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26 Nov 2018

Leaving Certificate Examination 1966 Honours Applied Mathematics

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Question 1

(a) Represent any two velocities quicklatex.com-00f19de60e164b642d64f33cdbfd4ed3_l3 | Leaving Certificate Examination 1966 Honours Applied Mathematics | Maths Grinds and quicklatex.com-734938bfc289d848b1ec19504fdb71b0_l3 | Leaving Certificate Examination 1966 Honours Applied Mathematics | Maths Grinds by a vector diagram, and illustrate the vectors quicklatex.com-2cf434ee3c92736cc2ac084f03363e40_l3 | Leaving Certificate Examination 1966 Honours Applied Mathematics | Maths Grinds and quicklatex.com-08b92d580d553866e64ebfed36257107_l3 | Leaving Certificate Examination 1966 Honours Applied Mathematics | Maths Grinds.

(b) When the sun, S, is quicklatex.com-f643069adb772a4896813c16b313ad05_l3 | Leaving Certificate Examination 1966 Honours Applied Mathematics | Maths Grinds above the horizon, an aeroplane quicklatex.com-cba4133f939edee31a8ce51d790a9cd0_l3 | Leaving Certificate Examination 1966 Honours Applied Mathematics | Maths Grinds glides in for landing with the sun behind, moving with a speed of quicklatex.com-5291abebe16995c90201b72e133d15d8_l3 | Leaving Certificate Examination 1966 Honours Applied Mathematics | Maths Grinds m.p.h. along a path sloping quicklatex.com-0c559742eb6542699185a371f9051330_l3 | Leaving Certificate Examination 1966 Honours Applied Mathematics | Maths Grinds down from the horizontal, as in the diagram. Find the speed of the plane’s shadow on level ground (quicklatex.com-e979ae05a61debd12da167a83e2720d6_l3 | Leaving Certificate Examination 1966 Honours Applied Mathematics | Maths Grinds).

HAM-1966-Q1-300x190 | Leaving Certificate Examination 1966 Honours Applied Mathematics | Maths Grinds

Question 2

Write a short note on linear momentum.

A uniform cube of wood of mass quicklatex.com-739f6781b65065fc445d5f13b544d6b2_l3 | Leaving Certificate Examination 1966 Honours Applied Mathematics | Maths Grinds lbs. hangs vertically by a long light string fixed in the centre of one face of the cube. A bullet of mass half an ounce moving horizontally with velocity quicklatex.com-ecffa0daef95d0e24bb22fc5711a6bc3_l3 | Leaving Certificate Examination 1966 Honours Applied Mathematics | Maths Grinds strikes the centre of another face in a direction perpendicular to the face, embeds itself in the wood and causes the combined mass to swing. If in the first swing the centre of gravity of the combined mass rises quicklatex.com-ee873c946a42a316d07b2d164e2b7d67_l3 | Leaving Certificate Examination 1966 Honours Applied Mathematics | Maths Grinds inches, calculate the magnitude of quicklatex.com-ecffa0daef95d0e24bb22fc5711a6bc3_l3 | Leaving Certificate Examination 1966 Honours Applied Mathematics | Maths Grinds.

Question 3

For a body moving in a straight line with constant acceleration quicklatex.com-d1b784e6a7a3c76809c155e9da81fc51_l3 | Leaving Certificate Examination 1966 Honours Applied Mathematics | Maths Grinds, initial velocity quicklatex.com-7037dec17f4a6620ec536567c455a084_l3 | Leaving Certificate Examination 1966 Honours Applied Mathematics | Maths Grinds and velocity quicklatex.com-ecffa0daef95d0e24bb22fc5711a6bc3_l3 | Leaving Certificate Examination 1966 Honours Applied Mathematics | Maths Grinds at time quicklatex.com-db2144d676b0c010fa61116d07db1982_l3 | Leaving Certificate Examination 1966 Honours Applied Mathematics | Maths Grinds, show that quicklatex.com-616989d4d3df4b6e5b34f6a7421160ac_l3 | Leaving Certificate Examination 1966 Honours Applied Mathematics | Maths Grinds and quicklatex.com-6e2883dfc3d4d98fb8868ba148e1b2c2_l3 | Leaving Certificate Examination 1966 Honours Applied Mathematics | Maths Grinds.

