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(ii) A man bought a certain number of articles of equal value for Rs. 375. By selling them at Rs 9 each he gained as mach as 10 of them cost him. How many did he buy? 4. If

Si=1+2+3+ and

89= 12 + 12 + 32 + ... + n2, find $; and Sg in terms of n, and show that

38+381 - (n+1)-(n+1)8.

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5. Find the number of ways in which a selection of r things can be made out of n things which are all different. If 0, denote this number, prove that

, +0,

2

n

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2

6. Prove the binomial theorem for a positive integral exponent. Show that the sum of the coefficients of the odd powers of x in the ex. pansion 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 A to the middle point of BC is equal to half of BC.

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

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

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

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CONIC SECTIONS AND TRIGONOMETRY.

Moderator- Dr. C. E. CULLIS, M.A., Ph.D.

Puper-Setters—{

SJ, A, CUNNINGHAM, Esq., B.A., A.R.C.Sc.I.
D. N. MALLIE, Esq., B.A., B.Sc.

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

Examiners

BABU BAIDYANATH BASU, M.A.

KSHETRAMOHAN BANERJEE, M.A.
SYAMADAS MUKERJI, M.A.
KRISHNAPRASAD DE, M.A.
KALIPRASANNA CHATTORAJ, M.A.

The figures in the margin indicate full marks.

1. Define sin A and tan A, and from your definitions prove that

(1) (1 + tan2A) (1-sin2A)-1;

1 1

(2) tan (45° + A)

1+tan A
1-tan A'

the angle A being acute.

sin A;

2. Prove that

(1) sin (180° + A) (2) sin (A + B) sin (A–B) = cos2B-cos?A. Find all the values of A which satisfy the equation sin 3A = sin 2A. 3. (1) Prove, in the triangle ABC, the relations

ANNON

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whero R is the radius of the circumcircle, and a is the area of the tri. angle.

(2) Deduce the relation c2 a2 +62 - 2ab cos C. 4. If, in the triangle ABC, b = 1000 feet, 4 = 45°, B = 66° 42' 20" find a, being given that

2 5

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

logio 3 = .4771213, log10e =°4342945.

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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 origin. ally 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 throngh 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 becond 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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LATIN POETRY.

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

The figures in the margin indicate full marks.

1. Translate :-
(a) Hano pro Palladio moniti, pro nomine laeso,

Effigiem statuere ; nefas quae triste piaret.
Hanc tamen immensam Calchas attollere molem
Roboribus textis, caeloque edocere, iussit:
Ne recipi portis, aut duci in moenia possit,
Nen populum antiqua sub religione tueri.
Nam, si vestra manus violasset dona Minervae,
Tum magnum exitium (quod di prios omen in ipsum
Convertant!) Priami imperio Phrygibusque futurom.
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 allasions. Account for the mood of piaret. Parselaeso, statuere, testis, and tueri, giving the principal parts. Derive and explain religio.

(6) Ille nihil ; nec me quaerentem vana moratur :

Sed graviter gemitas imo de pectore ducens,
Heu ! fage, nate dea, teque his, ait, eripe flammis.
Hostis babet 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.

Parse_quaere, pererrato, and effert, giving the principal parte

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(c) Vix ea fatus erat senior; subitoque fragore

Intonnit laevam, et de caelo lapsa per umbras
Stella facem ducens multa cnm luce cucorrit.
Illam, summa super labentem culmina tecti,
Cernimus Idaea claram se condere silva,
Signantemque vias ; tum longo limite salons
Dat lucem, et late circum loca sulfure fumant.
Hic vero victus genitor se tollit ad auras,
Affaturque deos, et sanctum sidas adorat.
Iam iam nulla mora est; sequor, et, qaa ducitis, adsum.
Dixerat ille : et iam per moenia clarior ignis
Auditor, propiusque aestos incendia volvant.
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 onder 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 : wbat virtres 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 opacam.
Hinc tibi, quae semper, vicino ab limite saepes
Hyblaeis apibus florem depasta salicti,
Saepe levi somnum suadebit inire susurro.
Hinc alta sub rope canet frondator ad auras,
Nec tamen intera rancae, tua cura, palumbes,

Nec gemere aëria cessabit turtur ab almo.
(6) Prima Syracosio dignata est ludere versu

Nostra, neque erubuit silvas habitare, Thalia.
Quum canerem reges et proelia, Cynthius aorem
Vellit, et admonuit: “Pastorem, Tityre, pingues
Pascere oportet oves, deductum dicere carmen.
Nunc ego (namque super tibi erunt qui dicere laudes,
Vare, tuas copiant, et tristia condere bella)
Agrestem tenui meditabor arundine musam.
Non iniunga cano. Si quis tamen haec quoque, si quis,
Captus amore, leget; te nostrae, Vare, myricae,
Te nemus omne canet; nec Phoebo dignior alla est

Quam sibi quae Vari praescripsit pagina nomen.
(c) Huo ipsi potom venient per prata invenci;

Hic virides tenera praetexit arundine ripas

Mincias, eque sacra resonant examina querca. 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 :

15

(Aeneasthe Sibyl- the Golden Bough.)

Vivacis antra Sibyllae
Intrat: et ut manes adeat per Averna paternos,
Orat. At illa diu valtus tellure moratos
Erexit; tandemque, deo furibunda recepto,

I

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 mandi,
Me dnce, cognosces simulacraqne cara parentis.
Invia virtuti nulla est via. Dixit : et auro
Folgentem ramnm silva Iunonis Avernae
Monstravit; inssitqne suo divellere trunco.
Paruit Aeneas ; et formidabilis Orci
Vidit opas, atavosque snos 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.

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1. Translate :

(a) Negant intneri lucem esse fas ei, qui a se hominem occisum esse fateatur. In qua tandem urbe hoc homines stultissime dispntant? Nempe in ea, quae primum indicium 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, quom de homine occiso quaerator, ant negari solere omnino esse factum, ant recte et iure factam esse defendi? Nisi vero existimatis dementem P. Africanum fuisse, qui, qnum seditiose in contione interrogaretur, quid de Ti. Gracchi morte sentiret, responderit, iare caesum videri. Itaqne hoc, indices, non sine causa etiam fictis fabulis doctissimi homines memoriae prodiderunt, eum, qui patris ulcis. cendi causa matrem necavisset, variatis hominum sententiis, non solom divina, sed etiam sapientissimae deae sententia liberatom. Quod si duodecim tabulae noctarnnm furem quoquo modo, diurnum antem, si se telo defenderet, intertici impune voluerunt, quis est, qui, quoqao modo quis interfectus sit, poniendam patet, 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 bistorical and literary allusions in the above passage.

(6) Interim quam sciret Clodins iter sollemne, legitimum, neces. sarium ante diem XIII Kalendas Feb. Miloni esse Lanuvium ad flami. dem prodendom, qnod erat dictator Lanuvii Milo, Roma subito ipse profectos pridie est, ut ante suum fundum Miloni insidias collocaret. Atque ita profectus est, ut contionem turbulentam, in qua eius furor desideratas est, quae illo ipso die habita est, relinqueret; quam, nisi obire facinoris locum tempusqne volnisset, uanquam reliquisset. Milo autem quum in senata fuisset eo die, quoad senatus est dimissn8, domom venit; calceos et vestimenta mutavit; paullisper, dum se uxor com. parat, commoratus est, deinde profectas id temporis, quam iam Clodius,

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