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30 Nov 2018

Leaving Certificate Examination 1970 Honours Applied Mathematics

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Question 1

A bullet of mass quicklatex.com-7a02bd7300eeadcf269af0186938941e_l3 | Leaving Certificate Examination 1970 Honours Applied Mathematics | Maths Grinds is fired with speed quicklatex.com-8bafe7d16e660869fc203e8bffd77abc_l3 | Leaving Certificate Examination 1970 Honours Applied Mathematics | Maths Grinds into a fixed block of wood and is brought to rest in a distance quicklatex.com-fc85ef309591bf8b549446f3ebd0b450_l3 | Leaving Certificate Examination 1970 Honours Applied Mathematics | Maths Grinds. Find the resistance to motion assuming it to be constant.

Another bullet also of mass quicklatex.com-7a02bd7300eeadcf269af0186938941e_l3 | Leaving Certificate Examination 1970 Honours Applied Mathematics | Maths Grinds is then fired with speed quicklatex.com-384747fb9ecf401a408cb966f883e630_l3 | Leaving Certificate Examination 1970 Honours Applied Mathematics | Maths Grinds into another fixed block of thickness quicklatex.com-b7b2a142cc98f32fb3949f841b7114c6_l3 | Leaving Certificate Examination 1970 Honours Applied Mathematics | Maths Grinds, which offers the same resistance as the first block. Find the speed with which the bullet emerges, and the time it takes to pass through the block.

Question 2

Prove that the formula quicklatex.com-6cf0cd9dbc2828b0eb1e4aabc37794a4_l3 | Leaving Certificate Examination 1970 Honours Applied Mathematics | Maths Grinds represents the distance quicklatex.com-ada3765a264ac03a253ee0ccbe61f602_l3 | Leaving Certificate Examination 1970 Honours Applied Mathematics | Maths Grinds travelled in time quicklatex.com-db2144d676b0c010fa61116d07db1982_l3 | Leaving Certificate Examination 1970 Honours Applied Mathematics | Maths Grinds by a body moving in a straight line with constant acceleration quicklatex.com-84fdf06eaf2b99889abd1c8acdc13d5f_l3 | Leaving Certificate Examination 1970 Honours Applied Mathematics | Maths Grinds.

A train takes quicklatex.com-d61119651ddf565da4198bb11ee60847_l3 | Leaving Certificate Examination 1970 Honours Applied Mathematics | Maths Grinds minutes to travel between two stations quicklatex.com-f6693e2d5e1632e5f01847eed3a1cb6c_l3 | Leaving Certificate Examination 1970 Honours Applied Mathematics | Maths Grinds and quicklatex.com-7aa6af2988da671d2d73b46d78786215_l3 | Leaving Certificate Examination 1970 Honours Applied Mathematics | Maths Grinds which are quicklatex.com-a285a70e7627dcea0fd6f980ddb0acf7_l3 | Leaving Certificate Examination 1970 Honours Applied Mathematics | Maths Grinds meters apart. It starts from rest at quicklatex.com-f6693e2d5e1632e5f01847eed3a1cb6c_l3 | Leaving Certificate Examination 1970 Honours Applied Mathematics | Maths Grinds and finishes at rest at quicklatex.com-7aa6af2988da671d2d73b46d78786215_l3 | Leaving Certificate Examination 1970 Honours Applied Mathematics | Maths Grinds, by travelling with uniform acceleration for the first minute and with uniform deceleration for the last quicklatex.com-294859fdbb1fdda439ec2363e11fdfa1_l3 | Leaving Certificate Examination 1970 Honours Applied Mathematics | Maths Grinds minute. Find the train’s constant speed during the remainder of the journey.

If a second train, travelling with a constant speed of quicklatex.com-832988d7b85b3784b072284b63fdfa60_l3 | Leaving Certificate Examination 1970 Honours Applied Mathematics | Maths Grinds m/min, in the same direction passes quicklatex.com-f6693e2d5e1632e5f01847eed3a1cb6c_l3 | Leaving Certificate Examination 1970 Honours Applied Mathematics | Maths Grinds as the first train leaves this station, find when overtaking occurs. (The lengths of the trains may be neglected).