A cage travels down a vertical mine-shaft quicklatex.com-9a98f9c89d48c0e9bdb897d109b66270_l3 | Leaving Certificate Examination 1966 Honours Applied Mathematics | Maths Grinds yards deep in quicklatex.com-9e7ec55fdcc4467668102a4d17def052_l3 | Leaving Certificate Examination 1966 Honours Applied Mathematics | Maths Grinds seconds, starting from rest. For the first quarter of the distance only the velocity undergoes a constant acceleration, and for the last quarter of the distance the velocity undergoes a constant retardation, the acceleration and retardation having equal magnitudes. If the velocity, quicklatex.com-00f19de60e164b642d64f33cdbfd4ed3_l3 | Leaving Certificate Examination 1966 Honours Applied Mathematics | Maths Grinds, is uniform while traversing the central portion of the shaft, find the magnitude of quicklatex.com-00f19de60e164b642d64f33cdbfd4ed3_l3 | Leaving Certificate Examination 1966 Honours Applied Mathematics | Maths Grinds.

Question 4

The co-ordinates quicklatex.com-10b68f67e4402fd35e152da9899ca73b_l3 | Leaving Certificate Examination 1966 Honours Applied Mathematics | Maths Grinds of a moving particle at any time quicklatex.com-db2144d676b0c010fa61116d07db1982_l3 | Leaving Certificate Examination 1966 Honours Applied Mathematics | Maths Grinds are given by:

quicklatex.com-7beb008cb31899d147db6cf0bce15ec0_l3 | Leaving Certificate Examination 1966 Honours Applied Mathematics | Maths Grinds, quicklatex.com-277e382a16e2e90d1f09c3ef40f5e5c3_l3 | Leaving Certificate Examination 1966 Honours Applied Mathematics | Maths Grinds.

Deduce that its motion is circular.

A particle is moving along the circumference of a circle with constant speed. Show that that components of the motion of the particle along any two perpendicular diameters are Simple Harmonic Motions.

Question 5

A particle is projected horizontally from a height with velocity quicklatex.com-ecffa0daef95d0e24bb22fc5711a6bc3_l3 | Leaving Certificate Examination 1966 Honours Applied Mathematics | Maths Grinds and falls under gravity. If quicklatex.com-10b68f67e4402fd35e152da9899ca73b_l3 | Leaving Certificate Examination 1966 Honours Applied Mathematics | Maths Grinds are the co-ordinates of the particle at any time, and quicklatex.com-1a71ac7f67f3b40ffb4c9fb889271a2d_l3 | Leaving Certificate Examination 1966 Honours Applied Mathematics | Maths Grinds the co-ordinates of the point of projection, show that quicklatex.com-ef8429fca85c2e86f784f2ed0364f3b4_l3 | Leaving Certificate Examination 1966 Honours Applied Mathematics | Maths Grinds us the equation of its path, the x-axis being horizontal and the y-axis vertical.

A marble rolls off the top of a stairway horizontally and in a direction perpendicular to the edge of each step, with a speed of quicklatex.com-2dda138cd42bede9cb97214fc5c8a205_l3 | Leaving Certificate Examination 1966 Honours Applied Mathematics | Maths Grinds feet per second. Each step is quicklatex.com-ee873c946a42a316d07b2d164e2b7d67_l3 | Leaving Certificate Examination 1966 Honours Applied Mathematics | Maths Grinds in. high and quicklatex.com-2b18fa15a2f284d990fd81b9cae524ce_l3 | Leaving Certificate Examination 1966 Honours Applied Mathematics | Maths Grinds in. wide. Which is the first step struck by the marble (assuming the stairway is sufficiently high)?

Question 6

Masses of quicklatex.com-d39c7a34226b6a7fc7b5d3ae51b6b5e4_l3 | Leaving Certificate Examination 1966 Honours Applied Mathematics | Maths Grinds lb., quicklatex.com-ce939b244df556d77c12e315c6165e7d_l3 | Leaving Certificate Examination 1966 Honours Applied Mathematics | Maths Grinds lbs., quicklatex.com-1ee1c2caec46fc384f9e0e830a2e8b84_l3 | Leaving Certificate Examination 1966 Honours Applied Mathematics | Maths Grinds lbs., and quicklatex.com-fd6542ae3d49d38a0f73fefe83b285af_l3 | Leaving Certificate Examination 1966 Honours Applied Mathematics | Maths Grinds lbs. are placed at points the co-ordinate of which are quicklatex.com-5cb6787c0e19022bda5029e9be24290b_l3 | Leaving Certificate Examination 1966 Honours Applied Mathematics | Maths Grinds, quicklatex.com-4ccbb9dea4bd0bbb52122bd51cdc757e_l3 | Leaving Certificate Examination 1966 Honours Applied Mathematics | Maths Grinds, quicklatex.com-9a04df0497f683fb3f19e59b341be8ba_l3 | Leaving Certificate Examination 1966 Honours Applied Mathematics | Maths Grinds, and quicklatex.com-f87e881ca211369b95179d57ea96c60a_l3 | Leaving Certificate Examination 1966 Honours Applied Mathematics | Maths Grinds respectively. Locate the centre of gravity (or the centre of mass) of the system.

