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1 Dec 2018

Leaving Cert Applied Maths Higher Level 1971

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Special Thanks: A copy of this Leaving Cert Applied Maths Higher Level 1971 exam was kindly provided by Noel Cunningham.

Question 1

a) Explain how a graph of velocity plotted against time can be used to calculate acceleration and distance travelled, with particular reference to motion with constant acceleration.

[Video Solution]

b) A pigeon in flight releases a small stone from its beak at a height of quicklatex.com-7260fe4f20eb47e11e44b79da133e21c_l3 | Leaving Cert Applied Maths Higher Level 1971 | Maths Grinds metres when its velocity is quicklatex.com-ba05715987b9d18252648b0ecfc8eb72_l3 | Leaving Cert Applied Maths Higher Level 1971 | Maths Grinds. If the stone takes quicklatex.com-6024d8b1a86917dc3144686a41ea70aa_l3 | Leaving Cert Applied Maths Higher Level 1971 | Maths Grinds seconds to reach the ground, show that the direction of quicklatex.com-ba05715987b9d18252648b0ecfc8eb72_l3 | Leaving Cert Applied Maths Higher Level 1971 | Maths Grinds is not horizontal and compute the greatest height reached by the stone after release. (Give your answer correct to the nearest tenth of a metre.)

[Video Solution]

Question 2

A particle is projected from a point quicklatex.com-5b090119e833ac90162777d07e3fe243_l3 | Leaving Cert Applied Maths Higher Level 1971 | Maths Grinds on a plane inclined at quicklatex.com-43254279a766115b4c8c77b4f9e086ff_l3 | Leaving Cert Applied Maths Higher Level 1971 | Maths Grinds to the horizontal with a velocity quicklatex.com-c29ec972f4ddd539fde7b3a0a669884f_l3 | Leaving Cert Applied Maths Higher Level 1971 | Maths Grinds metres/second where quicklatex.com-f453dfa2ac10361897075d9950ad4c04_l3 | Leaving Cert Applied Maths Higher Level 1971 | Maths Grinds is a unit vector through quicklatex.com-5b090119e833ac90162777d07e3fe243_l3 | Leaving Cert Applied Maths Higher Level 1971 | Maths Grinds pointing upward along the line of greatest slope in the plane and quicklatex.com-be4ffaf4d71dac859eb0823be8c6649f_l3 | Leaving Cert Applied Maths Higher Level 1971 | Maths Grinds is a unit vector perpendicular to the plane.

i) Show that after time quicklatex.com-db2144d676b0c010fa61116d07db1982_l3 | Leaving Cert Applied Maths Higher Level 1971 | Maths Grinds seconds the position vector, quicklatex.com-9d6cb6705ba8afb57cbf6daf3ee743dd_l3 | Leaving Cert Applied Maths Higher Level 1971 | Maths Grinds of the particle relative to quicklatex.com-5b090119e833ac90162777d07e3fe243_l3 | Leaving Cert Applied Maths Higher Level 1971 | Maths Grinds is given by

    quicklatex.com-3956dfe2321874cd5c2223f43acef1b2_l3 | Leaving Cert Applied Maths Higher Level 1971 | Maths Grinds

 

ii) Prove that the range on the inclined plane is quicklatex.com-e1fd848ceef13bf0e3f700ca4c5e5231_l3 | Leaving Cert Applied Maths Higher Level 1971 | Maths Grinds metres,

iii) and find the velocity of the particle when it strikes the plane.

[Video Solution]

