Leaving Certificate Examination 1943 Honours Applied Mathematics
Question 1
Two ships $A$ and $B$ are $10$ miles apart, $A$ being due west of $B$ and steaming due East at $12$ miles per hour. $B$ is steaming due North at $18$ miles per hour. When will they be nearest one another and what will be their distance apart at that time?
Question 2
A particle slides down a smooth inclined plane $12$ feet long in $4$ seconds : find its acceleration and the inclination of the plane to the horizontal.
How far from the bottom should a second particle be started on the plane one second after the first had started from the top so that the two particles will reach the bottom together?
Question 3
On a rectangular sheet of iron, $12$ inches by $9$ inches, marks are made on two adjacent edges at a distance of $3$ inches from one corner. These marks are joined by a straight line and the sheet folded down along this line. Find the centre of gravity of the sheet so folded.
Question 4
Describe any method by which the acceleration due to gravity may be determined experimentally. Mention the causes of error likely to affect the result.
Question 5
An object of $18$ lb. wt. is placed upon a sheet of glass resting on a horizontal table and a string attached to the object passes horizontally over a pulley at the end of the table. When a weight of $2$ lb. is hung on the free end of the string, the object is drawn from rest through a distance of $6$ feet in $3$ seconds. Find the coefficient of friction between the object and the glass and also the tension in the string.
Question 6
A uniform iron bar $AB$, $5$ ft. long, weighs $10$ lb., has a $5$ lb. weight fixed to it at a point $1$ ft. from $A$, and is supported by means of cords attached to its ends from a peg $C$. The cord $CA$ is $3\frac{1}{2}$ ft. long and the cord $CB$ is $4$ ft. long. Find the inclination of the bar to the horizontal in the position of equilibrium and find the tensions in the cords.
Question 7
The height of a lift from the ground $t$ seconds after starting is $h$ feet, given by the following table :-
| $t$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ | $11$ | $12$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| $h$ | $1.6$ | $4.5$ | $8.5$ | $14.4$ | $22.7$ | $33.3$ | $44$ | $54.8$ | $63$ | $69$ | $72.5$ | $74$ |
Draw the distance-time graph for the lift. Find the greatest velocity of the lift.
Draw a rough velocity-time graph, and use it to give a short account of the variation in the acceleration throughout the journey.
Question 8
An inverted cone, of angle $60^\circ$, has a smooth inside surface and rests with its axis vertical. A smooth particle moves in a horizontal circle on the inside of the cone with a uniform speed of $8$ feet per second. Find the distance of the particle from the vertex of the cone.
Question 9
A particle is moving in a straight line with Simple Harmonic Motion. When it is $5$ ft. and $12$ ft. from the centre of its path, its speed are $24$ ft./sec. and $10$ ft./sec. respectively. Find the period and the amplitude of the motion. Find also the maximum speed and the maximum acceleration of the particle.
Citation:
State Examinations Commission (2023). State Examination Commission. Accessed at: https://www.examinations.ie/?l=en&mc=au&sc=ru
Malone, D and Murray, H. (2023). Archive of Maths State Exams Papers. Accessed at: http://archive.maths.nuim.ie/staff/dmalone/StateExamPapers/
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