Two discs of radii quicklatex.com-95d7c225dd31ad7d6dd5fb9f0980b531_l3 | Leaving Certificate Examination 1966 Honours Applied Mathematics | Maths Grinds and quicklatex.com-144430d1cdfc95609f2001bd7e21e538_l3 | Leaving Certificate Examination 1966 Honours Applied Mathematics | Maths Grinds are placed flat on a horizontal plane with a distance quicklatex.com-331484459a9d4c7be13bcc3784ccaada_l3 | Leaving Certificate Examination 1966 Honours Applied Mathematics | Maths Grinds between their centres. If quicklatex.com-05d30bc5e19b413cea824383c6bce148_l3 | Leaving Certificate Examination 1966 Honours Applied Mathematics | Maths Grinds is their centre of gravity and quicklatex.com-5b090119e833ac90162777d07e3fe243_l3 | Leaving Certificate Examination 1966 Honours Applied Mathematics | Maths Grinds the centre of gravity of the large disc calculate the length of quicklatex.com-1565c0a45d8ee629092e9a8de79f559b_l3 | Leaving Certificate Examination 1966 Honours Applied Mathematics | Maths Grinds.

A square hole is punched out of a circular disc (radius quicklatex.com-0dce58ae1ea930bb9c39e85ab5e4c977_l3 | Leaving Certificate Examination 1966 Honours Applied Mathematics | Maths Grinds), a diagonal of the square being a radius of the circle. Show that the centre of gravity of the remainder of the disc is at a distance quicklatex.com-e9320f82194097dfc0434c62e8d36ba6_l3 | Leaving Certificate Examination 1966 Honours Applied Mathematics | Maths Grinds from the centre of the circle.

Question 7

Explain “normal reaction” and “angle of friction”.

Deduce quicklatex.com-86f4a1305fd4898a41e4bc53336e31f8_l3 | Leaving Certificate Examination 1966 Honours Applied Mathematics | Maths Grinds where quicklatex.com-4ac1cf9e7a40a63c9af04825a0b3c05c_l3 | Leaving Certificate Examination 1966 Honours Applied Mathematics | Maths Grinds is the angle of friction and quicklatex.com-d6ffefedf9d4e7e4f42ba65b9f3b9416_l3 | Leaving Certificate Examination 1966 Honours Applied Mathematics | Maths Grinds the angle between the normal reaction and the reaction (resultant reaction).

A uniform rod quicklatex.com-bb11fdb4c82e898ea8147471416d775f_l3 | Leaving Certificate Examination 1966 Honours Applied Mathematics | Maths Grinds quicklatex.com-fd6542ae3d49d38a0f73fefe83b285af_l3 | Leaving Certificate Examination 1966 Honours Applied Mathematics | Maths Grinds feet long leans against the smooth edge of a table three feet in height with one end of the rod on a rough horizontal floor (as in diagram). If the rod is one the point of slipping when inclined at an angle of quicklatex.com-43254279a766115b4c8c77b4f9e086ff_l3 | Leaving Certificate Examination 1966 Honours Applied Mathematics | Maths Grinds to the horizontal, find the coefficient of friction.

