EUCLID, ALGEBRA, AND MENSURATION. THREE HOURs allowed for this paper. 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.) The square EUCLID. on P Q may be written “ sq. on PQ," and the rectangle con. tained by P Q and RS, “rect. P Q R S.” Section I. Distinguish problems and theorems. Define an obtuse angle, à segment of a circle, and a rhomboid. Write out the 12th axiom. Which of the six parts of two equal triangles, taken three and three together, are given in the 4th, 8th, and 26th proposition of the first Book, to prove the equality of other parts of the triangles ? (These form one question.) SECTION II. 1. To make a triangle of which the sides shall be equal to three given straight lines. What addition is needed that the problem may be always possible? Give examples. Show that the area of a triangle formed by the diagonals of an oblong and a line equal to twice the length of either of its sides is equal to the area of the oblong. 2. If two triangles have two angles of the one equal to two angles of the other, each to each, and the side adjacent to the equal angles in each equal, the three angles of the one shall be equal to the three angles of the other. Draw a figure showing that two triangles may have two sides and one angle equal in each and yet may not be equal. 3. The straight lines which join the extremities of equal and parallel lines towards the same parts are also themselves equal and parallel. If two points be taken on the diameter of a semi-circle at equal distances from the centre, and perpendiculars a a be drawn from them to meet the circle, the line joining the points in which the two perpendiculars meet the semi-circle will be parallel to the diameter. SECTION III. 1. 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. On the base of a given triangle to construct an oblong equal to one-fourth of the given triangle. 2. If the square described on one of the sides of a triangle be equal to the squares described on the other two sides of it, the angle contained by these two sides is a right angle. The sum of the squares on the diagonals of a rhombus is equal to the sum of the squares on the sides. 3. In any triangle the lines drawn from the angular points to bisect the opposite sides meet in a point. 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 on which, when produced, the perpendicular falls, and the straight line intercepted without the triangle, between the perpendicular and the obtuse angle. The square on the side of an equilateral triangle is equal to three times the square on the radius of a circle which passes through its angular points. 2. The opposite angles of any quadrilateral figure inscribed in a circle are together equal to two right angles. Show that if two of the adjacent angles are right angles, the figure is a square, but that if two of the opposite angles are right angles, the figure is not necessarily a square. 3. If two straight lines cut one another at right angles within a circle, the rectangle contained by the segments of the one sball be equal to the rectangle contained by the segments of the other. From this proposition deduce a method for constructing a square equal to a given parallelogram. a ALGEBRA. SECTION V. Add together a + b - 2c, 8a + 4c + 26, 3c - 2a – 6b, and 26 – 2c + 3a; subtract from their sum 7a 5b; multiply their remainder by 3a – 66 +5c. Multiply a2 + y2 + 72 – 2yz by 22. ya — 72 + 2yz. Write down the factors of Que2 — 3xy — 42y, and of a +y -*+ 2/xy. (These form one question.) SECTION VI. 1. Find the L. C. M. of 2a+ - 4.23 – 6.2% + 16. - 8, and 3x4 – 15x3 + 60x — 48. Reduce to one fractionæ+ 3 a + 5 + x + 2 + 5 5 1 3y 2x + 2. Reduce to lower terms- 1500- + 13x + 2 4x and simplify + y 22 + y 64x2 243yž 22 3. Divide by 3y? 3yj223 Section VII. Solve the equations : 27 3x – 4 5. - 2 1. 2. 9 2. (x -a) (a - b)+(x-1) (a - c)-(2-c) (x -a)= 0. 3. 3x - 5y = x + 9y + 4 = 166. Form the equation whose roots are 5+2, and 5–2. (These form one question.) SECTION VIII. 1. A steamer takes two hours and forty minutes less time to travel from A to B than from B to A: the steamer travels at the rate of 14 miles per hour and the stream flows at the rate of 13 miles per hour; find the distance from A to B. 2. If each side of a square plot were enlarged by 10 yards, it would contain 1} acre more: find a side of : the plot MENSURATION. Section IX. 1. A map is drawn on a scale of 1 in. to the square mile, and is 6 feet 5 inches long and 3 feet 6 inches broad: find the area of a map drawn on a slate which is 15 inches long and 6 inches broad on the largest scale which the slate will permit; find also the scale on. which it is drawn. 2. What rent should be paid for a triangular field whose sides are 175, 220, and 375 yards, at £6 12s. per acre ? ARITHMETIC. Females. THREE HOURS allowed for this paper. Candidates are not permitted to answer more than one question in each Section. The solution must in every instance be given at such length as to be intelligible to the Examiner, otherwise the answer will be considered of no value. Section I. Add together seven millions seventy-six thousands and sixty-seven; three hundred and nine thousands nine hundred and thirty-nine; thirteen mil. lions thirty-one thousands seven hundred and thirteen; two hundred and ninety thousands nine hundred and nine; eighty-four thousands four hundred and fortyeight. Subtract seventy-one thousands and sixty-one from the above sum, and divide the remainder by three hundred and seventy-three. Section II. A landowner left by will a farm-in extent one square mile-to be divided into equal por. tions for his five children and eight grandchildren. Each child was to receive a double portion and each grandchild a single portion. What extent of land did cach child and each grandchild receive ? Prove your answer. . SECTION III. Work out the following bill of parcels : 37 yards of Holland at 7 d. per yard. K a SECTION IV. Find, by Practice, the value of three trucks of coal, each containing 7 tons 13 cwt. 2 qrs., at 13s. 4d. per ton; and prove your sum. SECTION V. 1. How many planks 17 feet long, 8 inches broad, and 34 inches deep, can be stored in a place 51 yards long by 6 yards broad, and 13 feet deep ? 2. How many gas shares at £206 10s. per share are equivalent to 708 railway shares at £136 1ās. per share? SECTION VI. 1. A piece of work is done by 90 masons working for 19 days, and 120 labourers for 21 days; the former receive 10d. per hour, the latter 9d.; what part of the whole cost' in wages (£2094 158.) should be assigned to each of the two parties of workmen ? 2. How much silver at 3s. 9d. per oz. should be ex. changed for a bar of gold weighing 31 lb. 3 oz., if 15 oz. of gold cost £79 8s. 9d. ? SECTION VII. 1. Subtract (of 3 of 7}) from (7} of 51 of 3), and divide the remainder by (3 of 13). 2. What fraction of £11 78. 6d. is equivalent to 81 of £15 68. ? SECTION VIII. 1. Express in vulgar fractions •04, ·004 and .0004. Find the average of 8:13, 9:195, 7:007, 3.668, and reduce £1 78. 9 d. to the decimal of 7.d. 2. How many times is 29.75 of 14 pole contained in 4.25 of 3 acres ? SECTION IX. 1. What capital would obtain £84 78.54d. in 2 years and 5 months at 3) per cent. P. 2. What would be the difference between the simple and compound interest of £9902 13s. 4d. for 2 years at 3} per cent. ? |