# 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 $s=9t-12t^3$. Calculate its average speed in the interval $t=2$ to $t=3$; its instantaneous speed at $t=4$. 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 $\theta$ to the horizontal up an inclined plane of inclination $\alpha$. Show that is hodograph is represented by a vertical line $XY$ where $OX$ is its initial speed, and $OY$ its speed on striking the plane. Show that if $C$ is the middle point of $XY$ that the angle $XOC$ is $\theta-\alpha$ and that $OC$ 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 $\theta$ with a speed $u$ sufficient to enable it to strike a point on a wall which is at a horizontal distance $a$ from the point of projection. Find the height of the point on the wall struck by the particle in terms of $\tan\theta$. 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 $P$ grams. The acceleration $f$ in cm. per sec. per sec. for different values of $P$ is given in the following table :-

$P$ … | $50$ | $60$ | $70$ | $80$ | $90$ | $100$ |

$f$ … | $15.5$ | $25.5$ | $32.5$ | $39.8$ | $49.1$ | $56.5$ |

Explain how you would treat these observations.

(1) to obtain the value of $P$ 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 $4$ft. $8\frac{1}{2}$in. and the radius of the curve is $440$yds., find how much the outer rail must be raised for an engine going round the curve at 30 miles an hour. [$g=32$ft./sec$^2$]

##### Question 5

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

##### Question 6

An oil-electric coach weights $45$ tons when fully loaded and is equipped with an engine of $250$ h.p. Taking the resistance to motion on the level as $30$lb. per ton; find the speeds which can be attained (1) on the horizontal, (2) up an incline with a gradient of $1$ in $100$.

##### Question 7

A regular hexagon is constructed of rods $16$in. long, each weighing $40$ grams. Each corner is loaded in the following order going round the hexagon with $40$, $50$, $50$, $80$, $90$ and $100$ grams, respectively. Find the centre of gravity of the whole.

##### Question 8

The beam of a balance weighs $w$ grams and its centre of gravity is at a distance $h$ 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 $a$ from the central knife-edge and each $x$ cm. higher than the central knife-edge. If weights of the scale-pans and their loads are $p$ and $q$ respectively, find an expression for the inclination of the beam to the horizontal.

##### Question 9

A string of length $s$ is fastened to the ends of a uniform rod of length $l$. Using a suitable length of string it is possible to find a point on it at a distance $x$ from one end so that if the rod is suspended by holding the string at the point the portion $x$ is horizontal. Find an equation for $x$ and show that the length of the string should not exceed $2l\sqrt{3}$.

##### Question 10

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

**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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