GEOMETRY AND TRIGONOMETRY. The Board of Examiners. In the first six questions the symbol - must not be used; and the only abbreviation admitted for "the square described on the straight line AB” is "sq. on AB,” and for "the rectangle contained by the straight lines AB,CD" is "rect. AB,CD." 1. Describe four circles to touch three given indefinite straight lines, and explain what becomes of the circles when two of the lines become parallel. 2. Prove that in the regular pentagon each diagonal is parallel to a side. 3. The areas of parallelograms which are equiangular to one another have to one another the ratio which is compounded of the ratios of their sides. Hence deduce that the areas of similar parallelograms are to one another in the duplicate ratio of their homologous sides. 4. Find a point inside a triangle at which the three sides shall subtend equal angles. 5. The rectangle contained by the diagonals of a quadrilateral figure inscribed in a circle is equal to the sum of the rectangles contained by its opposite sides. But if the quadrilateral is such that a circle cannot be described about it, then the rectangle contained by the diagonals is less than the sum of the rectangles contained by the opposite sides. 6. A point moves so that the ratio of its distances from two fixed points is constant; find its locus. 7. Prove geometrically that sin 242 sin A cos A and sin A+ sin B = 2 sin A+ B A-B COS 2 2 8. If the sum of the angles A, B, C is a right angle, prove that 4 cos A cos B cos C = cos (B + C A) 9. Given the four sides and one angle of a quadrilateral, obtain expressions for the lengths of the two diagonals in terms of them. 10. Establish the following expression for the area of a triangle bc + ca + ab 2 (cosec A + còsec B + cosec C)* 11. Prove that the radius of the circle inscribed in a triangle is equal to twice the area of the triangle divided by the sum of the sides; and establish the following expression for the area of the inscribed circle:- 12. Two observers 8 miles apart on the same horizontal plane measure the angular elevation above the horizon of a point on a cloud which lies in the vertical plane passing through their points of D observation and between them. The angles are 30° and 9° 36'. Calculate the height of the cloud above the horizontal plane by means of the following data: log sin 9° 36′ log sin 30° 1.22211 = 1·69897 1. Make a full analysis, on any recognized system, of the following passages: (a) The next moment the child had sprung into her arms, sobbing passionately at he knew not what, but, as his paroxysm of emotion subsided, whispering over and over, with shy urgency, Play! Play!" 66 (b) To pursue the allegory, custom being but a mere face, as echo is a mere voice, rests not in her unaccomplishment, until by secret inclination she accorporate herself with error, who being a blind and serpentine body without a head, willingly accepts what he wants, and supplies what her incompleteness went seeking. 2. Parse every word in italic in the following sentences. In parsing, state the part of speech and show the connexion of the word with the rest of the sentence: While raking with curious hand but pious heart among the mouldering remains of former days anxious to draw therefrom the honey of wisdom I may fare somewhat like that valiant worthy Samson who in meddling with the carcase of a dead lion drew a swarm of bees about his ears. These delights if thou canst give 3. (a) State and illustrate the principal rules of punctuation. (b) Write down six defective verbs, and point out the defect in each. 4. Write out four lines after each of the following openings: (a) The fisher left.... (b) Without one envious... (c) The broad sun above.... 5. Give the meaning and trace the origin of each of the following words:-Ægis, avatar, biceps, cope, disembogues, offing, pachydermatous, periwig, persiflage, pistoles, pharos, plummet, poet, proem. 6. Explain fully the allusions in the following:(a) So stalked he when he turned to flight on that famed Picard field Bohemia's plume and Genoa's bow and Cæsar's eagle shield. (b) On a far shore I smoothed with tender hand Through months of pain the sleepless bed of Hyde. (c) He heard simple folk talk of a Pope Angelico, who was to come by-and-by and bring in a new order of things, to purify the church from simony, and the lives of the clergy from scandal. (d) A series of crosses beginning with the Baden revolution. (e) Mr. Lincoln persisted in calling him Mr. Partington. 7. Explain the following passages from Browning:Thinking on Metternich our friend. Before the duomo shuts. Our Italy's own attitude. 8. Comment on the following passages from Milton:(a) A high enterprise and a hard, and such as every seventh son of a seventh son does not venture upon. (b) Set an Oligarchy of twenty ingrossers over it. (c) The careful search that Isis made for the mangled body of Osiris. (d) This I take to be the most rational and the most profitable way of learning languages. 9. Comment on the following passages from Dryden:(a) To be nobly born and of an ancient family is in the extremes of fortune, either good or bad. (b) They loved the prospect of this quiet in reversion. |