8. Define mean proportional, third proportional, fourth proportional; and show that if M be the mean proportional between the two numbers 3 and 12, and T their third proportional, then 256 M1: 74 is the duplicate ratio of 1: 4. 9. Extract the square root of and of 16 x6-16x+4x2+8x3 −4x2 + 1 ; a2x2-2 ax3+x+ a1+2a3x + a2x2 TUESDAY, JUNE 9, from 11 A.M. to 1 P.M. 1. Assuming the parallelogram of forces for direction, prove it for magnitude. 2. R is the resultant of two equal forces P, Q, acting at a point. When the direction of P is reversed show that the new direction of R is at right angles to its former direction. 3. The distances of any number of heavy particles in one plane from a straight line in the plane being given, determine the distance of the Centre of Gravity of the system from that straight line. 4. If a uniform triangular plate be kept horizontal by three vertical strings one at each corner, show that the tensions on the three strings are equal. 5. If two forces act at a point A in directions AB, AC, and O be a point within the angle BAC, show that the difference of the moments of the two forces round O is equal to the moment of their resultant. 6. If three forces keep a body at rest, prove that they must either be parallel or meet in a point. ACB is a uniform rod, of weight W, such that AC = } AB. The rod is supported, (B uppermost,) with its end A against a smooth vertical wall AD, by a string CD; DB being horizontal, and CD inclined to the wall at an angle of 30°. Find the tension of the string, and the pressure on the wall, 7. Describe the three systems of pulleys. Apply the principle of virtual velocities to determine the ratio of the Power to the Weight in the system in which each string is attached to the weight, and all the strings are parallel (the weights of the pulleys being neglected). 8. State the 1st and 2nd Laws of Motion. Establish the parallelogram of velocities. 9. Prove the formula s = gt2, for a falling body. If a body which has fallen from rest, fall in one particular second through 144 feet, find how long it has been falling; (g = 32). 10. Show that the range of a projectile on a horizontal plane is proportional to the product of the vertical and horizontal velocities at starting. An equilateral triangle ABC in a vertical plane, has its base BC horizontal, and its vertex A uppermost. From B two particles are projected with the same velocity, one in the direction of BA, the other in that of the perpendicular from B on AC. Show that the range will be the same in each case. TUESDAY, JUNE 9, from 2.30 to 4.30 P.M. D. Mathematics. EUCLID. 1. Give all Euclid's definitions of four sided figures, and the four definitions concerning segments of circles. 2. If two triangles have two angles of the one equal to two angles of the other, each to each, and one side equal to one side; viz. the sides adjacent to the equal angles in each; then shall the other sides be equal, each to each, and also the third angle of the one to the third angle of the other. 3. If a straight line be divided into two equal, and also into two unequal parts, the squares on the two unequal parts are together double of the square on half the line and of the square on the line between the points of section. 4. If in a circle two straight lines cut one another, which do not both pass through the centre, they do not bisect one another. 5. Inscribe in a circle a triangle equiangular to a given triangle. 6. The opposite sides and angles of a parallelogram are equal to one another, and the diameter bisects the parallelogram, that is, divides it into two equal parts. 7. Describe a square which shall be equal to a given rectilineal figure. 8. In a circle the angle in a semicircle is a right angle; and the angle in a segment greater than a semicircle is less than a right angle; and the angle in a segment less than a semicircle is greater than a right angle. 9. Describe an isosceles triangle, having each of the angles at the base double of the third angle. What part of a right angle is each of the angles of the figure in this proposition? WEDNESDAY, JUNE 10, from 9 to 11 A.M. F. Logic and Political Economy. (Candidates are advised to attempt every question.) 1. Explain with illustrations the meanings of Term, Proposition, Inference. 2. When is a definition a good one? What is the difference between a definition and a description? Give examples. 3. What is meant by Intension, Dilemma, Permutation, Irrelevancy, Cross-Division, Ambiguity, Classification? 4. On what principles does the validity of the Syllogistic Process depend? Show that a Petitio Principii is a violation of the Laws of the Syllogism. 5. State your own views regarding the practical value of some knowledge of Logic. 6. What are the requisites of any valid Induction? What general principles underlie all Induction, and how have they been established? 7. Explain Phenomenon, Observation, Empirical Law, Hypothesis, Method of Residues, Analogy. WEDNESDAY, JUNE 10, from 2.30 to 4.30 P.M. F. Logic and Political Economy. (Second Paper.) Political Economy. (Candidates are advised to attempt every question.) 1. Criticise the present efficiency of Land as an agent of Production. 2. How are Wages determined? What Remedies have been suggested for Low Wages? 3. For what purposes are Precious Metals required in this Country? From what sources are they supplied? 4. Distinguish between-Gross Profit, Net Profit; Convertible Currency, Inconvertible Currency; Real Price, Nominal Price; Fixed Capital, Circulating Capital; Conacre, Cottier, Crofter. 5. Why is the Income Tax a bad tax, and why is it a good Tax? What new Taxes would you suggest for increasing the Revenue? 6. When and with what objects were the following passedBank Charter Act, Repeal of Corn Laws, Statute of Apprenticeship, Tithes Commutation Act? 7. How would you explain the wealth of England as contrasted with the poverty of Ireland? MONDAY, JUNE 8, from 2.30 to 5.30 P.M. SECTION 5. English. I. 1. Shew, with instances from the various parts of speech, that modern English in its development from Anglo-Saxon has passed from the synthetical to the analytical stage. 2. Give examples of corruption of language caused by false analogy and phonetic decay. What causes operate to prevent dialectical variation in languages? 3. Can you give any account of the English dialects? Which finally became the literary language? which had nearly succeeded in becoming so? 4. Shew by citing instances that the English language is mixed both in its vocabulary and grammar. 5. Examine the following expressions, and criticise :— Darkling I listen (Keats). Many a man and many a maid (Milton). And knew not eating death (Milton). He was an-hungered. Woe worth the day! Methinks. Under a star-ypointing pyramid (Milton). When do we find the auxiliaries do and did first used in English? 6. Illustrate from the authors of various periods the tendency of verbs with strong past tenses to change them into corresponding weak forms. 7. Give the derivations of the following words-bargain, dungeon, danger, orchard, starvation, mob, chapel, bedridden, jeopardy, king, lobster, could, talk, and yule; and criticise any incorrect derivations which have been given of any of them. 8. What do you mean by gender in language? From what did it arise? How far does it exist in English? |