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9 Apr 2018

Leaving Certificate Examination 1927 Honours Applied Mathematics

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This is the Leaving Certificate Examination 1927 Honours Applied Mathematics Paper which I got a hold of from the Archives and decided to retype it up. The archived version was a scanned document (of presumably an original 1927 copy). I hope some of you find it useful and/or interesting. Student had to answer 5 questions in 2 hours. Note that questions “to the end of the paper” (whatever that means) carried more marks.
Copyright notice:

I, Stephen Easley-Walsh, retyped these Department of Education Examination questions under section 53.5 of the Copyright and Related Rights Act, 2000 i.e. “copyright in a work is not infringed by anything done for the purposes of an examination by way of setting questions, communicating questions to the candidates or answering questions”. As such I invite any student to use this page in their studies but to keep in mind I take no responsibility for any errors/typos/omissions/etc contained below and would appreciate any such errors brought to my attention in the comment section. 🙂 Thank you.

Question 1

A ship $A$ sailing due N. at 15 miles per hour is at a certain instant 41 miles due E. of another ship B which is sailing E. at 12 miles per hour. When will the ships be nearest together?

Question 2

Explain the variations of the forces between the floor of a lift and the feet of a man standing on it during the upward and downward journeys from rest to rest.

Question 3

A bullet weighing $1$ oz. is fired horizontally into a block of wood weighing 12 lbs. suspended so that the wood and the bullet embedded in it swing without rotation to a height of 2 ft. 6 in. Find the velocity of the bullet on entering the wood.

Question 4

What is meant by Simple Harmonic Motion?

Show that the bob of a simple pendulum moves with Simple Harmonic Motion when the angle of swing is small.

Find an expression for the periodic time of the pendulum.

Question 5

Show that the area under a force distance diagram represents work done.

The force exerted by a spring is proportional to the extension of the spring and a forces of 5 lbs. wt. produces an extension of 2 ins. Show graphically the relation between the tension and the extension and deuce the work done in extending the spring through 10 inches.

Show that the work done on such a spring when stretched $a$ feet is $\frac{Pa}{2}$ ft. lbs., where $P$ lbs. wt. is the tension for extension $a$ feet.

Question 6

A body of $W$ lbs. can just be maintained at rest on a rough inclined plane by a force $P$ lbs. wt. acting along the plane, of by a force $Q$ lbs. wt. acting horizontally.

Show that $\frac{\sec^2 \varphi}{P^2} = \frac{1}{Q^2} + \frac{1}{W^2}$ where $\varphi$ is the angle of friction.

Question 7

Find the acceleration of a particle which moves with uniform speed in a circle.

A trin is to travel round a curve of radius $r$ ft. If $a$ ft. is the distance between the rail, find the height $b$ ft. to which one rail should be raised above the level of the other  so as to eliminate side pressure on the rails for trains travelling at $v$ ft. per sec.

If a train of $W$ tons weight moves on such a track with speed $nv$ feet per sec. prove that there is now a side pressure on rails of approximately $W \frac{b}{a} \left(n^2-1\right)$ tons wt.

Question 8

A shot at the instant of projection breaks into two parts of masses $m_1$ and $m_2$ lbs., the first part starting with velocity $v_1$ at an angle $\theta_1$ to the horizontal and the second with veloicty $v_2$ at an angle $\theta_2$ to the horizontal. Show that the centre of gravity moves at if the whole shot started with velocity $v$ at an angle $\theta$ to the horizontal; the find $v$ and $\theta$ in terms of $m_1$, $m_2$, $v_1$, $v_2$, $\theta_1$, $\theta_2$.

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