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20Dec2018

Leaving Cert Applied Maths Higher Level 1973

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Question 1

A cyclist has a maximum acceleration of quicklatex.com-ce939b244df556d77c12e315c6165e7d_l3 | Leaving Cert Applied Maths Higher Level 1973 | Maths Grinds m/squicklatex.com-025303222356a63c2332a429c33d35a9_l3 | Leaving Cert Applied Maths Higher Level 1973 | Maths Grinds, a maximum speed of quicklatex.com-8a0f60b8ca3ec09a3984d53508863c9a_l3 | Leaving Cert Applied Maths Higher Level 1973 | Maths Grinds m/s and a maximum deceleration of quicklatex.com-fd6542ae3d49d38a0f73fefe83b285af_l3 | Leaving Cert Applied Maths Higher Level 1973 | Maths Grinds m/squicklatex.com-025303222356a63c2332a429c33d35a9_l3 | Leaving Cert Applied Maths Higher Level 1973 | Maths Grinds. The cyclsit wishes to travel a distance quicklatex.com-273ac051ff0a53582365352ee5b6950d_l3 | Leaving Cert Applied Maths Higher Level 1973 | Maths Grinds from rest to rest in the shorest time. Find the time taken in the two cases

(i) quicklatex.com-cd16a644a383c473af62b28f4e67723a_l3 | Leaving Cert Applied Maths Higher Level 1973 | Maths Grinds m and

(ii) quicklatex.com-f427f8602625bb448331df59d09d5189_l3 | Leaving Cert Applied Maths Higher Level 1973 | Maths Grinds m.

Draw a rough velocity-time graph for each case and explain why quicklatex.com-7b8eb84790f02441c5fd14d2d19e5abd_l3 | Leaving Cert Applied Maths Higher Level 1973 | Maths Grinds m is a critical distance.

[Video Solution]

Question 2

A perfectly elastic particle falls vertically with speed quicklatex.com-ba05715987b9d18252648b0ecfc8eb72_l3 | Leaving Cert Applied Maths Higher Level 1973 | Maths Grinds on to a smooth plane inclined at an angle quicklatex.com-0dce58ae1ea930bb9c39e85ab5e4c977_l3 | Leaving Cert Applied Maths Higher Level 1973 | Maths Grinds to the horizontal, and rebounds, hopping down the plane.

i) Write down its displacement quicklatex.com-9d6cb6705ba8afb57cbf6daf3ee743dd_l3 | Leaving Cert Applied Maths Higher Level 1973 | Maths Grinds from the point of contact after time quicklatex.com-db2144d676b0c010fa61116d07db1982_l3 | Leaving Cert Applied Maths Higher Level 1973 | Maths Grinds in terms of quicklatex.com-f453dfa2ac10361897075d9950ad4c04_l3 | Leaving Cert Applied Maths Higher Level 1973 | Maths Grinds and quicklatex.com-be4ffaf4d71dac859eb0823be8c6649f_l3 | Leaving Cert Applied Maths Higher Level 1973 | Maths Grinds, where quicklatex.com-f453dfa2ac10361897075d9950ad4c04_l3 | Leaving Cert Applied Maths Higher Level 1973 | Maths Grinds is drawn directly down the plane.

ii) Show that the length of the first hop is quicklatex.com-5ff0ebbec1a26f770d5615c4c65470a4_l3 | Leaving Cert Applied Maths Higher Level 1973 | Maths Grinds and that

iii) the length of the second hop is double this.

