{(ii)}$Given that the block is in limiting equilibrium, find the coefficient of friction between the block and the surface.$[2]$The force of magnitude$6.1 \mathrm{~N}$is now replaced by a force of magnitude$8.6 \mathrm{~N}$acting in the same direction, and the block begins to move.$\text{(iii)}$Find the magnitude and direction of the acceleration of the block.$[3]$Question 5 Code: 9709/41/M/J/13/5, Topic: - A light inextensible string has a particle$A$of mass$0.26 \mathrm{~kg}$attached to one end and a particle$B$of mass$0.54 \mathrm{~kg}$attached to the other end. The particle$A$is held at rest on a rough plane inclined at angle$\alpha$to the horizontal, where$\sin \alpha=\frac{5}{13}$. The string is taut and parallel to a line of greatest slope of the plane. The string passes over a small smooth pulley at the top of the plane. Particle$B$hangs at rest vertically below the pulley (see diagram). The coefficient of friction between$A$and the plane is$0.2$. Particle$A$is released and the particles start to move.$\text{(i)}$Find the magnitude of the acceleration of the particles and the tension in the string.$[6]$Particle$A$reaches the pulley$0.4 \mathrm{~s}$after starting to move.$\text{(ii)}$Find the distance moved by each of the particles.$[2]$Question 6 Code: 9709/43/M/J/17/5, Topic: - A particle is projected vertically upwards from a point$O$with a speed of$12 \mathrm{~ms}^{-1}$. Two seconds later a second particle is projected vertically upwards from$O$with a speed of$20 \mathrm{~m} \mathrm{~s}^{-1}.$At time$t \mathrm{~s}$after the second particle is projected, the two particles collide.$\text{(i)}$Find$t$.$[5]\text{(ii)}$Hence find the height above$O$at which the particles collide.$[1]$Question 7 Code: 9709/41/M/J/19/5, Topic: - A particle$P$moves in a straight line from a fixed point$O$. The velocity$v \mathrm{~m} \mathrm{~s}^{-1}$of$P$at time$t \mathrm{~s}$is given by $$v=t^{2}-8 t+12 \quad \text { for } 0 \leqslant t \leqslant 8$$$\text{(i)}$Find the minimum velocity of$P$.$[3]\text{(ii)}$Find the total distance travelled by$P$in the interval$0 \leqslant t \leqslant 8$.$[7]$Question 8 Code: 9709/43/M/J/11/6, Topic: - A lorry of mass$15000 \mathrm{~kg}$climbs a hill of length$500 \mathrm{~m}$at a constant speed. The hill is inclined at$2.5^{\circ}$to the horizontal. The resistance to the lorry's motion is constant and equal to$800 \mathrm{~N}$.$\text{(i)}$Find the work done by the lorry's driving force.$[4]$On its return journey the lorry reaches the top of the hill with speed$20 \mathrm{~m} \mathrm{~s}^{-1}$and continues down the hill with a constant driving force of$2000 \mathrm{~N}$. The resistance to the lorry's motion is again constant and equal to$800 \mathrm{~N}$.$\text{(ii)}$Find the speed of the lorry when it reaches the bottom of the hill.$[5]$Question 9 Code: 9709/43/M/J/17/6, Topic: - A car of mass$1200 \mathrm{~kg}$is travelling along a horizontal road.$\text{(i)}$It is given that there is a constant resistance to motion.$\text{(a)}$The engine of the car is working at$16 \mathrm{~kW}$while the car is travelling at a constant speed of$40 \mathrm{~m} \mathrm{~s}^{-1}$. Find the resistance to motion.$[2]\text{(b)}$The power is now increased to$22.5 \mathrm{~kW}$. Find the acceleration of the car at the instant it is travelling at a speed of$45 \mathrm{~m} \mathrm{~s}^{-1}$.$[3]\text{(ii)}$It is given instead that the resistance to motion of the car is$(590+2 v) \mathrm{N}$when the speed of the car is$v \mathrm{~m} \mathrm{~s}^{-1}$. The car travels at a constant speed with the engine working at$16 \mathrm{~kW}$. Find this speed.$[3]$Question 10 Code: 9709/42/M/J/18/6, Topic: - A particle$P$moves in a straight line passing through a point$O$. At time$t \mathrm{~s}$, the acceleration,$a \mathrm{~m} \mathrm{~s}^{-2}$, of$P$is given by$a=6-0.24 t$. The particle comes to instantaneous rest at time$t=20$.$\text{(i)}$Find the value of$t$at which the particle is again at instantaneous rest.$[5]\text{(ii)}$Find the distance the particle travels between the times of instantaneous rest.$[3]$Question 11 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 12 Code: 9709/42/M/J/18/7, Topic: - As shown in the diagram, a particle$A$of mass$1.6 \mathrm{~kg}$lies on a horizontal plane and a particle$B$of mass$2.4 \mathrm{~kg}$lies on a plane inclined at an angle of$30^{\circ}$to the horizontal. The particles are connected by a light inextensible string which passes over a small smooth pulley$P$fixed at the top of the inclined plane. The distance$A P$is$2.5 \mathrm{~m}$and the distance of$B$from the bottom of the inclined plane is$1 \mathrm{~m}$. There is a barrier at the bottom of the inclined plane preventing any further motion of$B$. The part$B P$of the string is parallel to a line of greatest slope of the inclined plane. The particles are released from rest with both parts of the string taut.$\text{(i)}$Given that both planes are smooth, find the acceleration of$A$and the tension in the string.$[5]\text{(ii)}$It is given instead that the horizontal plane is rough and that the coefficient of friction between$A$and the horizontal plane is$0.2$. The inclined plane is smooth. Find the total distance travelled by$A$.$[9]\$

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