Obrázky stránek
PDF
ePub

ses dettes, réparé ses brèches, acquitté les créances de Brême, fait face aux échéances de Saint-Malo. Il avait exonéré sa maison des Bravées des hypothèques qui la grevaient; il avait racheté toutes les petites rentes locales inscrites sur cette maison. Il était possesseur d'un

grand capital productif. (6). On remarquait sur la table une boussole et une

liasse de carnets; c'étaient sans doute la boussole de la Durande et les papiers de bord remis par Clubin à Imbrancam et à Tangrouille au moment du départ de la chaloupe : magnifique abnégation de cet homme sauvant jusqu'à des paperasses à l'instant où il se laisse mourir; petit détail plein

de grandeur; oubli sublime de soi-même. (c) Presque sans avoir l'air d'y toucher, cette

coalition latente le mettait en haillons, en sang, aux abois, et, pour ainsi dire, hors de combat avant le combat. Il n'en travaillait pas moins, et sans relâche; mais, à mesure que l'ouvrage se faisait, l'ouvrier se défaisait. On eût dit que cette fauve nature, redoutant l'âme, prenait le parti d'exténuer l'homme. Gilliatt tenait tête, et attendait. L'abîme commençait par l'user. Que

ferait l'abîme ensuite ? (d) Cela ressembla à un linge qui se détache. La

pompe aspirante détruite, le vide se défit. Les quatre cents ventouses lâchèrent à la fois le rocher et l'homme. Ce haillon coula au fond de l'eau.

Gilliatt, haletant du combat, put apercevoir à ses pieds sur les galets deux tas gélatineux informes, la tête d'un côté, le reste de l'autre. Nous disons le reste, car on ne pourrait dire le corps.

12. Translate and explain the words-semoun, spahi,

voile latine, moutier, aubépine, pavot, acajou, coutre, cul-de-sac, déniquoiseaux, timonier, habitacle.

PURE MATHEMATICS.-Part I.

The Board of Examiners. 1. Draw a tangent to a circle from a given point

either on or without the circle.

Shew that the two tangents from an external

point are equal. 2. Inscribe a circle in a given triangle.

Shew that the bisectors of the angles of a

triangle meet in a point. 3. If two triangles be equi-angular to one another,

the sides about the equal angles shall be proportionals, those sides which are opposite to equal angles being homologous.

Shew that the diagonals of a trapezium cut one another in the same ratio.

4. If two straight lines are perpendicular to the same

plane they shall be parallel to one another. 5. Shew how to solve two simultaneous equations of the form

ax* + bxy + cy* = d,

a'u + b'xy + d'y' = d'. Solve the equations

2 + y = a + bo, xy = ab.

6. What meanings are given to the expressions

in

12 a,

a, where m, n are positive integers, and why are these meanings given ? Simplify (yz)+ (2x)c+a (xy)a+

are y zo

7. Find the sum of any number of terms of an

arithmetical progression.

The sum of three numbers in arithmetical progression is 12, and the sum of their squares is 50. Find the numbers.

8. Find the number of permutations of n different things r at a time.

In how many ways can n children form a ring?

9. Prove that

sin A =

= cos (90° — A),

= sin (90° - A). Solve the equation

sin x = cos A.

COS A

10. Prove that

cos (A + B) = cos A cos B – sin A sin B, A, B being positive angles whose sum is less than a right angle.

If cos A 3, cos B = $, find all the values of cos (A + B).

11. Shew that in any triangle

C = a + b2 2ab cos C.
Prove that

С
c

12. Shew how to solve a triangle, having given two

sides and an angle opposite one of them.
If A =

60°, a = ✓3, b= V2, solve the triangle.

PURE MATHEMATICS.-PART II.

The Board of Examiners. 1. Find the equation of a straight line in terms of the intercepts which it makes on the axes.

If the sum of the reciprocals of the intercepts be constant, shew that the line passes through a

fixed point. 2. Find the general polar equation of a circle.

If from a fixed point a straight line be drawn to cut a given circle, the rectangle contained by

the segments is constant. 3. Find the equation of the straight line passing through two given points on a parabola.

If the sum of the reciprocals of the ordinates of the extremities of a chord of a parabola be constant, the chord passes through a fixed point. 4. The sum of the squares of two conjugate semi

diameters of an ellipse is constant.

Find the magnitude and position of the equiconjugate diameters of an ellipse.

5. Define a differential coefficient, and find that of r*. Differentiate

a 2 - X2
62 x2

6. Prove that under certain conditions

f(a+h)=f(a) + hf'(a + oh), and state the conditions.

Expand x cot x in ascending powers of x as far as the term in x4.

7. Shew how to find the value of an expression which

takes the indeterminate form co

Find the value when x = 0 of cot x - cosec X.

[ocr errors]

8. State and prove the rule for integration by substi

tution.
Integrate

1

1 wa? – x?,

a a + b cos

[ocr errors]

9. Shew how to integrate

Vax2 + bx + c

AX + B

« PředchozíPokračovat »