EXAMINATION PAPERS FOR SCIENCE SCHOOLS AND MAY 1869. SUBJECT I. PRACTICAL PLANE GEOMETRY. EXAMINER, PROFESSOR T. BRADLEY. First Stage or Elementary Examination. By well considering the question he is about to construct, the candidate will be able to get all his work, without any crowding or confusion, on the four pages of the paper given out to him, for he must understand that no second sheet will be allowed him. The candidate may construct as many of the questions as he has time for; but his constructions must be strictly and demonstrably geometrical, that is neither empirical nor the result of calculation or trial. The absence of those lines which would be essential to a correct construction immediately show that the figure is worthless, however correct the result may appear. The constructions may be left in pencil provided they are distinct and neat; lines, parallel or perpendicular, may be drawn by means of a "set square "without showing any constructions for that purpose. In all cases the number of the question must be placed before the answer on the worked paper. Four hours are allowed for this examination. 1. Divide a line A B=3′′ 3 inches long in the point C so that AC: BC: 3:1·75. 2. Construct a triangle A B C from one of the following conditions. a. Its perimeter (or sum of its sides)=7" 5: its angles as 2:3: 4. b. Equilateral, its area equal to that of a square of 2′′ 35 side. 3. Construct a regular pentagon A B E on a line of 2.35 inches. 4. The sides of a triangle are 2, 2.5, 3 inches. Construct a triangle similar to it but of twice its area. 5. Construct a triangle A B C having its angles 50°, 60°, 70°; and circumscribing a circle of 1 inch radius. 6. The distance between the centres (A) (B) of two circles is 2", their radii are 75′′ and 1" inch; draw a circle of 2" radius to touch both the former, but to contain the smaller within it; the points of contact to be correctly determined. 25011. A 2 7. Draw a circle the area of which is equal to the sum of those of two other circles of 1" and 2" diameters, and another circle the circumference of which is equal to the sum of those of the same given circles. Second Stage or Advanced Examination. INSTRUCTIONS. By well considering the question he is about to construct the candidate will be able to get all his work, without any crowding or confusion, on the four pages of the paper given out to him, for he must understand that no second sheet will be allowed him. The candidate may construct as many of the questions as he has time for, but his constructions must be strictly and demonstrably geometrical, that is neither empirical nor the result of calculation or trial, the absence of those lines which would be essential to a correct construction immediately shows that the figure is worthless, however correct the result may appear. The constructions may be left in pencil provided they are distinct and neat; lines parallel or perpendicular may be drawn by means of a "set square " without showing any constructions for that purpose. In all cases the letter distinguishing the question must be placed before the answer on the worked paper. Four hours allowed for this paper. a. Construct an irregular five-sided figure ABCDE from the following conditions: Sides. Angles. BCD=110° وو دو Write down the length of the remaining side EA, and the CDE=100° (magnitude of the two angles DEA, EAB. (The candidate is advised to deduce the angles from a circle of BA radius described from B as a centre; and not to trust to an ordinary "protractor.") b. Divide a line AB 3.3" long in the points C and D in AB produced so that AC AD:: BC: BD. AC being 2.85′′. c. Construct a triangle ABC from either of the following conditions. a. Its angles as 2, 3, 4, and its area 5 square inches. b. Its side AB=3·7′′, its altitude 2·75′′, and its angle C=60°. d. A triangle ABC has its sides 3′′, 2.5′′, 2" a. Construct a triangle similar to it but of 3 its area. b. Construct a parallelogram ADEF equal to the given triangle both in area and perimeter. (The perimeter of any figure is the sum of sides.) e. The centre (A) of a circle of 1.5′′ radius is 1.75′′ from a line; a point P in the line is 2" from A. Draw a circle to touch the line in P and to touch the circle, but to contain it within it. (The whole circle need not be drawn, but the point of contact must be correctly determined.) f. A curve line passes through the extremities of all tangents drawn to a circle of 1.5" radius, the length of each tangent from its point of contact being equal to the arc of the circle included between that point and another fixed point A on the circumference; the curve to be continued to the diameter produced passing through A. 9. Draw a scale of 3320 to show furlongs, poles, and yards, by diagonal division (1 furlong 220 yards). The scale to show not less than 5 furlongs. Honours Examination. INSTRUCTIONS. By well considering the question he is about to construct the candidate will be able to get all his work, without any crowding or confusion, on the two sides of the sheet of paper given out to him, for he must understand that no second sheet will be allowed him. The candidate may construct or answer as many of the following questions as he pleases, but his constructions must be strictly geometrical, showing by the lines employed the principles on which they are based, and no result arrived at by calculation or trial is admissible. The written answers to any questions that require such must be brief and made clear by algebraical symbols and equations in the demon strations. The constructions may be left in pencil and lines may be drawn parallel or at right angles by means of a set square." 