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31 Jul 2019

Leaving Cert Applied Maths Higher Level 1979

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Question 1

i) How may a velocity-time graph be used to find the distance travelled in a given time?

ii) An athlete runs quicklatex.com-4fbb80c3d7c3c9bdfaf780287e19f597_l3 | Leaving Cert Applied Maths Higher Level 1979 | Maths Grinds m in quicklatex.com-ed5decd492fdc6648538222379ceaf27_l3 | Leaving Cert Applied Maths Higher Level 1979 | Maths Grinds seconds. Starting from rest, he accelerates uniformly to a speed of quicklatex.com-739f6781b65065fc445d5f13b544d6b2_l3 | Leaving Cert Applied Maths Higher Level 1979 | Maths Grinds m/s. and then continues at that speed. Calculate the acceleration.

iii) A body starting from rest travels in a straight line, first with uniform acceleration quicklatex.com-84fdf06eaf2b99889abd1c8acdc13d5f_l3 | Leaving Cert Applied Maths Higher Level 1979 | Maths Grinds and then with uniform deceleration quicklatex.com-91dc9a245a0f25a550d41a6feb58273f_l3 | Leaving Cert Applied Maths Higher Level 1979 | Maths Grinds. It comes to rest when it has covered a total distance quicklatex.com-7e66f8d3fbb50779ceb68e5184deb32d_l3 | Leaving Cert Applied Maths Higher Level 1979 | Maths Grinds. If the overall time for journey is quicklatex.com-6460d10729af641188d49a36153c8def_l3 | Leaving Cert Applied Maths Higher Level 1979 | Maths Grinds, show that

    quicklatex.com-83d59dd4e7e4e7ef321b608619036688_l3 | Leaving Cert Applied Maths Higher Level 1979 | Maths Grinds

Question 2

A ship quicklatex.com-cba4133f939edee31a8ce51d790a9cd0_l3 | Leaving Cert Applied Maths Higher Level 1979 | Maths Grinds is travelling South-West at quicklatex.com-739f6781b65065fc445d5f13b544d6b2_l3 | Leaving Cert Applied Maths Higher Level 1979 | Maths Grinds knots. Another ship quicklatex.com-eaf6d3ddf3d9ff0d18eb3da055c8c94a_l3 | Leaving Cert Applied Maths Higher Level 1979 | Maths Grinds is travelling at quicklatex.com-2dda138cd42bede9cb97214fc5c8a205_l3 | Leaving Cert Applied Maths Higher Level 1979 | Maths Grinds knots in a direction quicklatex.com-f643069adb772a4896813c16b313ad05_l3 | Leaving Cert Applied Maths Higher Level 1979 | Maths Grinds North of West.

i) Draw a diagram to show the velocity of quicklatex.com-eaf6d3ddf3d9ff0d18eb3da055c8c94a_l3 | Leaving Cert Applied Maths Higher Level 1979 | Maths Grinds relative to quicklatex.com-cba4133f939edee31a8ce51d790a9cd0_l3 | Leaving Cert Applied Maths Higher Level 1979 | Maths Grinds.

ii) Calculate the magnitude of the relative velocity, correct to the nearest knot, and its direction correct to the nearest degree.

iii) By how much should quicklatex.com-cba4133f939edee31a8ce51d790a9cd0_l3 | Leaving Cert Applied Maths Higher Level 1979 | Maths Grinds increase its speed, without changing direction, so that quicklatex.com-eaf6d3ddf3d9ff0d18eb3da055c8c94a_l3 | Leaving Cert Applied Maths Higher Level 1979 | Maths Grinds would appear to quicklatex.com-cba4133f939edee31a8ce51d790a9cd0_l3 | Leaving Cert Applied Maths Higher Level 1979 | Maths Grinds to be travelling due North?

Question 3

a) Explain the terms: coefficient of friction, angle of frcition. What is the relationship between them?

b) A uniform ladder of weight quicklatex.com-054ef36c6e54f65ccdd586c0869b4fb0_l3 | Leaving Cert Applied Maths Higher Level 1979 | Maths Grinds rests with one end against a smooth vertical wall and teh other end on rough ground which slopes away from the wall at an angle quicklatex.com-0dce58ae1ea930bb9c39e85ab5e4c977_l3 | Leaving Cert Applied Maths Higher Level 1979 | Maths Grinds to the horizontal (seed diagram). The ladder makes an angle quicklatex.com-d6ffefedf9d4e7e4f42ba65b9f3b9416_l3 | Leaving Cert Applied Maths Higher Level 1979 | Maths Grinds with the wall.

i) Show that the reaction at the wall is quicklatex.com-9ee725e8945317775efc4ea4590b12b7_l3 | Leaving Cert Applied Maths Higher Level 1979 | Maths Grinds.

ii) If the ladder is on the point of slipping prove that

    quicklatex.com-e36ce114fdde05a1792590396c99ec6f_l3 | Leaving Cert Applied Maths Higher Level 1979 | Maths Grinds

where quicklatex.com-0dce58ae1ea930bb9c39e85ab5e4c977_l3 | Leaving Cert Applied Maths Higher Level 1979 | Maths Grinds is the angle of friction.

