2. Enunciate the exponential theorem, and deduce from it the logarith- 10 mic series, viz. — Prove that 3. 1 1.2 1 2.3 3.4 Prove that the sides of a triangle are proportional to the sines of the 11 opposite angles. In a triangle ABC, tan A = 2: shew that b c+a cosec B b+ a cosec C 4. Solve a triangle, having given two sides and the included angle. Given A = = 2.25 ft., c = 5. With the usual notation for a triangle prove that The diagonals of a quadrilateral ABCD meet in O perpendicularly: prove that the sum of the radii of the inscribed circles of the triangles 40B, BOC, COD, DOA, is equal to the difference between the sum of the diagonals and half the sum of the sides of the quadrilateral. 6. A circle of 14 inches diameter cuts a circle of 4 feet diamater at right 11 angles: shew that the length of their common chord is 13 25 inches. 7. Circles are described touching each other successively and also each of two given straight lines meeting in A; the outermost circle has its centre at C, and AC produced meets this circle again at B. Prove that the sum of the areas of all the circles bears to the area of the outerinost circle the ratio AB2: 44 C.CB. 8. Shew how the prismoidal formula may be applied to the measurement of irregular solids. A chord A04' of a circle bisects a radius CB at right angles, and a solid is generated by the revolution of the segment ABA' about AA': find the volume of the solid approximately, dividing AA' into six equal parts and drawing planes perpendicular to it at the points of division. 9. Find the volume of a regular octahedron whose surface is eight times that of a regular tetrahedron whose edge is 2 ft. 6 in., correctly to three places of decimals. 10. Find the volume of any segment of a sphere. On the same circle as base and on the same side of it are described a hemisphere and a cone of vertical angle 60°: find the volume of the cone outside the hemisphere. 7 13 12 FRIDAY, 8TH NOVEMBER, [10 A.M. TO 1 P.M.] EUCLID AND CONIC SECTIONS. 1. If two straight lines, which meet one another, be parallel to two other straight lines which meet one another, but are not in the same plane with the first two, the plane passing through these is parallel to the plane passing through the others. Draw two parallel planes, one through each of two straight lines which do not meet and are not parallel. 2. If two planes which cut one another be each of them perpendicular to a third plane, their common section shall be perpendicular to the same plane. Perpendiculars 4E, BF are drawn to a plane from two points A, B above it; a plane is drawn through 4 perpendicular to AB; shew that its line of intersection with the given plane is perpendicular to EF. 9 3. Construct the straight line which is perpendicular to two straight lines 10 which are not parellel and do not meet. Shew also that the common perpendicular is the shortest distance between the two given lines. 4. Every solid angle is contained by plane angles which are together less 12 than four right angles. Shew that there cannot be more than five regular polyhedra. 5. In any conic prove that SG: SP= eccentricity, If a chord PQ be drawn through a focus S and the normals at P and Q meet in O, shew that PO2 + QO2 varies as PQ2. 10 6. If any chord of a parabola be drawn through a fixed point 0, cutting 10 the curve in Q, Q' and the tangent at A, the extremity of the diameter through O, in the point T, and if QN, Q'N' be ordinates to the diameter 40, prove that AN. AN': = If PQ be any chord of a parabola and TOR a diameter meeting the tangent at P, the curve and PQ, in T, O, R, respectively, shew that 7. Shew that the tangent at any point of an ellipse is equally inclined to the focal distances of the point. An eilipse and two circles have the same tangent at P, and lie on the same side of the tangent; one circle passes through each focus and the circles cut the major axis in W and W. Prove that PWW' is an isosceles triangle. 9 8. If the tangent at any point P of an ellipse whose foci are S, S' be cut 10 by any pair of parallel tangents in the points T, T' and CD be the semidiameter conjugate to CP, prove that 9. If PN be the perpendicular from any point P of an hyperbola, on the 11 transverse axis AA', shew that the ratio PN: AN. A'N is constant. Pis any point on a circle of which AB is a fixed diameter; through Ba straight line is drawn to meet AP produced in Q, so that BP, BQ make equal angles with AB. Prove that the locus of Q is a rectangular hyperbola. 