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23 Oct 2018

Leaving Certificate Examination 1954 Honours Applied Mathematics

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Question 1

quicklatex.com-cba4133f939edee31a8ce51d790a9cd0_l3 | Leaving Certificate Examination 1954 Honours Applied Mathematics | Maths Grinds and quicklatex.com-eaf6d3ddf3d9ff0d18eb3da055c8c94a_l3 | Leaving Certificate Examination 1954 Honours Applied Mathematics | Maths Grinds are the ends of a uniform bar which is quicklatex.com-2b18fa15a2f284d990fd81b9cae524ce_l3 | Leaving Certificate Examination 1954 Honours Applied Mathematics | Maths Grinds inches in length and which weighs quicklatex.com-dc2e717421b16a9f080dba4c8c0e3437_l3 | Leaving Certificate Examination 1954 Honours Applied Mathematics | Maths Grinds lb. The bar is supported by two strings quicklatex.com-371fe833e9a89e2790fc66adb618b944_l3 | Leaving Certificate Examination 1954 Honours Applied Mathematics | Maths Grinds and quicklatex.com-3675ba693016b80ac93fbf734e15678b_l3 | Leaving Certificate Examination 1954 Honours Applied Mathematics | Maths Grinds attached to a fixed peg quicklatex.com-e86ae6aeacc8696c140ee1ca29d51f96_l3 | Leaving Certificate Examination 1954 Honours Applied Mathematics | Maths Grinds. If quicklatex.com-08a746ec26d5d11d21789f7b5c90bbe1_l3 | Leaving Certificate Examination 1954 Honours Applied Mathematics | Maths Grinds inches and quicklatex.com-93d30279aba4b0d1a831bc131b20e5e1_l3 | Leaving Certificate Examination 1954 Honours Applied Mathematics | Maths Grinds inches, find the tensions in the strings.

What weight attached at quicklatex.com-eaf6d3ddf3d9ff0d18eb3da055c8c94a_l3 | Leaving Certificate Examination 1954 Honours Applied Mathematics | Maths Grinds would maintain the bar in a horizontal position?

Question 2

A non-uniform beam rests with one end in contact with a rough horizontal plane (coefficient of friction quicklatex.com-a326c15ec641a65ba7ad05100f5fca5a_l3 | Leaving Certificate Examination 1954 Honours Applied Mathematics | Maths Grinds) and the other end in contact with a rough vertical wall (coefficient of friction quicklatex.com-294859fdbb1fdda439ec2363e11fdfa1_l3 | Leaving Certificate Examination 1954 Honours Applied Mathematics | Maths Grinds), the inclination of the beam to the horizontal being quicklatex.com-f643069adb772a4896813c16b313ad05_l3 | Leaving Certificate Examination 1954 Honours Applied Mathematics | Maths Grinds. If the beam is just about to slip, in a vertical plane, find the ratio in which the length of the beam is divided by the position of its centre of gravity.

Question 3

A car starts from rest and gathers speed. The following table gives the distance travelled by the car from its starting point in the corresponding time :-

Time (in seconds) quicklatex.com-dc2e717421b16a9f080dba4c8c0e3437_l3 | Leaving Certificate Examination 1954 Honours Applied Mathematics | Maths Grinds quicklatex.com-739f6781b65065fc445d5f13b544d6b2_l3 | Leaving Certificate Examination 1954 Honours Applied Mathematics | Maths Grinds quicklatex.com-8a0f60b8ca3ec09a3984d53508863c9a_l3 | Leaving Certificate Examination 1954 Honours Applied Mathematics | Maths Grinds quicklatex.com-2dda138cd42bede9cb97214fc5c8a205_l3 | Leaving Certificate Examination 1954 Honours Applied Mathematics | Maths Grinds quicklatex.com-66bafba23129fa8ab0a7d7fd2fc25d52_l3 | Leaving Certificate Examination 1954 Honours Applied Mathematics | Maths Grinds quicklatex.com-7bc4f1f0f4c11887bbe547ac256853e5_l3 | Leaving Certificate Examination 1954 Honours Applied Mathematics | Maths Grinds
Distance (in feet) quicklatex.com-7bc4f1f0f4c11887bbe547ac256853e5_l3 | Leaving Certificate Examination 1954 Honours Applied Mathematics | Maths Grinds quicklatex.com-cde6f09d3cc0e14c252edf4ac6855889_l3 | Leaving Certificate Examination 1954 Honours Applied Mathematics | Maths Grinds quicklatex.com-202f23f7e87c1cd3224295bc298e415d_l3 | Leaving Certificate Examination 1954 Honours Applied Mathematics | Maths Grinds quicklatex.com-00b9b07aba175f0367ac12c6b39491d3_l3 | Leaving Certificate Examination 1954 Honours Applied Mathematics | Maths Grinds quicklatex.com-0babb964ac0c509dcc9be3e56945cc50_l3 | Leaving Certificate Examination 1954 Honours Applied Mathematics | Maths Grinds quicklatex.com-5287c21a2871707c5dac212192e8d208_l3 | Leaving Certificate Examination 1954 Honours Applied Mathematics | Maths Grinds

