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ENGLISH COMPOSITION.

Each candidate is required to write a short English composition, correct in spelling, punctuation, grammar, and expression. This composition must be at least fifty lines long, and be properly divided into paragraphs. One of the following subjects must be taken : —

The Character of Dr. Primrose.

An Account of the Tent-scene between Brutus and Cassius. The Argument of Marmion.

FRENCH.

1. Translate Into English:

Je sortis, et me promenai toute la matinée dans la ville, en songeant sans cesse à la réception que mon oncle me ferait. Je crois, disais-je en moi-même, qu'il sera ravi de me voir. Je jugeais de ses sentiments pas les miens, et je me préparais à une reconnaissance fort touchante. Je retournai chez lui en diligence à l'heure qu'on m'avait marquée. Vous arrivez à propos, me dit son valet, mon maître va bientôt sortir. Attendez ici un instant, je vais vous annoncer. A ces mots, il me laissa dans l'antichambre. Il y revint un moment après, et me fit entrer dans la chambre de son maître, dont le visage me frappa d'abord par un air de famille. H me sembla que c'était mon oncle Thomas, tant ils se ressemblaient tous deux. Je le saluai avec un profond respect, et lui dis que j'étais fils de maître Nicolas: je lui appris aussi que j'exerçais à Madrid, depuis trois semaines, le métier de mon père en qualité de garçon, et que j'avais dessein de faire le tour de l'Espagne pour me profectionner. Tandis que je parlais, je m'aperçus que mon oncle rêvait. Il doutait apparemment s'il me désavouerait pour son neveu, ou s'il se déferait adroitement de moi: il choisit ce dernier parti. Il affecta de prendre un air riant et me dit: Eh bien! mon ami, comment se portent ton père et tes oncles ? dans quel état sont leurs affaires ? — Le Sage.

2. State the tense of the italicized verbs in the above and give it in full.

3. Give the principal tenses of savoir, acquérir, prendre, envoyer (thus, Inf., ac ; Pres. Part., étant; Past. Part., été; Pres. lxn.,je suis; Pram, ye fus).

4. Using mostly the words of 1, translate into French:

(a) Do you think that they are delighted to see him?

(b) They left me waiting more than an hour, (c) I fear that he has gone out.

PLANE TRIGONOMETRY.

1. Find the angles of the plane right triangle in which the hypothenuse is $ of one of the sides.

2. Obtain, without using the tables, the natural trigonometric functions of 60°.

3. Obtain, from fundamental formulas, the sine and cosine of 270°, 270° — x, 270° + a.

4. Obtain, from fundamental formulas, sin x — sin y= ....

5. In the plane oblique triangle A B C, B is 40°, b is 100. What values of a will give two solutions; one solution; no solution? Give the reason for each answer.

6. Obtain, from fundamental formulas,

. ». 1 — cos x

7. Solve the triangle whose sides are 0.1498,0.1596, 0.1943.

8. Prove the formula

cos (x -\-y) cos (x y) = cos*y— sina a;.

ANALYTIC GEOMETRY.

1. What is the equation of a line parallel to the axis of x, 3 units below it? At what point does this line intersects the line 3y + 4a; +1 = 0? What is the acute angle between these two lines?

2. What are the axes and the parameter of the curve iy2 + 3x2 = 36? What is the equation of the circle whose diameter coincides with the transverse axis of this curve?

3. State and prove the relation between any ordinate of an ellipse and the corresponding ordinate of the inscribed circle.

4. Deduce formulas for passing from a rectangular to a polar system.

5. The equation of the tangent to the parabola y1 = %px is yy' —p (x-\-x'). Find the equations of the tangent and the normal to y2=8x, at the extremity of the positive ordinate through the focus.

6. Is the point (— 2, 1) situated on the hyperbola iy*— Taft = — 24? Why?

7. Of what is xy + 4 = 0 the equation? Illustrate by a figure.

8. Find the points in which the curve y* <kc intersects the curve 3y2-\-2x2= 14.

SOLID GEOMETRY.

1. Prove that, if a solid angle is formed by three plane angles, the sum of either two of them is greater than the third. The sum of the three angles taken together cannot exceed a certain quantity: what is it?

2. A pyramid is cut by two planes parallel to the base: prove that the two sections are similar polygons. State in the form of a proportion the relation which holds between the areas of these sections and their respective distances from the vertex of the pyramid.

3. Prove that, if from the vertices of a given spherical triangle as poles arcs of great circles are described, another triangle is formed, the vertices of which are the poles of the sides of the given triangle.

4. A ball of lead is three inches in diameter: what is its weight? A cubic foot of lead weighs 712 pounds.

5. A certain cylindrical vessel is twelve inches in diameter and eight inches deep. What are the dimensions of a vessel, similar in form, which will hold only one sixty-fourth as much?

6. What is a degree of spherical surface? How is the area of a spherical triangle measured? State without proving.

ADVANCED ALGEBRA.

1. Dividea»—J»— c»— 2bc by .

o + i — c

2. What is the equation whose roots are 1, —— —'

2

3. Obtain the formulas for the last term and the sum of the series in a geometrical progression. Obtain also an expression for the sum of the series, in terms of the first term, the last term, and the common factor or ratio.

4. Solve the equation V x -+- V — VI —x) = 1.

5. How many words can be formed from seven letters taken all together, provided that 3 given letters are never separated?

6. Find the sum of n terms of the series 1, 3, 5, 7 .. .

7. Solve the equations x* y* = 26, x* -f- xy -\- y1 =z 13.

8. What is the sixth term of (1—z)-a?

9. A courier travels from P to Q in 14 hours: a second courier starts at the same time from a place ten miles behind P, and arrives at Q at the same time as the first courier. The times in which the couriers travel 20 miles differ by half an hour. Find the distance from P to Q.

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