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2 Aug 2019

Leaving Cert Applied Maths Higher Level 1980

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Question 1

A boat has to travel by the shortest route to the point quicklatex.com-e91b7d3a0bd86c01264f503c76bdba7f_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds km and then return immediately to its starting point at the origin. The velocity of the water is quicklatex.com-99d1c85e56c1ab1ec4f8e73a9c2c0914_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds km/hour and the boat has a speed of quicklatex.com-ced9adf500abe3e625f021cf18429a9e_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds km/hour in still water.

If quicklatex.com-57ac8a5549bfd627065ab8fa45e00586_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds is the velocity of the boat on the outward journey,

i) find quicklatex.com-84fdf06eaf2b99889abd1c8acdc13d5f_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds and quicklatex.com-91dc9a245a0f25a550d41a6feb58273f_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds and the time taken for the outward journey, leaving your answer in surd form.

ii) Find, also, the time taken for the whole journey.

Question 2

A body of weight quicklatex.com-054ef36c6e54f65ccdd586c0869b4fb0_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds is supported by two vertical inextensible strings at quicklatex.com-84fdf06eaf2b99889abd1c8acdc13d5f_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds and quicklatex.com-91dc9a245a0f25a550d41a6feb58273f_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds as in diagram where quicklatex.com-0e972dd838db0238488fb95544141593_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds cm. The tensions in the strings are quicklatex.com-4306914099647cc99417a7634f1c1fe5_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds and quicklatex.com-8e2fdf52bd560c8cb509bbddc43dc403_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds and the string of tension quicklatex.com-4306914099647cc99417a7634f1c1fe5_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds makes an angle of quicklatex.com-9c3fb28be0c20ddb0b2c4bd59c303de8_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds with quicklatex.com-819964b5440e1d7a1941259fada7d9c6_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds. The centre of gravity of the body is at quicklatex.com-32a1fff034751d4fc3e4edc15ca376af_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds, the centre of quicklatex.com-419d1b6bb8ada4b549be20df7204f8da_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds is quicklatex.com-5403eeae09f3e9689d2ad28aa6fe1599_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds and quicklatex.com-688eec0b6aca25a17c61214f05ed402f_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds.

Express quicklatex.com-a2aeeaf7658656e400e742893eb7778d_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds in terms of quicklatex.com-054ef36c6e54f65ccdd586c0869b4fb0_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds and quicklatex.com-4306914099647cc99417a7634f1c1fe5_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds and hence find the distance of quicklatex.com-32a1fff034751d4fc3e4edc15ca376af_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds from quicklatex.com-819964b5440e1d7a1941259fada7d9c6_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds in terms of quicklatex.com-054ef36c6e54f65ccdd586c0869b4fb0_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds and quicklatex.com-4306914099647cc99417a7634f1c1fe5_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds.

HAM-1980-Q2 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds

Question 3

A projectile is fired with initial velocity quicklatex.com-1f5f51e02d5b3e82545ed5ef6fcbf49c_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds, where quicklatex.com-f453dfa2ac10361897075d9950ad4c04_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds is along the horizontal. A plane quicklatex.com-d613350484b582bc83e2a6f9473437f8_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds passes through the point of projection and makes an angle quicklatex.com-402735c0bdab0b3477755624fc9b4d06_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds with the horizontal.

i) If the projectile strikes the plane quicklatex.com-d613350484b582bc83e2a6f9473437f8_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds at right angles to quicklatex.com-d613350484b582bc83e2a6f9473437f8_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds after time quicklatex.com-db2144d676b0c010fa61116d07db1982_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds, show that

    quicklatex.com-2a7ba35105dd9b44d10ef14e4f383c9d_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds

ii) and deduce that quicklatex.com-65e370081bf58b4dca696a15472d87df_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds.

iii) If quicklatex.com-a9a28444fddb988d1c5cca9ced5e8ab3_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds, find in terms of quicklatex.com-ba05715987b9d18252648b0ecfc8eb72_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds and quicklatex.com-32a1fff034751d4fc3e4edc15ca376af_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds the range of the projectile alone quicklatex.com-d613350484b582bc83e2a6f9473437f8_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds.

