Investigate an expression for the sum of n terms of an Arithmetic progression, and for the limit of the sum of an infinite Geometric progression. In what case is the latter possible? Sum the following series : When does one quantity vary as another? If A a B, C, D, &c., when only one of the quantities is changed, show that A a BCD when all change. Apply this principle to the following example If 10 men do a piece of work in 12 days of 12 hours each, in what time will 23 men do three times as much, each working 9 hours per day? Find (1) the number of permutations which can be formed from the letters of the word Sebastopol, taken all together. (2) the number of combinations when three letters are taken together. EUCLID, ALGEBRA, AND TRIGONOMETRY. For the Office of the COMMITTEE OF COUNCIL ON EDUCATION. [N.B.—In this Examination Mathematics are not prescribed, but may be selected by any candidate who has made them his especial study,” with the view of displaying his industry and intelligence.] If two triangles have two sides of the one equal to two sides of the other, each to each, but the angle contained by the two sides of one of them greater than the angle contained by the two sides equal to them, of the other; the base of that which has the greater angle shall be greater than the base of the other. BOOK III. PROP. 20.-The angle at the centre of a circle is double of the angle at the circumference upon the same base, that is upon the same part of the circumference. Book IV. PROP. 11. To inscribe an equilateral and equiangular pentagon in a given circle. BOOK VI PROP. 18.-Upon a given straight line to describe a rectilineal figure similar and similarly situated, to a given rectilineal figure. A common tangent is drawn to two circles which touch externally; prove that if a circle be described on that part of it which lies between the points of contact, as a diameter, it will pass through the point of contact of the two circles. Inscribe a circle in a given quadrant of a circle. Divide 4 bx3+ (4 c — ab) x2 · (4 d + ac) x + ad by 4 x 2 aNo 1 + x2 Find the value of x + 1 + x 2 when x = 2 { a N а. b a A and B have the same annual income, and occupy lodgings for 30 weeks in the year, the former at 14s., the latter at 21s. per week, all other expenses being exactly the same for both: B exceeds his income by as much as A comes short of his, and finds that he has spent one-tenth too much Required the annual income and the whole expenditure of each. Write down the expansion of (3 x — 4 y)o, and by means of the binomial Having given the numerical value of sin A find that of cos A 2 and show that there ought to be four corresponding values. Determine which is the proper value when A lies between 180° and 270°. In a plane triangle, having given two sides and the included angle, obtain the formulæ for solving the triangle. Ex. Given a = 205, b = 195, C4°, 10230103, L cot 2° 11 4569162, L cot 54° 20' 98559376, L cot 54° 21'=9·8556708; find the remaining angles. What are the advantages of employing the number 10 as the base for logarithms? Having given the logarithms of a number to the base e, show how to find the logarithms of the same number to the base 10. Given log, 71968 48571394; diff. for 1 60: find the value of 80719686 to seven places of decimals. ALGEBRA. To Candidates for the Admiralty, who selected Algebra as a subject of Examination. 4 x3 (x + y) + 4 x3 (x − y) + 2 x2(x2 + y2) Show that the product of two quantities equals that of their greatest common measure and least common multiple. Find the greatest common measure of 35 x3 +47 x2 + 13 x + 1 and 42 x + 41 x3 Solve the following equations 9x29x- 1 A and B have the same annual income, and occupy lodgings for 30 weeks, the former at 14s., the latter at 21s. per week, all other expenses being exactly the same for both: B exceeds his income by as much as A comes short of his, and finds that he has spent one-tenth too much : Required the annual income and the whole expenditure of each. 4 16 ... (2)-1+ &c. to 10 terms, and to infinity. 15 Write down the expansion of (3 x — 4 y), and by means of the binomial theorem approximate to 3√31. What are the advantages of employing the number 10 as the base for logarithms? Having given the logarithms of a number to the base e, show how to find the logarithms of the same number to the base 10. Given log, 71968 = 4.8571394; diff. for 160: find the value of B0719686 to seven places of decimals. Solve the following equations: x2 No62 + x2 (1) Va2 с Insert four harmonic means between 2 and 12. Find what number r out of n things must be taken together so that the number of combinations formed may be the greatest possible. When does one quantity vary directly as another, and when inversely as another? Given that y varies as the sum of two quantities, one of which varies as r directly, the other as a inversely; and that when x = 1, y=4, when x=2, y=6: Find the relation between x and y. In what scale of notation is sixteen hundred and sixty-four ten thousandths of unity represented by ⚫0404? GEOLOGY. For CANDIDATES for the COLONIAL OFFICE. Define the terms anticlinal, synclinal, unconformable, strike, and dip. What are the constituent minerals of granite, basalt, greenstone, gneiss, trachyte? Describe the divisions of the wealden formation, and give a sketch of its distribution in England, and the reasons for looking on it as a freshwater deposit. Where are the points of division placed by geologists to separate the hypozoie, the paleozoic, the mesozoic, and the cainozoic strata? Exemplify the principles on which these divisions have been founded. Coal has been accounted for sometimes as the result of drift by water of masses of vegetable matter; sometimes as an accumulation of such matter by growth in situ. What hypotheses do these views involve, and what circumstances lend probability to each view?. Give an accurate description of the stigmaria and the sigillaria, and of the facts that prove their mutual relation, with the most characteristic mode of their occurrence in the strata. Describe the mountain limestone formation. How is it distributed over the world? By what observations and arguments does the geologist seek to determine the period of elevation of a mountain chain? Illustrate this by some example. Show that by the amount and by the characters of the distribution of organic remains in one and the same rock in different localities we may predicate facts concerning its oceanic and littoral deposition, pointing to the limits of the sea in which it was formed. Give illustrations of this. Give a description of the most important characteristics common to the trilobites, and give the history of their distribution in time. What are the usual characters of a mineral vein? How far is its wealth found to depend on the rock it traverses? Describe the methods adopted by the practical miner for the discovery of a lode. Describe the structure of the ammonite, and give an account of the distribution of its species in time. Trace the changes in the character of the zoology during the oolitic period, CHEMISTRY. For Candidates for the COLONIAL OFFICE. Define the term element. What elements are gaseous, what are liquids, under the ordinary conditions of the globe? What changes do these undergo by considerable alteration of such conditions? State the law of multiple proportions; and illustrate it by means of the oxides (1) of nitrogen, (2) of manganese. Give the chemical names of, and write in formulæ, alum, common salt, green vitriol, calomel, corrosive sublimate, and chloride of lime. The equivalent of aluminium is 137: How much per cent. of oxygen, of sulphur, and of aluminium is contained in the anhydrous normal (or neutral) sulphate of alumina? Of what gases does the atmosphere consist? Give any accurate method of effecting its analysis, and state the results of this analysis. Explain the changes resulting from the action (1) of hydrochloric acid, (2) of strong nitric acid, (3) of very dilute nitric acid, on gold, iron, tin, and zinc respectively; and state any facts regarding the modification of the result by the purity or the alloying of any of these metals. Define the terms temperature, specific heat, and latent heat, and the term volume as applied to a gas. What law has been asserted connecting the specific heats of the several elements? How far is it universal? What is meant by the theoretical density of carbon vapour? Within what limits is its determination true, and on what assumptions is that determination based? Describe the oxides (1) of carbon, (2) of iron; and give a complete account of the most important oxides of chlorine. Phosphoric acid is tribasic. Give illustrations of each type of its salts. It undergoes modifications by the action of heat. Under what circumstances, and what types of salts result? |