Question 3

Two smooth spheres quicklatex.com-cba4133f939edee31a8ce51d790a9cd0_l3 | Leaving Cert Applied Maths Higher Level 1971 | Maths Grinds and quicklatex.com-eaf6d3ddf3d9ff0d18eb3da055c8c94a_l3 | Leaving Cert Applied Maths Higher Level 1971 | Maths Grinds of equal radii but of masses quicklatex.com-2dda138cd42bede9cb97214fc5c8a205_l3 | Leaving Cert Applied Maths Higher Level 1971 | Maths Grinds kg and quicklatex.com-739f6781b65065fc445d5f13b544d6b2_l3 | Leaving Cert Applied Maths Higher Level 1971 | Maths Grinds kg respectively collide on a smooth horizontal table. Before collision the velocity of quicklatex.com-cba4133f939edee31a8ce51d790a9cd0_l3 | Leaving Cert Applied Maths Higher Level 1971 | Maths Grinds is quicklatex.com-5d582d147ac71a244e7808d2aea4f422_l3 | Leaving Cert Applied Maths Higher Level 1971 | Maths Grinds m/s and the velocity of quicklatex.com-eaf6d3ddf3d9ff0d18eb3da055c8c94a_l3 | Leaving Cert Applied Maths Higher Level 1971 | Maths Grinds is quicklatex.com-48781440bc29193fb9bf51cfa0d3020c_l3 | Leaving Cert Applied Maths Higher Level 1971 | Maths Grinds m/s where quicklatex.com-f453dfa2ac10361897075d9950ad4c04_l3 | Leaving Cert Applied Maths Higher Level 1971 | Maths Grinds and quicklatex.com-be4ffaf4d71dac859eb0823be8c6649f_l3 | Leaving Cert Applied Maths Higher Level 1971 | Maths Grinds are unit perpendicular vectors in the plane of the table and quicklatex.com-f453dfa2ac10361897075d9950ad4c04_l3 | Leaving Cert Applied Maths Higher Level 1971 | Maths Grinds lies along the line of centres at impact. If the collision is perfectly elastic find the velocity of quicklatex.com-cba4133f939edee31a8ce51d790a9cd0_l3 | Leaving Cert Applied Maths Higher Level 1971 | Maths Grinds and the velocity of quicklatex.com-eaf6d3ddf3d9ff0d18eb3da055c8c94a_l3 | Leaving Cert Applied Maths Higher Level 1971 | Maths Grinds immediately afterwards.

[Video Solution]

Question 4

A light inextensible string which is fastened at one end to a point in the ceiling passes under a smooth movable pulley quicklatex.com-cba4133f939edee31a8ce51d790a9cd0_l3 | Leaving Cert Applied Maths Higher Level 1971 | Maths Grinds of mass quicklatex.com-e2160dde8a98fb45301e4e1674e830fa_l3 | Leaving Cert Applied Maths Higher Level 1971 | Maths Grinds kg, then over a smooth fixed pulley quicklatex.com-eaf6d3ddf3d9ff0d18eb3da055c8c94a_l3 | Leaving Cert Applied Maths Higher Level 1971 | Maths Grinds, and a particle quicklatex.com-e86ae6aeacc8696c140ee1ca29d51f96_l3 | Leaving Cert Applied Maths Higher Level 1971 | Maths Grinds of mass quicklatex.com-3883c8dfa7e1fd380cd2fa2674d982e2_l3 | Leaving Cert Applied Maths Higher Level 1971 | Maths Grinds kg hangs freely from its other side. All parts of the string which are not touching the pulleys are vertical.

LCAMH1971Q4-300x264 | Leaving Cert Applied Maths Higher Level 1971 | Maths Grinds

i) When the system is released from rest show that the acceleration of the particle is quicklatex.com-ce939b244df556d77c12e315c6165e7d_l3 | Leaving Cert Applied Maths Higher Level 1971 | Maths Grinds m/squicklatex.com-025303222356a63c2332a429c33d35a9_l3 | Leaving Cert Applied Maths Higher Level 1971 | Maths Grinds and that the acceleration of the pulley is half this.

ii) Find also the tension in the string.

[Video Solution]

Question 5

a) Define simple harmonic motion in a straight line and show it can be described by the differential equation quicklatex.com-7c7664841cf1e9a57df1b6c2e595f472_l3 | Leaving Cert Applied Maths Higher Level 1971 | Maths Grinds. Prove that quicklatex.com-d840d53b50d94e6d58818a4025f3eb6e_l3 | Leaving Cert Applied Maths Higher Level 1971 | Maths Grinds, where quicklatex.com-cba4133f939edee31a8ce51d790a9cd0_l3 | Leaving Cert Applied Maths Higher Level 1971 | Maths Grinds is a constant, is a solution of this equation.

