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### OTIA - AS MATHEMATICS

#### Cambridge International AS and A Level

 Name of student Date Adm. number Year/grade 11 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 4 6 7 6 7 6 7 7 6 11 10 9 86
Score

Get Mathematics 9709 Topical Questions (2010-2021) for $14.5 per Subject. Attempt all the 12 questions Question 1 A particle of mass$8 \mathrm{~kg}$is pulled at a constant speed a distance of$20 \mathrm{~m}$up a rough plane inclined at an angle of$30^{\circ}$to the horizontal by a force acting along a line of greatest slope.$\text{(i)}$Find the change in gravitational potential energy of the particle.$\text{(ii)}$The total work done against gravity and friction is$1146 \mathrm{~J}$. Find the frictional force acting on the particle.$$Question 2 Coplanar forces of magnitudes$12 \mathrm{~N}, 24 \mathrm{~N}$and$30 \mathrm{~N}$act at a point in the directions shown in the diagram.$\text{(i)}$Find the components of the resultant of the three forces in the$x$-direction and in the$y$-direction.$$Component in$x$-direction. Component in$y$-direction.$\text{(ii)}$Hence find the direction of the resultant.$$Question 3 Four coplanar forces act at a point. The magnitudes of the forces are$5 \mathrm{~N}, 4 \mathrm{~N}, 3 \mathrm{~N}$and$7 \mathrm{~N}$, and the directions in which the forces act are shown in the diagram. Find the magnitude and direction of the resultant of the four forces.$$Question 4 A particle of mass$8 \mathrm{~kg}$is projected with a speed of$5 \mathrm{~m} \mathrm{~s}^{-1}$up a line of greatest slope of a rough plane inclined at an angle$\alpha$to the horizontal, where$\sin \alpha=\frac{5}{13}$. The motion of the particle is resisted by a constant frictional force of magnitude$15 \mathrm{~N}$. The particle comes to instantaneous rest after travelling a distance$x \mathrm{~m}$up the plane.$\text{(i)}$Express the change in gravitational potential energy of the particle in terms of$x$.$\text{(ii)}$Use an energy method to find$x$.$$Question 5 A car of mass$1230 \mathrm{~kg}$increases its speed from$4 \mathrm{~m} \mathrm{~s}^{-1}$to$21 \mathrm{~m} \mathrm{~s}^{-1}$in$24.5 \mathrm{~s}$. The table below shows corresponding values of time$t \mathrm{~s}$and speed$v \mathrm{~m} \mathrm{~s}^{-1}$. $$\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}$$$\text{(i)}$Using the values in the table, find the average acceleration of the car for$0< t <0.5$and for$16.3< t <24.5.$While the car is increasing its speed the power output of its engine is constant and equal to$P \mathrm{~W}$, and the resistance to the car's motion is constant and equal to$R \mathrm{~N}$.$\text{(ii)}$Assuming that the values obtained in part$\text{(i)}$are approximately equal to the accelerations at$v=5$and at$v=20$, find approximations for$P$and$R$.$$Question 6 Coplanar forces of magnitudes$50 \mathrm{~N}, 48 \mathrm{~N}, 14 \mathrm{~N}$and$P \mathrm{~N}$act at a point in the directions shown in the diagram. The system is in equilibrium. Given that tan$\alpha=\frac{7}{24}$, find the values of$P$and$\theta$.$$Question 7 Question 8 A particle$P$moves in a straight line$A B C D$with constant deceleration. The velocities of$P$at$A, B$and$C$are$20 \mathrm{~m} \mathrm{~s}^{-1}, 12 \mathrm{~m} \mathrm{~s}^{-1}$and$6 \mathrm{~m} \mathrm{~s}^{-1}$respectively.$\text{(i)}$Find the ratio of distances$A B: B C$.$\text{(ii)}$The particle comes to rest at$D$. Given that the distance$A D$is$80 \mathrm{~m}$, find the distance$B C$.$$Question 9 A particle of mass$3 \mathrm{~kg}$is on a rough plane inclined at an angle of$20^{\circ}$to the horizontal. A force of magnitude$P \mathrm{~N}$acting parallel to a line of greatest slope of the plane is used to keep the particle in equilibrium. The coefficient of friction between the particle and the plane is$0.35$. Show that the least possible value of$P$is$0.394$, correct to 3 significant figures, and find the greatest possible value of$P$.$$Question 10 Question 11 Two particles$A$and$B$start to move at the same instant from a point$O$. The particles move in the same direction along the same straight line. The acceleration of$A$at time$t \mathrm{~s}$after starting to move is$a \mathrm{~m} \mathrm{~s}^{-2}$, where$a=0.05-0.0002 t$.$\text{(i)}$Find$A$'s velocity when$t=200$and when$t=500$.$B$moves with constant acceleration for the first$200 \mathrm{~s}$and has the same velocity as$A$when$t=200. B$moves with constant retardation from$t=200$to$t=500$and has the same velocity as$A$when$t=500$.$\text{(ii)}$Find the distance between$A$and$B$when$t=500$.$$Question 12 Two rectangular boxes$A$and$B$are of identical size. The boxes are at rest on a rough horizontal floor with$A$on top of$B$. Box$A$has mass$200 \mathrm{~kg}$and box$B$has mass$250 \mathrm{~kg}$. A horizontal force of magnitude$P \mathrm{~N}$is applied to$B$(see diagram). The boxes remain at rest if$P \leqslant 3150$and start to move if$P>3150$.$\text{(i)}$Find the coefficient of friction between$B$and the floor.$$The coefficient of friction between the two boxes is$0.2$. Given that$P>3150$and that no sliding takes place between the boxes,$\text{(ii)}$show that the acceleration of the boxes is not greater than$2 \mathrm{~m} \mathrm{~s}^{-2}$,$\text{(iii)}$find the maximum possible value of$P$.$\$

Worked solutions: P1, P3 & P6 (S1)

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