Leaving Certificate Examination 1953 Honours Applied Mathematics
Question 1
The ends of a light inextensible string 
feet long are attached to two pegs which are 
feet apart and in the same horizontal line. A mass of 
lb. is suspended from the string at a point which is 
feet from one peg and 
feet from the other. Find the tension in each part of the string.
Question 2
A block weighing 
lb. can just rest without slipping on a rough plane when the plane is inclined at an angle of 
to the horizontal. Find
(a) the magnitude of the least force in the horizontal direction,
(b) the magnitude and direction of the least force, that will keep the block from slipping when the plane is inclined at an angle of 
to the horizontal.
Question 3
An engine weighing 
tons is pulling a train of weight 
tons down an incline of 
in 
. The speed of the train is 
m.p.h. and it is accelerating at the rate of 
ft. per sec
. If the frictional resistances to motion are equivalent to lb. wt. per ton, find the horse-power at which the engine is working.
Question 4
A pile driver of mass 
cwt. falls freely from rest through a distance of 
feet and strikes a pile of mass cwt. The pile driver and the pile move together as a single body after the impact and the pile is driven 
inches into the ground. Find the average resistance of the ground, in tons weight.
Question 5
What is meant by the velocity of one body relative to another? Explain with the aid of a diagram how the velocity of 
relative to 
may be found if the velocities of and are known.
To a person on a ship travelling due East at m.p.h., another ship two miles due South appears to be travelling at 
m.p.h. in a direction 
West of North. Find the velocity of the second ship in magnitude and direction.
What is the distance between the ships when they are nearest to each other?
Question 6
If 
if the greatest height reached by a projectile, prove that
where 
is the initial velocity, 
the angle of projection, and 
the total time of flight.
Two vertical posts, each of height 
feet, stand 
feet apart on a horizontal plane. A particle is projected from a point on the plane at an angle of 
with the horizontal, and just clears the top of each post. If 
is the greatest height reached by the particle, and v
v^2=\frac{5}{2}gl
x=a \sin \omega t – b \cos \omega t.
Show that the motion is Simple Harmonic, and find an expression for (i) the amplitude, (ii) the maximum velocity.
Find also an expression for the least time taken by the particle to travel a distance 
from its mean position.
Question 9
A triangular lamina 
is immersed in a vertical position in water with its vertex at the surface and its base 
parallel to the surface. The base is inches in length, and the height of the triangle is inches. Find the total thrust of the water on .
, 
are two points on 
, 
respectively such that the straight line 
is horizontal. If the thrust on 
is one-eight of the thrust on , find the depth of below the surface.
[One cubic foot of water weight 
lb.]
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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