(ii) A man bought a certain number of articles of equal value for 3 Rs. 375. By selling them at Rs 9 each he gained as much as 10 of them cost him. How many did he buy?

5. Find the number of ways in which a selection of r things can be made out of n things which are all different.

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6. Prove the binomial theorem for a positive integral exponent. Show that the sum of the coefficients of the odd powers of a in the expansion of (1+x)" is 2n−1.

7. Prove that if a side of a triangle is produced, then the exterior angle is equal to the sum of the two interior and opposite angles; also that the three interior angles of a triangle are together equal to two right angles.

If ABC is a triangle having a right angle at A, show that the straight line joining 4 to the middle point of BC is equal to half of BC.

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8. Prove that the opposite angles of a quadrilateral inscribed in a circle are together equal to two right angles.

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5

If ABC is any triangle, and if three points D, E, F are taken on the sides BC, CA, AB respectively, show that the circles described round the triangles AEF, BFD, CDE pass through a common point.

9. Show how to circumscribe a circle about a given triangle.

10. If the vertical angle of a triangle is bisected by a straight line which cuts the base, prove that the segments of the base have to one another the same ratio as the remaining sides of the triangle.

If A, B, C, are three points in a straight line, and if D is a point at which AB and BC subtend equal angles, show that the locus of D is a circle.

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5

CONIC SECTIONS AND TRIGONOMETRY.

Moderator-DR. C. E. CULLIS, M.A., PH.D.

Puper-Setters

J. A. CUNNINGHAM, ESQ., B.A., A.R.C.Sc.I.
D. N. MALLIK, Esq., B.A., B.Sc.

Head Examiner-C. W. PEAKE, ESQ., M.A.

1. Define sin A and tan A, and from your definitions prove that (1) (1+ tan2A) (1-sin2A)=1;

(2) sin (A+B) sin (A-B) =cos2B-cos24.

Find all the values of A which satisfy the equation sin 34 = sin 24. 3. (1) Prove, in the triangle ABC, the relations

1

1

ลง

2224

where R is the radius of the circumcircle, and ▲ is the area of the triangle.

=

(2) Deduce the relation c2 4. If, in the triangle ABC, b find a, being given that

log 2 = 3010300; log 7.698622

L sin 66° 42′ = 9.9630538; difference for 1' = 544.

and apply this result to calculate log10 11 to five places of decimals, being given that

=

log10 3 4771213,

=

logioe 4342945.

6. Prove that the focal chord bisected by any diameter of a parabola is four times the focal distance of the extremity of the diameter.

7. If a parabola rolls on an equal fixed parabola, the vertices originally coinciding, prove that the focus describes a straight line.

8. Define the centre of an ellipse, and prove that every chord of the ellipse drawn through the centre is bisected at that point.

9. Prove that the locus of the middle points of any system of parallel chords of an ellipse is a straight line passing through the centre.

If one diameter bisects chords parallel to a second, show that the second diameter bisects chords parallel to the first.

10. Prove that the difference of the focal distances of any point on a hyperbola is constant and equal to the transverse axis.

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6

5

5

1.

LATIN POETRY.

Examiner-W. W. HORNELL, ESQ., B.A.

The figures in the margin indicate full marks.

Translate:

(a) Hanc pro Palladio moniti, pro numine laeso,
Effigiem statuere ; nefas quae triste piaret.
Hanc tamen immensam Calchas attollere molem
Roboribus textis, caeloque educere, iussit :
Ne recipi portis, aut duci in moenia possit,
Neu populum antiqua sub religione tueri.

Nam, si vestra manus violasset dona Minervae,
Tum magnum exitium (quod di prius omen in ipsum
Convertant!) Priami imperio Phrygibusque futurum.
Sin manibus vestris vestram ascendisset in urbem,
Ultro Asiam mango Pelopea ad moenia bello
Venturam, et nostros ea fata manere nepotes.

Explain the context and the allusions. Account for the mood of piaret. Parse-laeso, statuere, testis, and tueri, giving the principal parts. Derive and explain religio.

(b) Ille nihil; nec me quaerentem vana moratur :
Sed graviter gemitus imo de pectore ducens,
Heu ! fuge, nate dea, teque his, ait, eripe flammis.
Hostis habet muros; ruit alto a culmine Troia.
Sat patriae Priamoque datum. Si Pergama dextra
Defendi possent, etiam hac defensa fuissent.
Sacra suosque tibi commendat Troia Penates:
Hos cape fatorum comites; his moenia quaere,
Magna pererrato statues quae denique ponto.
Sic ait; et manibus vittas Vestamque potentem
Aeternumque adytis effert penetralibus ignem.

Explain the context and import of this passage. pererrato, and effert, giving the principal parts

Parse-quaere,

(c) Vix ea fatus erat senior; subitoque fragore
Intonnit laevum, et de caelo lapsa per umbras
Stella facem ducens multa cum luce cucurrit.
Illam, summa super labentem culmina tecti,
Cernimus Idaea claram se condere silva,
Signantemque vias ; tum longo limite sulcus
Dat lucem, et late circum loca sulfure fumant.
Hic vero victus genitor se tollit ad auras,
Affaturque deos, et sanctum sidus adorat.

