SCHOLARSHIP QUESTIONS, 1877. The time allowed for each paper is three hours, unless otherwise specified. Candidates are not permitted to answer more than one question in each section. The solution must be given at such length as to be intelligible to the Examiner, otherwise the answer will be considered of no value. ARITHMETIC. SECTION I. Express in figures, thirty billion, ninetyfive million, seventy thousand and five. Express the year 1988 A.D., in Roman numerals. How many times can 75 be subtracted from 6075 so as to leave a remainder equal to itself. Divide the continued product of 17, 18, 19, 23, 301, 113, by the difference of 259867, and 126145. (These form one question.) SECTION II. 1. In a bag are 90 shillings, 100 sixpences, and 120 fourpenny pieces; find the least number of threepenny pieces that must be added, so that the whole may be distributed in 'sums of eighteen pence, no such sum being made up of any one coin exclusively. 2. An orphan asylum contains 120 children, each of whom consumes 40 oz. of meat per week: find the weekly saving effected by using American beef at 6d. per lb., allowing also 2s. 4d. per cwt. for carriage, in place of English beef at 6s. 4d. per stone of 8 lbs. 3. The length of the solar year being 365 days, 5 hrs 48 min. 50 sec., and the Chinese civil year consisting of 12 months of 29 and 30 days alternately, how many months of 30 days must be added in the course of 60 years that the Chinese civil year and the solar year may agree most nearly? SECTION III. 1. Make out the duty payable on the following goods: 7,000 lbs. of coffee at 14s. per cwt. 6.252 packs of cards at 3s. 9d. per doz. packs. 90 bushels of malt at £1 4s. per qr. SECTION IV. Find, by Practice, the value of— 1. 15 cub. yds. 21 cub. ft. 648 cub. in, at £4 48. per cub. yd. Or 2. 75 miles 5 fur. 9 chains 23 yds. at £56 per mile. SECTION V. 1. A tradesman sold some goods at 15 per cent. above the price at which he bought them: the two prices together amounted to £733 3s.: find the price at which the goods were bought. 2. A block of granite 16 ft. long, 8 ft. broad, 4 ft. deep, stands on one of its broadest faces; the other faces are polished at a cost of £16: find the cost of polishing simiJarly another block 24 ft. long, 10 ft. broad, 5 ft. deep, similarly placed. 3. The earth's orbit is 446 millions of miles, and is traversed in 365 days; the distance of the earth from the sun may be reckoned as 71 millions of miles, and light travels at the rate of 186,000 miles per second: how far will the earth travel on its orbit, while a ray of light is passing from the sun to the earth? 2. Reduce 2 cwt. to the fraction of 4 tons 2 cwt. 14 lbs.; and 19 guineas to the fraction of £14 9s. 3d. Show that d. and every multiple of it will give a terminating decimal of one pound. 3. If of an estate be worth 200 guineas, and the value of the estate be increased 075 per cent. by improvements, find the value of 28 of the improved estate. 10 = 63 SECTION VII. 1. Express as decimals, 718, 1, 11. Show that +7 +7 + etc. ad infinitum 2. Find the value of (+) of four guineas + (·09 ̊ — ·01) of £2 1s. 3d. + (5·05 ÷005) of 5s. 3. The rainfall of London in the months of April, May, June, 1876, was 1·90, 0·94, 1·27 inches respectively, being in April 0.77 above, and in May and June 1:46 and 1.78 inches respectively below the average of those months for the previous six years: find the average for these three months, both for the year 1876 and for the whole seven years, to two places of decimals. SECTION VIII. 1. Which of the following stocks is most profitable for investment-the 3 per cents. at 91, the 3 per cents. at 108, the 4 per cents. at 118? Find the yearly income produced by investing £5217 16s. 3d. in the most advantageous of the three. 2. A tradesman induces customers who owe £750, and otherwise would not have paid him for 6 months, by an offer of 5 per cent. discount, to pay ready money; at the end of the six months, instead of £750, he had received £947 12s. 6d. What is his profit per cent. 4. A commission agent agrees to take 5 per cent. on all sales, or 3rd of net profit; 35 of his transactions realize 10 per cent. profit, the remainder 19 per cent. What would be the difference on sales of £600,000 according as all his customers adopt one or other agreement? SECTION IX. 1. Find the square root of 32.7142073 to four places of decimals, and the cube root of 941,192. Show that the cube root of every perfect cube less than a million can be determined by inspection. 2. If exchange be at the rate of 25.50 francs for one pound, and of 57.75 florins for 119 francs, find the value of £4,760 in florins. SECTION X. 1. Find the cost of turfing a ground 10 chains long and 5 chains broad, each turf being 15 in. long and 6 in. broad, and 100 turfs costing 18. 3d. 2. £8 11s. is spent upon the floor of a room 24 ft. long and 18 ft. wide; the centre of the room is covered with carpet 2 ft. wide, at 4s. 3d. per yard, leaving a margin of 3 ft. all round the carpet: how much per square foot does the margin cost to paint? 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. on PQ. may be written "sq. on PQ." and the rectangle con- SECTION I. Define a "straight line," a "rhombus." Write out the 12th axiom of Euclid. Show that the following definitions are incomplete: "Of quadrilateral figures, a square has all its sides equal." "An acute-angled triangle is that which has two acute angles." "Parallel straight lines are such as do not meet however far they may be produced." (These form one question.) SECTION II. 1. If the equal sides of an isosceles triangle be produced, the angles on the other side of the base shall be equal. Show that this property can be proved by a method similar to that employed in the 4th Proposition. 2. If from the ends of the side of a triangle there be drawn two straight lines to a point within the triangle, these shall be less than the other two sides of the triangle, but shall contain a greater angle. Write out the enunciations of any previous propositions employed in this proof. 3. Straight lines, which are parallel to the same straight line, are parallel to each other. If two adjacent sides of a parallelogram be parallel to two adjacent sides of another parallelogram, the other sides will also be parallel. SECTION III. 1. Triangles upon the same base and between the same parallels are equal. Construct a triangle equal to a given triangle and having a base three times as great. 2. To a given straight line to apply a parallelogram which shall be equal to a given triangle, and have one of its angles equal to a given rectilineal angle. To one of the sides of an equilateral triangle apply an equal parallelogram having one of its angles equal to that of the given triangle. 3. If a straight line be divided into any two parts, the square on the whole line is equal to the squares on the two parts, together with twice the rectangle contained by those parts. [It may be assumed that parallelograms about the diameter of a square, are likewise squares.] Show algebraically that the squares on the two parts are always greater than twice their rectangle except when the line is bisected. SECTION IV. 1. In obtuse-angled triangles, if a perpendicular be drawn from either of the acute angles to the opposite side produced, the square on the side subtending the obtuse angle is greater than the squares on the sides containing the obtuse angle, by twice the rectangle contained by the side upon which, when produced, the perpendicular falls and the straight line intercepted without the triangle between the perpendicular and the obtuse angle. If the triangle be isosceles, the square on the side subtending the obtuse angle is equal to twice the rectangle contained by either side and the straight line made up of that side and the straight line intercepted as in the proposition. 2. Equal straight lines in a circle are equally distant from the centre. AB, AC are two equal chords of a circle at right angles to each other. Show that they are sides of a square inscribed in the circle. 3. If a straight line touch a circle, and from the point of contact a straight line be drawn cutting the circle, the angle which this line makes with the line touching the circle shall be equal to the angles in the alternate segments of the circle. If a tangent be defined as the limiting position of a secant, show that the tangent to a circle is perpendicular to the radius drawn to the point of contact. |