Leaving Certificate Examination 1951 Honours Applied Mathematics
Question 1
Two pegs and
are at the same height above the ground, and are
feet apart. One end of an inextensible string
feet in length is attached to
, and the other end is attached to
. A
lb. weight is suspended from the string at a point
feet from
and
feet from
. Find the tension in each part of the string.
Question 2
Two roads cross at right angles at . Two men,
and
, are walking towards
.
is walking at
m.p.h. on one road,
at
m.p.h. on the other road. When
is
yards from
,
is
yards from
. Find the velocity of
relative to
.
Find the distance of from
and from
, when
and
are nearest to each other.
Question 3
Prove that the centre of gravity of any triangular lamina is the same as that of three equal masses situated at the vertices of the triangle.
is a quadrilateral lamina. A mass equal to one-third the mass of the lamina is placed at the intersection of the diagonals
and
. Show that the centre of gravity of the lamina and the mass together is the same as that of four equal masses situated at the vertices of the quadrilateral.
Question 4
Two masses of lb. and
lb. connected by a light inextensible string
feet long, are lying on a smooth horizontal table
feet high. The
lb. mass is at the edge of the table and the other mass is
feet away in a direction perpendicular to the edge. If the
lb. mass is pushed gently over the edge of the table, find how long it takes to reach the ground, and how much longer the
lb. mass takes to reach the edge of the table.
Find also the change in kinetic energy when the lb. mass is jerked into motion.
Question 5
A train ascending an incline of in
accelerates uniformly from
m.p.h. to
m.p.h. in a distance of
miles. Find this acceleration. If the train and the engine together weigh
tons, and if the frictional resistances to motion are equivalent to
lb. wt. per ton, find the horse-power at which the engine was working when the speed of the train was
m.p.h.
Question 6
A point is describing a circle, centre
, with uniform speed. If
is the foot of the perpendicular from
to a fixed diameter, show that
moves with simple harmonic motion.
When is
feet from
its velocity is
feet per second, and when it is
feet from
its velocity is
feet per second towards
. Find how long
takes to reach
from the latter position.
Question 7
A gun is fixed at a point at the top a vertical cliff
feet high.
is the foot of the perpendicular from
to the sea. A ship is travelling away from the cliff at
feet per second along a straight line that passes through
. When the ship is
feet from
a shell is fired from the gun at an angle of
and strikes the ship. Find the initial velocity of the shell.
Question 8
A mass of one ounce attached to a fixed point by an inextensible string feet long, describes a horizontal circle at a uniform rate of
revolutions per minute. Find the tension in the string and the vertical distance from the fixed point to the plane of the circle.
What vertical distance will the mass rise if its speed is increased to revolutions per minute?
Question 9
Prove that if a plane surface is immersed in a liquid the total thrust on it due to the liquid is equal to the area of the surface multiplied by the pressure at its center of gravity.
A triangular lamina is totally immersed in water so that the plane of the lamina makes an angle of
with the horizontal plane. The base
is
inches in length and the height of the triangle is
inches. Find the total thrust of the water of
(i) when its vertex is at the surface of the water and its base
is horizontal;
(ii) when its base is at the surface of the water.
[ cubic foot of water weighs
lbs.]
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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