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24 Jun 2019

Leaving Cert Applied Maths Higher Level 1977

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Question 1

(i) A car starts from rest at quicklatex.com-81c6b3e443a83c026f06b718b1dedcef_l3 | Leaving Cert Applied Maths Higher Level 1977 | Maths Grinds and moves with constant acceleration quicklatex.com-66c806215945e3dc90f658b54f6dcae9_l3 | Leaving Cert Applied Maths Higher Level 1977 | Maths Grinds metres/secondquicklatex.com-025303222356a63c2332a429c33d35a9_l3 | Leaving Cert Applied Maths Higher Level 1977 | Maths Grinds. Three seconds later another car passes through quicklatex.com-d613350484b582bc83e2a6f9473437f8_l3 | Leaving Cert Applied Maths Higher Level 1977 | Maths Grinds travelling in the same direction with constant speed quicklatex.com-ba05715987b9d18252648b0ecfc8eb72_l3 | Leaving Cert Applied Maths Higher Level 1977 | Maths Grinds metres/second, where quicklatex.com-05870eb3e99553ba2bf7b4bf18e78f8a_l3 | Leaving Cert Applied Maths Higher Level 1977 | Maths Grinds. Draw a velocity/time graph for the two cards, using the same axes and the same scales.

(ii) Hence, or otherwise, show that the second car will just catch up on the first if quicklatex.com-b8b69b33adc0a792c2c1d0e6f688c8e0_l3 | Leaving Cert Applied Maths Higher Level 1977 | Maths Grinds, and that it will not catch up on it if quicklatex.com-eaeb4a3d6189ac0e4a8ea1518725473e_l3 | Leaving Cert Applied Maths Higher Level 1977 | Maths Grinds.

(iii) If quicklatex.com-bd905ec2c807cdd5b8bd4241c9217d7a_l3 | Leaving Cert Applied Maths Higher Level 1977 | Maths Grinds, find the greatest distance the second car will be ahead of the first.

Question 2

(a) Explain, with the aid of a diagram, what is meant by the relative velocity of one body with respect to another.

(b) To a cyclist riding North at quicklatex.com-edeca52e97715225ceecec173423f15e_l3 | Leaving Cert Applied Maths Higher Level 1977 | Maths Grinds m/s the wind appears to blow from the North-West. To a pedestrian walking due West at quicklatex.com-d39c7a34226b6a7fc7b5d3ae51b6b5e4_l3 | Leaving Cert Applied Maths Higher Level 1977 | Maths Grinds m/s the same wind appears to come from the South-West. Find the magnitude and direction of the velocity of the wind, by expressing it in the form quicklatex.com-85f93ecec18e9c8c3ba6b916abe161b5_l3 | Leaving Cert Applied Maths Higher Level 1977 | Maths Grinds or otherwise.

Question 3

A particle is projected up a plane which is inclined at an angle quicklatex.com-0dce58ae1ea930bb9c39e85ab5e4c977_l3 | Leaving Cert Applied Maths Higher Level 1977 | Maths Grinds to the horizontal, where quicklatex.com-2c10d2ac611fc26afad5486576c286d8_l3 | Leaving Cert Applied Maths Higher Level 1977 | Maths Grinds. The direction of projection makes an angle of quicklatex.com-43254279a766115b4c8c77b4f9e086ff_l3 | Leaving Cert Applied Maths Higher Level 1977 | Maths Grinds with the inclined plane. The plane of projection is vertical and contains the line of greatest slope.

(i) Show that the particle strikes the inclined plane at right angles.

(ii) Verify that the total energy of the particle at the moment of striking the plane is the same as when the particle is first projected.

Question 4

A mass of quicklatex.com-ce939b244df556d77c12e315c6165e7d_l3 | Leaving Cert Applied Maths Higher Level 1977 | Maths Grinds kg is lying on a rough plane inclined at quicklatex.com-43254279a766115b4c8c77b4f9e086ff_l3 | Leaving Cert Applied Maths Higher Level 1977 | Maths Grinds to the horizontal, coefficient of friction quicklatex.com-294859fdbb1fdda439ec2363e11fdfa1_l3 | Leaving Cert Applied Maths Higher Level 1977 | Maths Grinds. The quicklatex.com-ce939b244df556d77c12e315c6165e7d_l3 | Leaving Cert Applied Maths Higher Level 1977 | Maths Grinds kg mass is connected, by a light inextensible string passing over a smooth fixed pulley at the top of the plane, to a mass of quicklatex.com-dc2e717421b16a9f080dba4c8c0e3437_l3 | Leaving Cert Applied Maths Higher Level 1977 | Maths Grinds kg hanging freely. When the system is set free the quicklatex.com-dc2e717421b16a9f080dba4c8c0e3437_l3 | Leaving Cert Applied Maths Higher Level 1977 | Maths Grinds kg mass moves downwards.

