# Leaving Certificate Examination 1963 Honours Applied Mathematics

##### Question 1

and are two fixed pegs in the same horizontal line. One end of a light string is attached to and the other end to . When two masses are attached to the string, one at a point and the other at a point , the angles , , are , , , respectively. Find the weight of each mass if the two together weigh lbs.

##### Question 2

is a quadrilateral lamina in which cms., cms., cms., cms. and . Find the perpendicular distance of the centre of gravity of from .

Find, also, the perpendicular distance of the centre of gravity of from .

##### Question 3

To a person on a ship travelling due East at m.p.h. another ship miles due North appears to be travelling at m.p.h. in a direction West of South. Find the velocity of the second ship in magnitude and direction as accurately as you can by using your tables.

Find the distance between the ships when they are nearest to one another.

##### Question 4

An engine develops a horse-power of as it accelerates at the rate of ft. per sec. when travelling at a speed of m.p.h. down an incline of in . If the eight of the engine is tons, calculate the frictional resistance to motion in lb. wt. per ton.

##### Question 5

A pile of mass one ton is driven vertically feet into the ground by blows of a hammer of mass tons which falls vertically through a height of feet.

Hence find the least mass which, when placed on top of the pile, would begin to drive the pile down. (Assume that the resistance of the ground is uniform and neglect the resistance-effects of side-friction on the pile.)

##### Question 6

A mass of ounces attached to a fixed point by a light inextensible string of length feet describes a horizontal circle at a uniform rate of radians per minute. Find the radius of the circle in feet and the tension in the string in lb. wt., correct to two significant figures in each case.

What angle would the string make with the vertical if the angular velocity were increased to radians per minute?

##### Question 7

A dense fog is assumed to cover a horizontal plane to a uniform height of feet. A mass projected from the plane rises above the fog after sec. and then travels a horizontal distance of feet before re-entering the fog. Find the initial velocity of the mass and its angle of projection.

##### Question 8

A particle moves in a straight line so that its displacement (cm.) from a fixed point at time (sec.) is given by the formula

Show that the motion is simple harmonic and find (i) the amplitude, (ii) the maximum velocity of the particle.

Find, also, the least time, correct to three significant figures, taken by the particle to travel a distance cm. from its mean position.

##### Question 9

is a triangular lamina in which , , and , where is the mid-point of . The lamina is immersed in a vertical position in a liquid of specific gravity so that the base is at the surface. Find the total thrust of the liquid on .

How far must the lamina be withdrawn vertically upwards out of the liquid, the base remaining parallel to the surface, if the thrust on the portion in the liquid is lb. wt. ?

(One cubic foot of water weighs lb.)

**Citation:**

**Citation:**

State Examinations Commission (2018). *State Examination Commission*. Accessed at: https://www.examinations.ie/?l=en&mc=au&sc=ru

Malone, D and Murray, H. (2016). *Archive of Maths State Exams Papers*. Accessed at: http://archive.maths.nuim.ie/staff/dmalone/StateExamPapers/

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