My Journal

All things Mathematical
27 Jan 2019

Leaving Cert Applied Maths Higher Level 1975

/
Posted By
/
Comments0

Question 1

A particle falls freely under gravity from rest at a point quicklatex.com-81c6b3e443a83c026f06b718b1dedcef_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds. After it has fallen for one second another particle is projected vertically downwards from quicklatex.com-81c6b3e443a83c026f06b718b1dedcef_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds with speed quicklatex.com-00331d3dc0909b451c2c2d6075b95af3_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds m/s. By considering the relative motion of the particles, or otherwise, find the time and distance from quicklatex.com-81c6b3e443a83c026f06b718b1dedcef_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds at which they collide. Show the motion of both on a velocity-time graph.

[Video Solution]

Question 2

A man wishes to swim across a rive quicklatex.com-5710ac6ade182d0ad850e6efd2e4a189_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds m wide. The river flows with a velocity of quicklatex.com-dc2e717421b16a9f080dba4c8c0e3437_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds m/s parallel to the straight banks, and the man swims at a speed of quicklatex.com-1ee1c2caec46fc384f9e0e830a2e8b84_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds m/s relative to the water. If he heads at an angle quicklatex.com-0dce58ae1ea930bb9c39e85ab5e4c977_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds to the upstream direction, and his actual velocity is at an angle quicklatex.com-d6ffefedf9d4e7e4f42ba65b9f3b9416_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds to the downstream direction,

i) show that

    quicklatex.com-3a061eb9bbf1aa9c021c7eba9acbf156_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds

.

ii) Prove that quicklatex.com-20ea7019abeaec5ff84bcbb3852fc44f_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds has a maximum value when quicklatex.com-c0c6efcdbd378db3f6878c293a21dcc9_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds. Deduce that the time taken for the man to cross by the shortest path is quicklatex.com-66bafba23129fa8ab0a7d7fd2fc25d52_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds s.

[Video Solution]

Question 3

A particle of mass quicklatex.com-701ba630cb8f3efdb2500ef85e0bdad6_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds rests on a rough horizontal table, where the coefficient of friction between the particle and the table is quicklatex.com-c1202576eb6006ba366f3dcac824fc44_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds, and is attached by two inelastic strings to particles of masses quicklatex.com-e20b49f19ccabffb1d8e0d926f339dc7_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds and quicklatex.com-941d689ad0e65c5d5733efadca408734_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds which hang over smooth light pulleys at opposite edges of the table. The particle and the two pulleys are collinear.

i) Show in separate diagrams the forces acting on each of the three particles when the system is released from rest.

ii) Find the distance fallen by the quicklatex.com-e20b49f19ccabffb1d8e0d926f339dc7_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds particle in time quicklatex.com-db2144d676b0c010fa61116d07db1982_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds.

[Video Solution]

Question 4

A particle of mass quicklatex.com-dc2e717421b16a9f080dba4c8c0e3437_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds kg rests on the highest point of a fixed sphere of radius quicklatex.com-6a9b8d931378ce72c36b1ded6afb8547_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds m. The particle is slightly displaced from the highest point and slides down the smooth outer surface of the sphere.

i) Show in a diagram the forces acting on the particle when the radius to it has turned through an angle quicklatex.com-d6ffefedf9d4e7e4f42ba65b9f3b9416_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds.

ii) Express the speed quicklatex.com-8bafe7d16e660869fc203e8bffd77abc_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds of the particle and the reaction quicklatex.com-b90475b22ab5c52e8edfa41828a34159_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds of the sphere in terms of quicklatex.com-d6ffefedf9d4e7e4f42ba65b9f3b9416_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds.

iii) Find where the particle leaves the sphere.

[Video Solution]

Question 5

A missile is projected from a point quicklatex.com-2a137ec0f9eba36b0ab49cf15f39f89e_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds with speed quicklatex.com-20b2359777f6479b766141ee1eb47708_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds m/s at an angle quicklatex.com-0dce58ae1ea930bb9c39e85ab5e4c977_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds to the horizontal.

i) Express its velocity quicklatex.com-ecffa0daef95d0e24bb22fc5711a6bc3_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds and its displacement quicklatex.com-9d6cb6705ba8afb57cbf6daf3ee743dd_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds from quicklatex.com-2a137ec0f9eba36b0ab49cf15f39f89e_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds after time quicklatex.com-db2144d676b0c010fa61116d07db1982_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds seconds in terms of unit vectors along the horizontal and vertical.

