# Leaving Certificate Examination 1958 Honours Applied Mathematics

#### Question 1

$ABC$ is a triangle in which $BC=5$ cm., $\angle BCA=30^\circ$, and $\cos ABC = \frac{3}{5}$. Forces of $10$, $8$, $6$ lb. wt. act along $BA$, $BC$, $CA$, respectively. If their resultant acts along the straight line $DE$, cutting $BC$ in $D$, calculate the length of $BD$ and the size of the angle $CDE$.

#### Question 2

Show that the centre of gravity of a triangular lamina is the same as that of three equal masses situated at the vertices of the triangle.

A lamina is in the shape of a trapezium $ABCD$ in which $AB$ and $CD$ are parallel. $AB=12$, $CD=3$, $AD=8$ and $\angle DAB=30^\circ$. Find the perpendicular distance of the centre of gravity of the lamina (i) from $AB$, and (ii) from $AD$.

#### Question 3

Explain the terms “limiting friction”, “angle of friction”.

A horizontal force of $3$ lb. wt. would just keep a block of mass $5$ lb. from sliding down a rough plane which is inclined to the horizontal at an angle of $60^\circ$. Calculate the coefficient of friction between the block and the plane.

Find the direction and magnitude of the least force that would keep the block from sliding down the plane.

#### Question 4

$A$, $B$, $C$ are three ships at sea. To an observer on $A$, $B$ appears to be travelling South-East at $6$ knots. To an observer on $B$, $C$ appears to be travelling due North at $5$ knots. Find the velocity of $C$ relative to $A$ in magnitude and direction.

#### Question 5

A car travelling on a level road was uniformly accelerated for $12$ seconds and was then uniformly retarded for $2$ seconds. In that $14$ seconds it travelled from $A$ to $B$, a distance of $180$ yards. Its velocity at $A$ was $15$ m.p.h. and its velocity at $B$ was $30$ m.p.h. Find the uniform acceleration and the uniform retardation, in feet per sec.$^2$.

Calculate the horse-power at which the car was working when its velocity was $20$ m.p.h., given that the car weighed one ton and that the frictional resistances to motion were equivalent to $50$ lb. wt.

#### Question 6

A $70$ gm. mass is projected vertically upwards with an initial velocity of $52$ ft. per sec. and half a second later a $40$ gm. mass is projected vertically upwards from the same point with an initial velocity of $80$ ft. per sec. Calculate the height at which the masses will collide.

If the masses coalesce on colliding, find the greatest height which the combined mass will reach and the velocity it will have on returning to the point of projection.

#### Question 7

If a body is describing a circle of radius $r$ with constant angular velocity $w$, show that its acceleration is $w^2 r \left( =\frac{v^2}{r} \right)$ directed towards the centre of the circle.

A particle of mass $m$, suspended from a fixed point by a string of length $l$, is describing a horizontal circle with uniform angular velocity $w$. Express in terms of $m$, $l$m $w$ (i) the tension in the string, (ii) the cosine of the angle which the string makes with the vertical.

Show that the time taken by the particle in making one revolution is less than the time of oscillation of a simple pendulum of length $l$.

#### Question 8

Define simple harmonic motion.

A particle is moving along a straight line. At time $t$ (secs.) its distance $x$ (cms.) from its mean position is given by the formula

$$x=3\sin 2t.$$

Show that the motion is simple harmonic and find the periodic time.

Calculate the velocity of the particle (i) when it is in its mean position, (ii) half a second later.

#### Question 9

There are $10$ feet of water on one side of a lock-gate and $6$ feet of water on the other. The gate is rectangular and is $12$ feet wide. Calculate the resultant thrust of the water on the date, in tons.

By how much should the level of the water on the deeper side be lowered so that the resultant thrust would be $7,125$ lb. less?

[A cubic foot of water weighs $62\frac{1}{2}$ 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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