E.—la.

6. A straight line drawn parallel to the base of a triangle cuts the two sides proportionally. In the sides AC, BC of a triangle ABC, points D and E are taken so that AD = -| AC and BE = J BC. Show that DE is parallel to AB and that DE = £ AB. What is the ratio of the area of the triangle CDE to the area of the triangle ABC ? 7. Explain clearly the two statements rr = - and it — 180°. Find the values of cos 26 and cos 36 in terms of cos 6. Find the values of tan 30°, tan 45°, tan 75°, tan 15°, and sin 18°. 8. Prove— (a.) sin A sin 2A + sin 2A sin 5A + sin 3A sin 10A =tan 7A. sin A cos 2A + sin 2A cos 5A + sin 3A cos 10A (b.) sin A + sin B + sin C= 4 cos - cos - cos -> (A, B, and C being the angles of a triangle). 9. Find all the values of 8 which satisfy the equation (1 — tan 0)(1 + sin 20) = I + tan 0). 10. Prove log aT- = x log a. Given log 10 2 = 0-3010300, log 10 3 = 0-4771213, find in how many years a sum of money will treble itself at 8 per cent, compound interest. 11. The sides of a triangle are 7, 8, 9 : find the smallest angle — Given L tan 24° s'= 9-6502809 L tan 24° 6'= 9-6506199 log 2 = 0-3010300. 12. A column 200 feet high supporting a statue 30 feet high stands on the bank of a river. To an observer immediately opposite on the other side of the river the statue subtends the same angle as a man 6 feet high standing at the base of the column. Find the breadth of the river.

Mechanics. — For Class D, and for Civil Service Junior. Time alloived: Three hours. [Draw diagrams wherever you can do so.] 1. Explain mass, weight, force, energy, momentum, specific gravity. 2. A stone is dropped into a well, and after 3 seconds it is observed to strike the water : find .the depth of the surface of the water. 3. Show how to find the magnitude and position of the resultant of two parallel forces. A uniform bar of iron 5 feet long and weighing 4 pounds has a weight of 6 pounds suspended from one extremity : find the point about which it will balance. If instead of the weight of 6 pounds, weights of 1, 2, 3, and 4 pounds were suspended 1, 2, 3, and 4 feet respectively from one end, about what point would it balance ? 4. Find the centre of gravity of a triangular board, and show that it is the same as the centre of gravity of three equal weights placed at the angles of the triangle. 5. A cylinder whose base is a circle 1 foot in diameter, and whose height is 3 feet, rests on a horizontal plane with its axis vertical : find how high one edge of the base can be raised without causing the cylinder to turn over. 6. In the system of pulleys in which all the pulleys are arranged in two blocks, one block fixed and the other movable, the same string going round all the pulleys, and all parts of the string between the blocks being parallel, find the number of pulleys at the lower block if 12 stone will sustain 18 hundredweight. 7. Prove that the pressure at any point in a fluid varies as the depth below the surface, the pressure at the surface being neglected. If the atmospheric pressure is 15 pounds on the square inch, find the pressure at a point 2.0 feet below the surface of water, assuming the weight of a cubic foot of water to be 1,000 ounces. 8. State and show how to prove practically the principle of Archimedes. 9. Explain the action of the common air-pump. 10. A piece of copper weighs in air 88 ounces ; if immersed in water it weighs 78 ounces, and in alcohol 80 ounces. Neglecting the weight of air, find (1) the specific gravity of copper, (2) the specific gravity of alcohol.

Theoretical Mechanics. — For Civil Service Senior. Time allowed: Three hours. [Draw diagrams in all cases where possible.] 1. A mass of 4 pounds starts from rest and moves with a uniformly increasing velocity. At the end of 6 seconds its velocity is 30 feet a second. Find its momentum, its kinetic energy, and the force which acted on it. 2. Explain the action of Atwood's machine, and show by an arithmetical example how to use it to calculate the value of g. 3. If eight forces acting on a particle be represented in magnitude and direction by the straight line drawn from the angular points of a quadrilateral to the middle points of the opposite sides, prove that they will form a system in equilibrium. ; 4. A line is drawn parallel to a diagonal of a square cutting off a quarter of the square. Find the distance of the centre of gravity of the remainder from the centre of gravity of the square. 5. The sides of a triangle are 3, 4, and 5 feet in length, and along them in order act forces of 6, 7, and 8 pounds respectively. Find the magnitude and direction of the resultant. 6. Find the ratio of the power to the weight in the wheel and axle. If the wheel is replaced -by a square frame having a side of 32 inches, the axle being round, and its diameter 8 inches, find the position of equilibrium when the weight is 10 pounds and the power 2 pounds.

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