(i) Show in separate diagrams the forces acting on each mass,

(ii) and calculate the common acceleration.

(iii) If a mass of quicklatex.com-ed5decd492fdc6648538222379ceaf27_l3 | Leaving Cert Applied Maths Higher Level 1977 | Maths Grinds kg were used instead of the quicklatex.com-ce939b244df556d77c12e315c6165e7d_l3 | Leaving Cert Applied Maths Higher Level 1977 | Maths Grinds kg mass, show by considering the forces acting that it would not move up the plane or down the plane.

Question 5

A pump raises water from a depth of quicklatex.com-dc2e717421b16a9f080dba4c8c0e3437_l3 | Leaving Cert Applied Maths Higher Level 1977 | Maths Grinds m and discharges it horizontally through a nozzle of diameter quicklatex.com-04772f5fdcda7a04794e42968e58d969_l3 | Leaving Cert Applied Maths Higher Level 1977 | Maths Grinds m at a speed of quicklatex.com-739f6781b65065fc445d5f13b544d6b2_l3 | Leaving Cert Applied Maths Higher Level 1977 | Maths Grinds m/s. Calculate

(i) the mass of water raised per second,

(ii) the kinetic energy given to this mass,

(iii) the power at which the pump is working.

(iv) If the water strikes a fixed vertical wall directly in front of the nozzle, find the forces exerted by the water on the wall, on the assumption that no water bounces back.

[Mass of quicklatex.com-d39c7a34226b6a7fc7b5d3ae51b6b5e4_l3 | Leaving Cert Applied Maths Higher Level 1977 | Maths Grinds mquicklatex.com-52afa795fdc1bd40ad8e16bd4cd15661_l3 | Leaving Cert Applied Maths Higher Level 1977 | Maths Grinds of water is quicklatex.com-832988d7b85b3784b072284b63fdfa60_l3 | Leaving Cert Applied Maths Higher Level 1977 | Maths Grinds kg. Take quicklatex.com-0dacb6df4e0543dc2ca8a9b18b5b6e46_l3 | Leaving Cert Applied Maths Higher Level 1977 | Maths Grinds.]

Question 6

One end of a uniform ladder of weight quicklatex.com-054ef36c6e54f65ccdd586c0869b4fb0_l3 | Leaving Cert Applied Maths Higher Level 1977 | Maths Grinds rests against a smooth vertical wall and the other rests on a rough horizontal ground so that it makes an angle quicklatex.com-deaa154925fac96c09b4c1ab01c39e08_l3 | Leaving Cert Applied Maths Higher Level 1977 | Maths Grinds with the horizontal.

(i) Show that the ladder will start to slip outwards if the coefficient of friction quicklatex.com-74ddc0a71cf47b34f1f6eb6eda14a2ea_l3 | Leaving Cert Applied Maths Higher Level 1977 | Maths Grinds is less than quicklatex.com-a94f7602fb0c8f60ddf34329136f92e1_l3 | Leaving Cert Applied Maths Higher Level 1977 | Maths Grinds.

(ii) When quicklatex.com-67b275569c8833168b70d4dbba96dd26_l3 | Leaving Cert Applied Maths Higher Level 1977 | Maths Grinds the ladder is just prevented from slipping by a vertical string attached to the ladder at a point quicklatex.com-c1202576eb6006ba366f3dcac824fc44_l3 | Leaving Cert Applied Maths Higher Level 1977 | Maths Grinds of its length from the top. Calculate the tension in the string in terms of quicklatex.com-054ef36c6e54f65ccdd586c0869b4fb0_l3 | Leaving Cert Applied Maths Higher Level 1977 | Maths Grinds.

