3. How may the least common multiple of three algebraical expressions be found ? Find that of 2x2 1, 2x2 + 3x + 1, I. 4. Prove the rule for extracting the cube root of an algebraical quantity, and extract the cube root of 8a6 12a5b + 42a482 – 37a3b3 + 63a2b4 – 27ab5 + 2760. 5. The sides of 3 cubes have equal differences, and their sum is 15 inches: the solid content of the 3 is 495 cubic inches : find their dimensions and volume. 6. If a and ß be the roots of the equation ax2 + bx + c = 0, b prove that a t B and aß What forms will the equation assume, if the roots be equal, and (1) of the same sign, (2) of opposite signs? a a 92 7. Solve the equations a2c 6 = 14. 8. Prove that the value of the fraction a +b+c lies between Þtator a b c those of the greatest and least of the fractions : and show in Þar what cases m-ax b has a real and positive, a real and negative, or no real value. 9. Two men leave two places, A and B, distant (d) miles from each other, and travel (a) and (6) miles a day respectively in the same straight line AB: what is their distance apart at the end of (t) days, and after what time will they come together ? Explain the results (1) when a = b, (2) when a = 6 and d 10. If three equal circles, whose radii are each 7 inches, touch each other, find the area enclosed between them to three places of decimals (TT 22). = 0. II. Two thin vessels, without lids, each contain a cubic foot : the one is a rectangular parallelopiped on a square base whose height is half its length: the other a right circular cylinder whose height is equal to the radius of the base: compare the amounts of material which it would require to make them, the thickness being the same for both (TT 3-1416). 12. A solid consisting of a right cone standing on a hemisphere is placed in a right cylinder full of water, and touches the bottom. Find the weight of water displaced; having given that the radius of the cylinder is 3 feet, and its height 4, the radius of the hemisphere 2, and the height of the cone 4; and that a cubic foot of water weighs 1,000 oz. 13. Prove that log o a X log ab I, and find the value of ovzdz. Given log 2 = *30103 47712 3*21112 log 1627 = 3·21139. 14. The number of births in a town is 25 in every thousand of the population annually, and the deaths 20 in every thousand; in how many years will the population double itself? Given log 67 1.82607. al. a'. ao.... a" = p. find (in terms of a and P) an expression for the number of factors al, a, a, &c. 6 log 3 log 1626 II.- EUCLID AND TRIGONOMETRY. 1. Show that if a straight line be divided into any two parts, the rectangle contained by the whole and one of the parts, is equal to the rectangle contained by the two parts, together with the square on the aforesaid part. To what algebraical formula does this proposition correspond ? 2. Describe a square that shall be equal to a given rectilineal figure. A given square is divided into four equal squares by two straight lines drawn through its centre. If one of these squares be supposed to be removed, show. how to construct a square equal to the remaining gnomon. 3. Prove that the angle at the centre of a circle is double of the angle at the circumference on the same base, that is, on the same part of the circumference. Deduce from this proposition the fact that the angle in a semicircle is a right angle. 4. From a given circle cut off a segment containing an angle equal to a given rectilineal angle. 5. Describe a circle about a given triangle. 6. Define the terms“ alternando” and “convertendo”; “similar figures" and "reciprocal figures." If four magnitudes be proportionals, they shall also be proportionals, when taken inversely. 7. If a straight line be drawn parallel to one of the sides of a triangle, it shall cut the other sides, or those sides produced, proportionally. In a triangle ABC, BC is bisected in D, AD is bisected in E, and BE produced to meet AC in F: show that AF = {AC, and EF = 1BF. 8. Equal triangles which have one angle of the one equal to one angle of the other, have their sides about the equal angles reciprocally proportional. 9. Name the trigonometrical ratios, and define them. Prove that sec ? A + cosec 2A sec 2 A cosec 2 A. u. Find the value of sin 15°, and prove that sin 3(A – 159) = 4 cos (A - 45°) cos(A + 15%) sin (A – 15%). 12. Show that the sines of the angles of a triangle are proportional to the opposite sides. If a, b, y are the perpendiculars from the angular points of a triangle upon the opposite sides, prove that a sin A + b sin B + c sin C 2(a cos A + B cos B + y cos C). Find 13. The sides BC, CA, AB of a triangle are as 4:5:6. the angle B. L cos 27° 53' = 99464040 log 2 = *3010300. L cos 27° 54' = 9*9463371 14. In a triangle of given perimeter prove that the radii of the escribed circles are proportional to the tangents of the semi-angles opposite to the sides to which they are escribed. Show that the area of the triangle formed by joining the centres of the escribed circles of the original triangle when s is the semiperimeter sin A' of the original triangle. 15. The angular elevation of a steeple at a place due south of it is 45°, and at another place due west of the former station the elevation is 15° : show that the height of the steeple is - (31 – 3-1), 2 a being the distance between the places. December 1881. I.-ALGEBRA AND MESURATION. 1. Simplify the following expressions x2 22 + 22y I I + (2) N3 - N 2 15 - VW (3) [a-154a-4}}{cob Wab)2}}]-! 2. Multiply 23-2axt + a2 by x} +a and divide x3 + 33/2.x+1 by x + 32 I. 3. Assuming that am X an = - am+n, for all values of m and n, show what meaning must be assigned to the following expressions : (2) a-1, (3) ai, and reduce to its simplest form (a + b - 1)* + (a - b - 14 (1) ao, 322 4. Find the Least Common Multiple of IIX + 6, 272 7x + 3, and 6x? — 7x + 2. 5. Extract the square root of zada a2 3 da 410 16 and the cube root of 76 - 6x5 + 1574 – 20x3 + 1522 - 6x + 1. 2 2 7. At what time between 6 and 7 o'clock are the hands of a clock at right angles ? 8. A circular plot is surrounded by a ring of gravel 6 feet wide; if the radius of the circle, including the ring, be a feet, find the relation between a and b, so that the areas of grass and gravel may be equal. 9. A man divides his property between his wife, his three sons, and his three brothers, in such a manner that after the legacy duty on each share is paid, the widow's share is one-third of that of each of the sons, and equal to the whole received by the brothers ; find the proportion which each receives, the legacy duty for a son being one per cent., for a brother three per cent., and for a widow nothing, and the brothers' shares being equal. |