# Leaving Certificate Examination 1968 Honours Applied Mathematics

##### Question 1

Show how to find the centre of gravity of a non-rectangular parallelogram.

A thin uniform square sheet of metal of area square inches weighs ounces. At the corners , , , weights of oz., oz., oz., and oz., respectively are placed. Locate the centre of gravity of the system.

##### Question 2

Establish the formulae and for the velocity and acceleration of a particle moving with uniform speed in a circle or radius .

Assuming that the moon travels in a circular orbit whose radius is cm. and whose centre ie the center of the earth, and completes a revolution every seconds, calculate the ratio of its acceleration towards the centre of the earth to gravity at the earth’s surface.

##### Question 3

Two indentical cars and each feet in length travel with constant velocity feet per second along a straight level road, the front of being feet directly behind the rear of . Immediately on reaching a line drawn across the road perpendicular to the direction of motion of the cars each car decelerates at feet per sec.

(i) Show that crashes into .

(ii) At what distance from the line does the crash occur?

##### Question 4

An aeroplane about to drop a bomb on a small stationary target . on the sea surface descends along a line which is inclined at an angle of to the horizontal and which passes directly over the target at a vertical height of mile. If the aeroplane travels at m.p.h. what duration of time must elapse between the release of the bomb and its impact on the target. (See diagram).

##### Question 5

Show that the work done by a body in coming to rest is the kinetic energy of the moving body. A body of mass lbs. sliding on a rough horizontal plane with initial velocity feet sec. is brought to rest by friction in feet. Calculate the coefficient of friction.

At what average rate in horse-power was work done on the body in bringing it to rest?

##### Question 6

A particle executes simple harmonic motion with a frequency of vibrations per second. The amplitude is millimetres.

Find (i) the maximum speed of the particle,

(ii) the maximum acceleration of the particle.

If the cutting-head of an electric razor has mass grams and oscillates with simple harmonic motion at a frequency of vibrations per second, and with an amplitude of millimetres, calculate the maximum value of the driving force.

##### Question 7

A train is rounding a circular curve of radius yards. A bob hangs at one end of a light string which is fixed in the ceiling of a carriage and which is inclined at an angle of to the vertical. Estimate the speed of the train.

##### Question 8

A fire-engine raises gallons of water per minute from a well feet deep and then pumps it as a jet issuing with a nozzle velocity of feet per sec. At what horsepower is the engine working?

##### Question 9

The density of lead is gram cm. and that of mercury is grams cm. Explain why a block of lead will float on mercury.

A rectangular uniform block of wood floats on sea water with of its volume above the water-line. What is the volume of the smallest rectangular block of the wood which will float without being totally submerged while supporting a weight of lb. placed vertically above its centre of gravity?

(Take $1.03 as the specific gravity of sea-water.)

**Citation:**

**Citation:**

State Examinations Commission (2018). *State Examination Commission*. Accessed at: https://www.examinations.ie/?l=en&mc=au&sc=ru

Malone, D and Murray, H. (2016). *Archive of Maths State Exams Papers*. Accessed at: http://archive.maths.nuim.ie/staff/dmalone/StateExamPapers/

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