12. Prove that in a triangle of sides a, b, and c, and with R as the radius of its circumscribing circle The Board of Examiners. 1. Write an Essay on this saying of Bacon's"A man that is young in years may be VIC IL hours." 2. Analyse- (a) "To her fair works did Nature link The human soul that through me ras, "Who would true valour see Let him come hither! One here will constant be, Sed quid ego haec autem nequiquam ingrata Quidve moror, si omnis uno ordine habetis Idque audire sat est? Iamdudum sumite poenas; Hoc Ithacus velit, et magno mercentur Atridae. (d) Set postquam Cn. Pompeius ad bellum maritumum atque Mithridaticum missus est, plebis opes inminutae, paucorum potentia crevit. Ei magistratus provincias aliaque omnia tenere: ipsi innoxii florentes sine metu aetatem agere, ceteros, qui plebem in magistratu placidius tractarent, iudiciis terrere. Set ubi primum dubiis rebus novandi spes oblata est, vetus certamen animos eorum adrexit. Quodsi primo proelio Catilina superior aut aequa manu discessisset, profecto magna clades atque calamitas rem publicam obpressisset: neque illis, si victoriam adepti forent, diutius ea uti licuisset, quin defessis et exsanguibus qui plus posset imperium atque libertatem extorqueret. (e) Namque animus inpurus, dis hominibusque infestus, neque vigiliis neque quietibus sedari poterat: ita conscientia mentem excitam vastabat. Igitur colos ei exsanguis, foedi oculi, citus modo modo tardus incessus: prorsus in facie voltuque vecordia inerat. Set iuventutem, quam, ut supra diximus, inlexerat, multis modis mala facinora edocebat. Ex illis testis signatoresque falsos commodare: fidem fortunas pericula vilia habere, post, ubi eorum famam atque pudorem adtriverat, maiora alia inperabat. (f) Ecce Iovis monitu, deceptus imagine somni, Rex iubet incepti curam dimittere belli. Ille potest auctore suam defendere vocem. Non sinat hoc Aiax delendaque Pergama poscat, ituros? Cur non arma capit, dat quod vaga turba sequatur? Non erat hoc nimium numquam nisi magna loquenti. (g) Atque ego contemptus essem patientior huius, Si fugeres omnes. Sed cur Cyclope repulso Acin amas praefersque meis conplexibus Acin? Ille tamen placeatque sibi placeatque licebit, Quod nollem, Galatea, tibi; modo copia detur, Sentiet esse mihi tanto pro corpore vires. 2. Translate and comment on the construction ofQuos, ni mea cura resistat, (a) Iam flammae tulerint inimicus et hauserit ensis. (b) Inde domum, si forte pedem, si forte tulisset, Me refero. (c) Hoc erat, alma parens, quod me per tela per ignes Eripis, ut mediis hostem in penetralibus cernam? (d) Perque pedes traiectus lora tumentes. 3. Explain (a) Catilina pecuniarum repetundarum reus prohibitus erat consulatum petere quod intra legitumos dies profiteri nequiverit. (b) Isque postea permotus oratione C. Caesaris pedibus in sententiam Ti Neronis iturum se dixerat, quod de ea re praesidiis additis referun dum censuerat. 4. Discuss the propriety of Cicero's action in ordering the execution of the Catilinarian conspirators. ALGEBRA. The Board of Examiners. 1 Resolve into five factors a3 (bc) + b3 (c − a) + c3(a - b) + abc(b −c) (c — a) (a — b). ayz + bzx + cxy = 0, a'yz + b'zx + c'xy = 0, lx + my + nz = 0, and x, y, z are not all zero, prove that Imn (IBC + mCA + nAB) = 0, where A, B, C denote bc' 5. Eliminate x, y, x', y' from b'c, ca'-c'a, ax2+2hxy + by2+2gx + 2fy + c = 0, ax'2 + 2hx'y' + by'2 + 2gx' + 2fy' + c = 0, lx+my+ n = 0, lx' + my' + n = 0, xy=x'y'. 6. Find xyz, having given a2x + b2y + c3z = 0, 8. If three independent homogeneous quadratic equations in four variables x, y, z, n have one indeterminate solution of the form 9. If ax + by + cz + dw = 0, a'x + b'y + c'z + d'w = 0, shew that they have in general four determinate solutions. But if the quadratics have a second indeterminate solution of the same character they have no other determinate solution unless the two indeterminate solutions have a common solution, in which case the quadratics have in general two determinate solutions. a+b+c=0, x + y + z = 0, prove that (a3x + b3y + c3z) (a3y + b3z + c3x) (a3z + b3x+c3y) (a2 + b2 + c2)3 ̧(x3u+y3b+z3c) (x3b + y3c+ z3a)(x3c+y3a + z3b) (x2 + y2 + z2)3 |