[Video Solution]

Question 3

A ship quicklatex.com-cba4133f939edee31a8ce51d790a9cd0_l3 | Leaving Cert Applied Maths Higher Level 1973 | Maths Grinds is steaming with velocity quicklatex.com-8bafe7d16e660869fc203e8bffd77abc_l3 | Leaving Cert Applied Maths Higher Level 1973 | Maths Grinds m/s where quicklatex.com-a101fb2c06d0a690fc618d43b05fb151_l3 | Leaving Cert Applied Maths Higher Level 1973 | Maths Grinds where quicklatex.com-f453dfa2ac10361897075d9950ad4c04_l3 | Leaving Cert Applied Maths Higher Level 1973 | Maths Grinds and quicklatex.com-be4ffaf4d71dac859eb0823be8c6649f_l3 | Leaving Cert Applied Maths Higher Level 1973 | Maths Grinds are point East and North, respectively. At midday a second ship quicklatex.com-eaf6d3ddf3d9ff0d18eb3da055c8c94a_l3 | Leaving Cert Applied Maths Higher Level 1973 | Maths Grinds has a position quicklatex.com-5cf50f9f381de34ad5143f646a5eb972_l3 | Leaving Cert Applied Maths Higher Level 1973 | Maths Grinds km with respect to quicklatex.com-cba4133f939edee31a8ce51d790a9cd0_l3 | Leaving Cert Applied Maths Higher Level 1973 | Maths Grinds.

i) Find the minimum speed quicklatex.com-eaf6d3ddf3d9ff0d18eb3da055c8c94a_l3 | Leaving Cert Applied Maths Higher Level 1973 | Maths Grinds must have if it is to intercept quicklatex.com-cba4133f939edee31a8ce51d790a9cd0_l3 | Leaving Cert Applied Maths Higher Level 1973 | Maths Grinds.

ii) If the maximum speed of quicklatex.com-eaf6d3ddf3d9ff0d18eb3da055c8c94a_l3 | Leaving Cert Applied Maths Higher Level 1973 | Maths Grinds is in fact quicklatex.com-ee873c946a42a316d07b2d164e2b7d67_l3 | Leaving Cert Applied Maths Higher Level 1973 | Maths Grinds m/s, show that it can steer in either of two directions to intercept quicklatex.com-cba4133f939edee31a8ce51d790a9cd0_l3 | Leaving Cert Applied Maths Higher Level 1973 | Maths Grinds and

iii) calculate the two times of interception.

[Video Solution]

Question 4

HAM-1973-Q4-300x251 | Leaving Cert Applied Maths Higher Level 1973 | Maths Grinds

The diagram shows a light inelastic string with one end connected to a fixed point quicklatex.com-84fdf06eaf2b99889abd1c8acdc13d5f_l3 | Leaving Cert Applied Maths Higher Level 1973 | Maths Grinds of a ceiling, passing under a heavy movable pulley quicklatex.com-91dc9a245a0f25a550d41a6feb58273f_l3 | Leaving Cert Applied Maths Higher Level 1973 | Maths Grinds of mass quicklatex.com-739f6781b65065fc445d5f13b544d6b2_l3 | Leaving Cert Applied Maths Higher Level 1973 | Maths Grinds kg and then over a fixed pulley quicklatex.com-5403eeae09f3e9689d2ad28aa6fe1599_l3 | Leaving Cert Applied Maths Higher Level 1973 | Maths Grinds attached to the ceiling. To the other end quicklatex.com-fc85ef309591bf8b549446f3ebd0b450_l3 | Leaving Cert Applied Maths Higher Level 1973 | Maths Grinds of the string is attached a particle of mass quicklatex.com-739f6781b65065fc445d5f13b544d6b2_l3 | Leaving Cert Applied Maths Higher Level 1973 | Maths Grinds kg hanging freely.

i) Show in separate diagrams the forces acting on the particle and on the pulley quicklatex.com-91dc9a245a0f25a550d41a6feb58273f_l3 | Leaving Cert Applied Maths Higher Level 1973 | Maths Grinds when they are released from rest.

ii) Show that the acceleration of quicklatex.com-fc85ef309591bf8b549446f3ebd0b450_l3 | Leaving Cert Applied Maths Higher Level 1973 | Maths Grinds is double that of quicklatex.com-91dc9a245a0f25a550d41a6feb58273f_l3 | Leaving Cert Applied Maths Higher Level 1973 | Maths Grinds,

iii) and calculate the acceleration of quicklatex.com-fc85ef309591bf8b549446f3ebd0b450_l3 | Leaving Cert Applied Maths Higher Level 1973 | Maths Grinds and

iv) the tension in the string. (Neglect the inertia of both pulleys).

