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6. The horizontal trace of a plane A makes an angle of 50° with the ground line to the left, and its inclination to the horizontal plane is 67°; the horizontal trace of a plane B makes an angle of 40° with the ground line to the right and its inclination to the horizontal plane is 60°: find the angle between the planes A and B, the intercept between their traces on the ground line being 4 inches.
7. In the accompanying diagram, fig. 3, construct the projections of a line passing through the point PP' and meeting the lines AA', BB'.
8. Two adjacent sides AB and AD of a parallelogram are 1 and 2 inches respectively, and the contained angle 45°: find the plan of the parallelogram and its vertical projection, when AB is inclined at 30° and AD inclined at 50°.
9. The directrix of an oblique cone is a circle lying in the horizontal plane, its radius is 1 inch and its centre is 2 inches from the ground line, the horizontal projection of the vertex is an inch from the ground line, and 24 inches from the centre of the circle, and the height of the vertex is 2 inches, the horizontal projection of a point p on the surface of the cone is 1 inch from the ground line and 14 inches from the centre of the circle: find the vertical projection of the point p, and construct the traces of a plane touching the cone at that point.
10. A right pentagonal prism lies with one of its faces on the horizontal plane, the length of a side of the base is 14 inches, and the horizontal projection of its axis makes an angle of 35° with the ground line to the right, the horizontal trace of a plane makes an angle of 45° with the ground line to the left, and its vertical trace makes an angle with the ground line of 50° to the left: construct the horizontal and vertical projections of the intersection of the plane with the prism, and find the real form of the curve of intersection.
DESCRIPTIVE ARCHITECTURE, DESCRIPTIVE
GEOMETRY, CURVES, &c.
1. Sketch and describe the shafts of the columns in the Greek Doric Order, noting points in which they differ from those of other orders.
2. Give illustrative sketches, showing a typical arrangement of triumphal arch, as used by the Romans.
3. Sketch and name the modifications of the arch introduced in later Saracenic and Moorish architecture.
4. Sketch and name some of the principal mouldings and ornaments occurring in Norman Gothic.
5. Trace, with illustrative sketches, the successive forms of doorways employed from Norman to Tudor, inclusive, in English Gothic.
DESCRIPTIVE GEOMETRY, CURVES, &c.
6. The conjugate diameters of an ellipse are 4 inches and 3 inches, and the angle between them 55°: construct the curve and find the lengths of the axes.
7. The distance between two points on a French map is 180 millimetres, and the representative fraction of the map is r find the distance between the points in English miles.
8. A thin hemispherical shell, 3 inches in diameter, rests on the horizontal plane: construct the horizontal projection of the shadow thrown by the rim on the internal surface of the shell, the horizontal and vertical projections of the rays of light making angles of 45° with the ground-line.
9. The horizontal trace of a plane makes an angle of 65° with the ground-line to the right, and the vertical trace an angle of 50° with the ground-line to the right, a square whose side is 1 inches lies in the plane, and the horizontal projection of one of its sides makes an angle of 60° with the ground-line to the left: construct the horizontal and vertical projections of the square.
10. A right hexagonal pyramid lies with one of its faces on the horizontal plane, the side of the base of the pyramid lying in the horizontal plane makes an angle of 60° with the ground-line, and its length is 14 inches: construct the horizontal and vertical projections of the pyramid, its height being 3 inches.
[All chemical changes must be expressed both in words and by equations. Candidates who neglect this instruction will not receive full credit for their answers.]
1. What varieties of silica occur in nature? What chemical changes occur (a) when silica is fused with sodium carbonate, (b) when the product so obtained is treated with weak hydrochloric acid, and (c) when it is treated with strong hydrochloric acid?
2. Describe the commercial manufacture of oil of vitriol. 3. Starting from metallic iron, how would you prepare the following?—
Anhydrous ferrous chloride.
4. Give the mineralogical names and the chemical formulæ of the chief ores of zinc. How is the metal extracted?
5. Calculate the volume occupied by 1 gram of steam at 182° C. and 836 mm. and the weight of 1 litre under the same conditions of temperature and pressure.
6. Give the chief reasons for placing calcium, strontium, and barium in the same group of metals, and mention the most striking points which distinguish them from each other.
7. What do you understand by the following terms:a salt'; colloidal substance'; 'permanent hardness' (of water); allotropism'; 'anhydride'; basic salt'; oxidizing agent'?
Give a typical example in each case.
8. What reactions take place when chlorine gas is passed into solutions of the following at ordinary temperature ?— (a) Silver nitrate. (d) Caustic potash. (e) Ammonia.
(b) Sulphuretted hydrogen.
9. Name the chief carbonates and sulphates occurring in nature, remarking those which are technically important.
10. How is phosphorus obtained? State briefly its properties and uses.
3. Find, in terms of a, b, c, the lengths of the lines joining the angles of a triangle to the middle points of the opposite sides; if these lengths are p, q, r, prove that the area of the triangle is
$[o(o − p) (o − q) (o − r)]3,
p + q + r.
Show that the sum of any two of the three p, q, r is greater than the third.
4. A, B, C, D are the angular points of a quadrilateral taken in order; AB and CD meet in F, and it is found that AF = 122, BF = 102, CF = 61, DF = 164, BC = 109, all the lengths being measured in feet. Find the length of AD, correct to the nearest foot, either graphically or by calculation.
5. Prove that in a spherical triangle
sin a cos B = cos b sin c - sin b cos c cos 4.
6. In a spherical triangle one side is 90° and the opposite angle is 70°, while another angle is 23° 27': find the remaining parts, correct to the nearest minute of arc.