Draw a distance-time curve and use it to estimate the velocity of the car at the end of (i) quicklatex.com-8a0f60b8ca3ec09a3984d53508863c9a_l3 | Leaving Certificate Examination 1954 Honours Applied Mathematics | Maths Grinds seconds (ii) quicklatex.com-2dda138cd42bede9cb97214fc5c8a205_l3 | Leaving Certificate Examination 1954 Honours Applied Mathematics | Maths Grinds seconds (iii) quicklatex.com-66bafba23129fa8ab0a7d7fd2fc25d52_l3 | Leaving Certificate Examination 1954 Honours Applied Mathematics | Maths Grinds seconds.

Plot those three values on a velocity-time curve, and so find approximately the acceleration of the car at the end of quicklatex.com-2dda138cd42bede9cb97214fc5c8a205_l3 | Leaving Certificate Examination 1954 Honours Applied Mathematics | Maths Grinds seconds.

Question 4

A circular disc of radius quicklatex.com-95d7c225dd31ad7d6dd5fb9f0980b531_l3 | Leaving Certificate Examination 1954 Honours Applied Mathematics | Maths Grinds is rolling in a straight line on a fixed horizontal plane, the centre of the disc having a uniform velocity quicklatex.com-8bafe7d16e660869fc203e8bffd77abc_l3 | Leaving Certificate Examination 1954 Honours Applied Mathematics | Maths Grinds. Assuming that no sliding takes place, find the angular velocity of the disc about its centre.

If a point quicklatex.com-d613350484b582bc83e2a6f9473437f8_l3 | Leaving Certificate Examination 1954 Honours Applied Mathematics | Maths Grinds is initially at the highest point of the disc, find the linear velocity of quicklatex.com-d613350484b582bc83e2a6f9473437f8_l3 | Leaving Certificate Examination 1954 Honours Applied Mathematics | Maths Grinds at time quicklatex.com-db2144d676b0c010fa61116d07db1982_l3 | Leaving Certificate Examination 1954 Honours Applied Mathematics | Maths Grinds in magnitude and direction – giving the inclination to the horizontal.

Show that at any instant there are two points on the rim of the disc whose linear velocities are both equal to quicklatex.com-8bafe7d16e660869fc203e8bffd77abc_l3 | Leaving Certificate Examination 1954 Honours Applied Mathematics | Maths Grinds in magnitude.

Question 5

Starting from rest at the foot of an incline of quicklatex.com-d39c7a34226b6a7fc7b5d3ae51b6b5e4_l3 | Leaving Certificate Examination 1954 Honours Applied Mathematics | Maths Grinds in quicklatex.com-4fbb80c3d7c3c9bdfaf780287e19f597_l3 | Leaving Certificate Examination 1954 Honours Applied Mathematics | Maths Grinds, a cyclist travels up the incline with uniform acceleration until he has reached a speed of quicklatex.com-8a0f60b8ca3ec09a3984d53508863c9a_l3 | Leaving Certificate Examination 1954 Honours Applied Mathematics | Maths Grinds m.p.h.; he then continues at a steady speed of quicklatex.com-8a0f60b8ca3ec09a3984d53508863c9a_l3 | Leaving Certificate Examination 1954 Honours Applied Mathematics | Maths Grinds m.p.h., so that quicklatex.com-7bc4f1f0f4c11887bbe547ac256853e5_l3 | Leaving Certificate Examination 1954 Honours Applied Mathematics | Maths Grinds seconds after starting from rest he has travelled quicklatex.com-f8d719ce8e2cfbd1840cc41fa7d54324_l3 | Leaving Certificate Examination 1954 Honours Applied Mathematics | Maths Grinds feet up the incline. Find his uniform acceleration.

If the cyclist and the cycle together weigh quicklatex.com-e2421346fe4da17e1d74a7cb53af7968_l3 | Leaving Certificate Examination 1954 Honours Applied Mathematics | Maths Grinds lb., and if the frictional resistances to motion are equivalent to quicklatex.com-1ee1c2caec46fc384f9e0e830a2e8b84_l3 | Leaving Certificate Examination 1954 Honours Applied Mathematics | Maths Grinds lb. wt., find the horse-power at which the cyclist was working when his speed was quicklatex.com-739f6781b65065fc445d5f13b544d6b2_l3 | Leaving Certificate Examination 1954 Honours Applied Mathematics | Maths Grinds m.p.h.