[Video Solution]

b) A particle describing simple harmonic motion on a straight line has velocities quicklatex.com-fd6542ae3d49d38a0f73fefe83b285af_l3 | Leaving Cert Applied Maths Higher Level 1971 | Maths Grinds m/s and quicklatex.com-ce939b244df556d77c12e315c6165e7d_l3 | Leaving Cert Applied Maths Higher Level 1971 | Maths Grinds m/s when at a distance of quicklatex.com-d39c7a34226b6a7fc7b5d3ae51b6b5e4_l3 | Leaving Cert Applied Maths Higher Level 1971 | Maths Grinds m and quicklatex.com-ce939b244df556d77c12e315c6165e7d_l3 | Leaving Cert Applied Maths Higher Level 1971 | Maths Grinds m respectively from the centre of oscillation.

i) Find the amplitude and periodic time of the motion.

ii) Calculate the least time taken for the particle to travel from the position of rest to the point where the velocity was quicklatex.com-ce939b244df556d77c12e315c6165e7d_l3 | Leaving Cert Applied Maths Higher Level 1971 | Maths Grinds m/s.

[Video Solution]

Question 6

a) i) Show that quicklatex.com-7ac8767a3aff8b50c78b6016c3eef5fd_l3 | Leaving Cert Applied Maths Higher Level 1971 | Maths Grinds, where quicklatex.com-cba4133f939edee31a8ce51d790a9cd0_l3 | Leaving Cert Applied Maths Higher Level 1971 | Maths Grinds, quicklatex.com-6fd4a229a60f10645306b68a0a0ee2e6_l3 | Leaving Cert Applied Maths Higher Level 1971 | Maths Grinds are constants and quicklatex.com-f453dfa2ac10361897075d9950ad4c04_l3 | Leaving Cert Applied Maths Higher Level 1971 | Maths Grinds, quicklatex.com-be4ffaf4d71dac859eb0823be8c6649f_l3 | Leaving Cert Applied Maths Higher Level 1971 | Maths Grinds are constant unit perpendicular vectors, is the position vector at time quicklatex.com-db2144d676b0c010fa61116d07db1982_l3 | Leaving Cert Applied Maths Higher Level 1971 | Maths Grinds of a particle moving in a circle.

ii) Show that the velocity quicklatex.com-8bafe7d16e660869fc203e8bffd77abc_l3 | Leaving Cert Applied Maths Higher Level 1971 | Maths Grinds is of magnitude quicklatex.com-48da23e041d79460ae9bc6fec3b47d70_l3 | Leaving Cert Applied Maths Higher Level 1971 | Maths Grinds and is at right angles to quicklatex.com-9d6cb6705ba8afb57cbf6daf3ee743dd_l3 | Leaving Cert Applied Maths Higher Level 1971 | Maths Grinds.

iii) Prove that the acceleration is of magnitude quicklatex.com-a6ecabb580bbf8ddf1671651722197e7_l3 | Leaving Cert Applied Maths Higher Level 1971 | Maths Grinds and acts towards the centre of the circle.

[Video Solution]

b) A particle of mass quicklatex.com-739f6781b65065fc445d5f13b544d6b2_l3 | Leaving Cert Applied Maths Higher Level 1971 | Maths Grinds kg is describing a circle on a smooth horizontal table. It is connected by a light inelastic string of length quicklatex.com-9a61841f2abe3e6f5e5b707bbfade7f4_l3 | Leaving Cert Applied Maths Higher Level 1971 | Maths Grinds m to a point which is quicklatex.com-d87274adcc3e9b397c1d81918125c810_l3 | Leaving Cert Applied Maths Higher Level 1971 | Maths Grinds m vertically above the centre of the circle. If the reaction of the table on the mass is quicklatex.com-ced9adf500abe3e625f021cf18429a9e_l3 | Leaving Cert Applied Maths Higher Level 1971 | Maths Grinds N, calculate:

i) the constant angular velocity of the particle,

(ii) the tension of the string.

