6. If p+q = r +8 = 1 1 prove that a(b-c) (op + bq) + b(c- a) (ar + cs) + (a - b) (ar + c3) (cp + bq) = (c – a) (c- b) bas – apr). 7. A merchant imports goods from England. The freight amounts to one-tenth of the English cost, and the duty is reckoned at so much per cent. ou the English cost. The merchant reckons the cost price to himself as made up of English cost, freight and duty, and then adds on a certain percentage of profit on the total. The result is that his selling price is 75 per cent. above the cost price in England. If the duty were increased by 5 per cent., and the merchant were content with 5 per cent. less profit, his selling price would be 75 per cent. above English cost. Find the rate of duty and the percentage of profit exacted by the merchant. 8. Five authors each write four books, one on each of the subjects Algebra, Arithmetic, Trigonometry, and Geometry. Out of the twenty books in how many ways is it possible to select four so that no two are by the same author or on the same subject ? In how many ways is it possible to make the selection so as to include a given book? 9. Find the sum of n terms of a Geometrical Progression whose first term is a and common ratio r. In an Arithmetical Progression the p', q", and pe terms are in Geometrical Progression. If p, q, and r are also in G.P., prove that the first term of the A.P. is equal to the common difference. 10. Prove that the coefficient of x" in the expansion of (1 – 3)* is equal to the difference of the coefficients of " and x-? in the expansion of (1 – x)"-). Hence or otherwise show the algebraic sum of the coefficients of **+1, x*+2, xr+3, w"+' in the expansion of (1 – x)"+1 is equal to the difference between the coefficients of w*+4 and x" in the expansion of (1 – x)". APPLIED MATHEMATICS. Time : three hours. 1. The sides AB, BC, CA, of a triangle are in the proportion 3, 4, 5. Three forces which act at a point are represented in direction and magnitude by the sides of the triangle. State the proposition which asserts that such a set of forces is in equilibrium. Three forces acting at a point are represented in direction by the sides AB, BC, CA of this triangle, but in magnitude by the sides BC, CA, AB respectively. Find the direction and relative magnitude of the single force which will balance them. 2. Show how to find the resultant of two unlike parallel forces acting upon a rigid body. A beam 8 ft. long, weighing 20 lbs. rests on two supports, distant 2 ft. and 3 feet respectively from the ends. Where must a weight of 10 lbs. be placed on the beam so as to make the pressures on the supports equal ? 3. A solid hemisphere of uniform density rests with its curved surface in contact with two inclined planes, which meet in a horizontal line, and are inclined to the horizontal plane at angles of 30° and 60° respectively. The hemisphere weighs 10 lbs. Find the pressures on the planes. If a weight of 1 lb. be fastened to the rim of the hemisphere, what new position will it take? 4. Show that the work done in raising a number of particles from one position to another is wh, where w is the total weight of the particles, and h is the distance through which the centre of gravity of the particles has been raised. A chain weighing 10 lbs. and 6 ft. long lies on the ground. One end is picked up and raised to a height of 10 ft. How much work is done? 5. A ship is moving at 12 miles an hour. A gun is to be fired at an object in a direction at right angles to the ship’s course. The velocity of the shot being supposed uniform and equal to 1,000 ft. a second, at what angle with the apparent direction of the object should the gun be fired ? Give the answer in degrees, minutes, and seconds. 6. Define the terms momentum and impulse. Illustrate by reference to the case of the relative motion of a gun and a shot fired from it. A man holds in his hand two balls of equal mass, connected by a string 3 ft. long. He drops one, and lets go the other just as the string becomes tight. With what velocity will the latter ball begin to move ? 7. What is the value of the horsepower required to cause a jet of water to stream from a nozzle at the rate of 30 ft. a second, 100 gallons issuing every minute? 8. Show that the resultant vertical pressure on any surface immersed in any heavy fluid is equal to the weight of the superincumbent fluid and acts through the centre of gravity of this superincumbent fluid. How is this proposition to be modified if the liquid presses upward ? A hemispherical bowl, of 6 in. radius, is placed rim downwards on a flat plate which it closely fits. Water is poured in at a small hole at what is now the highest point of the bowl. When the bowl is just full it lifts from the plate. What is the weight of the bowl ? 9. A cylindrical diving bell is lowered into water, no air being supplied from above. Find the compression of the air at s given depth. PHYSICS. Time : three hours. 1. Explain the principle of the “Conservation of Energy.' Illustrate it in the case of (a) a swinging pendulum ; (b) the motion of the earth. 2. Describe Torricelli's experiment upon the flow of liquid through an orifice, and state the results he obtained. What influence has the sbape of the orifice upon the flow ? 3. State a comprehensive law embodying Boyle's and Charles's laws. P.V 3. Find the numerical value of T for one gramme of air at a temperature 100°C and a pressure of 1,000 atmospheres. The volume of one gramme of air at standard atmospheric 1 pressure and at a temperature of 0°C=. ·001293 The density of mercury = 13.6. C.cs. 4. What is the effect of pressure upon the melting point of a substance ? Explain clearly the process of regelation. 5. Explain the occurrence of beats. discord and harmony? 6. Explain fully the phenomenon of the rainbow; also of the secondary bow. 7. Define the terms surface density, potential, unit quantity of electricity. Describe the action of the Wimshurst machine. 8. Describe the tangent galvanometer, and explain what precautions must be used in setting it up. Show that the current is proportional to the tangent of the angle of deflection. In a tangent galvanometer, if the mean radius of the coils be 9 cms., the number of coils 50, and H :25, find the constant of the instrument. Wben a deflection of 30° is given, what current is passing? 9. A galvanometer of resistance 6,000 ohms is shunted by a resistance of 50 ohms. It is then connected to a battery of 30 cells, each of E.MF. 1.5 volts, and internal resistance 5 ohms. Find the values of the current passing from the cells; also the current through the galvanometer coils and shunt respectively. = INORGANIC CHEMISTRY. IN.B.-Candidates are expected to write formulae and equations wherever possible.] Time : three hours. 1. Describe carefully the structure and use of a eudiometer, stating the necessary corrections in the measurement of the gaseous volumes and how you would apply them. 2. Given potassium carbonate, silver nitrate, nitrogen, charcoal, and ordinary laboratory appliances, show how you could prepare cyanogen. 3. Write what you know of nitrous acid and its salts. 4. Explain the various steps necessary in determining the formula for water. 5. Write what you know of the preparation, properties, decomposition, and uses of sulphuretted hydrogen. 6. Mention any resemblances between the compounds of carbon and silicon. Write the formula for “normal” silicic acid, and show how other forms of silicic acid are derivable from it. 7. How would you prepare a. Potassium iodide. on a solution of silver nitrate. solution ? d. Plaster of Paris. What is the effect of overheating it! e. Anhydrous magnesium chloride. j. Potassium dichromate. Why is it more correctly called dichromate than bichromate ? 8. State the results of adding solutions of ammonia, caustic potash (in small quantity and in excess), sodium carbonate, and ammonium sulphide to a solution of aluminium sulphate. Explain the effects obtained. PHYSICAL GEOGRAPHY AND GEOLOGY. Time: three hours. a 1. What is a volcano ? Give some account of the distribution of volcanoes at the present day. Can any reason be assigned for the manner of their distribution ? 2. Explain the origin and effects of the trade winds, anti-trades, and monsoons. Supply illustrations from the occurrence of these winds in the Australian region. 3. What is meant by the terms acidic and basic as applied to rocks? Name examples of each class and state their respective modes of occurrence. 4. Give a description of the Protozoa, and name the rocks and geological periods in which their remains are particularly abundant. |