Values. 7 4. (a) Find an expression for the sum of any number of terms in Arithmetic Progression. 9 (b) If S1, S2, S3, Sp are the sums of n terms of Arithmetic series whose first terms are 1, 2, 3, 4, and whose common differences are 1, 3, 5, show that S1, +S2+S3 + 7. +Sp 5 5. (a) Prove that the geometric mean of any two positive quantities is the geometric mean of the arithmetic and harmonic means of the same quantities. 9 (b) On the ground lie n stones at intervals of 1 yard, 3 yards, 5 yards, 7 yards, etc. How far will a person at the first stone have to travel to go and bring them one by one to the first stone? 7 6. (a) Find the number of combinations of n different things r at a time. 77 6 (b) Find the number of permutations of n things taken all together, when the things are not all different, p being of one kind, q of another kind, r of another kind, and the rest all different. (c) How many different permutations can be made out of the letters of the word "assassination," taken all together? 8 7. From 4 officers and 8 privates, in how many ways can 6 be chosen (1) to include exactly one officer, (2) to include at least one officer? 9 8. (a) Prove the Binomial Theorem for a positive integral 7 exponent. (b) In the expansion of (1+r)n show that the sum of the coefficients is 2n. 8 9. Find the coefficient of a2 in the expansion of 5 10. Change 3152 from the scale of 9 to the scale of 12. 8 NOTE-Particular enunciations and authorities are required. 1. If a straight line fall on two parallel straight lines, then it shall make the alternate angles equal to one another, and the exterior angle equal to the interior opposite angle on the same side; and also the two interior angles on the same side together equal to two right angles. 29-I. 8 2. Prove that the straight line joining the middle points of the sides of a triangle is parallel to the base and equal to one half of it. 3. If a straight line be bisected and produced to any point, the sum of the squares on the whole line thus produced and on the part produced is twice the sum of the squares on half the line bisected and on the line made up of the half and the part produced. 10-II. 8 4. If a straight line be divided in medial section as in 11-II, show that the rectangle contained by the sum and difference of the segments is equal to the rectangle contained by the segments. 8 5. If from a point within a circle more than two equal straight lines can be drawn to the circumference, that point is the centre of the circle. 9-III. 7 6. Describe a circle to pass through a given point and touch a given circle at a given point. 7 7. In the diameter of a circle produced, determine a point such that the two tangents drawn from it may contain a given angle. 8 8. If from a point without a circle, two straight lines be 7 9. ABC is a triangle right-angled at C, and from any point Pin AC, a perpendicular PQ is drawn to the hypothenuse; show that the rectangle AC AP is equal to the rectangle AB AQ. S 10. ABC is a given triangle, draw the escribed circle which shall touch AB and AC produced and BC. Values. 8 8 8 8 11. To inscribe a regular pentagon in a given circle. 11-IV. 12. ABC is an isosceles triangle in which each of the angles at B and C is double of the angle at A; show that the square on AB is equal to the rectangle AB BC and the square on BC. 13. If the vertical angle of a triangle be bisected by a straight line which cuts the base, the segments of the base shall have to one another, the same ratio as the remaining sides of the triangle. 3-VI. 14. State and prove the converse of the theorem in number 13. 15. If the line bisecting the vertical angle of a triangle is divided into parts which are to one another as the base is to the sum, of the sides, the point of division is the centre of the inscribed circle. 