[Video Solution]

Question 5

A particle of mass quicklatex.com-739f6781b65065fc445d5f13b544d6b2_l3 | Leaving Cert Applied Maths Higher Level 1973 | Maths Grinds kg hangs freely from the end quicklatex.com-91dc9a245a0f25a550d41a6feb58273f_l3 | Leaving Cert Applied Maths Higher Level 1973 | Maths Grinds of a light inextensible string of length quicklatex.com-61cb8bf66d4f8d9b7f43c80c50913760_l3 | Leaving Cert Applied Maths Higher Level 1973 | Maths Grinds m which is attached at the other end to a fixed point quicklatex.com-2a137ec0f9eba36b0ab49cf15f39f89e_l3 | Leaving Cert Applied Maths Higher Level 1973 | Maths Grinds. The particle is then projected horizontally with speed quicklatex.com-fa8b3f237886602084d290b45f0944d1_l3 | Leaving Cert Applied Maths Higher Level 1973 | Maths Grinds m/s.

i) Show in a diagram the forces acting on the particle when quicklatex.com-6a89565a8f71e05c36efc14397e68165_l3 | Leaving Cert Applied Maths Higher Level 1973 | Maths Grinds is inclined at an angle quicklatex.com-d6ffefedf9d4e7e4f42ba65b9f3b9416_l3 | Leaving Cert Applied Maths Higher Level 1973 | Maths Grinds to the downward vertical assuming that it has a speed quicklatex.com-8bafe7d16e660869fc203e8bffd77abc_l3 | Leaving Cert Applied Maths Higher Level 1973 | Maths Grinds m/s at that point.

ii) Use conservation of energy to determine quicklatex.com-8bafe7d16e660869fc203e8bffd77abc_l3 | Leaving Cert Applied Maths Higher Level 1973 | Maths Grinds and

iii) express the tension in the string in terms of quicklatex.com-d6ffefedf9d4e7e4f42ba65b9f3b9416_l3 | Leaving Cert Applied Maths Higher Level 1973 | Maths Grinds.

iv) Find where the particle comes to instantaneous rest and

v) show that the tension in the string is then quicklatex.com-4f1d3075721bd7b8c0e6652c92aeb23e_l3 | Leaving Cert Applied Maths Higher Level 1973 | Maths Grinds N.

[Video Solution]

Question 6

A light string quicklatex.com-22557733158ceb5e857160898787bf8e_l3 | Leaving Cert Applied Maths Higher Level 1973 | Maths Grinds is attached to fixed points at its ends quicklatex.com-95d7c225dd31ad7d6dd5fb9f0980b531_l3 | Leaving Cert Applied Maths Higher Level 1973 | Maths Grinds and quicklatex.com-ba05715987b9d18252648b0ecfc8eb72_l3 | Leaving Cert Applied Maths Higher Level 1973 | Maths Grinds, so that quicklatex.com-ba05715987b9d18252648b0ecfc8eb72_l3 | Leaving Cert Applied Maths Higher Level 1973 | Maths Grinds is vertically below quicklatex.com-95d7c225dd31ad7d6dd5fb9f0980b531_l3 | Leaving Cert Applied Maths Higher Level 1973 | Maths Grinds. Particles of weights quicklatex.com-7260fe4f20eb47e11e44b79da133e21c_l3 | Leaving Cert Applied Maths Higher Level 1973 | Maths Grinds N and quicklatex.com-4fbb80c3d7c3c9bdfaf780287e19f597_l3 | Leaving Cert Applied Maths Higher Level 1973 | Maths Grinds N are attached to the string at quicklatex.com-273ac051ff0a53582365352ee5b6950d_l3 | Leaving Cert Applied Maths Higher Level 1973 | Maths Grinds and quicklatex.com-db2144d676b0c010fa61116d07db1982_l3 | Leaving Cert Applied Maths Higher Level 1973 | Maths Grinds, respectively, and a horizontal force quicklatex.com-ada3765a264ac03a253ee0ccbe61f602_l3 | Leaving Cert Applied Maths Higher Level 1973 | Maths Grinds newtons is applied to the particle at quicklatex.com-273ac051ff0a53582365352ee5b6950d_l3 | Leaving Cert Applied Maths Higher Level 1973 | Maths Grinds so that the string is in equilibrium in a vertical plane through quicklatex.com-eb84229bb7357bac3cb8e4676b79f5a5_l3 | Leaving Cert Applied Maths Higher Level 1973 | Maths Grinds with quicklatex.com-bae677cc60fcf5f7cbc24b6711a8f16c_l3 | Leaving Cert Applied Maths Higher Level 1973 | Maths Grinds and quicklatex.com-f65a23daaca3fe355aa2f321fd46eb1c_l3 | Leaving Cert Applied Maths Higher Level 1973 | Maths Grinds. Show in a separate diagrams the forces acting on the two particles and prove that quicklatex.com-dfad222c75feafae4e772a912de93d25_l3 | Leaving Cert Applied Maths Higher Level 1973 | Maths Grinds N.