Iam iam nulla mora est; sequor, et, qua ducitis, adsum.
Dixerat ille: et iam per moenia clarior ignis
Auditur, propiusque aestus incendia volvunt.
Ergo age, care pater, cervici imponere nostrae ;
Ipse subibo humeris, nec me labor iste gravabit.

Parse, and give the principal parts of-fatus, cucurrit, imponere, sidus, and aestus.

2. By whom, and under what circumstances is the story of the fall of Troy related in Aeneid II? It is undertood that Virgil in his epic aimed at reviving the ancient virtues of the Roman people: what virtues especially are represented in this book? Quote, or state in your own words, the simile of the mountain ash, indicating the points of comparison. 3. Translate :

(a) Fortunate senex! hic inter flumina nota

Et fontes sacros, frigus captabis opacum.

Hinc tibi, quae semper, vicino ab limite saepes
Hyblaeis apibus florem depasta salicti,
Saepe levi somnum suadebit inire susurro.
Hinc alta sub rupe canet frondator ad auras,
Nec tamen intera raucae, tua cura, palumbes,
Nec gemere aëria cessabit turtur ab ulmo.
(b) Prima Syracosio dignata est ludere versu

Nostra, neque erubuit silvas habitare, Thalia.
Quum canerem reges et proelia, Cynthius aurem
Vellit, et admonuit: "Pastorem, Tityre, pingues
Pascere oportet oves, deductum dicere carmen.'
Nunc ego (namque super tibi erunt qui dicere laudes,
Vare, tuas cupiant, et tristia condere bella)
Agrestem tenui meditabor arundine musam.
Non iniussa cano. Si quis tamen haec quoque, si quis,
Captus amore, leget; te nostrae, Vare, myricae,
Te nemus omne canet; nec Phoebo dignior ulla est
Quam sibi quae Vari praescripsit pagina nomen.
(c) Huc ipsi potum venient per prata iuvenci ;

Hic virides tenera praetexit arundine ripas
Mincius, eque sacra resonant examina quercu.

Explain the allusions in (b). Write the scansion of (c).

4. Explain the terms Bucolica and Eclogae. Mention the principal writers of this kind of poetry.

5. Translate :

(Aeneas the Sibyl-the Golden Bough.)

Vivacis antra Sibyllae

Intrat: et ut manes adeat per Averna paternos,
Orat. Atilla diu vultus tellure moratos
Erexit; tandemque, deo furibunda recepto,

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Magna petis, dixit, vir factis maxime, cuius
Dextera per ferrum, pietas spectata per ignes.
Pone tamen: Troiane, metam: potiere petitis;
Elysiasque domos et regna novissima mundi,
Me duce, cognosces simulacraque cara parentis.
Invia virtuti nulla est via. Dixit: et auro

Fulgentem ramnm silva Iunonis Avernae
Monstravit ; iussitque suo divellere trunco.
Paruit Aeneas; et formidabilis Orci

Vidit opes, atavosque suos umbramque senilem
Magnanimi Anchisae; didicit quoque iura locorum,
Quaeque novis essent adeunda pericula bellis.

LATIN PROSE.

Paper set by-H. STEPHEN, Esq., M.A.
Examiner - REV. R. GEE, M.A.

The figures in the margin indicate full marks.

1. Translate :

(a) Negant intneri lucem esse fas ei, qui a se hominem occisum 10 esse fateatur. In qua tandem urbe hoc homines stultissime disputant? Nempe in ea, quae primum iudicium de capite vidit M. Horatii, qui, nondum libera civitate, tamen Populi Romani comitiis liberatus est, quum sua mann sororem esse interfectum fateretur. An est quisquam qui hoc ignoret, quam de homine occiso quaeratur, ant negari solere omnino esse factum, aut recte et iure factum esse defendi? Nisi vero existimatis dementem P. Africanum fuisse, qui, quum seditiose in contione interrogaretur, quid de Ti. Gracchi morte sentiret, responderit, iare caesum videri. Itaque hoc, indices, non sine causa etiam fictis fabulis doctissimi homines memoriae prodiderunt, eum, qui patris ulciscendi causa matrem necavisset, variatis hominum sententiis, non solum divina, sed etiam sapientissimae deae sententia liberatum. Quod si duodecim tabulae nocturnam furem quoquo modo, diurnum autem, si se telo defenderet, interfici impune voluerunt, quis est, qui, quoquo modo quis interfectus sit, puniendum putet, quum vident aliquando gladium nobis ad hominem occidendum ab ipsis porrigi legibus ?

Account for the moods of fateatur, ignoret, responderit, and necavisset. Parse and derive comitiis, and explain its meaning. Explain the historical and literary allusions in the above passage.

(b) Interim quam sciret Clodius iter sollemne, legitimum, necessarium ante diem XIII Kalendas Feb. Miloni esse Lanuvium ad flaminem prodendam, quod erat dictator Lanuvii Milo, Roma subito ipse profectus pridie est, ut ante suum fundum Miloni insidias collocaret. Atque ita profectus est, ut contionem turbulentam, in qua eius furor desideratus est, quae illo ipso die habita est, relinqueret; quam, nisi obire facinoris locam tempusque voluisset, nunquam reliquisset. Milo autem quum in senatu fuisset eo die, quoad senatus est dimissus, domum venit; calceos et vestimenta mutavit; paullisper, dum se uxor comparat, commoratus est, deinde profectus id temporis, quum iam Clodius,

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