HAM-1979-Q3 | Leaving Cert Applied Maths Higher Level 1979 | Maths Grinds

Question 4

a) A see-saw consits of a uniform plank freely pivoted at its mid-point on a support of vertical height quicklatex.com-63ab7de62920c09a3246a8ce084ab3ce_l3 | Leaving Cert Applied Maths Higher Level 1979 | Maths Grinds. It carries a mass quicklatex.com-7a02bd7300eeadcf269af0186938941e_l3 | Leaving Cert Applied Maths Higher Level 1979 | Maths Grinds at one end and a mass quicklatex.com-bc90a078f248740eba6949305c963ffc_l3 | Leaving Cert Applied Maths Higher Level 1979 | Maths Grinds at the other. The see-saw is released from rest with the mass quicklatex.com-bc90a078f248740eba6949305c963ffc_l3 | Leaving Cert Applied Maths Higher Level 1979 | Maths Grinds at its highest point. Find, in terms of quicklatex.com-63ab7de62920c09a3246a8ce084ab3ce_l3 | Leaving Cert Applied Maths Higher Level 1979 | Maths Grinds, the velocity with which the see-saw strikes the ground.

b) State the Principle of Archimedes.

c) A uniform road of length quicklatex.com-7e66f8d3fbb50779ceb68e5184deb32d_l3 | Leaving Cert Applied Maths Higher Level 1979 | Maths Grinds and relative density quicklatex.com-273ac051ff0a53582365352ee5b6950d_l3 | Leaving Cert Applied Maths Higher Level 1979 | Maths Grinds is freely hinged to the base of a tank containing a liquid to a depth quicklatex.com-63ab7de62920c09a3246a8ce084ab3ce_l3 | Leaving Cert Applied Maths Higher Level 1979 | Maths Grinds. The relative density of the liquid is quicklatex.com-66c806215945e3dc90f658b54f6dcae9_l3 | Leaving Cert Applied Maths Higher Level 1979 | Maths Grinds. As a result, the rod is inclined but not fully submerged. Dervive an expression for the angle the rod makes with the horizontal.

Question 5

A plane is inclined at an angle quicklatex.com-0dce58ae1ea930bb9c39e85ab5e4c977_l3 | Leaving Cert Applied Maths Higher Level 1979 | Maths Grinds to the horizontal. A particle is projected up the plane with a velocity quicklatex.com-ba05715987b9d18252648b0ecfc8eb72_l3 | Leaving Cert Applied Maths Higher Level 1979 | Maths Grinds at an angle quicklatex.com-d6ffefedf9d4e7e4f42ba65b9f3b9416_l3 | Leaving Cert Applied Maths Higher Level 1979 | Maths Grinds to the plane. The plane of projection is vertical and contains the line of greatest slope.

i) Show that the time of flight is quicklatex.com-7b745caa617bee72be63050c3bbbf6ba_l3 | Leaving Cert Applied Maths Higher Level 1979 | Maths Grinds.

ii) Prove that the range up the plane is a maximum when quicklatex.com-a9a0d73c3aace09f466765118141c5bf_l3 | Leaving Cert Applied Maths Higher Level 1979 | Maths Grinds.

iii) Prove that the particle will strike the plane horizontally if quicklatex.com-15f7809a49605a620ffd88d47294a3cb_l3 | Leaving Cert Applied Maths Higher Level 1979 | Maths Grinds.

Question 6

a) A particle, moving at constant speed, is describing a horizontal circle on the inside surface of a smooth sphere of radius quicklatex.com-95d7c225dd31ad7d6dd5fb9f0980b531_l3 | Leaving Cert Applied Maths Higher Level 1979 | Maths Grinds. The centre of the circle is a distance quicklatex.com-97de682d736be83e69dc03503f627e52_l3 | Leaving Cert Applied Maths Higher Level 1979 | Maths Grinds below the centre of the sphere. Prove that the speed of the particle is quicklatex.com-ad1aeb8ed2801bbf32e6b53e840fbcca_l3 | Leaving Cert Applied Maths Higher Level 1979 | Maths Grinds.

