before six, a.m., on March 15, 1878, and ten minutes after two, p.m., on June 6, 1878. (These form one question.) SECTION II. 1. An Indian officer whose annual pay was estimated in rupees, lost £145 5s. 10d. in one year by a fall in the value of the rupee from 1s. 10d. to 1s. 7 d.: what was his salary, estimated in rupees ? 2. Find the profit on 1000 tons of meat, bought at £70 per ton, and retailed at 5s. 10d. per stone of 8lbs., after paying 30s. per ton for carriage. 3. A railway, 1320 chains long, has, besides the two terminal stations, other stations at distances of 12 furlongs: find the annual cost of repairs, at the rate of £16 per annum for each 120 yards of road, and £57 for each station. SECTION III. Find, by Practice, the value (a) of 742 articles at £171 98. 9 d. each, and (b) of 21 acres 2 roods 9 poles, at £183 10s. per rood. (Each step should be intelligible to children learning Practice.) SECTION IV. 1. A room 16 feet square, requires for its walls six pieces of paper 18 yds. 2 ft. long, and 2 ft. broad: find the height of the room. 2. A room 16 feet wide, costs £14 88. to carpet at 4s. 6d. per yard; find the length of the room, the width of the pieces of carpet being 18 inches. SECTION V. 1. If a human heart expels 5 oz. of blood at each beat, and 28,000 lbs. in the course of a day, how many beats would it make in a life of 72 years long, each year consisting of 365 days? 2. A schoolroom accommodates 72 children at the rate of 12 square feet of area for each child; if it had been 12 feet longer it would have accommodated 90 children: find the length and breadth of the room. 3. If 99 men can be fed for 70 days for £430), how much will it cost to feed an army of 770,000 men for 18 months of 30 days each ? 2. Reduce of 6s. 3d. to the fraction of of 28. 1d.; and 2 roods to the fraction of 17 acres 15 poles. What quantity must be added to the difference of 5% and 911, such that if the sum be multiplied by 24 the product may be 28 ? 3. A grocer buys some goods, of which he retails at a gain of 5 per cent., at a gain of 10 per cent., and the remainder at a gain of 20 per cent.; the whole of his sales amounting to £67 158.: find the price at which he bought them. 75 SECTION VII. 1. Express as decimals 15, of '00016 Divide 2939166-69 by 54 1283. 91 256 Show that will not produce a recurring decimal. 2. A warehouse consists of 7 floors; the rent of each floor is 875 times that of the floor below; the rent of the fourth is £17 38.: compare the rents of the highest and lowest floors, and find that of the lowest. 3. If 124-25 francs are equivalent to £5 18. 8d., find the gain or loss on £56.375, if exchanged for 1360 francs. SECTION VIII. 1. What is the annual income derived from investing £3025 in the 3 per cents. at 903, after deducting income tax of 4d. in the pound? 2. A clerk is directed to calculate the discount on £1620 due 3 months hence at 5 per cent.: find the error that would be produced if he were to calculate interest instead of discount. 3. The 3 per cents. are at 93; the stock falls by a certain amount. Two sums of £8184 are invested before and after the fall: what is the extent of the fall, if the difference of income of the two investments is £15? SECTION IX. 1. Find the edge of a cubical box which contains 197137-368 cubic inches. 2. The areas of three squares are in the ratio 1: 9: 16; the area of the second contains 944784 square inches: find the length of a side of each of the others in yards. 3. Three schools educate 75, 33, and 31 children, at an annual average cost of £2 10s., £3 10s., £3 15s. respectively; they are united, and an annual saving is thereby effected of £141 58.: find the annual average cost after their union. SECTION X. 1. A tradesman who makes 15 per cent. on his capital, makes a profit three times as great as another who makes 12 per cent. in the same time; they unite their capital, and the joint capital is £867 4s. 3d.: what amount did each contribute? 