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30Aug2019

Leaving Cert Applied Maths Higher Level 1982

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Question 1

a) A car A passes a point p on a straight road at a constant speed of 10 m/s. At the same time another car B starts from rest at p with uniform acceleration 2.5 m/s^2.

i) When and how far from p with overtake A?

ii) If B ceases to accelerate on overtaking, what time elapses between the two cars passing a point q three kilometres from p?

b) A particle of mass 3 grammes falls from rest from a height of 0.4 m on to a soft material into which it sinks 0.0245 m. Neglecting air resistance, calculate the constant resistance of the material.

Question 2

Two straight roads intersect at an angle of 60^\circ. Cyclist A and B move towards the point of intersection at 30 km/h and 40 km/h respectively.

i) Calculate the velocity of A relative to B.

ii) If A is 3.5 km and B is 2 km from the intersection at a given moment, calculate the shortest distance between them in their subsequent motion.

Question 3

A particle is projected with a speed of 10 m/s at an angle \alpha^\circ to the horizontal up a plane inclined at 30^\circ to the horizontal.

i) If the particle strikes the plane at right angles, show that the time of flight can be represented by the two expressions \frac{10\cos\left(\alpha-30\right)}{g\sin 30} and \frac{20\sin\left(\alpha-30\right)}{g\cos 30}.

ii) Hence deduce a value for \tan\left(\alpha-30\right).

iii) Calculate the range of the particle along the plane.

Question 4

a) A smooth sphere of mass 10 kg moving at 10 m/s impinges directly on another smooth sphere of mass 50 kg moving in the opposite direction at 5 m/s. If the coefficient of restitution is \frac{1}{2}, calculate the speeds after impact and the magnitude of the impulse during impact.

b) A smooth metal sphere falls vertically and strikes a fixed smooth plane inclined at an angle of \theta to the horizontal. If the coefficient of restitution is \frac{2}{3} and the sphere rebounds horizontally, calculate the fraction of kinetic energy lost during impact.

Question 5

a) The diagram shows a string prq which is fixed at p and q where q vertically below p. r is a small ring threaded on the string which is made to rotate at an angular velocity, \omega rad/s, in a horizontal circle, centre q, the string being taut. If \left|pq\right|=0.12 m, \left|pr\right|+\left|rq\right|=0.18 m, show that \omega=\sqrt{294} rad/s.

b) A small bead of mass m is threaded on a smooth circular wire of radius a, fixed with its plane vertical. The bead is projected from the lowest poitn of the wire with speed u. Show that the reaction between the bead and the wire, when the radius to the bead makes an angle of 60^\circ with the downward vertical is

    \[m \left(\frac{u^2}{a}-\frac{g}{2}\right)\]

Question 6

a) Define (i) limiting friction, (ii) coefficient of friction.

b) A lamina of weight W in the shape of a thin equilateral triangle pqr is positioned vertically with the vertex p against a smooth vertical wall, and q on rough horizonal floor. [qr] is parallel to the wall.

i) Find in terms of W, the horizontal and vertical reactions at q.

ii) Find the least value of \mu, the coefficient of friction, so that slipping will not occur.

Question 7

a) Show that the moment of inertia of a uniform square lamina of side 2l and mass m about an axis perpendicular to the lamina through its centre of mass is \frac{2ml^2}{3}.

b) i) A thin uniform rod of length 2l and of mass m is attached to the mid point of the rim of the square. Find the moment of inertia of the system about an axis through p perpendicular to the common plane of lamina and rod.

ii) When this system makes small oscillations in a vertical plane about the axis through p, show that the period of the oscillations is 2\pi\sqrt{\frac{31l}{7g}}.

Question 8

i) Define simple harmonic motion.

ii) The distance, x, of a particle from a fixed point, o, is given by

    \[x = A \cos \left( \omega t + \alpha \right)\]

where A, \omega, \alpha are positive constants. Show that the particle is describing simple harmonic motion about o and calculate \omega and \alpha if the velocity v=-2a and x=\frac{3a}{5} where t=0.

iii) After how many seconds from the start of the motion is x=0 for the first time?

Question 9

a) A bucket has the form of a frustum of a right circular cone. When it is completely filled with water, find

i) the pressure at a point on the base

ii) the thrust, T, on the base

iii) the ratio \frac{\mbox{weight of water}}{T}.

b) A cubical block of wood of mass 50 kg floats in water with three quarters of its volume immersed. In oil, when a mass of 10 kg is placed on the same block, it floats just totally immersed, the 10 kg mass being above the oil. Find the specific gravity of the oil.

Question 10

a) Find the solution of the differential equation

    \[\left( 1 + x^3 \right) \frac{dy}{dx} = x^2 y\]

where y=2 at x=1.

b) i) Find the solution of the differential equation

    \[\frac{d^2 s}{dt^2} = - \left(\frac{ds}{dt}\right)^2\]

when \frac{ds}{dt}=1 at t=0 and s=0 at t=0.

ii) If its initial velocity is 1 m/s, calculate the distance travelled one second later.


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“Contains Irish Public Sector Information licensed under a Creative Commons Attribution 4.0 International (CC BY 4.0) licence”.

Important Exception to the above Licence:

The State Examination Commission is the copyright holder which is providing the material under the above license (as per current directives and regulations from the relevant government bodies). However the State Examination Commission as an Irish examination body is able to use copyrighted material in its exams without infringing copyright but this right is not extended to third parties when those exams are re-used.

(For example: the State Examination Commission may include in their exam a copyrighted poem and this action does not require the permission of the poet but the poet’s permission must be sought when the exam is re-used by someone other than the State Examination Commission.)

Also, all derived and related work (such as video solutions, lessons, notes etc) are the copyrighted material of Stephen Easley-Walsh (unless stated otherwise). And that the above licence is for only the exam itself and nothing further.

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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