« ElőzőTovább »
17. If gr2 = pas, the equation
34+ px3 +222 + rx +s=0 can be solved as a quadratic. 18. The roots of the cubic
203 + px2 + 2x +r=0 are a, b, c; transform it into another whose roots are
1. Express the equation
sin 30 = cos 20 in terms of sin 0; and observing that I is a root of the resulting cubic, reduce and solve the equation.
2. Find the value of the symmetric function ? (a – B)2 of the roots of the equation
28 – 422 +52 – 6=0. 3. What must be written instead of 6 for the last term of the preceding equation, if its roots are to be connected by the relation a +B=2y?
4. Find the co-ordinates of the intersection of perpendiculars of the triangle the co-ordinates of whose vertices are
(1, 2), (3, -4), (-5,6). 5. A circle passes through the point (1, 1), and meets one axis in the point x=2, and the other in the point y=3; find where it meets again the axes which are supposed to be rectangular.
6. Given sin a =p sin ß, (n + 1) cos a=j cos ß, find cos a.
7. Given Pp+(1-P)=P'p'+(1-P')=P"p" + (1 - P"), prove that
p-p p. p"-
P" P P' 8. A messenger starts with an errand at the rate of 3 miles an hour; another is sent half an hour afterwards to overtake him, which he does in two hours; at what rate did he ride? Find also in what time he will do it, if he rides 12 miles an hour.
9. One of the diagonals of a given quadrilateral is bisected by the other; prove from the equations of the lines that if the opposite sides of the quadrilateral be produced to meet, and the points of intersection joined, the joining line is parallel to the bisected diagonal.
10. The base angles of a right-angled triangle are joined by right lines to the opposite corners of the squares described on the sides; show from the equations of the joining lines that they meet on the perpendicular of the triangle.
11. If the sines of the angles of a triangle be in arithmetical progression, so are the cotangents of the half angles also ?
12. The sides of a plane triangle are 3, 5, 6; compare the radii of the inscribed and circumscribed circles.
13. Find the co-ordinates of the vertices, and the area of a triangle, the equations of whose sides are
4x+y=11, 23 – 3y=23, and 6x + 5y = 27. 14. Find the equation of the pair of tangents from the origin to the circle,
22 + y2 + 2x – 4y = 16. 15. Resolve into its partial fractions the expression
2.263 – 522 – 33 16. In a plane triangle, prove
a2 - 62
2ab sin c where o is the angle the bisector of the base makes with the base. 17. Show that
sin 3A sin 3 A + cos 3 A cos 3 A=cos 32 A. 18. One of the roots of the equation
24+423 + 3x2 – 23 – 6=0 is – I+V-T; find all the roots.
1. Find the value of the square root of 343 divided by the fifth root of 0.027.
2. In how many years will a sum of money double itself at 4 per cent. compound interest, payable half-yearly
3. Find x from the equation 2*+ 2x1 = 10.
4. Find the area of a quadrilateral inscribed in a circle whose sides are 521, 348, 419, 623.
5. Two variable right lines pass respectively through the two points 3, 4; – 2, 7, and intersect at tan-15.4; determine the radius and centre of the circle which is the locus of their intersection.
6. Find the coefficient of 5 in the expansion (2 - 52–722).
7. If in a triangle, 3c=76, and A = 6°37'24", find the other angles.
8. Apply Horner's method to find the positive root to four places of decimals, lying between 5 and 6, of the equation
23 + 2x2 – 23% – 70=0.
9. Find expressions for sin 15° and tan 15°, and calculate either of them to 5 decimal places. 10. Apply the method of divisors to find a root of the equation
3:04 – 17.303 + 41x2 – 59% + 40 =0. 11. The sides of a triangle are 2357, 4312, 3675; find its greatest angle.
12. Find the value of the symmetric function Ea253 of the roots of the equation
24 +723 – 9x2 + 12 = 0.
Translate the following passages into English :-
Ending, και σταθμοί τε και υπέρθυρα ώσαύτως.
Lib. i. c. 179.
2. Beginning, Έμοι δοκέει, ένα μεν ημέων μούναρχον, κ. τ. λ. Ending, έν γάρ τω πολλώ ένι τα πάντα.
Lib. iii. c. 8o. 3. Beginning, Ταύτης ών οι Σκύθαι της καννάβιος, κ. τ.λ. Ending, καταπλαστών γίνονται καθαραι και λαμπραι.