16. ABC and DEF are two circles in which the angle BAC, formed by the chords BA and AC, is supplementary to the angle EDF formed by the chords ED and DF; prove that the circle ABC is equal to the circle EDF. Si te parentes timerent atque odissent tui neque eos ulla ratione placare posses, ut opinor, ab eorum oculis aliquo concederes: nunc te patria, quae communis est parens omnium nostrum, odit ac metuit et iam diu nihil te iudicat nisi de parricidio suo cogitare: huius tu neque auctoritatem verebere nec iudicium sequere nec vim pertimesces? Quae tecum, Catilina, sic agit, et quodam modo tacita loquitur: 'Nullum iam aliquot annis facinus exstitit, nisi per te, nullum flagitium sine te; tibi uni multorum civium neces, tibi vexatio direptioque sociorum impunita fuit ac libera; tu non solum ad neglegendas leges ac quaestiones, verum etiam ad evertendas perfringendasque valuisti. Superiora illa, quamquam ferenda non fuerunt, tamen, ut potui, tuli: nunc vero me totam esse in metu propter unum te, quidquid increpuerit, Catilinam timeri, nullum videri contra me consilium iniri posse, quod a tuo scelere abhorreat, non est ferendum. Quam ob rem discede atque hunc mihi timorem eripe; si est verus, ne opprimar, sin falsus, ut tandem aliquando timere desinam.' (b) What kind of conditional sentence is found at the beginning of the above extract? Change to forms. expressing (i) probability in the present, (ii) supposition in the past. (c) Account for the forms: nostrum, verebere, tibi uni, perfringendas, videri, abhorreat. (d) Translate the following passages and comment on peculiarities in construction: 1. Videant consules ne quid res publica detrimenti capeat. 2. Novis rebus studere. 3. An leges, quae de civium Romanorum supplicio rogatae sunt? 2. (a) Translate: Quae cum ita sint, Quirites, vos, quem ad modum iam antea dixi, vestra tecta vigiliis custodiisque defendite; mihi, ut urbi sine vestro motu ac sine ullo tumultu satis esset praesidii, consultum atque provisum est. Coloni omnes municipesque vestri, certiores a me facti de hac nocturna excursione Catilinae, facile urbes suas finesque defendent; gladiatores, quam sibi ille manum certissimam fore putavit, quamquam animo meliore sunt quam pars patriciorum, potestate tamen nostra continebuntur. Q. Metellus, quem ego hoc prospiciens in agrum Gallicum Picenumque praemisi, aut opprimet hominem aut eius omnes motus conatusque prohibebit. Reliquis autem de rebus constituendis, maturandis, agendis iam ad sena tum referemus, quem vocari videtis. (b) Account for the following forms: mihi, praesidii, facti, fore, animo, meliore. (c) What was the exact occasion and purpose of this second oration against Catiline? Give a brief outline of it. 3. (a) Translate: O fons Bandusiae, splendidior vitro, dulci digne mero son sine floribus, cras donaberis haedo, cui frons turgida cornibus. primis et Venerem et proelia destinat; lascivi suboles gregis. te flagrantis atrox hora Caniculae praebes et pecori vago. fies nobilium tu quoque fontium, lymphae desiliunt tuae. (b) What are Horace's favorite themes in his Odes? Tell what you have gathered, from the Odes you have read, of Horace's private life and tastes. (c) Parse: digne, haedo, vomere, dicente. 4. (a) Translate: CAELO supinas si tuleris monus Si thure placaris te horna Nec pestilentem sentiet Africum Nam quae nivali pascitur Algido Cervice tinget: te nihil attinet Rore Deos fragilique myrto. Immunis aram si tetigit manus, Farre pio et saliente mica. (b) Explain the case-relations of caelo, alumni, coronantem, caede. (c) Name the metre, and scan the first four lines. STANDARD VIII. LATIN GRAMMAR AND PROSE COMPOSITION. TIME-TWO AND ONE-HALF Values. HOURS. EXAMINERS E. W. COFFIN, PH.D. D. A. MCKERRICHER, B.A. 1. Decline together: aequor asperum, senex prudens, quidam dies. 2. Give the meaning, in sing. and plur. of: finis, auxilium, copia, gratia, impedimentum. 3. (a) Outline the rules for comparison of adjectives, with examples of each class. (b) Give the other degree-forms of: felix, multus, summus, vetus, ultimus. |