[Video Solution]

Question 7

a) Two rods quicklatex.com-cacf86ec54fe47688ea7a15a4ca0d7d0_l3 | Leaving Cert Applied Maths Higher Level 1971 | Maths Grinds, quicklatex.com-f8c6b4193243000bcbfe888197b871ca_l3 | Leaving Cert Applied Maths Higher Level 1971 | Maths Grinds, of equal lengths but of weights quicklatex.com-b0d27cf6cd65cce748ac61ffa5c19bbe_l3 | Leaving Cert Applied Maths Higher Level 1971 | Maths Grinds and quicklatex.com-054ef36c6e54f65ccdd586c0869b4fb0_l3 | Leaving Cert Applied Maths Higher Level 1971 | Maths Grinds, respectively, are freely hinged together at quicklatex.com-c8abc5e5c1ec96e74c27c739f796cedb_l3 | Leaving Cert Applied Maths Higher Level 1971 | Maths Grinds. They stand in equilibrium in a vertical plane with the ends quicklatex.com-05d30bc5e19b413cea824383c6bce148_l3 | Leaving Cert Applied Maths Higher Level 1971 | Maths Grinds and quicklatex.com-a6a7269cfeb6bd1b31d6d6ca7a75489b_l3 | Leaving Cert Applied Maths Higher Level 1971 | Maths Grinds on a rough horizontal plane with the angle quicklatex.com-6fcb9a5bbe28efc6b830fe33ef42b9a2_l3 | Leaving Cert Applied Maths Higher Level 1971 | Maths Grinds equal to quicklatex.com-a167991e6500d652c6496c53e44145a7_l3 | Leaving Cert Applied Maths Higher Level 1971 | Maths Grinds.

i) Show that the vertical components of the action of the plane at quicklatex.com-05d30bc5e19b413cea824383c6bce148_l3 | Leaving Cert Applied Maths Higher Level 1971 | Maths Grinds and quicklatex.com-a6a7269cfeb6bd1b31d6d6ca7a75489b_l3 | Leaving Cert Applied Maths Higher Level 1971 | Maths Grinds are quicklatex.com-405bccbf94c76f35ebf8a5412eea5266_l3 | Leaving Cert Applied Maths Higher Level 1971 | Maths Grinds and quicklatex.com-1f578bfea3e0d82ea207605471dcd7c8_l3 | Leaving Cert Applied Maths Higher Level 1971 | Maths Grinds respectively.

ii) Show in a separate diagram the forces acting on each rod, using vertical and horizontal components for the reaction at quicklatex.com-c8abc5e5c1ec96e74c27c739f796cedb_l3 | Leaving Cert Applied Maths Higher Level 1971 | Maths Grinds. Find all these forces and prove that if one rod is on the point of slipping, the coefficient of friction is quicklatex.com-de421505bec6889a37d0084cff4ee399_l3 | Leaving Cert Applied Maths Higher Level 1971 | Maths Grinds.

[Video Solution]

Question 8

i) Prove that the moment of inertia of a uniform circular lamina of mass quicklatex.com-941d689ad0e65c5d5733efadca408734_l3 | Leaving Cert Applied Maths Higher Level 1971 | Maths Grinds and radius quicklatex.com-95d7c225dd31ad7d6dd5fb9f0980b531_l3 | Leaving Cert Applied Maths Higher Level 1971 | Maths Grinds about an axis  through its centre quicklatex.com-5403eeae09f3e9689d2ad28aa6fe1599_l3 | Leaving Cert Applied Maths Higher Level 1971 | Maths Grinds, perpendicular to the plane of the lamina, is quicklatex.com-0b86d03dd9f401c8c95d16c246e15c9a_l3 | Leaving Cert Applied Maths Higher Level 1971 | Maths Grinds.

[Video Solution]

ii) Deduce that the moment of inertia about a parallel axis through a point of the circumference is quicklatex.com-096e624d5fe993a62b7c55fcd8b5142a_l3 | Leaving Cert Applied Maths Higher Level 1971 | Maths Grinds.

