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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, P43 Start time Duration Stop time

Qtn No. 1 2 3 4 5 6 7 Total
Marks 3 7 6 6 6 9 12 49
Score

Get Mathematics 9709 Topical Questions (2010-2021) $14.5 per Subject. Attempt all the 7 questions Question 1 Code: 9709/41/M/J/21/1, Topic: - A winch operates by means of a force applied by a rope. The winch is used to pull a load of mass$50 \mathrm{~kg}$up a line of greatest slope of a plane inclined at$60^{\circ}$to the horizontal. The winch pulls the load a distance of$5 \mathrm{~m}$up the plane at constant speed. There is a constant resistance to motion of$100 \mathrm{~N}$. Find the work done by the winch.$[3]$Question 2 Code: 9709/43/M/J/10/3, Topic: - A load is pulled along a horizontal straight track, from$A$to$B$, by a force of magnitude$P \mathrm{~N}$which acts at an angle of$30^{\circ}$upwards from the horizontal. The distance$A B$is$80 \mathrm{~m}$. The speed of the load is constant and equal to$1.2 \mathrm{~m} \mathrm{~s}^{-1}$as it moves from$A$to the mid-point$M$of$A B$.$\text{(i)}$For the motion from$A$to$M$the value of$P$is$25.$Calculate the work done by the force as the load moves from$A$to$M$.$[2]$The speed of the load increases from$1.2 \mathrm{~m} \mathrm{~s}^{-1}$as it moves from$M$towards$B$. For the motion from$M$to$B$the value of$P$is 50 and the work done against resistance is the same as that for the motion from$A$to$M$. The mass of the load is$35 \mathrm{~kg}$.$\text{(ii)}$Find the gain in kinetic energy of the load as it moves from$M$to$B$and hence find the speed with which it reaches$B$.$[5]$Question 3 Code: 9709/43/O/N/16/3, Topic: - Particles$P$and$Q$, of masses$7 \mathrm{~kg}$and$3 \mathrm{~kg}$respectively, are attached to the two ends of a light inextensible string. The string passes over two small smooth pulleys attached to the two ends of a horizontal table. The two particles hang vertically below the two pulleys. The two particles are both initially at rest,$0.5 \mathrm{~m}$below the level of the table, and$0.4 \mathrm{~m}$above the horizontal floor (see diagram).$\text{(i)}$Find the acceleration of the particles and the speed of$P$immediately before it reaches the floor.$[4]\text{(ii)}$Determine whether$Q$comes to instantaneous rest before it reaches the pulley directly above it.$[2]$Question 4 Code: 9709/41/O/N/15/4, Topic: - Blocks$P$and$Q$, of mass$m \mathrm{~kg}$and$5 \mathrm{~kg}$respectively, are attached to the ends of a light inextensible string. The string passes over a small smooth pulley which is fixed at the top of a rough plane inclined at$35^{\circ}$to the horizontal. Block$P$is at rest on the plane and block$Q$hangs vertically below the pulley (see diagram). The coefficient of friction between block$P$and the plane is$0.2.$Find the set of values of$m$for which the two blocks remain at rest.$[6]$Question 5 Code: 9709/41/O/N/20/4, Topic: - A particle$P$moves in a straight line. It starts from rest at a point$O$on the line and at time$t \mathrm{~s}$after leaving$O$it has acceleration$a \mathrm{~m} \mathrm{~s}^{-2}$, where$a=6 t-18$. Find the distance$P$moves before it comes to instantaneous rest.$[6]$Question 6 Code: 9709/43/M/J/20/6, Topic: - A particle travels in a straight line$P Q$. The velocity of the particle$t \mathrm{~s}$after leaving$P$is$v \mathrm{~m} \mathrm{~s}^{-1}$, where $$v=4.5+4 t-0.5 t^{2}$$$\text{(a)}$Find the velocity of the particle at the instant when its acceleration is zero.$[3]$The particle comes to instantaneous rest at$Q$.$\text{(b)}$Find the distance$P Q$.$[6]$Question 7 Code: 9709/41/O/N/20/7, Topic: - Three points$A, B$and$C$lie on a line of greatest slope of a plane inclined at an angle of$30^{\circ}$to the horizontal, with$A B=1 \mathrm{~m}$and$B C=1 \mathrm{~m}$, as shown in the diagram. A particle of mass$0.2 \mathrm{~kg}$is released from rest at$A$and slides down the plane. The part of the plane from$A$to$B$is smooth. The part of the plane from$B$to$C$is rough, with coefficient of friction$\mu$between the plane and the particle.$\text{(a)}$Given that$\mu=\frac{1}{2} \sqrt{3}$, find the speed of the particle at$C$.$[8]\text{(b)}$Given instead that the particle comes to rest at$C$, find the exact value of$\mu$.$[4]\$

Worked solutions: P1, P3 & P6 (S1)

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