Question 3

A particle is projected under gravity with an initial velocity quicklatex.com-8557001db9d8aa5a5a62f7a57a2f0356_l3 | Leaving Certificate Examination 1970 Honours Applied Mathematics | Maths Grinds at an angle quicklatex.com-d6ffefedf9d4e7e4f42ba65b9f3b9416_l3 | Leaving Certificate Examination 1970 Honours Applied Mathematics | Maths Grinds to the horizontal. Find its position and the direction of motion after time quicklatex.com-6460d10729af641188d49a36153c8def_l3 | Leaving Certificate Examination 1970 Honours Applied Mathematics | Maths Grinds in terms of quicklatex.com-6460d10729af641188d49a36153c8def_l3 | Leaving Certificate Examination 1970 Honours Applied Mathematics | Maths Grinds in terms of quicklatex.com-8557001db9d8aa5a5a62f7a57a2f0356_l3 | Leaving Certificate Examination 1970 Honours Applied Mathematics | Maths Grinds, quicklatex.com-d6ffefedf9d4e7e4f42ba65b9f3b9416_l3 | Leaving Certificate Examination 1970 Honours Applied Mathematics | Maths Grinds, quicklatex.com-32a1fff034751d4fc3e4edc15ca376af_l3 | Leaving Certificate Examination 1970 Honours Applied Mathematics | Maths Grinds and quicklatex.com-6460d10729af641188d49a36153c8def_l3 | Leaving Certificate Examination 1970 Honours Applied Mathematics | Maths Grinds.

A particle is projected from the top of a cliff which is quicklatex.com-b7a42c6b2d798853df3cd9ae5539a187_l3 | Leaving Certificate Examination 1970 Honours Applied Mathematics | Maths Grinds ft. above sea level and the angle of projection is quicklatex.com-9c3fb28be0c20ddb0b2c4bd59c303de8_l3 | Leaving Certificate Examination 1970 Honours Applied Mathematics | Maths Grinds to the horizontal. If the greatest height reached above the point of projection is quicklatex.com-0a7d1908dc17ba7c399644cfaecdaeba_l3 | Leaving Certificate Examination 1970 Honours Applied Mathematics | Maths Grinds ft, find the speed of projection and the time taken to reach this greatest height.

Find when and where the particles strikes the sea.

(Take quicklatex.com-32a1fff034751d4fc3e4edc15ca376af_l3 | Leaving Certificate Examination 1970 Honours Applied Mathematics | Maths Grinds to be quicklatex.com-094ee6d517670326a569d5b0dc00b494_l3 | Leaving Certificate Examination 1970 Honours Applied Mathematics | Maths Grinds ft/secquicklatex.com-025303222356a63c2332a429c33d35a9_l3 | Leaving Certificate Examination 1970 Honours Applied Mathematics | Maths Grinds).

Question 4

Prove that the bob of a simple pendulum moves in simple harmonic motion – stating any assumptions made.

The string of such a pendulum is quicklatex.com-ce939b244df556d77c12e315c6165e7d_l3 | Leaving Certificate Examination 1970 Honours Applied Mathematics | Maths Grinds ft. long and the bob is released from rest when at a distance quicklatex.com-a5990b654956436786b5311c7aada3b3_l3 | Leaving Certificate Examination 1970 Honours Applied Mathematics | Maths Grinds ft from the equilibrium position. Calculate the time taken to travel halfway to the equilibrium position and the speed of the bob then.

(Take quicklatex.com-32a1fff034751d4fc3e4edc15ca376af_l3 | Leaving Certificate Examination 1970 Honours Applied Mathematics | Maths Grinds to be quicklatex.com-094ee6d517670326a569d5b0dc00b494_l3 | Leaving Certificate Examination 1970 Honours Applied Mathematics | Maths Grinds ft/secquicklatex.com-025303222356a63c2332a429c33d35a9_l3 | Leaving Certificate Examination 1970 Honours Applied Mathematics | Maths Grinds).

Question 5

By deriving an expression for the necessary acceleration, prove that a particle of mass quicklatex.com-7a02bd7300eeadcf269af0186938941e_l3 | Leaving Certificate Examination 1970 Honours Applied Mathematics | Maths Grinds moving in a circle of radius quicklatex.com-95d7c225dd31ad7d6dd5fb9f0980b531_l3 | Leaving Certificate Examination 1970 Honours Applied Mathematics | Maths Grinds with speed quicklatex.com-8bafe7d16e660869fc203e8bffd77abc_l3 | Leaving Certificate Examination 1970 Honours Applied Mathematics | Maths Grinds must have a force of magnitude quicklatex.com-d2d92cd4eb0ac946cc55f916dbe43b35_l3 | Leaving Certificate Examination 1970 Honours Applied Mathematics | Maths Grinds pointing towards the centre acting on it.