Question 7

A smooth sphere of mass quicklatex.com-1ee1c2caec46fc384f9e0e830a2e8b84_l3 | Leaving Cert Applied Maths Higher Level 1977 | Maths Grinds kg moving at quicklatex.com-951bfbd153c44e8e43522b90361c128e_l3 | Leaving Cert Applied Maths Higher Level 1977 | Maths Grinds m/s collides with a second sphere of mass quicklatex.com-ee873c946a42a316d07b2d164e2b7d67_l3 | Leaving Cert Applied Maths Higher Level 1977 | Maths Grinds kg moving at quicklatex.com-dc2e717421b16a9f080dba4c8c0e3437_l3 | Leaving Cert Applied Maths Higher Level 1977 | Maths Grinds m/s. The direction of motion of the spheres make angles of quicklatex.com-26e3abbffc82f8caad0cdd1496d7f30d_l3 | Leaving Cert Applied Maths Higher Level 1977 | Maths Grinds and quicklatex.com-a8e3789b1bca7c83a1bbce739f6254bd_l3 | Leaving Cert Applied Maths Higher Level 1977 | Maths Grinds, respectively, with the line of centres, both angles being measured in the same sense. The coefficient of restitution is quicklatex.com-a94f7602fb0c8f60ddf34329136f92e1_l3 | Leaving Cert Applied Maths Higher Level 1977 | Maths Grinds.

(i) Find the speeds and direction of motion of the spheres after impact and

(ii) calculate the kinetic energy lost in the collision.

Question 8

(a) The position vector of a particle moving in a circle of radius quicklatex.com-95d7c225dd31ad7d6dd5fb9f0980b531_l3 | Leaving Cert Applied Maths Higher Level 1977 | Maths Grinds with constant angular velocity quicklatex.com-6fd4a229a60f10645306b68a0a0ee2e6_l3 | Leaving Cert Applied Maths Higher Level 1977 | Maths Grinds can be expressed in the form

quicklatex.com-4fe35499ec7171c2ecacbb4e3da4d2a6_l3 | Leaving Cert Applied Maths Higher Level 1977 | Maths Grinds.

Find the acceleration of the particle and show that it is directed towards the centre.

(b) Three light rods quicklatex.com-819964b5440e1d7a1941259fada7d9c6_l3 | Leaving Cert Applied Maths Higher Level 1977 | Maths Grinds, quicklatex.com-93cd9fa3dc459ef3d77cd955c6c49da5_l3 | Leaving Cert Applied Maths Higher Level 1977 | Maths Grinds, quicklatex.com-59ae75b27d1f7afe01b5031d4316820a_l3 | Leaving Cert Applied Maths Higher Level 1977 | Maths Grinds, each of length quicklatex.com-7e66f8d3fbb50779ceb68e5184deb32d_l3 | Leaving Cert Applied Maths Higher Level 1977 | Maths Grinds, are freely jointed to form a triangle quicklatex.com-11c6fd2690294ef6f792ee277004059b_l3 | Leaving Cert Applied Maths Higher Level 1977 | Maths Grinds. Two particles of mass quicklatex.com-7a02bd7300eeadcf269af0186938941e_l3 | Leaving Cert Applied Maths Higher Level 1977 | Maths Grinds are attached, one at quicklatex.com-91dc9a245a0f25a550d41a6feb58273f_l3 | Leaving Cert Applied Maths Higher Level 1977 | Maths Grinds and one at quicklatex.com-5403eeae09f3e9689d2ad28aa6fe1599_l3 | Leaving Cert Applied Maths Higher Level 1977 | Maths Grinds. The system rotates about a vertical axis through quicklatex.com-84fdf06eaf2b99889abd1c8acdc13d5f_l3 | Leaving Cert Applied Maths Higher Level 1977 | Maths Grinds with constant angular velocity quicklatex.com-6fd4a229a60f10645306b68a0a0ee2e6_l3 | Leaving Cert Applied Maths Higher Level 1977 | Maths Grinds such that quicklatex.com-819964b5440e1d7a1941259fada7d9c6_l3 | Leaving Cert Applied Maths Higher Level 1977 | Maths Grinds is horizontal and quicklatex.com-5403eeae09f3e9689d2ad28aa6fe1599_l3 | Leaving Cert Applied Maths Higher Level 1977 | Maths Grinds is vertically below quicklatex.com-819964b5440e1d7a1941259fada7d9c6_l3 | Leaving Cert Applied Maths Higher Level 1977 | Maths Grinds. (see diagram)

HAM-1977-Q8-300x273 | Leaving Cert Applied Maths Higher Level 1977 | Maths Grinds

Show in separate diagrams the forces acting on the particles (the forces exerted by the rods act along the rods). Calculate the forces in the rods and prove that quicklatex.com-029128965844bc40282c5e6493f73823_l3 | Leaving Cert Applied Maths Higher Level 1977 | Maths Grinds.