Question 6

A mass of quicklatex.com-1ee1c2caec46fc384f9e0e830a2e8b84_l3 | Leaving Certificate Examination 1954 Honours Applied Mathematics | Maths Grinds ounces is projected vertically upwards with an initial velocity of quicklatex.com-5291abebe16995c90201b72e133d15d8_l3 | Leaving Certificate Examination 1954 Honours Applied Mathematics | Maths Grinds ft. per sec. and quicklatex.com-1ee1c2caec46fc384f9e0e830a2e8b84_l3 | Leaving Certificate Examination 1954 Honours Applied Mathematics | Maths Grinds seconds later a mass of quicklatex.com-d39c7a34226b6a7fc7b5d3ae51b6b5e4_l3 | Leaving Certificate Examination 1954 Honours Applied Mathematics | Maths Grinds ounce is projected vertically upwards from the same point with an initial velocity of quicklatex.com-1c2411091e17c31612a16015839f2c94_l3 | Leaving Certificate Examination 1954 Honours Applied Mathematics | Maths Grinds ft. per sec. If the masses coalesce on colliding, in how many seconds after the collision will the combined mass return to the point of projection?

If the masses were projected as stated, except that each mass, at its time of projection, is given also a component horizontal velocity, show that the masses would still collide provided that the horizontal velocities were in the same direction and were in the ratio quicklatex.com-b22d62ce5a3f061d284e9e6e1da87e0f_l3 | Leaving Certificate Examination 1954 Honours Applied Mathematics | Maths Grinds.

Question 7

A motor cyclist is travelling at quicklatex.com-5535a4d4af99cfbac2f2ee888cea3393_l3 | Leaving Certificate Examination 1954 Honours Applied Mathematics | Maths Grinds m.p.h. round a curve of quicklatex.com-0a7d1908dc17ba7c399644cfaecdaeba_l3 | Leaving Certificate Examination 1954 Honours Applied Mathematics | Maths Grinds feet radius on a track which is inclined to the horizontal at an angle of quicklatex.com-76d5119d26c2f5f005c273dfa20b464e_l3 | Leaving Certificate Examination 1954 Honours Applied Mathematics | Maths Grinds (the plane of the curve being horizontal). Find the angle which the cyclist makes with the normal to the track.

If he can travel round the curve at quicklatex.com-9e7ec55fdcc4467668102a4d17def052_l3 | Leaving Certificate Examination 1954 Honours Applied Mathematics | Maths Grinds m.p.h. without skidding, find the least possible value of the coefficient of friction between the track and the tyres.

Question 8

Define Simple Harmonic Motion.

The tension in a spiral spring is directly proportional to the distance through which it is extended beyond its natural length. When a mass quicklatex.com-7a02bd7300eeadcf269af0186938941e_l3 | Leaving Certificate Examination 1954 Honours Applied Mathematics | Maths Grinds is suspended from it, the spring is extended a distance quicklatex.com-84fdf06eaf2b99889abd1c8acdc13d5f_l3 | Leaving Certificate Examination 1954 Honours Applied Mathematics | Maths Grinds beyond its natural length. If the mass is then pulled downwards from its equilibrium position and released, show that its subsequent motion is Simple Harmonic, and find the periodic time in terms of quicklatex.com-84fdf06eaf2b99889abd1c8acdc13d5f_l3 | Leaving Certificate Examination 1954 Honours Applied Mathematics | Maths Grinds.

Question 9

Prove that the pressure is the same at all points at the same depth in a liquid at rest.

A solid cylinder is quicklatex.com-fd6542ae3d49d38a0f73fefe83b285af_l3 | Leaving Certificate Examination 1954 Honours Applied Mathematics | Maths Grinds inches in height and has a diameter of quicklatex.com-1ee1c2caec46fc384f9e0e830a2e8b84_l3 | Leaving Certificate Examination 1954 Honours Applied Mathematics | Maths Grinds inches, and its specific gravity is quicklatex.com-5d27a2fcfd584eed2c1aae9861271ec3_l3 | Leaving Certificate Examination 1954 Honours Applied Mathematics | Maths Grinds. The cylinder is suspended by a vertical string so that it is totally immersed in a liquid of specific gravity quicklatex.com-598fb5a16d3def68572898357672eea3_l3 | Leaving Certificate Examination 1954 Honours Applied Mathematics | Maths Grinds. The axis of the cylinder is vertical and the top of the cylinder is quicklatex.com-ce939b244df556d77c12e315c6165e7d_l3 | Leaving Certificate Examination 1954 Honours Applied Mathematics | Maths Grinds inches below the surface of the liquid. Find the total thrust of the liquid (i) on the base of the cylinder, (ii) on the top of the cylinder; and find also the tension in the string.

 


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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