b) A conical pendulum consists of a light elastic string with a mass quicklatex.com-7a02bd7300eeadcf269af0186938941e_l3 | Leaving Cert Applied Maths Higher Level 1979 | Maths Grinds attached to it which is rotating with uniform angular velocity quicklatex.com-6fd4a229a60f10645306b68a0a0ee2e6_l3 | Leaving Cert Applied Maths Higher Level 1979 | Maths Grinds. The natural length of the string is quicklatex.com-7e66f8d3fbb50779ceb68e5184deb32d_l3 | Leaving Cert Applied Maths Higher Level 1979 | Maths Grinds and its elastic constant is quicklatex.com-66c806215945e3dc90f658b54f6dcae9_l3 | Leaving Cert Applied Maths Higher Level 1979 | Maths Grinds, i.e. a force quicklatex.com-66c806215945e3dc90f658b54f6dcae9_l3 | Leaving Cert Applied Maths Higher Level 1979 | Maths Grinds produces unit extension. The extended length of the string is quicklatex.com-f83f92e258f12aa2c43f971a9ef974e5_l3 | Leaving Cert Applied Maths Higher Level 1979 | Maths Grinds and it makes an angle quicklatex.com-d6ffefedf9d4e7e4f42ba65b9f3b9416_l3 | Leaving Cert Applied Maths Higher Level 1979 | Maths Grinds with the vertical. Prove that quicklatex.com-bb4e3edaf390b5f754053fb11b56e7f6_l3 | Leaving Cert Applied Maths Higher Level 1979 | Maths Grinds and that

    quicklatex.com-0262ce4a879eb8c8b0cc33c06de0c9ed_l3 | Leaving Cert Applied Maths Higher Level 1979 | Maths Grinds

Question 7

a) Two smooth spheres of masses quicklatex.com-7a02bd7300eeadcf269af0186938941e_l3 | Leaving Cert Applied Maths Higher Level 1979 | Maths Grinds and quicklatex.com-bc90a078f248740eba6949305c963ffc_l3 | Leaving Cert Applied Maths Higher Level 1979 | Maths Grinds collide directly when moving in opposite directions with speeds quicklatex.com-ba05715987b9d18252648b0ecfc8eb72_l3 | Leaving Cert Applied Maths Higher Level 1979 | Maths Grinds and quicklatex.com-8bafe7d16e660869fc203e8bffd77abc_l3 | Leaving Cert Applied Maths Higher Level 1979 | Maths Grinds, respectively. The sphere of mass quicklatex.com-bc90a078f248740eba6949305c963ffc_l3 | Leaving Cert Applied Maths Higher Level 1979 | Maths Grinds is brought to rest by the impact. Prove that

    quicklatex.com-860d4d19670696b93a862e98a29b18e8_l3 | Leaving Cert Applied Maths Higher Level 1979 | Maths Grinds

where quicklatex.com-23159d6255ee873d52dfd5b58bf2bf57_l3 | Leaving Cert Applied Maths Higher Level 1979 | Maths Grinds is the coefficient of restitution.

b) A smooth sphere quicklatex.com-cba4133f939edee31a8ce51d790a9cd0_l3 | Leaving Cert Applied Maths Higher Level 1979 | Maths Grinds collides obliquely with another smooth sphere of equal mass which is at rest. Before impact the direction of motion of quicklatex.com-cba4133f939edee31a8ce51d790a9cd0_l3 | Leaving Cert Applied Maths Higher Level 1979 | Maths Grinds makes an angle quicklatex.com-0dce58ae1ea930bb9c39e85ab5e4c977_l3 | Leaving Cert Applied Maths Higher Level 1979 | Maths Grinds with the line of centres at impact (see diagram). After impact it makes an angle quicklatex.com-402735c0bdab0b3477755624fc9b4d06_l3 | Leaving Cert Applied Maths Higher Level 1979 | Maths Grinds with that line. If the coefficient of restitution is quicklatex.com-294859fdbb1fdda439ec2363e11fdfa1_l3 | Leaving Cert Applied Maths Higher Level 1979 | Maths Grinds, prove that

    quicklatex.com-19e849bd8ef9093a252c5fff5e5366d9_l3 | Leaving Cert Applied Maths Higher Level 1979 | Maths Grinds

HAM-1979-Q7-300x261 | Leaving Cert Applied Maths Higher Level 1979 | Maths Grinds

Question 8

a) Define simple harmonic motion. Using the usual notation, show that the equation quicklatex.com-c7bf8c7f0d08d370ddd01317ba215fd7_l3 | Leaving Cert Applied Maths Higher Level 1979 | Maths Grinds represents simple harmonic motion.

