# Leaving Certificate Examination 1961 Honours Applied Mathematics

##### Question 1

$ABC$ is a triangle in which $BC=8″$, $\cos B = \frac{1}{2}$, $\cos C = \frac{11}{14}$. Forces of $3$, $12$, $7$ lb. wt. act along $BA$, $BC$, $CA$, respectively. How far from $B$ does the line of action of their resultant cut $BC$?.

Find the magnitude of the resultant in lb. wt., correct to one place of decimals, and find the angle which its line of action make with $BC$, correct to the nearest degree.

##### Question 2

Explain the terms “limiting friction,” “coefficient of friction.”

When a truck is ascending an incline of $28$ in $100$ with a uniform acceleration of $1.6$ ft. per sec. per sec., a box on the floor of the truck is just about to slide backwards. Show by diagram the forces acting on the box, and find the coefficient of friction between the box and the floor of the truck.

##### Question 3

A man is cycling at $10$ m.p.h. in a steady wind. When he cycles in a direction $30^\circ$ north of east the wind appears to him to blow directly from the east. When he cycles due east the wind appears to him to blow from the south-east. Find the velocity of the wind in magnitude and direction.

##### Question 4

Two bodies, of mass $50$ gm. and $350$ gm. respectively, are lying on a smooth horizontal bench which is $10\frac{1}{2}$ feet high. The $50$ gm. body is at the edge of the bench and the $350$ gm. body is $4$ feet away in a direction perpendicular to the edge, the bodies being connected by a light inextensible string $8$ feet long. If the $50$ gm. body is pushed gently over the edge, find how many seconds later it will reach the ground.

##### Question 5

Derive an expression for the total time of flight of a projectile in terms of its initial velocity and angle of projection.

$A$, $B$, $C$, are three collinear points on a horizontal plane. A projectile fired from $A$ passes over $B$ at a height of $21$ feet and reaches its greatest height as it passes over $C$. If $BC=48$ feet and the total time of flight of the projectile is $2\frac{1}{2}$ seconds, find its initial velocity.

##### Question 6

A mass of $4$ lb. suspended from a fixed point by a light inextensible string $2$ feet long acts as a conical pendulum, the mass describing a horizontal circle at a uniform rate of $60$ revolutions per minute. Find the tension in the string, in lb., wt., and the inclination of the string to the vertical.

##### Question 7

A particle is moving along a straight line so that is distance $x$ (cms.) from a fixed point $O$ at the time $t$ (secs.) is given by the formula

$$x=5\sin2t$$.

Show that the motion is simple harmonic and find the periodic time and the maximum velocity.

If the velocity of the particle at $P$ is $6$ cm. per sec., find its acceleration at $P$, and find the least time the particle takes to travel to $P$ from $O$.

[See Tables, p. 30].

##### Question 8

A triangular lamina $ABC$ is immersed in a liquid of specific gravity $0.9$ so that the vertex $A$ is at the surface and the sides $AC$ is vertical. If $AB=5″$, $BC=3″$, $CA=4″$, find the thrust of the liquid on $ABC$ in ounces.

$P$ is a point on $AC$ such that the thrust of the liquid on $ABP$ is give-ninths of the thrust on $ABC$. Find the length of $AP$.

[A cubic foot of water weights $62\frac{1}{2}$ lb.]

##### Question 9

$ABCD$ is a quadrilateral lamina in which $AB=10″$, $AC=6″$, $BC=8″$, $AD=DC=5″$. Find the perpendicular distance of the centre of gravity of $ABCD$ from $AC$.

Find, also, the perpendicular distance of the centre of gravity of $BCD$ from $AB$.

**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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