Download Leaving Certificate Examination - Applied Mathematics Higher Level, 2004 and more Exams Applied Mathematics in PDF only on Docsity!
M
Coimisi˙n na Scr˙duithe St·it
State Examinations Commission
__________________________
LEAVING CERTIFICATE EXAMINATION, 2004
___________________________
APPLIED MATHEMATICS − HIGHER LEVEL
___________________________
FRIDAY, 25 JUNE − AFTERNOON, 2.00 to 4.
____________________________
Six questions to be answered. All questions carry equal marks. Mathematics Tables 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 ball is thrown vertically upwards with an initial velocity of 20 m/s. One second later, another ball is thrown vertically upwards from the same point with an initial velocity of u m/s. The balls collide after a further 2 seconds.
(i) Show that u = 17.75.
(ii) Find the distance travelled by each ball before the collision, giving your answers correct to the nearest metre.
(b) A car of mass 1200 kg tows a caravan of mass 900 kg first along a horizontal road with acceleration f and then up an incline α to the horizontal road at uniform speed. The force exerted by the engine is 2700 N. Friction and air resistance amount to 150 N on the car and 240 N on the caravan.
Calculate
(i) the acceleration, f , of the car along the horizontal road
(ii) the value of α , to the nearest degree.
2. (a) A bird flies at a uniform speed of 22 m/s. It wishes to fly to its nest which is 250 m due north of its present position. There is a wind blowing from the southeast at 18 m/s.
Find (i) the direction, to the nearest degree, in which the bird must fly to reach its nest
(ii) the time taken to reach the nest, correct to two decimal places.
(b) At time t = 0, two particles P and Q are set in motion. At time t = 0, Q has position vector i j
r r 20 + 40 metres relative to P. P has a constant velocity of i j
r r 3 + 5 m/s and Q has a constant velocity of i j
r r 4 − 3 m/s.
Find
(i) the velocity of Q relative to P
(ii) the shortest distance between P and Q, to the nearest metre
(iii) the time when P and Q are closest together, correct to one decimal place.
5. (a) A smooth sphere P, of mass 3 m , moving with speed u , collides directly with a smooth sphere Q, of mass 5 m , which is at rest. The coefficient of restitution for the collision is e.
Find (i) the speed, in terms of u and e , of each sphere after the collision
(ii) the condition to be satisified by e in order that the spheres move in opposite directions after the collision.
(b) A smooth sphere A, of mass m , moving with speed u , collides with an identical smooth sphere B which is at rest. The direction of motion of A, before impact, makes an angle 30∫ with the line of centres at impact. After impact the direction of A makes an angle θ with the line of centres, where 0∫ ≤ θ < 90∫. The coefficient of restitution between the spheres is e. The speeds of A and B immediately after impact are equal.
(i) Calculate the value of θ.
(ii) Find e.
6. (a) A particle can move on the smooth outer surface of a fixed sphere of radius r. The particle is released from rest on the
smooth surface of the sphere at a height 5
4 r
above the horizontal plane through the centre o of the sphere.
Find, in terms of r , the height above this plane at which the particle leaves the sphere.
(b) A particle moves in a straight line such that its displacement from a fixed point o at time t is given by x = a cos (ω t −β) where a , ω and β are positive constants.
(i) Show that the motion of the particle is simple harmonic motion.
The period of the motion is 16 seconds. At time t = 4 s, the particle is 12 m from o and 4 s later the particle is on the other side of o and at a distance of 5 m from o.
(ii) Find a , ω and β.
7. (a) A uniform rod [ pq ], of length 2 l and weight W is in equilibrium with the end p on a rough horizontal floor and the end q against a smooth vertical wall. The rod makes an angle 45˚ with the horizontal and is in a vertical plane which is perpendicular to the wall. The coefficient of friction between the floor and the
rod is
Find the distance from p of the highest point of the rod, in terms of l , from which a particle of weight W can be attached without disturbing equilibrium.
(b) A uniform rod [ ab ], of length h and weight W is in equilibrium with the end a resting on a rough horizontal plane. The rod is maintained in equilibrium by means of a light inextensible string which passes over a small smooth peg, of negligible diameter, at c , with one end of the string attached to b and with a weight Q attached to the other end of the string. The peg at c is at a height 2 h vertically above a and |∠ abc | = 90∫.
Find (i) Q in terms of W (ii) the magnitude of the force acting on the peg at c , in terms of W and correct to two decimal places.
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 21 mr^2.
(b) A smooth pulley wheel has a mass of 3 kg and a radius of 0.3 m. One end of a light inextensible rope is attached to a point p on the rim of the wheel. A particle of mass 0.2 kg attached to the other end of the rope hangs freely. The axis of rotation of the wheel is horizontal, perpendicular to the wheel, and passes through the centre of the wheel. The particle is released from rest and moves vertically downwards.
When the particle has acquired a speed of 1.2 m/s, find
(i) the kinetic energy gained by the wheel (ii) the distance descended by the particle, correct to two decimal places.
Blank Page
Blank Page
