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8. The square on the distance between the centres of two given circles is equal to twice the sum of the squares on their radii. Prove that the eight points of contact of the four common tangents of the two circles lie four by four on two right lines.
ARITHMETIC AND TRIGONOMETRY.
1. Find, to the nearest penny, the present value of £1000 to be paid in four equal yearly instalments of which the first falls due a year hence, reckoning compound interest at 5 per cent. per annum.
2. Given that the volume and surface of a sphere of radius rare, respectively, 3 and 4r2, where = 31416; find, to π3 π the nearest tenth of a cubic foot, the volume of a sphere whose surface is 1000 sq. feet.
3. A train 86 yds. long, moving at a uniform rate, meets a train 90 yds. long moving in the opposite direction at the rate of 60 miles an hour, and then overtakes a train moving in the same direction at the rate of 30 miles an hour. It takes 3 secs. to pass the first train and 32 secs. to pass the other. Find the length of this other train.
7. Given in a triangle
a = 2+1, b = 1, C= 45°,
find the remaining parts.
8. In a triangle right-angled at C the intersection of the medians is G. Prove that
tan AGB =- sin 24.
1. Will a spring-balance that weighs accurately in Dublin weigh accurately everywhere else? Give reasons for your answer.
2. State and prove the conditions that must be satisfied by a number of forces in a plane in order that the forces may be in equilibrium.
3. An engine drives machinery by means of a belt working on a pulley of radius r feet. Show that the rate of working in horse-power' is
when the pulley makes n revolutions per second, and t1 and t2 are the tensions in the two sides of the belt.
4. Two bullets are projected horizontally in the same vertical plane from points in the same vertical line with velocities of 2000 and 3000 feet per second: if the latter bullet is projected one second after the former, what must be the distance between the points of projection that the bullets may collide? [g = 32.]
5. Find the centre of gravity of a rod of uniform crosssection, the density of which varies as the distance from one end.
6. State the laws of friction, and apply them to find the acceleration of a body moving on a horizontal sheet of ice, given μ = 1/10.
1. What is meant by pressure at a point' in a liquid ? Show that the resultant thrust on a plane surface immersed in a liquid depends on the area of the surface, and the depth of its centroid. What do you know of the line of action of this resultant?
2. By what theoretical considerations would you provs that a body immersed wholly or partially in a liquid ie pressed upwards by a force equal to the weight of the displaced liquid.
Verify this for the case of a cube immersed in a liquid with two opposite corners in the same vertical, by finding the resolved parts of the pressures on each face.
3. A cylindrical vessel of any shape with a horizontal plane base is filled with a number of different liquids, show that the pressure on the base is the same whether they mix
4. What is meant by the expression 'height of the homogeneous atmosphere'? The weight of a litre of air at 0° C. and 760 mm. pressure is 1.29 grammes, and the pressure of the atmosphere is observed to be 1033 grammes per square cm. Find the height of the homogeneous atmosphere.
5. Describe a compressed-air manometer or siphon gauge. How would you graduate it by calculation?
6. Describe and explain the action of a suction-pump. Why does the addition of an air-chamber secure a more continuous flow?
FIRST UNIVERSITY EXAMINATION.
1. Translate into English:
Nec mihi cum Teucris ullum post eruta bellum
Terra viros, ultro Inachias venisset ad urbes
Quidquid apud durae cessatum est moenia Troiae,
2. (a) Form adverbs from-firmus, facilis, grex, cavere, caelum.
(b) Give five perfect participles from verbs which are active in form but not transitive.
(c) Conjugate the verbs-pango, revertor, divido, ordior,
(a) He was insolvent, and no one believes him.
(b) At the time of the gladiatorial shows, the law of extortion was proposed by the officers for the admnistration of justice.
(c) If you come to my house I shall promise to put to your credit three-fourths of the legacy.
4. Write grammatical notes on:
(a) Vereor ne aut celatum me illis ipsis non honestum, aut invitatum refugisse mihi sit turpissimum.
(b) Memoriae proditur ex Italia expulsum Cn. Pompeium quod imperii populi Romani decus ac lumen fuit.
Per ima cucurrit
Ossa tremor cui fata parent quem poscat Apollo.
Fulgentem imperio fertilis Africae
Fallit sorte beatior.
5. Translate with short notes:
(a) An mare, quod supra, memorem, quodque alluit infra?
(b) Talem dives arat Capua et vicina Vesevo
(d) Quo Chium pretio cadum
Mercemur, quis aquam temperet ignibus,
Pelignis caream frigoribus, taces.
Da Lunae propere novae,
Da noctis mediae, da, puer, auguris
Miscentor cyathis pocula commodis.