2. A man invests £2500 in a business paying 25 per cent. profit; at the end of each year he invests his profits in railway shares at par paying 6 per cent. The business fails after the end of the sixth year, and he loses the whole of his original capital: by how much greater is his present capital than it would have been if he had invested the whole in railway shares at first, calculating simple interest only ? EUCLID, ALGEBRA, AND MENSURATION. Candidates in Scotland may answer two questions out of Section IV. if they omit Section IX. With this exception Candidates are not permitted to answer more than one question in each section. (Marks are given for portions of questions.) EUCLID. Capital letters, not numbers, must be used in the diagrams. The only signs allowed in Geometry are + and. The square on PQ may be written " sq. on PQ," and the rectangle contained by PQ and RS, rect. PQ, RS." SECTION I. Define a right angle, an oblong. (a) Parallel straight lines are such as being produced ever so far both ways do not meet. (b) A square is a four-sided figure whose sides are all equal. (c) A diameter of a circle is a straight line drawn through the centre. Show that the following two statements are incorrect: If unequals be added to unequals the wholes are unequal. All equal angles fill the same space. (These form one question.) SECTION II. 1. If two straight lines cut one another, the vertical or opposite angles will be equal. Two equal perpendiculars, PA, QB, are drawn to the line AB from points P, Q, on opposite sides of AB; AB is bisected in 0. Show that POQ is a straight line. 2. The greater angle of every triangle is subtended by the greater side. BAC is a right-angled triangle, right-angled at A; CD is drawn to any point D of the side AB. Show that CD is greater than CA, but less than CB. 3. The three interior angles of every triangle are together equal to two right angles. If two right-angled isosceles triangles have their bases equal, they will be equal in all respects. SECTION III. 1. Triangles upon equal bases and between the same parallels are equal to each other. If AB, one of the sides of the triangle ABC, be bisected in O, and AC divided into three equal parts in E, D, show that the triangle EOD is one-sixth of the whole triangle ABC. 2. If the square described upon one of the sides of a triangle be equal to the squares described upon the other two sides of it, the angle contained by these two sides is a right angle. BAC is an isosceles triangle, AB being equal to AC, AD is drawn perpendicular to BC. Show that if the square on BC is less than four times the square on AD, the triangle BAC is an acute-angled triangle. 3. If a straight line be divided into two equal parts and also into two unequal parts, the rectangle contained by the unequal parts, together with the square on the line between the points of section, is equal to the square on half the line. Show that the square on the whole line is equal to four times the rectangle and smaller square above mentioned. SECTION IV. 1. In every triangle the square on the side subtending either of the acute angles is less than the squares on the sides containing that angle by twice the rectangle contained by either of these sides and the straight line intercepted between the acute angle and the perpendicular let fall upon it from the opposite angle. Show that the rectangle contained by the sides of a right-angled triangle can never be greater than half the square on the hypotenuse. 2. In any circle only two equal straight lines can be drawn from the same point, not being the centre, to the circumference, one on each side of the diameter drawn through the point. If in any diameter two points be taken equidistant from the centre, and four equal lines be drawn from these points to the circumference, the figure formed by joining the extremities of these four lines taken in order will be a rectangle. 3. If a straight line touch a circle and from the point of contact a straight line be drawn meeting the circle, the angles which this line makes with the line touching the circle shall be equal to the angles in the alternate segments of the circle. Two circles whose centres are A, B, touch externally in C; a common tangent meets the circles in P, Q. Show that the square on PQ is equal to four times the rectangle AC, CB. ALGEBRA. The solution must be given at such length as to be intelligible to the Examiner, otherwise the answer will be considered of no value. SECTION V. (a.) If a = 3, b = 4, c 27, find the value of 3ab + ac + 4√√/a2b -- 33⁄4/l3c + √ a2 + b2. (b.) Express in the ordinary notation: 7 x 107 +7 x 105 +7 x 103 +7 x 10. (c.) Simplify the expression: 5c6a-{3 (2bc) + 4 (a−2b)-6(2a-c)}. (d.) Prove the formula am X an = am+n (e.) Multiply a3-ay-2y3 by x3 + x2y-2y3. (These form one question.) SECTION VI. 1. Resolve into factors a2 + 4x — 21; 12x+11a2y-15y1; a + a2b2 + b1. 2. Simplify the expression: (x + z)2-y2 (y + x) 2 - 22 + 3. Prove the following identity: (y + z) - a2. (bc)3 + (ca)3 + (a−b)3 = 3 { a2 (c — b) + b2 (a—c) + c2 (b− a)} |