[Video Solution]

Question 7

An equilateral triangle is quicklatex.com-4d1d2c6ad90757df37c3d5634f1b203f_l3 | Leaving Cert Applied Maths Higher Level 1973 | Maths Grinds is formed from three uniform rods, each of length quicklatex.com-7ff41b23cb36059c55f89f7da3a641a8_l3 | Leaving Cert Applied Maths Higher Level 1973 | Maths Grinds and weight quicklatex.com-054ef36c6e54f65ccdd586c0869b4fb0_l3 | Leaving Cert Applied Maths Higher Level 1973 | Maths Grinds, freely jointed at their ends. The triangle is freely suspended by a string attached to the midpoint quicklatex.com-2a137ec0f9eba36b0ab49cf15f39f89e_l3 | Leaving Cert Applied Maths Higher Level 1973 | Maths Grinds of quicklatex.com-e3bb89a77ef6d64bbfb26135dc1c497b_l3 | Leaving Cert Applied Maths Higher Level 1973 | Maths Grinds so that it hangs symmetrically under gravity with quicklatex.com-95d7c225dd31ad7d6dd5fb9f0980b531_l3 | Leaving Cert Applied Maths Higher Level 1973 | Maths Grinds vertically below quicklatex.com-2a137ec0f9eba36b0ab49cf15f39f89e_l3 | Leaving Cert Applied Maths Higher Level 1973 | Maths Grinds. Show in separate diagrams the forces acting on quicklatex.com-e3bb89a77ef6d64bbfb26135dc1c497b_l3 | Leaving Cert Applied Maths Higher Level 1973 | Maths Grinds and quicklatex.com-df3f82fdfa45b33b7d180ac9c32929ad_l3 | Leaving Cert Applied Maths Higher Level 1973 | Maths Grinds, and calculate the horizontal and vertical components of the reactions at quicklatex.com-95d7c225dd31ad7d6dd5fb9f0980b531_l3 | Leaving Cert Applied Maths Higher Level 1973 | Maths Grinds.

[Video Solution]

Question 8

i) Prove that the moments of inertia of a uniform circular disc of radius quicklatex.com-d87274adcc3e9b397c1d81918125c810_l3 | Leaving Cert Applied Maths Higher Level 1973 | Maths Grinds m and mass quicklatex.com-dc2e717421b16a9f080dba4c8c0e3437_l3 | Leaving Cert Applied Maths Higher Level 1973 | Maths Grinds kg about an axis quicklatex.com-cbc7d317fca5cc4213f58e1c8940e8ff_l3 | Leaving Cert Applied Maths Higher Level 1973 | Maths Grinds through its centre quicklatex.com-2a137ec0f9eba36b0ab49cf15f39f89e_l3 | Leaving Cert Applied Maths Higher Level 1973 | Maths Grinds and perpendicular to the disc is quicklatex.com-d87274adcc3e9b397c1d81918125c810_l3 | Leaving Cert Applied Maths Higher Level 1973 | Maths Grinds kg mquicklatex.com-025303222356a63c2332a429c33d35a9_l3 | Leaving Cert Applied Maths Higher Level 1973 | Maths Grinds.

