Leaving Certificate Examination 1951 Honours Applied Mathematics
Question 1
Two pegs $A$ and $B$ are at the same height above the ground, and are $7$ feet apart. One end of an inextensible string $8$ feet in length is attached to $A$, and the other end is attached to $B$. A $28$ lb. weight is suspended from the string at a point $3$ feet from $A$ and $5$ feet from $B$. Find the tension in each part of the string.
Question 2
Two roads cross at right angles at $O$. Two men, $A$ and $B$, are walking towards $O$. $A$ is walking at $4$ m.p.h. on one road, $B$ at $3$ m.p.h. on the other road. When $A$ is $100$ yards from $O$, $B$ is $200$ yards from $O$. Find the velocity of $A$ relative to $B$.
Find the distance of $B$ from $O$ and from $A$, when $A$ and $B$ are nearest to each other.
Question 3
Prove that the centre of gravity of any triangular lamina is the same as that of three equal masses situated at the vertices of the triangle.
$ABCD$ is a quadrilateral lamina. A mass equal to one-third the mass of the lamina is placed at the intersection of the diagonals $AC$ and $BD$. Show that the centre of gravity of the lamina and the mass together is the same as that of four equal masses situated at the vertices of the quadrilateral.
Question 4
Two masses of $3$ lb. and $5$ lb. connected by a light inextensible string $9$ feet long, are lying on a smooth horizontal table $6$ feet high. The $5$ lb. mass is at the edge of the table and the other mass is $8$ feet away in a direction perpendicular to the edge. If the $5$ lb. mass is pushed gently over the edge of the table, find how long it takes to reach the ground, and how much longer the $3$ lb. mass takes to reach the edge of the table.
Find also the change in kinetic energy when the $3$ lb. mass is jerked into motion.
Question 5
A train ascending an incline of $1$ in $100$ accelerates uniformly from $20$ m.p.h. to $40$ m.p.h. in a distance of $2$ miles. Find this acceleration. If the train and the engine together weigh $200$ tons, and if the frictional resistances to motion are equivalent to $7$ lb. wt. per ton, find the horse-power at which the engine was working when the speed of the train was $30$ m.p.h.
Question 6
A point $P$ is describing a circle, centre $O$, with uniform speed. If $Q$ is the foot of the perpendicular from $P$ to a fixed diameter, show that $Q$ moves with simple harmonic motion.
When $Q$ is $4$ feet from $O$ its velocity is $1\frac{1}{2}$ feet per second, and when it is $3$ feet from $O$ its velocity is $2$ feet per second towards $O$. Find how long $Q$ takes to reach $O$ from the latter position.
Question 7
A gun is fixed at a point $P$ at the top a vertical cliff $400$ feet high. $Q$ is the foot of the perpendicular from $P$ to the sea. A ship is travelling away from the cliff at $20$ feet per second along a straight line that passes through $Q$. When the ship is $1000$ feet from $Q$ a shell is fired from the gun at an angle of $45^\circ$ and strikes the ship. Find the initial velocity of the shell.
Question 8
A mass of one ounce attached to a fixed point by an inextensible string $3$ feet long, describes a horizontal circle at a uniform rate of $40$ revolutions per minute. Find the tension in the string and the vertical distance from the fixed point to the plane of the circle.
What vertical distance will the mass rise if its speed is increased to $60$ revolutions per minute?
Question 9
Prove that if a plane surface is immersed in a liquid the total thrust on it due to the liquid is equal to the area of the surface multiplied by the pressure at its center of gravity.
A triangular lamina $ABC$ is totally immersed in water so that the plane of the lamina makes an angle of $30^\circ$ with the horizontal plane. The base $BC$ is $6$ inches in length and the height of the triangle is $3$ inches. Find the total thrust of the water of $ABC$
(i) when its vertex $A$ is at the surface of the water and its base $BC$ is horizontal;
(ii) when its base $BC$ is at the surface of the water.
[$1$ cubic foot of water weighs $62\frac{1}{4}$ lbs.]
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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