Question 4

a) State and prove the relationship between the coefficient of friction quicklatex.com-74ddc0a71cf47b34f1f6eb6eda14a2ea_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds and the angle of friction quicklatex.com-4ac1cf9e7a40a63c9af04825a0b3c05c_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds.

b) The diagram shows a particle of weight quicklatex.com-054ef36c6e54f65ccdd586c0869b4fb0_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds on a rough plane making an angle quicklatex.com-0dce58ae1ea930bb9c39e85ab5e4c977_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds with the horizontal. The particle is acted upon by a force quicklatex.com-8c28ca9825a5e0ff6a01cc15dd4efff1_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds whose lien of action makes an angle quicklatex.com-d6ffefedf9d4e7e4f42ba65b9f3b9416_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds with the line of greatest slope. The particle is just one the point of moving up the plane.

i) Draw a diagram showing the forces acting on the particle

ii) and prove that

    quicklatex.com-1127c8b252e94583d85a705694cb80f0_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds

If the particle is just on the point of moving up the plane, deduce

iii) the forces acting up along the plane that would achieve this

iv) the horizontal force that would achieve it

v) the minimum force that would achieve it.

HAM-1980-Q4 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds

Question 5

a) Two imperfectly elastic spheres of equal mass moving horizontally along the same straight line impinge and, as a result, one of them is brought to rest. Show that whatever be the value of the coefficient of restitution, quicklatex.com-9f1b35dbc8bc47aaf8635969e17d56e0_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds, they must have been moving in opposite directions.

b) A sphere quicklatex.com-cba4133f939edee31a8ce51d790a9cd0_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds of mass quicklatex.com-7a02bd7300eeadcf269af0186938941e_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds kg moving with a speed quicklatex.com-ba05715987b9d18252648b0ecfc8eb72_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds m/s on a smooth horizontal table impinges on a smooth plane quicklatex.com-93cd9fa3dc459ef3d77cd955c6c49da5_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds. This plane is inclined to the table at ang angle quicklatex.com-0dce58ae1ea930bb9c39e85ab5e4c977_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds and the line of intersection of it with the table is at right angles to the direction of motion of the sphere.

i) Write down the components of the velocity of quicklatex.com-cba4133f939edee31a8ce51d790a9cd0_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds perpendicular to the plane and parallel to the plane before impact and

ii) show that quicklatex.com-5ef2e26dbab3e7a8e6e12ea3dd90ce92_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds is the velocity of quicklatex.com-cba4133f939edee31a8ce51d790a9cd0_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds perpendicular to the plane after impact where quicklatex.com-23159d6255ee873d52dfd5b58bf2bf57_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds is the coefficient of restitution between the sphere and the plane.

iii) Find the magnitude of the impulse due to the impact.

HAM-1980-Q5 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds

Question 6

a) If a string whose elastic constant is quicklatex.com-66c806215945e3dc90f658b54f6dcae9_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds is stretched a distance quicklatex.com-ada3765a264ac03a253ee0ccbe61f602_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds beyond its natural length, show that the work done is quicklatex.com-5285b7e136075bf7dc0eb91b5acc2842_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds.

b) A particle of mass quicklatex.com-7a02bd7300eeadcf269af0186938941e_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds is on a rough horizontal plane is connected to a fixed point quicklatex.com-81c6b3e443a83c026f06b718b1dedcef_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds in the plane by a lgiht string of elastic constant quicklatex.com-66c806215945e3dc90f658b54f6dcae9_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds. Initially the string is just taut and the particle is projected along the plane directly away from quicklatex.com-81c6b3e443a83c026f06b718b1dedcef_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds with initial speed quicklatex.com-ba05715987b9d18252648b0ecfc8eb72_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds against quicklatex.com-84fdf06eaf2b99889abd1c8acdc13d5f_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds cosntant resistance quicklatex.com-8c28ca9825a5e0ff6a01cc15dd4efff1_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds.

i) Find an expression for the distance quicklatex.com-ada3765a264ac03a253ee0ccbe61f602_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds travelled by the particle.

ii) Noting that the particle will just return to its point of projection if the potential energy at any point is equal to the work done up to that point in overcoming quicklatex.com-8c28ca9825a5e0ff6a01cc15dd4efff1_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds, show that

    quicklatex.com-b9dd9e404db93c3b589fd51c9475b971_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds

Question 7

a) Establish the moment of inertia of a uniform rod about an axis through its centre perpendicular to the rod.

b) State the parallel axes theorem.

c) A thin uniform rod of length quicklatex.com-0e440a33f92d8dd692a299e70ffc00e6_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds and of mass quicklatex.com-7a02bd7300eeadcf269af0186938941e_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds has a mass of quicklatex.com-bc90a078f248740eba6949305c963ffc_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds attached at its mid-point. Find the positions of a point in the rod about which the rod (with attached mass) may oscillate as a compound pendulum, having period equal to that of a simple pendulum of length quicklatex.com-7e66f8d3fbb50779ceb68e5184deb32d_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds.