[Video Solution]

ii) Such a disc can rotate freely about the axis quicklatex.com-cbc7d317fca5cc4213f58e1c8940e8ff_l3 | Leaving Cert Applied Maths Higher Level 1973 | Maths Grinds which is fixed horizontally. A light inextensible string is wound around the rim of the disc with one end attached to it, and to the other end is tied a particle quicklatex.com-81c6b3e443a83c026f06b718b1dedcef_l3 | Leaving Cert Applied Maths Higher Level 1973 | Maths Grinds of mass quicklatex.com-ce939b244df556d77c12e315c6165e7d_l3 | Leaving Cert Applied Maths Higher Level 1973 | Maths Grinds kg which hangs vertically. If the system is released from rest, show that the speed of quicklatex.com-81c6b3e443a83c026f06b718b1dedcef_l3 | Leaving Cert Applied Maths Higher Level 1973 | Maths Grinds is quicklatex.com-3e581ca7cdb128e6a4ac01eb84024b06_l3 | Leaving Cert Applied Maths Higher Level 1973 | Maths Grinds m/s after it has descended a distance quicklatex.com-6ebce5037cfda094c0dcc90ccac6d10f_l3 | Leaving Cert Applied Maths Higher Level 1973 | Maths Grinds m.

[Video Solution]

Question 9

An engine pulls a train along a level track against a resistance which at any time is quicklatex.com-dcb759b8d4a6dfe1538f628aa07cc94f_l3 | Leaving Cert Applied Maths Higher Level 1973 | Maths Grinds times the momentum. The engine works at a constant power quicklatex.com-be1bcbb5b847d5fe7033a31b2bffd525_l3 | Leaving Cert Applied Maths Higher Level 1973 | Maths Grinds, where quicklatex.com-941d689ad0e65c5d5733efadca408734_l3 | Leaving Cert Applied Maths Higher Level 1973 | Maths Grinds is the total mass of the train and engine and quicklatex.com-dcb759b8d4a6dfe1538f628aa07cc94f_l3 | Leaving Cert Applied Maths Higher Level 1973 | Maths Grinds, quicklatex.com-38e7af809a644a7a830693fc58d60bae_l3 | Leaving Cert Applied Maths Higher Level 1973 | Maths Grinds are constants. Show that the equation of motion of the train is

    quicklatex.com-df165d040443fdd71c3cc896441d433a_l3 | Leaving Cert Applied Maths Higher Level 1973 | Maths Grinds

and find the time taken to increase speed from quicklatex.com-38e7af809a644a7a830693fc58d60bae_l3 | Leaving Cert Applied Maths Higher Level 1973 | Maths Grinds to quicklatex.com-9c2f55d48139086d69afd876eed2611b_l3 | Leaving Cert Applied Maths Higher Level 1973 | Maths Grinds.

[Video Solution]

Question 10

a) State the principle of Archimedes.

[Video Solution]

b) A solid hemisphere of radius quicklatex.com-84fdf06eaf2b99889abd1c8acdc13d5f_l3 | Leaving Cert Applied Maths Higher Level 1973 | Maths Grinds is help submerged in a liquid of density quicklatex.com-ebfa669ee18b88d9ca3d1258b6c3d322_l3 | Leaving Cert Applied Maths Higher Level 1973 | Maths Grinds with its plane face horizontal and uppermost at a distance quicklatex.com-7ff41b23cb36059c55f89f7da3a641a8_l3 | Leaving Cert Applied Maths Higher Level 1973 | Maths Grinds below the free surface of the liquid. Calculate the magnitude, direction and line of action of

i) the force exerted by the liquid on the plane face,

ii) the total force exerted by the liquid on the surface of the solid.

[Video Solution]


Latest PSI Licence:

“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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