HAM-1966-Q7 | Leaving Certificate Examination 1966 Honours Applied Mathematics | Maths Grinds

Question 8

Two particles quicklatex.com-d613350484b582bc83e2a6f9473437f8_l3 | Leaving Certificate Examination 1966 Honours Applied Mathematics | Maths Grinds and quicklatex.com-6ce7c226153d55058c10348c6739a4c5_l3 | Leaving Certificate Examination 1966 Honours Applied Mathematics | Maths Grinds move with velocities quicklatex.com-00f19de60e164b642d64f33cdbfd4ed3_l3 | Leaving Certificate Examination 1966 Honours Applied Mathematics | Maths Grinds and quicklatex.com-734938bfc289d848b1ec19504fdb71b0_l3 | Leaving Certificate Examination 1966 Honours Applied Mathematics | Maths Grinds respectively. Use diagrams to illustrate clearly the velocity of quicklatex.com-d613350484b582bc83e2a6f9473437f8_l3 | Leaving Certificate Examination 1966 Honours Applied Mathematics | Maths Grinds relative to quicklatex.com-6ce7c226153d55058c10348c6739a4c5_l3 | Leaving Certificate Examination 1966 Honours Applied Mathematics | Maths Grinds when quicklatex.com-00f19de60e164b642d64f33cdbfd4ed3_l3 | Leaving Certificate Examination 1966 Honours Applied Mathematics | Maths Grinds and quicklatex.com-734938bfc289d848b1ec19504fdb71b0_l3 | Leaving Certificate Examination 1966 Honours Applied Mathematics | Maths Grinds are

(i) parallel and in the same direction,

(ii) parallel and in opposite directions,

(iii) non-parallel.

A steamer going north-east at quicklatex.com-e2ceb65b3a6ab9dc7212d896621f4092_l3 | Leaving Certificate Examination 1966 Honours Applied Mathematics | Maths Grinds knots observes a cruiser which is quicklatex.com-739f6781b65065fc445d5f13b544d6b2_l3 | Leaving Certificate Examination 1966 Honours Applied Mathematics | Maths Grinds miles south-east from the steamer and which is travelling in a direction quicklatex.com-f643069adb772a4896813c16b313ad05_l3 | Leaving Certificate Examination 1966 Honours Applied Mathematics | Maths Grinds east of north at quicklatex.com-66bafba23129fa8ab0a7d7fd2fc25d52_l3 | Leaving Certificate Examination 1966 Honours Applied Mathematics | Maths Grinds knots.

Illustrate by diagram the cruiser’s course as i appears to the steamer. Find the velocity of the cruiser relative to the steamer, in magnitude and direction, and find also the shortest distance between the two ships.

Question 9

Establish an expression for the pressure at a depth quicklatex.com-fc85ef309591bf8b549446f3ebd0b450_l3 | Leaving Certificate Examination 1966 Honours Applied Mathematics | Maths Grinds in a liquid of density quicklatex.com-ebfa669ee18b88d9ca3d1258b6c3d322_l3 | Leaving Certificate Examination 1966 Honours Applied Mathematics | Maths Grinds at rest under gravity. Hence show that the resultant thrust on a plane surface immersed vertically in the liquid is the product of the area of the plane surface and the depth of its centre of gravity.

Find the resultant horizontal thrust on a rectangular lockgate quicklatex.com-2dda138cd42bede9cb97214fc5c8a205_l3 | Leaving Certificate Examination 1966 Honours Applied Mathematics | Maths Grinds feet wide when the water stands at a depth of quicklatex.com-2b18fa15a2f284d990fd81b9cae524ce_l3 | Leaving Certificate Examination 1966 Honours Applied Mathematics | Maths Grinds feet on one side and quicklatex.com-fd6542ae3d49d38a0f73fefe83b285af_l3 | Leaving Certificate Examination 1966 Honours Applied Mathematics | Maths Grinds feet on the other.

(1 cu. ft. of water weighs quicklatex.com-287a55b2d8cfd44f5f1f05ad3a5a70db_l3 | Leaving Certificate Examination 1966 Honours Applied Mathematics | Maths Grinds 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/

Licence:

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

The EU Directive 2003/98/EC on the re-use of public sector information, its amendment EU Directive 2013/37/EC, its transposed Irish Statutory Instruments S.I. No. 279/2005, S.I No. 103/2008, and S.I. No. 525/2015, and related Circulars issued by the Department of Finance (Circular 32/05), and Department of Public Expenditure and Reform (Circular 16/15 and Circular 12/16).

Note. Circular 12/2016: Licence for Re-Use of Public Sector Information adopts CC-BY as the standard PSI licence, and notes that the open standard licence identified in this Circular supersedes PSI General Licence No: 2005/08/01.

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https://circulars.gov.ie/pdf/circular/per/2016/12.pdf

https://creativecommons.org/licenses/by/4.0/legalcod

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