Question 8

a) A particle is moving in a straight line such that its distance quicklatex.com-ada3765a264ac03a253ee0ccbe61f602_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds from a fixed point at time quicklatex.com-db2144d676b0c010fa61116d07db1982_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds is given by

    quicklatex.com-70c21c21e702f425daff6dd7aac1297a_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds

Show that the particle is movign with simple harmonic motion.

b) A particle is moving in a straight line with simple harmonic motion. When it is a point quicklatex.com-5040715b7f9daeaf2efba5465e6e8993_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds of distance quicklatex.com-598fb5a16d3def68572898357672eea3_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds m from the mean-centre, its speed is quicklatex.com-ee873c946a42a316d07b2d164e2b7d67_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds m/s and when it is at a point quicklatex.com-28918dce14964b674ea0e1e2f3fd73f5_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds of distance quicklatex.com-61cb8bf66d4f8d9b7f43c80c50913760_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds m from the end-position on the same side of the mean-centre as quicklatex.com-5040715b7f9daeaf2efba5465e6e8993_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds, its acceleration is of magnitude quicklatex.com-4c8846d90c9db2c7945bc56cc2df5c3f_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds m/squicklatex.com-025303222356a63c2332a429c33d35a9_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds. If quicklatex.com-95d7c225dd31ad7d6dd5fb9f0980b531_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds is the amplitude of the motion,

i) show that

    quicklatex.com-2ee694210003fbc83c5ee49f5a4fe8b0_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds

and hence find the value of quicklatex.com-95d7c225dd31ad7d6dd5fb9f0980b531_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds.

ii) Find also the period of the motion and the shortest time taken between quicklatex.com-5040715b7f9daeaf2efba5465e6e8993_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds and quicklatex.com-28918dce14964b674ea0e1e2f3fd73f5_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds correct to two places of decimals.

Question 9

a) Solve the differential equation

    quicklatex.com-c15617de0963bfb30167dddf22d01426_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds

given that quicklatex.com-a43308efc53e60ea7d2bfb9fffae6256_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds and quicklatex.com-0a02f8bd74280f3ec1ab779a97897855_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds when quicklatex.com-287cbf25eabaa5b859b30410e8454f09_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds.

b) A car starts from rest. When it is at a distance quicklatex.com-273ac051ff0a53582365352ee5b6950d_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds from its starting point, its speed is quicklatex.com-8bafe7d16e660869fc203e8bffd77abc_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds and its accleration is quicklatex.com-24f70a7b838e1eb970e2495f38d1be6a_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds, show that

    quicklatex.com-954f2e43bd6fa6873cf911ce0793b1c9_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds

and find as accurately as the tables allow its speed when quicklatex.com-eeaa66331fc2eecc5d129c9172b8145a_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds.

Question 10

a) A vessel is in the form of a frustum of a right circular cone. It contains liquid to a depth quicklatex.com-63ab7de62920c09a3246a8ce084ab3ce_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds and at that depth the area of the free surfance of the liquid is quicklatex.com-a5990b654956436786b5311c7aada3b3_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds of the area of the base. Find in simplest surd form the ratio of the thrust on the base due to the liquid to the weight of the liquid.

b) A piece of wood and a piece of metal weigh quicklatex.com-e2ceb65b3a6ab9dc7212d896621f4092_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds N and quicklatex.com-ee873c946a42a316d07b2d164e2b7d67_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds N, respectively. When combined together the compound body weighs quicklatex.com-cc7109bb168a5ed4c70df2fe98bf3d1a_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds N in water. Given that the specific gravity of the metal is quicklatex.com-739f6781b65065fc445d5f13b544d6b2_l3 | Leaving Cert Applied Maths Higher Level 1980 | Maths Grinds, find the specific gravity of the wood.


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“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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