A particle of mass quicklatex.com-fd6542ae3d49d38a0f73fefe83b285af_l3 | Leaving Certificate Examination 1970 Honours Applied Mathematics | Maths Grinds lbs. moving on the inside smooth surface of a fixed spherical bowl of radius quicklatex.com-ce939b244df556d77c12e315c6165e7d_l3 | Leaving Certificate Examination 1970 Honours Applied Mathematics | Maths Grinds ft is describing a horizontal circle of radius quicklatex.com-e260861ef0368856de1d8a1af35ee11f_l3 | Leaving Certificate Examination 1970 Honours Applied Mathematics | Maths Grinds ft. Find the constant speed of rotation and the reaction of the sphere on the particle.

(Take quicklatex.com-32a1fff034751d4fc3e4edc15ca376af_l3 | Leaving Certificate Examination 1970 Honours Applied Mathematics | Maths Grinds to be quicklatex.com-094ee6d517670326a569d5b0dc00b494_l3 | Leaving Certificate Examination 1970 Honours Applied Mathematics | Maths Grinds ft/secquicklatex.com-025303222356a63c2332a429c33d35a9_l3 | Leaving Certificate Examination 1970 Honours Applied Mathematics | Maths Grinds).

Question 6

Show that the centre of gravity of a uniform triangular lamina coincides with the centre of gravity of three equal particles placed at the vertices of the triangle. Hence find the centre of gravity of a uniform trapezium quicklatex.com-0bc2407dfe67fc49000363238a97afd0_l3 | Leaving Certificate Examination 1970 Honours Applied Mathematics | Maths Grinds, of weight quicklatex.com-054ef36c6e54f65ccdd586c0869b4fb0_l3 | Leaving Certificate Examination 1970 Honours Applied Mathematics | Maths Grinds, in which quicklatex.com-b065ab3e1cbbe7d6e6f746582013b335_l3 | Leaving Certificate Examination 1970 Honours Applied Mathematics | Maths Grinds and quicklatex.com-ec3f315b831cbfc7502233551b2ef23b_l3 | Leaving Certificate Examination 1970 Honours Applied Mathematics | Maths Grinds.

A particle of weight quicklatex.com-d66dbaeb0eebc210a255b3fd356eacd0_l3 | Leaving Certificate Examination 1970 Honours Applied Mathematics | Maths Grinds is attached at quicklatex.com-e24826b0fac28861acd034428af346f0_l3 | Leaving Certificate Examination 1970 Honours Applied Mathematics | Maths Grinds and the system is suspended by a string attached to the midpoint quicklatex.com-8c28ca9825a5e0ff6a01cc15dd4efff1_l3 | Leaving Certificate Examination 1970 Honours Applied Mathematics | Maths Grinds of quicklatex.com-090252563a7d95933f4046a6c256de5b_l3 | Leaving Certificate Examination 1970 Honours Applied Mathematics | Maths Grinds. If in the position of equilibrium quicklatex.com-8c28ca9825a5e0ff6a01cc15dd4efff1_l3 | Leaving Certificate Examination 1970 Honours Applied Mathematics | Maths Grinds is vertically above quicklatex.com-cba4133f939edee31a8ce51d790a9cd0_l3 | Leaving Certificate Examination 1970 Honours Applied Mathematics | Maths Grinds show that quicklatex.com-3969bd6b3e9100028b063b3a9622c87d_l3 | Leaving Certificate Examination 1970 Honours Applied Mathematics | Maths Grinds.