Question 9

(a) For a compound pendulum (a rigid body performing small oscillations in a vertical plane about a horizontal axis) prove that the period time quicklatex.com-6460d10729af641188d49a36153c8def_l3 | Leaving Cert Applied Maths Higher Level 1977 | Maths Grinds is given by

quicklatex.com-50eef73e5e066092a882ce4c1be7dd4d_l3 | Leaving Cert Applied Maths Higher Level 1977 | Maths Grinds

where quicklatex.com-7a02bd7300eeadcf269af0186938941e_l3 | Leaving Cert Applied Maths Higher Level 1977 | Maths Grinds is the mass of the pendulum, quicklatex.com-9f999f50fd271969c888f3bfa1615590_l3 | Leaving Cert Applied Maths Higher Level 1977 | Maths Grinds the moment of inertia about the axis, and quicklatex.com-63ab7de62920c09a3246a8ce084ab3ce_l3 | Leaving Cert Applied Maths Higher Level 1977 | Maths Grinds the perpendicular distance from the centre of gravity to the axis.

(b) If the compound pendulum is a uniform rod of length quicklatex.com-0e440a33f92d8dd692a299e70ffc00e6_l3 | Leaving Cert Applied Maths Higher Level 1977 | Maths Grinds, show that quicklatex.com-71b6e2b500458a3dca0eccbd71f72e17_l3 | Leaving Cert Applied Maths Higher Level 1977 | Maths Grinds and calculate the value of quicklatex.com-76f5914c0214a7a690b1731932e73199_l3 | Leaving Cert Applied Maths Higher Level 1977 | Maths Grinds for which quicklatex.com-6460d10729af641188d49a36153c8def_l3 | Leaving Cert Applied Maths Higher Level 1977 | Maths Grinds is a minimum.

Question 10

(a) State the Principle of Archimedes.

(b) A tank contains a later of water and a later of oil of relative density quicklatex.com-598fb5a16d3def68572898357672eea3_l3 | Leaving Cert Applied Maths Higher Level 1977 | Maths Grinds). A uniform rod of relative density quicklatex.com-9f0ef54e6fb9479e1132560a852293d5_l3 | Leaving Cert Applied Maths Higher Level 1977 | Maths Grinds is totally immersed with one third of its volume in the water and two thirds in the oil. It is maintained in that position by two vertical strings attached to the ends of the rod and to the bottom of the tank.

(i) Show in a diagram the forces acting on the rod and

(ii) calculate the tensions in the strings in terms of quicklatex.com-054ef36c6e54f65ccdd586c0869b4fb0_l3 | Leaving Cert Applied Maths Higher Level 1977 | Maths Grinds, the weight of the rod.

Question 11

Answer any three of (a), (b), (c), (d) below.

(a) Using Taylor’s theorem find the first three terms in the Taylor series for quicklatex.com-085c5d21a6c66b42cccfeb54cbecc64b_l3 | Leaving Cert Applied Maths Higher Level 1977 | Maths Grinds in the neighbourhood of quicklatex.com-4b059b7e799e27edffd890d114f605d4_l3 | Leaving Cert Applied Maths Higher Level 1977 | Maths Grinds i.e. the Maclaurin series of quicklatex.com-085c5d21a6c66b42cccfeb54cbecc64b_l3 | Leaving Cert Applied Maths Higher Level 1977 | Maths Grinds.

(b) Determine if the series

quicklatex.com-4aae0b5c769c590506ffdfc79671606f_l3 | Leaving Cert Applied Maths Higher Level 1977 | Maths Grinds

is absolutely convergent for quicklatex.com-89358f3d6b219fe69b42a27c8a89dc2d_l3 | Leaving Cert Applied Maths Higher Level 1977 | Maths Grinds

(c) Solve the differential equation

quicklatex.com-13f39226aef174e8ab209fafd40a1768_l3 | Leaving Cert Applied Maths Higher Level 1977 | Maths Grinds

if quicklatex.com-9fc5afccf942783a4e5e307d4e17d184_l3 | Leaving Cert Applied Maths Higher Level 1977 | Maths Grinds when quicklatex.com-031ea0303b919864e09f6d6f3fb9a641_l3 | Leaving Cert Applied Maths Higher Level 1977 | Maths Grinds.

(d) Solve the equation

quicklatex.com-6e392c0a9c333d0887fb888500581514_l3 | Leaving Cert Applied Maths Higher Level 1977 | Maths Grinds

if quicklatex.com-3dafc7221936c5ec0f3bec99507fb9ed_l3 | Leaving Cert Applied Maths Higher Level 1977 | Maths Grinds and quicklatex.com-60a44924401c43d915d142d4d5ee1398_l3 | Leaving Cert Applied Maths Higher Level 1977 | Maths Grinds when quicklatex.com-4b059b7e799e27edffd890d114f605d4_l3 | Leaving Cert Applied Maths Higher Level 1977 | Maths Grinds.


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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