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All things Mathematical
13 Nov 2018

Leaving Certificate Examination 1956 Honours Applied Mathematics

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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 go6feet up the ladder? <h4>Question 4</h4> A3lb. mass is held at rest on a smooth plane inclined to the horizontal at an angle of30^\circ. A light string passing over a smooth pulley at the top of the plane connects the3lb. mass to a2lb. mass which is hanging freely and which is2\frac{1}{2}feet from the ground. The system is released and aftert_1seconds the2lb. mass is brought to rest on hitting the ground ; after a furthert_2seconds it is jerked into motion again. Find the values oft_1andt_2.  What fraction of the kinetic energy is lost when the2lb. mass is jerked into motion? <h4>Question 5</h4> A lamina is in the shapeABCDE(see diagram) in whichABEis an equilateral triangle of sides4cms. andBCDEis a rectangle in whichBC=1cm. Find the distance of the centre of gravity of the lamina from the lineCD<img class="size-medium wp-image-656 aligncenter" src="" alt="" width="300" height="274" />  If the triangular portionALMis removed (AL=AM=2cms.), find the distance fromCDof the centre of gravity of the remainder. <h4>Question 6</h4> An engine raises water from a depth of18feet and delivers it at the rate of300gallons per minute with a velocity ofvfeet per second. If the engine is working at2\frac{3}{4}horse-power, find the value ofv.  [One gallon of water weighs10lb.] <h4>Question 7</h4> A ball is lying on the ground atC, a point15feet away fromB, the base of a vertical pole. The ball is to be kicked fromCwith an initial velocity of20\sqrt{2}feet per second so as to strike the pole at a pointhfeet aboveB.  (i) Ifh=6, find the two possible angles of projection.  (ii) Find the value ofhfor which there is only one angle of projection. <h4>Question 8</h4> Define Simple Harmonic Motion.  A particle is moving in a straight line. Its distance,x(cms.), from a fixed point in the line at timet

    (seconds) is given by the formula <span class="ql-right-eqno">   </span><span class="ql-left-eqno">   </span><img src="" height="36" width="89" class="ql-img-displayed-equation " alt="\[x=5\sin\frac{1}{2}t\]" title="Rendered by"/> 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. <h4>Question 9</h4> A lamina is in the shape of a trapezium

ABCDin whichAD=6in.,AB=5in.,CD=11in., and the angles\hat{BAC}=\hat{CDA}=90^\circ. The lamina is immersed in a vertical position in water,ADbeing at the surface. Find the total thrust of the water on the lamina.  When the lamina is pushed down vertically so thatADisxinches below the surface and parallel to it, a horizontal line throughBwill divide the lamina into two parts the thrusts on which are equal.  Find the value ofx.  [One cubic foot of water weighs62.5$ lb.]


State Examinations Commission (2018). State Examination Commission. Accessed at:

Malone, D and Murray, H. (2016). Archive of Maths State Exams Papers. Accessed at:


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