10. In the hyperbola prove that the focal perpendiculars on a tangent 10 met the tangent on the auxiliary circle, and that the semi-conjugate axis is a mean proportional between their lengths. An ellipse and an hyperbola have the same foci and meet in P. PYZ is a tangent to the hyperbola at P, and SY, S'Z the focal perpendiculars on it. Prove that PY. PZ = BC2, where BCB' is the minor axis of the cllipse. WEDNESDAY, 6TH NOVEMBER. [2 P.M. TO 5 P.M.] PHYSICS. [General Physics, Sound and Heat.] 1. Describe Atwood's machine. In an Atwood's machine, the two weights are 500 gr. and 550 gr., respectively, the acceleration produced in one second 46.7 cm. Find the value of g. (neglect the fraction in the answer). Use the C. G. 3. system in your calculation. 2. Describe Fortin's barometer. The mercury stands in a faulty barometer 71:35 cm. high, when the true barometric pressure is 77 cm. Find the amount of air at standard pressure in the Torricellian vacuum of the barometer. Length of the tube of the barometer =85 cm. = 1 cm. (neglect the fraction in the answer.) 3. On what does the pressure in a liquid depend? Find the total pressure on the outside of a hollow sphere which is exhausted and is kept in water with its centre 50 cm. below the surface of the water. Radius of the 22 7 sphere 7 cm,, σ = 12 12 12 4. Explain Doppler's principle, and show that, all the other circum- 14 stances being the same, another sound is heard when the observer moves towards the source of sound, than when he moves away from the source of sound. A man at rest hears the sound of a whistle, the note of which is c. By what velocity should he move towards the whistle to hear c'. (Velocity of sound 340 m.) = 5. Explain how by Kundt's experiment the velocity of sound in solids and gases may be determined. SECTION II. 6. In a differential thermometer, consisting of two bulbs joined by a bent tube which contains liquid, one of the bulbs is filled with oxygen and the other with nitrogen. 10 What will happen if both bulbs are raised to the temperature of boiling water? What takes place when one bulb is heated while the other is kept at the temperature of the air? 7. Explain how water may be cooled, and even frozen, by its own rapid 1 evaporation. How much water would have to be evaporated from a vessel containing water at 0°C. in a room at the same temperature, in order to obtain 600 grams of ice? Latent heat of evaporation of water at 0°C.=606'5. S. Fifty grams of a metal, the specific heat of which is 0.09, are heated 10 to 100°C. and dropped into a calorimeter containing 1,000 grams of a liquid at 24°C. After mixing the temperature is found to be 30°C. The water value of the calorimeter, thermometer and stirrer is 10. The air temperatrue is 27°C. Why is this important? Find the specific heat of the liquid. 9. Give your reasons for thinking that heat is one of the forms of energy. 10 How can the number of ergs or of foot-pounds in a caloric be determined? THURSDAY, 7TH NOVEMBER. [10 A.M. TO 1 P.M.] SURVEYING AND LEVELLING. [The same as that set for the L. Ag. Examination; see page cccliii] WEDNESDAY, 6TH NOVEMBER. MATERIALS OF CONSTRUCTION. 1. Describe the operations of quarrying stone by means of blasting with 12 powder. Name the various implements used and give a rough rule to regulate the charges of powder to be used. 2. What is the difference of iron and steel both in its chemical composi- 12 tion and in its character? De:cribe briefly the manufacture of Blister steel, Shear steel, and Bessemer steel. 3. For what reason is oil paint applied to timber? Name and describe a 12 process for the protection of timber from dry rot and from the attacks of insects. Describe the method of seasoning timber by hot air. 4. State Vicats classification of lines, giving the proportions of con 11 stituents and the properties of each kind SECOND EXAM. IN CIVIL ENGINEERING, 1901-1902. ccclxi 5. Describe the manufacture of Portland cement. Mention the tests 16 which you would apply to any such cement before accepting it for use on an important work. 6. Describe the manufacture of bricks as ordinarily carried out in 16 this country. In selecting earth for brick-making what sort of clay would you look for? 7. What are the 3 classes into which building stones may be divided? 10 Name the principal kinds of stone belonging to each class. 8. What ingredients are used in the preparation of oil paints and how 10 is the paint prepared? Describe briefly the operation of painting woodwork. [The same as that set for the Intermediate Science Examination; B 1964-31 ex |