48. Supposing the receipts on a railway to vary as the increase of speed above 20 miles an hour, whilst the cost of working the trains varies as the square of that increase, and that at 40 miles an hour the expenses are just paid; find the speed at which the profits will be the greatest. 49. Determine whether the infinite series is convergent or divergent. a Ans. 30 miles. m+p m+2p m+3p Ans. Convergent or divergent as a<or>1. 50. Prove that every quadratic surd, supposing the unit to be a line, may have an exact geometrical representation. 51. A person continually walks at an uniform speed, and always in the same direction, round the boundary of a square field: and another continually walks, at the same uniform speed, from one end to the other of a diagonal of the field. Prove that according to their relative initial positions, they will either (1) never meet at all, or (2) meet once; but they can never meet more than once however long they continue to walk. 52. A sum of money in £. s. d. is multiplied by a certain number; the pence are now half what they were before, and the shillings and pounds each what the shillings were at first. What is the sum, and the multiplier ? (1) Ans. £9. 19s. 8d. (2) Ans. 2. 53. If N1 N, are any two consecutive convergents to D' D1 N, a the error in taking for < D ¿D1D2 α 1 but > D1(D2+D1) * 54. If p=√u2+ß be defined to be the 'modulus' of a binomial of the form a+ß-1, where a, ẞ are rational; and p12 P2, P3,...P. be the moduli ofn such binomials; prove that the modulus of the symbolical product of these n binomials is P1P2Pз••• Pn. 55. Explain the notation of functions'; and shew that, if F(x)=a*, F(x)×F(y)=F(x+y). and thence deduce the sum of the series for F(n, m) when n is a positive integer. 1.2.3...n Ans. p". m(m+p)...(m+np) ' 57. The Julian Period consists of 7980 years; the cycles of the sun, moon, and the Indictions, of 28, 19, and 15, years respectively. If any particular year be the nth of the Julian period, the remainders after the division of n by 28, 19, 15, are the years of the Solar and Lunar cycles, and of the Indictions, respectively. Prove the following Rule for determining the year of the Julian period when the years of the Solar and Lunar cycles and of the Indictions are given. Multiply the number of the year in the Solar Cycle by 4845, in the Lunar by 4200, and in the cycle of Indiction by 6916; divide the sum of these products by 7980, and the remainder is the year of the Julian pericd sought. 58. If an import duty of r shillings a quarter be laid on foreign corn when the price of corn in the English market is p shillings a quarter, and e, f, are the numbers of quarters of English and Foreign corn consumed in a year; and if the imposition of such a duty causes the price to rise to p+ shillings a quarter, and the consumption to sink, so that the same sum of money is still expended, the English produce remaining constant; find the value of r most productive of revenue. Ans. np{√/1+£−1}. 59. A square sign-board is divided into 16 equal squares by vertical and horizontal lines. In how many ways can 4 of these squares be painted white, 4 black, 4 red, and 4 blue, without repeating the same colour in the same vertical or horizontal row? Ans. 576. 60. Shew that the greatest coefficient, formed from the index n, in the expansion of (a1+a2+a,...+am)" is where q is the quotient Ln and r the remainder, when n is divided by m. COLLEGE EXAMINATION PAPERS. [Solutions of all the following Equations and Problems will be found in the Companion.] 5. A railway train travels from A to C passing through B where it stops 7 minutes; two minutes after leaving B it meets an express train which started from C when the former was 28 miles on the other side of B: the express travels at double the rate of the other, and performs the journey from C to B in 14 hours; and if on reaching A it returned at once to C it would arrive 3 minutes after the first train. Find the distances between A, B and C, and the speed of each train. Ans. AB=31 miles, BC=63 miles: speed of the ordinary train 21 miles, express......42.... 