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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 − HIGHER 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 9·8 m s−^2. Marks may be lost if necessary work is not clearly shown.
1. (a) A particle falls from rest from a point P. When it has fallen a distance 19·6 m a second particle is projected vertically downwards from P with initial velocity 39·2 m s−^1. The particles collide at a distance d from P.
Find the value of d.
(b) A car, starts from rest at A , and accelerates uniformly at 1 m s−^2 along a straight level road towards B , where AB = 1914 m. When the car reaches its maximum speed of 32 m s−^1 , it continues at this speed for the rest of the journey.
At the same time as the car starts from A a bus passes B travelling towards A with a constant speed of 36 m s−^1. Twelve seconds later the bus starts to decelerate uniformly at 0·75 m s−^2.
(i) The car and the bus meet after t seconds. Find the value of t.
(ii) Find the distance between the car and the bus after 48 seconds.
2. (a) Rain is falling with a speed of 25 m s−^1 at an angle of 20 °^ to the vertical. A car is travelling along a horizontal road into the rain. The windscreen of the car makes an angle of 32 ° with the vertical. The car is travelling at 20 m s−^1. Find the angle at which the rain appears to strike the windscreen.
(b) At noon ship A is 50 km north of ship B. Ship A is travelling southwest at 24 2 kmh−^1. Ship B is travelling due west at 17 kmh−^1.
(i) Find the magnitude and direction of the velocity of B relative to A.
A and B can exchange signals when they are not more than 20 km apart.
(ii) At what time can they begin to exchange signals?
(iii) How long can they continue to exchange signals?
32°
25 m s−^1
5. (a) Three smooth spheres, A, B and C, of mass 3 m , 2 m and m lie at rest on a smooth horizontal table with their centres in a straight line. Sphere A is projected towards B with speed 5 m s −^1. Sphere A collides directly with B and then B collides directly with C.
The coefficient of restitution between the spheres is e.
Show that if 2
e > there will be no further collisions.
(b) A smooth sphere P collides with an identical smooth sphere Q which is at rest. The velocity of P before impact makes an
angle α with the line of centres at impact,
where 0 ° ≤ α< 90 .°
The velocity of P is deflected through an angle θ by the collision.
The coefficient of restitution between the spheres is
Show that (^2) 2 tan tan. 1 3 tan
6. (a) A particle of mass 0·5 kg is suspended from a fixed point P by a spring which executes simple harmonic motion with amplitude 0·2 m. The period of the motion is 2 seconds.
Find (i) the maximum acceleration of the particle
(ii) the greatest force exerted by the spring correct to one place of decimals.
(b) A particle of mass m kg lies on the top of a smooth fixed sphere of radius 30 cm.
The particle is slightly displaced and slides down the sphere. The particle leaves the sphere at B.
(i) Find the speed of the particle at B.
(ii) The horizontal distance, in metres, of the particle from the centre of the
sphere t seconds after it has left the surface of the sphere is
Find the value of k correct to two places of decimals.
B
P^ θ Q
7. (a) A uniform wire ABC is bent at right angles at B. When it is suspended from B the parts AB and BC make angles of 30˚ and 60˚ respectively with the vertical.
If the mass per unit length of the wire is m and AB = hBC find the value of h.
(b) Two rough rings of equal weight W are a distance d apart on a horizontal rod. A light smooth inelastic string of length 2 connects the rings. Another ring of weight 2 W slides on the string. The coefficient of friction between the rough rings and the rod is μ.
Show that the system remains at rest if 2
d
μ μ
8. (a) Prove that the moment of inertia of a uniform circular disc, of mass m and
radius r , about an axis through its centre perpendicular to its plane is 2
m r
(b) A string is wrapped around a smooth pulley wheel of radius r. A particle of mass m is attached to the string.
The axis of rotation of the wheel is horizontal, perpendicular to the wheel, and passes through the centre of the wheel.
The moment of inertia of the wheel about the axis is I.
The particle is released from rest and moves vertically downwards.
(i) Find, in terms of I , m and r , the tension in the string.
(ii) If the acceleration of the particle is , 5
g find the mass of the pulley
wheel in terms of m.
A
B
C
d
m
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