# Leaving Certificate Examination 1956 Honours Applied Mathematics

#### Question 1

A warship is travelling on a course $30^\circ$ East of North. To an observer on a liner travelling due East at $15$ knots the warship appears to be moving due North. Find the speed of the warship.

In what direction should the warship travel at that speed so that it would appear to the observe to be moving North-Eastwards?

#### Question 2

$ABC$ is an equilateral triangle. Forces of $13$, $3$, $7$ lb. wt act along the lines $BA$, $BC$, $AC$ respectively. Find the magnitude of the their resultant.

Show that the line of action of their resultant cuts $BA$ and $AC$ internally and $CB$ externally, and find the angle which it makes with $CB$ produced.

#### Question 3

A uniform ladder is $10$ feet long and weighs $30$ lb. It rests with one end in contact with a rough horizontal plane (coefficient of friction $0.5$) and the other end in contact with a rough vertical wall (coefficient of friction $0.4$). If the bottom of the ladder is $8$ feet out from the wall, how far can a man who weighs $150% lb. go up the ladder without causing it to slip?

How far in must the bottom of the ladder be moved to enable him to go $6$ feet up the ladder?

#### Question 4

A $3$ lb. mass is held at rest on a smooth plane inclined to the horizontal at an angle of $30^\circ$. A light string passing over a smooth pulley at the top of the plane connects the $3$ lb. mass to a $2$ lb. mass which is hanging freely and which is $2\frac{1}{2}$ feet from the ground. The system is released and after $t_1$ seconds the $2$ lb. mass is brought to rest on hitting the ground ; after a further $t_2$ seconds it is jerked into motion again. Find the values of $t_1$ and $t_2$.

What fraction of the kinetic energy is lost when the $2$ lb. mass is jerked into motion?

#### Question 5

A lamina is in the shape $ABCDE$ (see diagram) in which $ABE$ is an equilateral triangle of sides $4$ cms. and $BCDE$ is a rectangle in which $BC=1$ cm. Find the distance of the centre of gravity of the lamina from the line $CD$

If the triangular portion $ALM$ is removed ($AL=AM=2$ cms.), find the distance from $CD$ of the centre of gravity of the remainder.

#### Question 6

An engine raises water from a depth of $18$ feet and delivers it at the rate of $300$ gallons per minute with a velocity of $v$ feet per second. If the engine is working at $2\frac{3}{4}$ horse-power, find the value of $v$.

[One gallon of water weighs $10$ lb.]

#### Question 7

A ball is lying on the ground at $C$, a point $15$ feet away from $B$, the base of a vertical pole. The ball is to be kicked from $C$ with an initial velocity of $20\sqrt{2}$ feet per second so as to strike the pole at a point $h$ feet above $B$.

(i) If $h=6$, find the two possible angles of projection.

(ii) Find the value of $h$ for which there is only one angle of projection.

#### Question 8

Define Simple Harmonic Motion.

A particle is moving in a straight line. Its distance, $x$ (cms.), from a fixed point in the line at time $t$ (seconds) is given by the formula

$$x=5\sin\frac{1}{2}t$$

Show that its motion is simple harmonic ; find its maximum velocity and the periodic time.

Find how far it is from its mean position when its velocity is half its maximum velocity.

#### Question 9

A lamina is in the shape of a trapezium $ABCD$ in which $AD=6$ in., $AB=5$ in., $CD=11$ in., and the angles $\hat{BAC}=\hat{CDA}=90^\circ$. The lamina is immersed in a vertical position in water, $AD$ being at the surface. Find the total thrust of the water on the lamina.

When the lamina is pushed down vertically so that $AD$ is $x$ inches below the surface and parallel to it, a horizontal line through $B$ will divide the lamina into two parts the thrusts on which are equal.

Find the value of $x$.

[One cubic foot of water weighs $62.5$ lb.]

**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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