(α) ἀλλ' ὅτε δὴ ὄγδοόν μοι ἐπιπλόμενον ἔτος ἦλθε, ARITHMETIC AND ALGEBRA. (TWO HOURS AND A-HALF.) PASS. end resting lower end on the ground The bottom of the ladder 1. A ladder 13 feet long is placed with its upper against a vertical wall and its level with the foot of the wall. is distant 5 feet from the wall. If the ladder slip down till its lower end is 6 feet from the wall, find through what distance the upper end has slipped. 2. If £100 amounts to £109.2 in two years at compound interest, what will it amount to in three years at the same rate of interest? 5. If a, ẞ are the roots of the quadratic equation ax2—bx+c=0, express a+ß, aß in terms of a, b, c. Hence or otherwise shew the sum of the squares of the roots of the equation 9.2+27+20=0 is equal to the sum of the squares of the roots of 9.2+33x+40=0. 8. Find the sum to n terms of a series in geometrical progression. The first term of a G.P. is 2, the second term is 2-√2; find the sum of the series to infinity. 9. If a, b, c are in harmonic progression, shew that 2a—b, b and 2c-b are in geometric progression. 10. By selling a parcel of shares for £80, I make a gross profit of a pounds: had I sold them for £78 15s. I should have made a profit of x per cent. on the transaction; find x. GEOMETRY AND MENSURATION. (TWO HOURS AND A-HALF.) PASS. 1. If three parallel straight lines make equal intercepts on a straight line which cuts them, they will make equal intercepts on every straight line which cuts them. 2. Divide a given straight line so that the rectangle contained by the whole and one of the parts shall be equal to the square on the other part. 3. The tangents drawn to two circles from any point in the straight line joining their points of intersection are equal to one another. 4. Inscribe a regular pentagon in a given circle. 5. If ABCDE be a regular pentagon, prove that the straight lines AC, AD, BE, BD, CE will by their intersection also form a regular pentagon. 6. Give Euclid's definition of proportionals and illustrate it by shewing that the ratio of 3 pence to 4 pence is not equal to the ratio of forty seconds to one minute. 7. Triangles of the same altitude are to one another as their bases. 8. Similar triangles are to one another in the duplicate ratio of their homologous sides. 9. Find the diameter of a cylindrical pipe, two feet long, containing a volume of 600 cubic inches. 10. A regular hexagon is inscribed in a circle of fifteen inches radius; find the area contained between them in square inches correct to three places of decimals. 1. Define a radian. TRIGONOMETRY. TWO HOURS AND A-HALF. PASS. Which is greater, 126° or 2-3 radians? 2. Define the tangent, cotangent and cosecant of an angle. Find the tangent and cotangent of an angle whose cosecant is 1.25. 3. Find by geometrical constructions the sine of 45° and the cosine of 30°. Prove that (sin 45°-sin 60°) (cos 150° - cos 45°)=sin2150°. 4. Prove the following, and in the case of (i.) state clearly what restrictions you make as to the magnitude of the angles. (i.) cos(A-B) = cos A cos B+sin A sin B, (ii.) tan A-sin2A=tan'A sin2A, (iii.) sin(x+3y)+sin(3x+y) −2 cos(x+y). sin 2r+ sin 2y 5. Solve the equations (i.) 3-2 sin2x-3 cos x=0, (ii.) sin 5a cos 3x=sin 9x cos 7æx. 6. In any triangle prove the following, proving (i.) from a figure (i.) a sin B-b sin A=0, (ii.) cos 2A Cos 2B 1 1 7. Two sides of a triangle are 1 foot and 2 feet respectively, and the angle opposite to the shorter side is 30°. Solve the triangle completely. 8. The shadow of a stick placed vertically in a horizontal piece of ground is observed at one instant to be equal in length to the stick. How much must the sun go down before the shadow will have increased in the ratio of √/3 to 1? 9. A man travelling due west along a straight road observes that when he is due south of a windmill, the bearing of a distant church spire makes an angle of 30° with the direction of the road. A mile further on the bearings of the windmill and spire are N.E. and N.W. respectively. Find correct to 1 yard the distance between the spire and windmill. JUNIOR FRENCH PROSE COMPOSITION AND UNSEEN TRANSLATION. 1. Translate into French PASS. (a) The Visigoths were marching through Italy to Africa, when their King Alarich, whom they loved exceedingly, died. Determined that his burial-place should not be profaned by the tread of strange feet, they testified in a singular manner their love and admiration for him. They diverted the course of the river Barent as it flowed from the foot of the mountain near the town of Constantina. Here, in the middle of the dry bed, they had a grave dug by a number of prisoners, and buried their king there, together with many valuables. This done, the river was brought back again to its former course, and that the place might be betrayed by no one, all the prisoners were put to death. (b) When in the year 1714 one of the sharp rocky peaks, called "Les Diablerets," fell down, a herdsman belonging to the village of Aven, in Valais, was among those who had not returned home, and was considered as having lost his life. His children were declared orphans by the court. Three months afterwards he suddenly appeared in his village-pale, thin, covered with rags, resembling a spectre. All the inhabitants of the village were frightened. The doors of his own house were shut to him. After some delay, the man succeeded in convincing the people that he was alive, and then he told them that the moment at which the mountain-slip took place, he had been on his knees praying to the Preserver of life, when an enormous fragment of rock in descending struck the ground before his hut, and resting, leant over against the rocky wall at the base of which his hut was built. xii. 2. Translate (at sight)— (a) (b) même ton. UN DEBAT EN ORIENT. On citait Aristote, Goethe, Bacon,-Voltaire surtout, très en honneur chez les Turcs éclairés. On répondait sur le C'était une défense de la politique générale de la Turquie, dans sa conduite vis-à-vis des masses. "Vous autres Européens occidentaux,' pacha, 66 vous marchez très vite; vous avez des inventions nous disait le merveilleuses que nous admirons; mais ne craignez-vous pas qu'en répandant dans les masses des idées pour lesquelles elles ne sont pas mûres, vous n'y semiez les du mal? Nous, nous allons plus lentement, mais nous germes nous appliquons à conserver le sentiment religieux, qui est la sauvegarde des Etats. Nous vous prenons vos découvertes quand nous les avons éprouvées, le chemin de fer, le télégraphe; mais la dynamite, par exemple, ne croyez-vous pas qu'elle a fait plus de mal que les hommes insuffisament éclairés à qui vous donnez que de bien, et les moyens de s'en servir en laisseront échapper le bon côté pour n'en garder que le mauvais ?” lence," lui répondit mon frère, "vient de rendre très "Votre Exceljustement une pensée qu'avait déjà exprimée un des anciens sages de la Chine. Confucius divise les étudiants en quatre catégories: les entonnoirs, les éponges, les tamis et les cribles. Les entonnoirs reçoivent tout et perdent tout. Les éponges reçoivent tout et conservent tout indistinctement, le bon comme le mauvais. Les tamis laissent échapper le bon et retiennent le mauvais; enfin les cribles laissent passer le mauvais et ne conservent que ce qui est bon; mais ce sont les plus rares." LE PHENIX. Le Phénix, venant d'Arabie, Dans nos bois parut un beau jour ; Grand bruit chez les oiseaux, leur troupe réunie Chacun l'observe, l'examine : Son plumage, sa voix, son chant mélodieux, Tout charme l'oreille et les yeux. Pour la première fois on vit céder l'envie |