6. To meet a deficiency of m millions in the revenue of a country, an additional tax of a per cent. was laid upon articles exported, and the tax upon imports was diminished c per cent.: in consequence of these alter ations the value of the imports was increased so as to be n times as great as the exports, and the deficiency was made up. It was afterwards found that if the additional tax upon the exports had been a' per cent., and the tax upon imports diminished c' per cent., the values of the articles being altered as before, the deficiency would not have been made up by m' millions. Find the values of the exports and imports after the alteration of the tax. 7. Fifty thousand voters, who have to return a member to an assembly, are divided into sections of equal size, and each section chooses an elector, the member being returned by the majority of such electors. There are two candidates, A and B. In those sections which return electors favourable to A, the majority is double the minority, while in those favourable to B, the minority forms only a tenth of the whole. After the primary elections a third candidate C comes forward, and is joined by so many electors of each party, that he is returned by a majority of 3 over 4, and 14 over B. If C had not come forward, A would have been returned by a majority 19 less than the whole number of votes actually polled by C, and if the elections had been by the 50,000 voters directly between 4 and B, B would have had a majority of 6000. Find the number of sections. 5. The distance between the two termini A and Z of a railway is 100 miles. A train starting from A runs up-hill during the first 30 miles of its journey, the next 50 miles are on a level, and the remaining 20 are up-hill. The train may be supposed to travel 5 miles an hour faster on the horizontal road than when it is ascending a hill. There are to be stoppages at stations B, C, D, and E, at distances 20, 42, 674, and 90 miles respectively from A, and each stoppage may be supposed to cause a detention of 3 minutes. Find the time of arrival at B, C, D, and E, of the train which starts from A at 8h. Om. and arrives at Z at 12h. 42m Ans. At B 9h, C 10h. 3m, D 11. 6m, E 12h. 9m. 6. A number of vessels A,, A2, A3, .......Ar, A‚+1,... Am are arranged in a A, contains a quantity of wine, A, a quantity of water, and the remaining vessels A3,...A,, Ar+1,... Am, contain any quantity of any other fluids. row. 1 of the wine in A, is taken from A, and added to the contents of 4,, of n the mixture is taken from A, and poured into A,, of the contents of 2 A, is poured into A,, and so on to the end of the series of vessels. Again, 1 n 3 n 1 of wine remaining in A, is poured into 4,, of the contents of 4, into A,, and so on. A, is supposed never to receive any addition. It is found that 60 times the quantity of wine in the vessel A, after r−1 abstractions of fluid from that vessel = 31 times the quantity of wine in the same vessel after r abstractions. Also 59 times the quantity of water in A, after r-1 abstractions of fluid from that vessel = 31 times the quantity of water in the same vessel after r abstractions. Find the numerical values of r and n. Ans. r=n=31. 7. Two points P and Q are connected by a wire (4), 4th of an inch in diameter and 50 miles in length, which is used for transmitting a galvanic current. It is required to replace the wire (4) by three others (a), (b), and (c), composed of different metals, and of lengths 50, 60 and 70 miles respectively. These new wires must be of such diameters that the current, which previously passed along (4) may be divided so that the quantities which pass along (a), (b), and (c) may be as 3, 4, and 5. The quantity of galvanic fluid that will pass along a wire is supposed to vary inversely as the resistance, and the resistance to vary directly as the length of wire to be traversed, inversely as the sectional area of the wire, and inversely as the conductibility. Also the sectional area of a wire varies as the square of its diameter. It is found by experiment that the quantity of galvanic fluid which will pass along a portion of the wire (4), 4th of an inch in diameter and 15 yds. long, may be denoted by 1000 k. Also portions of wire th of an inch in diameter, 1 10 composed of the same metals as (a), (b), and (c), of lengths 20, 10, and 40 yds. respectively, are capable of transmitting quantities of galvanic fluid 750 k, 5400 k, and 3500 k respectively. Find the least possible diameters of wires (a), (b), and (c), in order that the above conditions may be satisfied. Ans. 1 1 1 20' 30' 40 |