Lib. iv. C. 75 4. Beginning, Παρίoισι μεν δή ταύτα η Πυθίη έχρησε, κ. τ.λ. Fnding, τα δε πεντήκοντα τάλαντα εξέτισε ο παίς αυτού Κίμων.
Lib. vi. c. 136.
1. Describe the reforms introduced by Cleisthenes. What precaution did he take to prevent any tribe forming itself into a political faction ?
2. Give a short history of each of the Greek colonies in Italy. To what causes is their decline in power to be attributed ?
3. What war is alluded to by Herodotus in the words, oūros ó móleuos συστάς έσωσε τότε την Ελλάδα; and what circumstances led to it?
4. Describe accurately the state of Asia before the rise of the Persian monarchy.
5. Mention, in order, the principal towns and rivers which were passed by Xerxes on his march from Sardis to Greece.
6. Give an account of the physical and political geography of Asia Minor.
7. What are the principal peculiarities of the Ionic dialect in the case of (1) verbs in Mi, (2) barytone verbs, and (3) contracted verbs ?
8. The Ionians adopt contraction in certain cases where the Attic writers reject it?
9. In what cases does Herodotus omit the augment ?
10. What is the difference between the imperfect tense and aorist in the narration of past events; and what is the effect produced by their being interchanged and contrasted in the same passage ?
11. What were the sources from which Herodotus compiled his history? 12. Describe the battle of Marathon.
Translate the following passages into English 1. Beginning, Dum consul placandis Romæ diis trahendoque...... Ending, et quia medendi nec locus nec tempus erat, altero oculo capitur.
Lib. xxii. c. 2. 2. Beginning, Adversus hunc navalem apparatum Archimedes .. Ending, arcere terra marique commeatibus hostem, placuit.
Lib. xxiv. c. 34. 3. Beginning, Tum Virginius, Oratio, inquit, rebus dubiis inventa est. Ending, quum causæ dicendæ data facultas sit, tum se experturun.
Lib. iii. c. 56. 4. Beginning, Ad hæc consultanda procurandaque, ...... Ending, tum temperata et belli et pacis artibus, erat civitas.
Lib. i. c. 22.
1. Give a brief account of the origin and rise of Syracuse.
2. In what struggles were the Romans engaged between the first and second Punic wars?
3. Trace Hannibal's route from Spain into Italy.
4. Draw a plan of the battle field at Cannæ, and point out the exact disposition of the opposing armies.
5. What was the Constitution of Carthage ?
6. Describe the origin of the offices of the Dictator and Censor, the mode in which they were elected, and the extent and limitations of their powers.
7. Name the chief Romans who fell in battle during the second Punic
8. Draw a map of Sicily, and give both the ancient and modern names of each place.
9. What were the chief rivers of Spain ? 10. When do relative propositions stand in the conjunctive mood ? 1. How are questions expressed in the oratio obliqua ? 12. When may a preposition be separated from its case ?
13. Explain the formation of the perfects cessi, vexi, ussi, pegi, sevi, and of the presents sero, cerno, facio, gigno.
14. What were the original forms of the person-endings of the Latin verb? Identify these with the corresponding Greek terminations.
15. What historical fact do the forms of the words Carthago, Afri, and Sarranus teach us?
16. What is the origin of the name Iberi ?
17. What traces are there of Phænician settlements on the continent of Italy?
18. Explain the formation of the words Diana, regina, Pontius, Petreius, and quantus.
19. How does Merivale account for Livy's failure as an historian?
20. Give a brief sketch of the life of Livy. What indications are there that the earlier portions of his history were published before the later ?
Translate the following passage into Greek Prose :-
Spectator, No. 582.
Translate the following passage into Greek Verse :Beginning, Go, bear Patroclus' body to Achilles ; Ending, That proof is call'd impossibility.
SHAKSPEARE, Troilus and Cressida, act v. sc. 5.
Translate the following passage into Latin Prose :Beginning, Every one knows how well Voltaire...... Ending, the one may be a consequence, not a contradiction, of the other.
BULWER, The Student.
Translate the following passage into Latin Hexameters :Beginning, Men. There was a time, when all the body's members,... Ending, They are not such as you.
SHAKSPEARE, Coriolanus, act i. sc. 1.