# Leaving Certificate Examination 1932 Honours Applied Mathematics

##### Question 1

A particle moves along a straight line and its distance from a fixed point in the straight line is given by . Calculate its average speed in the interval to ; its instantaneous speed at . Find when and where its instantaneous speed is zero. Describe in a general way the motion of the particle.

##### Question 2 (a)

A particle is projected at angle to the horizontal up an inclined plane of inclination . Show that is hodograph is represented by a vertical line where is its initial speed, and its speed on striking the plane. Show that if is the middle point of that the angle is and that multiplied by the time of flight is equal to the range on the inclined plane.

**OR**

##### Question 2 (b)

A particle is projected at an inclination with a speed sufficient to enable it to strike a point on a wall which is at a horizontal distance from the point of projection. Find the height of the point on the wall struck by the particle in terms of . Hence find an expression for the maximum height on the wall which can be reached by the particle.

##### Question 3

A trolley is pulled along a horizontal table by a string passing over a pulley and carrying a scale pan and weights so that the total load on the string is grams. The acceleration in cm. per sec. per sec. for different values of is given in the following table :-

… | ||||||

… |

Explain how you would treat these observations.

(1) to obtain the value of required to make the system move with uniform speed.

(2) to exemplify the second law of motion.

(3) to deduce approximately the mass of the trolley.

##### Question 4

Why is the outer rail of a railway track raised on a curve? If the gauge is ft. in. and the radius of the curve is yds., find how much the outer rail must be raised for an engine going round the curve at 30 miles an hour. [ft./sec]

##### Question 5

A nail is driven into a wall vertically below a point to which is attached a pendulum bob by a string of length ( being greater than ). The bob is raised to the level of with the string taut and then released. The string strikes the nail at , and the bob just makes one complete revolution about . Find the value of in terms of .

##### Question 6

An oil-electric coach weights tons when fully loaded and is equipped with an engine of h.p. Taking the resistance to motion on the level as lb. per ton; find the speeds which can be attained (1) on the horizontal, (2) up an incline with a gradient of in .

##### Question 7

A regular hexagon is constructed of rods in. long, each weighing grams. Each corner is loaded in the following order going round the hexagon with , , , , and grams, respectively. Find the centre of gravity of the whole.

##### Question 8

The beam of a balance weighs grams and its centre of gravity is at a distance cm. below the central knife-edge. When the beam is in equilibrium with the scale pans removed, the outer knife-edges, from which hand the scale-pans, are at a horizontal distance from the central knife-edge and each cm. higher than the central knife-edge. If weights of the scale-pans and their loads are and respectively, find an expression for the inclination of the beam to the horizontal.

##### Question 9

A string of length is fastened to the ends of a uniform rod of length . Using a suitable length of string it is possible to find a point on it at a distance from one end so that if the rod is suspended by holding the string at the point the portion is horizontal. Find an equation for and show that the length of the string should not exceed .

##### Question 10

A body of weight is dragged with uniform speed up a plane inclined at an angle to the horizontal by a force applied to it in a direction inclined to the plane at an angle . Assuming the ordinary laws of friction, determine the magnitude of the force and the work done dragging the body up the plane.

**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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