5. Translate, with notes on the meaning or grammatical construction (α) ἀλλὰ τὰ μὲν πολίων ἐξεπράθομεν, τὰ δέδασται. (6) ἢ ἐθέλεις, ὄφρ' αὐτὸς ἔχῃς γέρας, αὐτὰρ ἐμ' αὕτως γνοίης χ ̓ οἷου φωτὸς ἔχεις θαλερὴν παράκοιτιν· 6. Mark the metre of the following lines (α) ως φάτο Πηλείδης, ποτὶ δὲ σκῆπτρον βάλε γαίῃ. 7. What can be inferred from the Iliad as to the form of government and the relations between rulers and subjects among the early Greeks? ARITHMETIC AND ALGEBRA. TWO HOURS AND A HALF. 1. Find the least sum of money which can be exactly divided among 13 men, 17 women, and 19 children, giving 5 men as much as 9 women, and 7 women as much as 11 children, and employing no coin of less value than one penny. 2. One clock gains at the rate of two seconds per day, and another loses at the rate of 5 seconds in three days. They are correct at noon on Monday. What time will it be by the second clock when it is 4.30 by the first clock on the following Friday afternoon ? (ii.) 23-11 x+5 + + 6x2-19x+15' 21x2-47x+20' 14x2 - 29x+12 ̊ 5. Prove that, if an algebraical expression vanishes identically when a is substituted for x, then (i.) the expression is divisible by x-a, (ii.) x=a is one root of the equation formed by equating the expression to zero. Hence or otherwise, solve the equation (a+b)(x3 — ax2)= b2 (x2 — a2). 6. If A varies as B when C is constant, and A varies as C when B is constant, prove that A varies as the product of B and C, when neither B nor C is constant. The volume of a certain solid varies as the product of its height and the square of the radius of its base. When the height is 1 foot, and the radius of the base 2 feet, the volume is 25.136 cubic feet. Find the volume when the height is 15 feet and the radius of the base 9 feet. 7. Find two numbers in the ratio 5:11, such that when 4 is added to each the ratio is doubled. 3(5√/7—1)_4(√5—1) __ 12+5√35 9. Sum the arithmetical progression whose second term is 7, its sixth term 19, and its last term 61. GEOMETRY AND MENSURATION. TWO HOURS AND A HALF. PASS. 1. On the same base, and on the same side of it, there cannot be two triangles, etc. Prove a Theorem of the First Book, the enunciation of which commences thus. 2. Show how to bisect a given triangle by a line drawn through a given point in one side of it. 3. If a straight line be divided into any two parts, the rectangle contained by the whole line and one of the parts is equal to the rectangle contained by the two parts together with the square on the aforesaid part. 4. If two straight lines cut one another within a circle, the rectangle contained by the segments of one of them shall be equal to the rectangle contained by the segments of the other. Prove this in the case in which one line passes through the centre, and they do not cut at right angles. 5. ABC is a triangle, and a circle passes through B and C, and cuts AB in F and AC in E. If another circle be described round AEF, prove that its tangent at A is parallel to BC. 6. Describe a regular hexagon in a given circle. 7. Give Euclid's definition of Equal Ratios, and shew that the following ratios do not satisfy the test 15 acres: 16 acres, 17 pints: 18 pints. 8. Three sides of a quadrilateral are a, b, c in length, and the two angles which they contain are half right angles. Prove that the area of the quadrilateral is (a+c)b/22-ac/2, and find the length of the fourth side. 9. The inner diameter of a hollow metal cylinder is two feet, the thickness of the metal is three-quarters of an inch, and the cylinder is two feet long. If the cylinder weighs two cwt., find the weight of a cubic inch of the metal. TRIGONOMETRY. TWO HOURS AND A HALF. PASS. 1. Define the Tangent and Secant of an angle, and investigate the value of one in terms of the other. If sin0=13, find the difference between tane and sec◊. 2. A train runs round a circular curve of length 73 chains, and thereby changes its direction of motion from due north to north 71° east. Find the radius of the curve, taking #=34. 3. Find tan 0°, cos 30°, and shew that sin 22}°= 4. Prove that cos(A+B)=cos A cos B-sin A sin B, also that cos (A+B+C)=cos A cos B cos C-cos A sin B sin C -cos B sin Csin A-cos Csin A sin B. 5. Shew from a figure that sin(270°+A)=sin(90°+A)=-cos A. 6. From the top of a house, on one side of a street 66 feet wide, the angle of elevation of the top of a building 150 feet high on the opposite side is 60°. height of the first house? 7. Solve the equations (i.) cos 30=sin 20. (ii.) sin 50+ sin 30+ sin 0=0. 8. In any triangle shew that sin A sin B__sin C What is the (i.) (ii.) =tan: cos C-cos B 2 9. If A=45° and B=75°, prove that a+c√2=2b. A 17 10. If b=91, c=125 and tan- = -, prove that a=204. 2 6 11. If A=30°, b=16 and a=12, find c, and shew by a figure that c may have two values. JUNIOR FRENCH PROSE COMPOSITION AND UNSEEN 1. Translate TRANSLATION. (a) Neither the fortitude of Caractacus, nor the despair of Boadicea, nor the fanaticism of the Druids, could avert the slavery of their country, or resist the steady progress of the imperial generals who maintained the national glory, when the throne was disgraced by the weakest orthe most vicious of mankind. At the very time when Domitian, confined to his palace, felt the terrors which he inspired, his legions under the command of the virtuous Agriola, defeated the collected force of the Caledonians, and his fleet, venturing to explore unknown and dangerous seas, displayed the Roman arms round every part of our island. (b) Quintus Curtius tells us that in certain seasons Bactria was darkened by whirlwinds of dust which completely covered and concealed the roads. Left thus without their usual landmarks, the wanderers awaited the rising of the stars, "to light them on their dim and perilous way." May we not say the same of literature? From time to time its pathways are so obscured beneath the rubbish of the age that many a weary pilgrim seeks in vain the hidden route. In such times it may be well to imitate the Bactrians. Ceasing to look upon the confusions of the day, and turning our gaze upon the great immortals who have gone before, we may seek guidance by their light. 2. Translate (a) Le 13 mai, le premier consul faisait défiler devant lui, à (b) Sois-moi fidèle, ô pauvre habit que j'aime! |