ii) The missiles strikes a small target whose horizontal and vertical distances from the point of projection are quicklatex.com-7bc4f1f0f4c11887bbe547ac256853e5_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds m an quicklatex.com-739f6781b65065fc445d5f13b544d6b2_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds m respectively. Write down two equations in quicklatex.com-0dce58ae1ea930bb9c39e85ab5e4c977_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds and quicklatex.com-db2144d676b0c010fa61116d07db1982_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds, the time taken to reach the target. Hence find the two possible values of quicklatex.com-ec89552f8b4d9fc55e241b52a66cea16_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds, and the times taken on the corresponding trajectories.

[Video Solution]

Question 6

quicklatex.com-84fdf06eaf2b99889abd1c8acdc13d5f_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds, quicklatex.com-91dc9a245a0f25a550d41a6feb58273f_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds are two points quicklatex.com-fd6542ae3d49d38a0f73fefe83b285af_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds m apart on a smooth horizontal table and quicklatex.com-fd983a118a39c1da7f969dad01b752c9_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds is the midpoint of quicklatex.com-419d1b6bb8ada4b549be20df7204f8da_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds. A particle of mass quicklatex.com-0ec045393ff984641d86b4ee444e6208_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds kg is held at quicklatex.com-fd983a118a39c1da7f969dad01b752c9_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds by two elastic strings the other ends of which are attached to quicklatex.com-84fdf06eaf2b99889abd1c8acdc13d5f_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds, quicklatex.com-91dc9a245a0f25a550d41a6feb58273f_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds respectively. Each of the strings is of natural length quicklatex.com-d39c7a34226b6a7fc7b5d3ae51b6b5e4_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds m and elastic constant quicklatex.com-dc2e717421b16a9f080dba4c8c0e3437_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds N/m. If the particle is then drawn aside along quicklatex.com-819964b5440e1d7a1941259fada7d9c6_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds and released from rest when quicklatex.com-a740037f818c3f0104a245035116aaa8_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds m from quicklatex.com-91dc9a245a0f25a550d41a6feb58273f_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds,

i) show that it moves with simple harmonic motion.

ii) Find the period, and the least time taken to reach a point quicklatex.com-8e57144d1baeafbe0f27f393263b3e86_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds m from quicklatex.com-91dc9a245a0f25a550d41a6feb58273f_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds.

[Video Solution]

Question 7

A light rod quicklatex.com-819964b5440e1d7a1941259fada7d9c6_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds of length quicklatex.com-291db5aa796790338244d26af8292adf_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds connects a smooth ring of mass quicklatex.com-55b2ff446b4b078800d74115f1f1dd03_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds, attached to it at quicklatex.com-84fdf06eaf2b99889abd1c8acdc13d5f_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds to a particle of mass quicklatex.com-941d689ad0e65c5d5733efadca408734_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds attached to it at quicklatex.com-91dc9a245a0f25a550d41a6feb58273f_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds. The ring is free to slide along a fixed thin smooth horizontal wire. The road quicklatex.com-819964b5440e1d7a1941259fada7d9c6_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds is held in a horizontal position underneath the wire and released from rest.

i) Show in a diagram the forces acting on the ring and the mass quicklatex.com-941d689ad0e65c5d5733efadca408734_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds during the motion.

ii) Explain why the conservation of energy and the conservation of linear momentum in a horizontal direction can be applied to the system.

iii) Show that the speed of quicklatex.com-941d689ad0e65c5d5733efadca408734_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds when quicklatex.com-91dc9a245a0f25a550d41a6feb58273f_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds is vertically below quicklatex.com-84fdf06eaf2b99889abd1c8acdc13d5f_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds is quicklatex.com-f1610dec0c2569fd1c352f25be62dea5_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds.

[Video Solution]

Question 8

a) A lamina is rotating with angular velocity quicklatex.com-6fd4a229a60f10645306b68a0a0ee2e6_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds about an axis perpendicular to its plane. If the moment of inertia of the lamina about the axis is quicklatex.com-9f999f50fd271969c888f3bfa1615590_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds, prove that is kinetic energy is quicklatex.com-0622242ecab5676528f3a95a471d7a49_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds.