A body is moving with simple harmonic motion, of amplitude quicklatex.com-dc2e717421b16a9f080dba4c8c0e3437_l3 | Leaving Cert Applied Maths Higher Level 1979 | Maths Grinds m. When it is quicklatex.com-fd6542ae3d49d38a0f73fefe83b285af_l3 | Leaving Cert Applied Maths Higher Level 1979 | Maths Grinds m from the mid-point of its path its speed is quicklatex.com-ee873c946a42a316d07b2d164e2b7d67_l3 | Leaving Cert Applied Maths Higher Level 1979 | Maths Grinds m/s. Find its speed when it is quicklatex.com-165ae8e53b023536cd30bc2f6a21fa2f_l3 | Leaving Cert Applied Maths Higher Level 1979 | Maths Grinds m from the mid-point.

b) A block rests on a rough platform which moves to and fro horizontally with simple harmonic motion. The amplitude is quicklatex.com-9475a7832812f0f97f3420b898bcbfc7_l3 | Leaving Cert Applied Maths Higher Level 1979 | Maths Grinds m and quicklatex.com-2dda138cd42bede9cb97214fc5c8a205_l3 | Leaving Cert Applied Maths Higher Level 1979 | Maths Grinds complete oscillations occur per minute. If the block remains at rest relative to the platform throughhout the motion, find the least possible value the coefficient of friction can have.

Question 9

a) State the theorem of parallel axes.

b) Prove that the moment of inertia of a uniform rod of mass quicklatex.com-7a02bd7300eeadcf269af0186938941e_l3 | Leaving Cert Applied Maths Higher Level 1979 | Maths Grinds and length quicklatex.com-0e440a33f92d8dd692a299e70ffc00e6_l3 | Leaving Cert Applied Maths Higher Level 1979 | Maths Grinds about a perpendicular axis through one of its end is quicklatex.com-6ae4ffafaa5ddc55720005f37ed7c0b7_l3 | Leaving Cert Applied Maths Higher Level 1979 | Maths Grinds.

c) A uniform rod of length quicklatex.com-2912efae9bab6a509e798bbd356587e9_l3 | Leaving Cert Applied Maths Higher Level 1979 | Maths Grinds is attached to the rim of a uniform disc of diameter quicklatex.com-0e440a33f92d8dd692a299e70ffc00e6_l3 | Leaving Cert Applied Maths Higher Level 1979 | Maths Grinds. The rod is collinear with a diameter of the disc (see diagram). The disc and the rod are both of mass quicklatex.com-7a02bd7300eeadcf269af0186938941e_l3 | Leaving Cert Applied Maths Higher Level 1979 | Maths Grinds. i) Calculate the moment of inertia of the compound body about a perpendicular axis through the end quicklatex.com-cba4133f939edee31a8ce51d790a9cd0_l3 | Leaving Cert Applied Maths Higher Level 1979 | Maths Grinds.

ii) If the compound body makes small oscillations in a vertical plane about a horizontal axis through quicklatex.com-cba4133f939edee31a8ce51d790a9cd0_l3 | Leaving Cert Applied Maths Higher Level 1979 | Maths Grinds, show that the periodic time is quicklatex.com-89424e1a2f4f041880a169a94f7b5c38_l3 | Leaving Cert Applied Maths Higher Level 1979 | Maths Grinds.

HAM-1979-Q9 | Leaving Cert Applied Maths Higher Level 1979 | Maths Grinds

Question 10

a) Solve the differential equation

    quicklatex.com-9345ea3396d277d47b23f6496d7fed3f_l3 | Leaving Cert Applied Maths Higher Level 1979 | Maths Grinds

Hence or otherwise solve

    quicklatex.com-83ede94a417b189f641fc1b596a7bdc0_l3 | Leaving Cert Applied Maths Higher Level 1979 | Maths Grinds

where quicklatex.com-55132ac813a32dab57735b6282f1843e_l3 | Leaving Cert Applied Maths Higher Level 1979 | Maths Grinds when quicklatex.com-8ed93fef0596d8393d81a24553e60308_l3 | Leaving Cert Applied Maths Higher Level 1979 | Maths Grinds and quicklatex.com-2b0aa608587e67b7c094729b12e2989b_l3 | Leaving Cert Applied Maths Higher Level 1979 | Maths Grinds when quicklatex.com-74f83f9fe2a1c04bb6ec97094c9c3122_l3 | Leaving Cert Applied Maths Higher Level 1979 | Maths Grinds.

b) A body is moving in a straight line subject to a deceleration which is equal to quicklatex.com-d3c2e4b0d428d25c71f178d6d856e6ec_l3 | Leaving Cert Applied Maths Higher Level 1979 | Maths Grinds, where quicklatex.com-8bafe7d16e660869fc203e8bffd77abc_l3 | Leaving Cert Applied Maths Higher Level 1979 | Maths Grinds is the velocity. The initial velocity is quicklatex.com-dc2e717421b16a9f080dba4c8c0e3437_l3 | Leaving Cert Applied Maths Higher Level 1979 | Maths Grinds m/s. In how many seconds will the velocity of the body be quicklatex.com-ce939b244df556d77c12e315c6165e7d_l3 | Leaving Cert Applied Maths Higher Level 1979 | Maths Grinds m/s and how far will it travel in that time?


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

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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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