[Video Solution]

iii) Such a lamina is free to rotate in a vertical plane about a horizontal axis perpendicular to its plane through a point quicklatex.com-84fdf06eaf2b99889abd1c8acdc13d5f_l3 | Leaving Cert Applied Maths Higher Level 1971 | Maths Grinds on its circumference. It is released from rest with quicklatex.com-dbc51c69debe1b39c6347aca12bad038_l3 | Leaving Cert Applied Maths Higher Level 1971 | Maths Grinds horizontal. Find the angular velocity of the lamina where quicklatex.com-dbc51c69debe1b39c6347aca12bad038_l3 | Leaving Cert Applied Maths Higher Level 1971 | Maths Grinds makes an angle quicklatex.com-d6ffefedf9d4e7e4f42ba65b9f3b9416_l3 | Leaving Cert Applied Maths Higher Level 1971 | Maths Grinds with the download vertical, show that the speed of quicklatex.com-5403eeae09f3e9689d2ad28aa6fe1599_l3 | Leaving Cert Applied Maths Higher Level 1971 | Maths Grinds at its lower point is quicklatex.com-42f383f88a18c74bb834bd4daf0c2ec0_l3 | Leaving Cert Applied Maths Higher Level 1971 | Maths Grinds.

[Video Solution]

Question 9

a ) State Archimedes Principle for a body immersed in a liquid.

[Video Solution]

b) A uniform rod quicklatex.com-819964b5440e1d7a1941259fada7d9c6_l3 | Leaving Cert Applied Maths Higher Level 1971 | Maths Grinds in equilibrium is inclined to the vertical with one quarter of its length immersed under water and its upper end quicklatex.com-84fdf06eaf2b99889abd1c8acdc13d5f_l3 | Leaving Cert Applied Maths Higher Level 1971 | Maths Grinds supported by a vertical force quicklatex.com-8c28ca9825a5e0ff6a01cc15dd4efff1_l3 | Leaving Cert Applied Maths Higher Level 1971 | Maths Grinds.

i) Show that in a diagram the three force acting on the rod, and prove that the specific gravity of the rod is quicklatex.com-610512f905feccdb839d51164f556fc4_l3 | Leaving Cert Applied Maths Higher Level 1971 | Maths Grinds.

ii) Express quicklatex.com-8c28ca9825a5e0ff6a01cc15dd4efff1_l3 | Leaving Cert Applied Maths Higher Level 1971 | Maths Grinds in terms of quicklatex.com-054ef36c6e54f65ccdd586c0869b4fb0_l3 | Leaving Cert Applied Maths Higher Level 1971 | Maths Grinds, the weight of the rod.

[Video Solution]

Question 10

A particle of mass quicklatex.com-941d689ad0e65c5d5733efadca408734_l3 | Leaving Cert Applied Maths Higher Level 1971 | Maths Grinds is projected with speed quicklatex.com-ba05715987b9d18252648b0ecfc8eb72_l3 | Leaving Cert Applied Maths Higher Level 1971 | Maths Grinds along a smooth horizontal table. The air resistance to motion when the speed of the particle is quicklatex.com-8bafe7d16e660869fc203e8bffd77abc_l3 | Leaving Cert Applied Maths Higher Level 1971 | Maths Grinds, is quicklatex.com-0b0013e2672a276eb0889d8f3bb33299_l3 | Leaving Cert Applied Maths Higher Level 1971 | Maths Grinds where quicklatex.com-66c806215945e3dc90f658b54f6dcae9_l3 | Leaving Cert Applied Maths Higher Level 1971 | Maths Grinds is a constant.

i) By solving the equation of motion for the particle, show that quicklatex.com-d620c3df0e893e9ef137d4882e14111e_l3 | Leaving Cert Applied Maths Higher Level 1971 | Maths Grinds and

ii) prove that as time increases indefinitely, the distance travelled ultimately approaches quicklatex.com-ebdffcee55a353258ea568ca1843231d_l3 | Leaving Cert Applied Maths Higher Level 1971 | Maths Grinds.

iii) Find the time taken for the particle to travel half this distance.

[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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