# Leaving Certificate Examination 1948 Honours Applied Mathematics

##### Question 1

A liner proceeding on a course $25^\circ$ North of East appears to be moving N.E. as seen from a tug moving at $10$ knots in an easterly direction. Find the speed of the liner.

How much must the tug increase its speed so that the liner will now appear to be moving due North?

##### Question 2

A piece of uniform wire is bent so as to form three sides $AB$, $BC$, $CD$ of a square $ABCD$. Find the position of the centre of gravity.

If a particle of mass equal to that of the wire is attached at the point $D$, and the whole is freely suspended at $A$, find the inclination of $AB$ to the vertical.

##### Question 3

An engine pulls a train at a uniform speed of $30$ miles per hour up an incline of $1$ of $100$. If the couplings between the train and the engine break, how far will the train continue to run up the incline, if the resistance to its motion is $20$ lb. wt. per ton?

##### Question 4

A football is kicked from a spot on the ground with a velocity of $60$ feet per second and strikes a vertical wall $60$ feet away at a point $12$ feet above the ground. Find the two possible angles of projection. Using the same initial velocity, what angle of projection will give the greatest height up the wall?

##### Question 5

A body of mass $W$ lb. is resting on a rough plane inclined to the horizontal at an angle $\alpha$, the coefficient of friction being $\mu$. Find an expression for the force acting along the line of greatest slope which will just drag the body up the plane.

Find also an expression for the least force required (when applied in the most suitable direction) to drag the body up the plane, and find the value of this force when $W=2$, $\alpha=30^\circ$, $\mu=0.5$.

##### Question 6

Two masses of $3$ and $5$ lb. are connected by a light inextensible string of length $2$ ft. Both masses rest on a smooth horizontal table, the $3$ lb. mass being one foot from the edge of the table and from the $5$ lb. mass which is resting at the edge. If the $5$ lb. mass is pushed gently over the edge, find the speed of $3$ lb. mass (i) when it is jerked into motion, and (ii) when it reaches the edge of the table.

##### Question 7

A triangular lamina $ABC$ is immersed in a vertical position in water with its vertex $A$ at the surface and its base $BC$ parallel to the surface. If $BC=4$ inches and the height of the triangle is equal to $3$ inches, find the thrust on $ABC$ due to the water. If $D$, $E$ are the middle points of $AB$, $AC$, respectively, compare the total thrusts on $ADE$ and $DECB$.

In what ratio must a parallel to $BC$ cut the other sides of the triangle $ABC$ so that the thrusts on the two portions of triangles may be equal?

[$1$ cubic ft. of water $=$ $62\frac{1}{4}$ lbs.]

##### Question 8

A mass of $2$ lb. fastened to a string $4$ ft. long revolves as a conical pendulum, the mass moving in a horizontal circle with a speed of $12$ feet per second. Find the tension of the string and its inclination to the vertical.

##### Question 9

What is Simple Harmonic Motion? What is meant by the *period*, *amplitude*, and* frequency* of a S.H.M.?

A particle is moving in S.H.M. of period $6$ seconds and amplitude $5$ cm. Find the displacement, speed an acceleration of the particle $1$ second after it reaches its maximum positive displacement.

**Citation:**

**Citation:**

State Examinations Commission (2023). *State Examination Commission*. Accessed at: https://www.examinations.ie/?l=en&mc=au&sc=ru

Malone, D and Murray, H. (2023). *Archive of Maths State Exams Papers*. Accessed at: http://archive.maths.nuim.ie/staff/dmalone/StateExamPapers/

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