[Video Solution]

b) Show that the moment of inertia of a uniform square lamina quicklatex.com-532a8f530b02adee3e441c75769e908c_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds, of mass quicklatex.com-941d689ad0e65c5d5733efadca408734_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds and side quicklatex.com-0e440a33f92d8dd692a299e70ffc00e6_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds, about an axis through quicklatex.com-84fdf06eaf2b99889abd1c8acdc13d5f_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds perpendicular to the lamina is quicklatex.com-92e7f0e4eab0f3ce8fe724d162cbbc25_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds.

[Video Solution]

c) The lamina [from part b] is free to rotate in a vertical plane under gravity about the axis, which is fixed horizontally. It is released from rest with quicklatex.com-819964b5440e1d7a1941259fada7d9c6_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds horizontal and above quicklatex.com-69b3f68db784b53d071e829732e8a755_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds. Find the speed of quicklatex.com-5403eeae09f3e9689d2ad28aa6fe1599_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds when quicklatex.com-dbc51c69debe1b39c6347aca12bad038_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds reaches the vertical.

[Video Solution]

Question 9

The force of attraction of the earth on a particle of mass quicklatex.com-941d689ad0e65c5d5733efadca408734_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds distance quicklatex.com-ada3765a264ac03a253ee0ccbe61f602_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds from the centre of the earth is quicklatex.com-061a41232009f7cc718b55de4fcd36a1_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds, where quicklatex.com-84fdf06eaf2b99889abd1c8acdc13d5f_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds is the radius of the earth quicklatex.com-29d7adbfa0dbb4640c26a8e47229b03d_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds.

i) Write down the equation of motion for a particle moving under this force alone and calculate the speed of the particle at distance quicklatex.com-ada3765a264ac03a253ee0ccbe61f602_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds if it was projected vertically upwards from the earth’s surface with speed quicklatex.com-7051f52053bf19e9eb0ac6d557e79654_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds.

ii) Prove that the time taken to reach a height quicklatex.com-c496f0cf7f4e06082e4692bede503efa_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds above the earth’s surface is quicklatex.com-eab322f27c42bba9546b072cce6d1ff9_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds.

[Video Solution]

Question 10

a) State the conditions for the equilibrium of a body immersed in a fluid.

[Video Solution]

b) A thin uniform rod quicklatex.com-819964b5440e1d7a1941259fada7d9c6_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds of length quicklatex.com-0e440a33f92d8dd692a299e70ffc00e6_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds and weight quicklatex.com-054ef36c6e54f65ccdd586c0869b4fb0_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds, can turn freely about the end quicklatex.com-84fdf06eaf2b99889abd1c8acdc13d5f_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds, which is fixed at a height quicklatex.com-7e66f8d3fbb50779ceb68e5184deb32d_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds above the surface of water into which the other end dips. Show in a diagram the forces acting on the rod. If the rod is in equilibrium when inclined at quicklatex.com-9c3fb28be0c20ddb0b2c4bd59c303de8_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds to the vertical, show that the specific weight of the rod is quicklatex.com-294859fdbb1fdda439ec2363e11fdfa1_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds.

[Video Solution]

Question 11

a) Find the first two terms in the Taylor’s series for quicklatex.com-43480173121e093b27d9209f5bac1ba9_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds in the neighbourhood of quicklatex.com-4b059b7e799e27edffd890d114f605d4_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds, i.e. the Maclaurin series for quicklatex.com-43480173121e093b27d9209f5bac1ba9_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds.

[Video Solution]

b) Find the domain of quicklatex.com-b1dfe67145fa683498d19e61279fba48_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds for which the series of complex terms.

    quicklatex.com-1f2ca3860b98fdbb179f893a7a78fe9c_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds

is absolutely convergent. Is the series absolutely convergent for quicklatex.com-6e9e2573018836ef5ee226d22c5ff790_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds?

[Video Solution]

c) By substituting quicklatex.com-b5e6dd5d531dca988f2ac772f0f6abc7_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds, show that the differential equation

    quicklatex.com-db3df9c5c633f5649620926fadb79aef_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds

becomes

    quicklatex.com-cade8b7c93e42324e4fdffb4374d362e_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds

Solve the equation, given that quicklatex.com-8ed93fef0596d8393d81a24553e60308_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds when quicklatex.com-55132ac813a32dab57735b6282f1843e_l3 | Leaving Cert Applied Maths Higher Level 1975 | Maths Grinds.

[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/

print

Leave a Reply

This site uses Akismet to reduce spam. Learn how your comment data is processed.