GEOMETRY AND TRIGONOMETRY. The Board of Examiners. 1. If the perimeter (sum of the sides) of a quadrilateral is given, show that its area is greatest when it is a square. 2. If circles are described on two sides of a triangle as diameters they intersect on the line of the third side. 3. Show how to find a point such that the tangents drawn from it to two given circles may be equal, and prove that the locus of the point is a straight line. 4. Describe a regular hexagon. Prove that the longest straight line which can be drawn inside a regular polygon is a diagonal. 5. If the base of a triangle is divided at a point into two parts in the ratio of the adjacent sides, then the line joining the point to the vertex bisects the vertical angle. State the corresponding proposition for the case where the base is divided externally, and interpret it when the triangle is isosceles. 6. Prove that the areas of similar rectilineal figures are to one another as the squares on their homologous sides. 7. Find an expression for all the angles which have the same cosine as the angle A, and explain how there come to be four values of sin value of sin A. A for one 8. Establish geometrically an expression for tan (A + B) in terms of tan A and tan B. 9. Given two adjacent sides and the included angle of a parallelogram, find an expression for some trigonometrical ratio of the angle between the diagonals, and reduce it to as simple a form as you can. 10. Prove that sin (x − a) sin (bc) + sin(x - b) sin (ca) + sin(x — c) sin (a c) sin (ab) is independent of x, a, b, and c. 11. Establish an expression for the area of a triangle in terms of the lengths of its sides, and deduce an expression in terms of only the perimeter and the angles. 12. If the earth is regarded as a perfect sphere of 7,900 miles diameter, show that the distance of the horizon in miles from an eye whose height above the surface is very small compared to the earth's diameter may be found by multiplying the square root of that height in feet by 1.22. ENGLISH. The Board of Examiners. 1. Write an essay on the following quotation from Bacon: Reading maketh a full man; conference a ready man; and writing an exact man. 2. Analyse the following passages: (a) A beautiful story, which has survived some centuries, is usually sure to have suffered much the same fate as a vessel of pure gold, which has been again and again remelted and remoulded to suit the taste of the immediate generation. (b) This apparently studious pursuit and preference of the most terse and elliptic expression which could be found for anything he might have to say could not but occasionally make even sovereign a master of two great languages appear dark with excess of light. 3. Parse the words in italic in the following stanza:Rest! 'twas the gift he gave; and peace! the shade He spread, for spirits fevered with the sun. To him his bounties are come back-here laid 4. Write out the couplet that follows each of the following openings: 5. Explain the following from Milton: (a) They are as lively and vigorously productive as those fabulous dragons' teeth. (b) He who takes up arms for cote and conduct and his four nobles of Danegelt. (c) The temple of Janus with his two controversal faces. 6. Write a note on each of the following words used by Milton: Anglicisms, combust, muing, praxis, syntagma. 7. Explain the following from Dryden: (a) I dare answer he would be more uneasy in their company, than he was with Crispinus, their forefather, in the Holy Way. (b) Who would always be a Heracleitus? (c) A gunsmith of the Minories. 8. Mention any use by Shakspeare of the following words different from the ordinary use of the present day: Battle, emulation, knave, kerchief, merely, nice, physical, security. What is the meaning of alchemy, cautelous, hurtle? 9. Comment on the grammar of (a) I do entreat you, not a man depart, Save I alone, till Antony have spoke. (b) Three parts of him is ours. (c) The posture of your blows are yet unknown. D 10. Where and what is each of the following:Alaska, Alsatia, Appenzel, Aurigny, Belvoir, Boom, Darwin, Gettysburg, Guelders, Longleat, Mendip, Tyre? 11. Explain the following passages: (a) Ho! Philip, send, for charity, thy Mexican pistoles, That Antwerp monks may send a mass for thy poor spearmen's souls. (b) And She of the seven hills shall mourn her children's ills. (c) Where the gigantic King of Day On his own Rhodes looks down. (d) Go to Paris: rank on rank Search the heroes flung pell-mell 12. Explain the following: (a) Were not gems medicinal, though they only pressed the finger? Were not all things charged with occult virtues ? (b) The Pope Angelico is not yet. (c) Alas! man of genius, to whom we owe so much, could you see nothing more than the burning of a foul chimney in that clash of Michael and Satan which flamed up under very eyes? your (d) A French traveller told me he had been a good deal in the British colonies, and had been astonished to see |