H.—3

13

4. Determine 6 from the equation—Cos 6 + Sin 6= </2 5. What is a logarithm ? Prove that log 10 a b — b log iO a Prove that log lo ai=log io a + log la b otvj i. i a i c 3 /13.715X 3.14159 6. Find by logs, the value of ' v .010/o 7. In any triangle prove that the sides are proportional to the sines of the opposite angles. 8. The three sides of a triangle are respectively 415G, 1327, and 4421 links: find the area. 9. Find, by means of the tables, the value of log. Sin 34° 15' 37" and log. Cosec 14° 38' 22", explaining carefully the several steps of the work. 10. Solve the triangle— C=l24°34' AC=25.2 A8=52.8 ALGEBRA. 1. What is the value of J—[J— (a + J)—{&—(&— a—&)}]? 2. What is the sum of the squares of two quantities increased by their product and multiplied by their difference equal to ? Also sum of their squares diminished by their product, and multiplied by their sum ? Prove your answers. 3. Divide a: 3+y s+3*y— lbyy+x-1. 4. Resolve into factors x 2+4x —45 ; also #4—# 2—110. 5. What value of x will make the difference between (# +1)(x + 2) and fa:—l) (x— 2) equal to 54? . - . d* + 2aix+aa? ~, . . ~ 6. Eeduce » s to its simplest form. 7. Solve the following equations : — x— 2 _, #+10 x X —3~=si 5- + I-fi*-*y+i* = 8. 1 Bx+2y—iz = 10. ( x—2y +dz = 11. i~+i + x =s--8. Extract the square root of 4a; 2 O 2— y) + f (y-2) +y 2 (4s 3 + 1). 9. A, B, and C had £375 divided amongst them, so that A had four-fifths of B's share, and C had a fifth part of B's share more than B. How much had each ? 10. A man working from 8 a.m. to 5 p.m. asks " What o'clock is it ?" and receives for answer, " Half the time since you commenced working is equal to one-third of the time remaining." What o'clock was it ? 11. At what time between 3 and 4 o'clock are the hands of a clock— (1.) Opposite to each other; (2.) At right angles; (3.) Coincident. GEOMETRY. 1. From the greater of two given straight lines cut off a part equal to the less. 2. The complements of the parallelograms which are about the diameter of any parallelogram are equal to one another. 3. If a straight line be divided into two equal, and also into two unequal parts, the squares on the two unequal parts are together double of the square on half the line, and of the square on the line between the points of section. 4. Draw a straight line from a given point, either without or in the circumference, which shall touch a given circle. 5. If a straight line touch a circle, and from the point of contact a straight line be drawn cutting the circle, the angles which this line makes with the line touching the circle shall be equal to the angles which are in the alternate segments of the circle. 6. Inscribe a square in a given circle. PHYSICAL SCIENCE. Zoology. 1. Name the classes of invertebrate animals, with a short description of the characters of each. 2. Describe the Paper Nautilus {Argonauta argo), and the Pearly Nautilus (Nautiluspompilius). 3. Classify the scales of fishes, and give examples of fishes possessing the different forms of scales. 4. Name the eight principal natural orders of birds, with examples of each order. 5. To what natural orders do the following animals belong: —Seal, Porpoise, Elephant, Camel, Bat, Anteater, Kangaroo, Beaver, Lynx, Porcupine, Antelope, Hedgehog ? Chemistry. 1. How can potassium and sodium compounds be recognized before the blow-pipe flame? 2. How is spongy platinum prepared, and what are its properties ? 3. What processes are usually employed for refining sulphur, and from what minerals besides native sulphur is sulphuric acid obtained ? State whether pure sulphuric acid can be obtained from these minerals ; and, if not, what is the principal impurity likely to be found. 4. Give the reason why two molecules of hydrochloric acid (HCI) are required to act upon an atom of zinc, and give the formula of the reaction.