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2012. M
Coimisiún na Scrúduithe Stáit
State Examinations Commission
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LEAVING CERTIFICATE EXAMINATION, 2012
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APPLIED MATHEMATICS − ORDINARY LEVEL
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FRIDAY, 22 JUNE − MORNING 9.30 to 12.
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Six questions to be answered. All questions carry equal marks.
A Formulae and Tables booklet may be obtained from the Superintendent.
Take the value of g to be 10 m s –^.
→ i and
→ j are unit perpendicular vectors in the horizontal and vertical directions,
respectively, or eastwards and northwards, respectively, as appropriate to the question.
Marks may be lost if necessary work is not clearly shown.
1. A car travels along a straight level road.
It passes a point P with a speed of 8 m s−^1 and accelerates uniformly for 12 seconds to a speed of 32 m s−^1. It then travels at a constant speed of 32 m s−^1 for 7 seconds. Finally the car decelerates uniformly from 32 m s−^1 to rest at a point Q. The car travels 128 metres while decelerating.
Find (i) the acceleration
(ii) the deceleration
(iii) | PQ |, the distance from P to Q
(iv) the speed of the car when it is 72 m from Q.
2. Ship A is positioned 80 km south of ship B. A is moving north-east at a constant speed of 30 2 km h−^1.
B is moving due west at a constant speed of 15 kmh−^1.
Find (i) the velocity of A in terms of
→ i and
→ j
(ii) the velocity of B in terms of
→ i and
→ j
(iii) the velocity of A relative to B in terms of
→ i and
→ j
(iv) the shortest distance between A and B in the subsequent motion.
15 km h–1^ B
A
80 km 30 2 km h^
4. (a) Two particles of masses 2 kg and 3 kg are connected by a taut, light, inextensible string which passes over a smooth light pulley.
The system is released from rest.
Find (i) the common acceleration of the particles
(ii) the tension in the string.
(b) Masses of 9 kg and 12 kg are connected by a taut, light, inextensible string which passes over a smooth light pulley as shown in the diagram.
The 9 kg mass lies on a rough horizontal plane and the coefficient of friction between the 9 kg mass and the plane is 3
The 12 kg mass lies on a smooth plane which is inclined at 30° to the horizontal.
The system is released from rest.
(i) Show on separate diagrams the forces acting on each particle. (ii) Find the common acceleration of the masses. (iii) Find the tension in the string.
5. A smooth sphere A, of mass 5 kg, collides directly with another smooth sphere B, of mass 2 kg, on a smooth horizontal table.
A and B are moving in the same direction with speeds of 4 m s−^1 and 1 m s−^1 respectively.
The coefficient of restitution for the collision is
Find (i) the speed of A and the speed of B after the collision (ii) the loss in kinetic energy due to the collision (iii) the magnitude of the impulse imparted to A due to the collision.
12 kg
9 kg
30°
2 kg 3 kg
A 5 kg B 2 kg
4 m s−^1 1 m s−^1
6. (a) Particles of weight 4 N, 7 N, 3 N and 5 N are placed at the points
( p , 2 ,^ ) ( −6, 1 ,^ ) ( 9,^ q ) and^ ( 12, 13), respectively.
The co-ordinates of the centre of gravity of the system are ( p , q ).
Find (i) the value of p
(ii) the value of q.
(b) A triangular lamina with vertices A , B and C has the portion inside its incircle (the circle that touches the three sides of the triangle) removed. D is the centre of the incircle. The co-ordinates of the points
are A ( 0, 0 ,) B ( 0, 27 ,) C ( 36, 0)
and D ( 9, 9).
Find the co-ordinates of the centre of gravity of the remaining lamina.
7. A uniform rod, [ AB ], of length 4 m and weight 80 N is smoothly hinged at end A to a horizontal floor.
One end of a light inelastic string is attached to B and the other end of the string is attached to a horizontal ceiling.
The string makes an angle of 60° with the ceiling and the rod makes an angle of 30° with the floor, as shown in the diagram.
The rod is in equilibrium.
(i) Show on a diagram all the forces acting on the rod [ AB ].
(ii) Write down the two equations that arise from resolving the forces horizontally and vertically.
(iii) Write down the equation that arises from taking moments about the point A.
(iv) Find the tension in the string.
(v) Find the magnitude of the reaction at the hinge, A.
A
B
C
D
A
B
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