Leaving Cert Applied Maths Higher Level 1978
Question 1
A driver starts from rest at and travels with a uniform acceleration of
m/s
for
seconds. He continues with uniform velocity for
seconds, and then decelerates uniformly to rest at
in a further
seconds. Express the distance
in terms of
and
.
Another driver can accelerate at m/s
and can decelerate at
m/s
. Find, in terms of
, the least time in which this driver can cover the distance
from rest to rest
(i) subject to a speed limit of m/s,
(ii) subject to a speed limit of m/s.
Question 2
A plane is inclined at an angle to the horizontal. A particle is projected up the plane with initial velocity
at an angle
to the plane. The plane of projection is vertical and contains the line of greatest slope.
(i) Write down the displacement and velocity of the particle parallel and perpendicular to the plane at time .
(ii) Show that the time taken by the particle to reach its maximum perpendicular height above the plane is half the time of flight up the plane.
(iii) When the particle is at its maximum perpendicular height above the plane, the distance travelled parallel to the plane is of the range up the plane. Show that in that case
.
Question 3
(i) Two vectors and
are at right angles. Write down the condition satisfied by the scalars
.
(ii) Two smooth spheres of masses and
and velocities
and
, respectively, collide as shown in the diagram, where
. The sphere of mass
is deflected through an angle of
by the collision. If the coefficient of restitution is
, show that
.
(iii) Find the direction of motion of the other sphere after the collision.
Question 4
A body of mass lies on a smooth horizontal table. It is connected by means of a light string passing over a smooth light pulley at the edge of the table, to a second smooth pulley of mass
hanging freely. Over this second pulley passes another light string carrying masses of
and
(see diagram).
(i) Show in separate diagrams the forces acting on each of the masses.
(ii) Write down the equations of motion involving the tensions and
in the strings, the common acceleration
of the
and
masses and the common acceleration
of the
and
masses relative to the
mass.
(iii) Show that .
Question 5
Two uniform rods and
of equal length and of masses
kg and
kg, respectively, are freely hinged at
.
and
are in a vertical plane and the ends
and
are on a rough horizontal plane. The coefficient of friction between each rod and the plane is the same.
(i) Find the normal reactions at and
.
(ii) The angle is increased until one of the rods begins to slip. Show that slipping will first occur at
rather than at
.
(iii) Find the least value of the coefficient of friction if slipping has not occurred before .
Question 6
Two small smooth rings and
, each of mass
are threaded on a fixed smooth horizontal wire. They are connected by means of two light inextensible strings
and
, each of length
metres, to a particle of mass
hanging freely at
.
,
,
are in the same vertical plane. The system is released from rest with the angles
.
(i) If travels a horizontal distance
while
falls a vertical distance
, show from geometry that
.
(ii) By differentiating find in terms of
and
, where
means
, and using the conservation of energy find
in terms of
.
(iii) Show that the velocity of is
where
.
Question 7
(a) Prove that the moment of inertia of a uniform circular disc about a perpendicular axis through its centre is , where
is the radius of the disc and
is its mass.
(b) A light string is wound around the rim of a uniform disc of radius and mass
. One end of the string is attached to the rim of the disc and the other end is attached to a fixed point above the disc, with the plane of the disc vertical (see diagram). When the disc is released from rest it falls vertically and the string unwinds.
If the disc falls a distance while it turns through an angle
, show that
and deduce that
. where
is the angular velocity of the disc. (
means
,
means
)
Using the principle of angular momentum, find the tension in the string and the vertical acceleration of the disc.
Question 8
Solve the following differential equations:
(i) if
when
(ii) if
and
when
.
(iii) A particle of mass is acted on by a force
directed away from a fixed point
, where
is the distance of the particle from
. The particle starts from rest at a distance
from
. Show that the velocity of the particle tends to a limit
.
Question 9
(a) Two particles and
are moving along two perpendicular lines towards a point
with constant velocities of
m/s and
m/s respectively. When
is
metres from
,
is
metres from
. Find the distance between them when they are nearest to each other.
(b) State the Principle of Archimedes.
(c) A uniform circular cylinder of height and relative density
floats with its axis vertical in a liquid of relative density
.
(i) Find the length of the axis of the cylinder immersed.
(ii) The cylinder is depressed vertically a further small distance and release. Show that it will preform simple harmonic motion, and find the period.
Question 10
(a) A portion in the shape of an equilateral triangle is removed from a circular lamina of radius . A vertex of the triangle was at the centre of the lamina and the sides of the triangle are of length
. Find the position of the centre of gravity of the remainder.
(b) A train of mass tonnes is maintaining a steady speed of
m/s up an incline of
in
against frictional forces amounting to
kN. Calculate the power at which the engine is working. (
tonne =
kg).
(c) A corner on a level track has a radius of m. Calculate the maximum speed at which a cyclist could take the corner if the coefficient of friction were
.
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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/