Question 7

A particle of mass quicklatex.com-739f6781b65065fc445d5f13b544d6b2_l3 | Leaving Certificate Examination 1970 Honours Applied Mathematics | Maths Grinds lbs. is placed on a rough inclined plane. The least force acting up the plane which will prevent the particle slipping down the plane is quicklatex.com-ce939b244df556d77c12e315c6165e7d_l3 | Leaving Certificate Examination 1970 Honours Applied Mathematics | Maths Grinds lbs. weight. The least force acting up the plane which will make the particle slip upwards is quicklatex.com-739f6781b65065fc445d5f13b544d6b2_l3 | Leaving Certificate Examination 1970 Honours Applied Mathematics | Maths Grinds lbs wt. Show that the coefficient of friction is quicklatex.com-294859fdbb1fdda439ec2363e11fdfa1_l3 | Leaving Certificate Examination 1970 Honours Applied Mathematics | Maths Grinds and that the inclination of the plane is quicklatex.com-0dce58ae1ea930bb9c39e85ab5e4c977_l3 | Leaving Certificate Examination 1970 Honours Applied Mathematics | Maths Grinds where quicklatex.com-c7f7d13f535e2596d7524fc19d75978e_l3 | Leaving Certificate Examination 1970 Honours Applied Mathematics | Maths Grinds.

Find the least force required to move the particle up the plane.

Question 8

Two equal uniform rods quicklatex.com-bb11fdb4c82e898ea8147471416d775f_l3 | Leaving Certificate Examination 1970 Honours Applied Mathematics | Maths Grinds and quicklatex.com-3675ba693016b80ac93fbf734e15678b_l3 | Leaving Certificate Examination 1970 Honours Applied Mathematics | Maths Grinds each of weight quicklatex.com-054ef36c6e54f65ccdd586c0869b4fb0_l3 | Leaving Certificate Examination 1970 Honours Applied Mathematics | Maths Grinds are freely jointed at quicklatex.com-eaf6d3ddf3d9ff0d18eb3da055c8c94a_l3 | Leaving Certificate Examination 1970 Honours Applied Mathematics | Maths Grinds. The system is suspended freely from quicklatex.com-cba4133f939edee31a8ce51d790a9cd0_l3 | Leaving Certificate Examination 1970 Honours Applied Mathematics | Maths Grinds and a horizontal force quicklatex.com-76bddd49e51d803b2711eb3faf8ed30a_l3 | Leaving Certificate Examination 1970 Honours Applied Mathematics | Maths Grinds is applied at the lowest point quicklatex.com-e86ae6aeacc8696c140ee1ca29d51f96_l3 | Leaving Certificate Examination 1970 Honours Applied Mathematics | Maths Grinds. If in the equilibrium position the inclination of quicklatex.com-bb11fdb4c82e898ea8147471416d775f_l3 | Leaving Certificate Examination 1970 Honours Applied Mathematics | Maths Grinds to the downward vertical is quicklatex.com-f643069adb772a4896813c16b313ad05_l3 | Leaving Certificate Examination 1970 Honours Applied Mathematics | Maths Grinds, find the corresponding inclination of quicklatex.com-3675ba693016b80ac93fbf734e15678b_l3 | Leaving Certificate Examination 1970 Honours Applied Mathematics | Maths Grinds and the supporting force at quicklatex.com-cba4133f939edee31a8ce51d790a9cd0_l3 | Leaving Certificate Examination 1970 Honours Applied Mathematics | Maths Grinds.

Question 9

A small uniform cylinder of density quicklatex.com-ebfa669ee18b88d9ca3d1258b6c3d322_l3 | Leaving Certificate Examination 1970 Honours Applied Mathematics | Maths Grinds, mass quicklatex.com-7a02bd7300eeadcf269af0186938941e_l3 | Leaving Certificate Examination 1970 Honours Applied Mathematics | Maths Grinds, total length quicklatex.com-7e66f8d3fbb50779ceb68e5184deb32d_l3 | Leaving Certificate Examination 1970 Honours Applied Mathematics | Maths Grinds and uniform cross section floats in a liquid of density quicklatex.com-0a1b2d313df6cd8516834a0a38e5fc9b_l3 | Leaving Certificate Examination 1970 Honours Applied Mathematics | Maths Grinds with its axis vertical. Find the thrust on the cylinder when it is displaced vertically in the liquid, without being completely immersed, through a distance quicklatex.com-ada3765a264ac03a253ee0ccbe61f602_l3 | Leaving Certificate Examination 1970 Honours Applied Mathematics | Maths Grinds from the equilibrium position. Show that if it is released in this position, it will oscillate with simple harmonic motion of period quicklatex.com-31e9044f8b7abadfb681fd6fbfcf546d_l3 | Leaving Certificate Examination 1970 Honours Applied Mathematics | Maths Grinds.


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.

Links:

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

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

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