66 In all cases the letter distinguishing the question must be placed before the answer on the worked paper. Four hours are allowed for this examination. p. Find an arithmetical, a geometrical, and an harmonical mean, and a third proportional, to two lines AB, CD 2.75"; 1.25 inches long. q. Determine by construction the value of x from any two of the following equations (the linear unit =1 inch). r. Construct a triangle, the base AB being 3" and the sines of the adjacent angles √3 1 2√2 s. A triangle ABC has two side 4", 3.5′′, and the included angle 67° construct a parallelogram equal to it both in area and perimeter. t. The perimeter of a plane figure is 9 425", construct the figure so that its area may be greater than that of any other plane figure with an equal perimeter. u. Inscribe a rectangle of 5′′ area in a circle of 2" radius. v. A line is 3" from the centre of a circle of 1.8" radius, determine a point P within the circle such that the tangents at the extremities of all chords drawn through P may meet in the given line. w. Construct a triangle ABC, the base AB being 2-5′′, its height 1.75", and the sum of the sides AC+BC=4.5". x. Prove that if from any point P out of a circle a tangent PT be drawn to the circle, and any line PS be drawn to cut the circle in the points Q and S, then shall PQ: PT: PS be in continued proportion. y. Of what geometrical propositions are the following equations the expressions? State what kind of magnitude does A represent in each. a. A2=a2+b2±2ab B. A2=sX (s-a) x (s—b) x (s—c) Υ. Cos A= b2+c2-a2 z. Prove that the square on the side of a regular pentagon is equal to the sum of the squares on the sides of a hexagon and decagon inscribed in the same circle. SUBJECT I. PRACTICAL SOLID GEOMETRY. EXAMINER, PROFESSOR T. BRADLEY. First Stage or Elementary Examination. INSTRUCTIONS. By well considering the question he is about to construct the canditate will be able to get all his work, without any crowding or confusion, on the four pages of the paper given out to him, for he must understand that no second sheet will be allowed him. The candidate may make as many of the constructions of the following questions as he pleases; but no credit will be allowed for any that are not completed, and do not strictly conform in every respect to the conditions. No point, line, plane, or figure will be considered as determined unless its projections or traces are drawn in conformity with the principles of orthographic projection. In all cases the number of the question must be placed before the answer on the worked paper. Four hours allowed for this paper. 1. A hexagon of 1.5 inches side is the base of a pyramid 3′′ high; show this solid by an elevation and two plans a. When one face (ABV) is horizontal, b. When that face is vertical. N.B.-Both these plans must be deduced from the elevation first drawn. 2. Construct the plan and two elevations of a square of 2.5" side when its plane is inclined at 50°, and one edge is inclined at 30° (one of the elevations to be on a plane parallel to one diagonal). 3. Draw the plan and two elevations of an equilateral triangle ABC of 3" side when the sides AB, BC are inclined at 35° and 55° to the paper (one elevation to be on a plane parallel to the line AC). 4. A prism 3" long has an equilateral triangle of 3" side for its base; draw the plan and an elevation of this solid when its edges are inclined at 333°, and the plane of one face BCFG is inclined at 77°. 5. A circle of 3" diameter lies in a plane inclined at 55° to the paper; draw a plan and an elevation of it, the ground line making an angle of 60° with the horizontal of the plane. 6. A cylinder 3" long and 3" diameter has its axis inclined to the paper at such an angle that the plans of the two ends touch each other in a point represent it in this position, an elevatlon to be made on a ground line making an angle of 60° with the plan of the axis of the cylinder. Second Stage or Advanced Examination. By well considering the question he is about to construct the candidate will be able to get all his work, without any crowding or confusion, on the four pages of the paper given out to him, for he must understand that no second sheet will be allowed him. The candidate may make as many of the constructions of the following questions as he pleases, but no credit will be allowed for any that are not completed, and do not strictly conform in every respect to the conditions. No point, line, plane, or figure will be considered as determined unless its projections or traces are drawn in conformity with the principles of orthographic projection. In all cases the letter distinguishing the question must be placed before the answer on the worked paper. Four hours allowed for this paper. a. An indefinite line inclined at 40° is the intersection of two planes, one inclined at 60°, and the other perpendicular to the first; show them by their traces. b. A cube of 2.5" edge has the planes of two of its faces inclined at 50° and 70°. Represent it by a plan, and an elevation on a plane parallel to a diagonal of the solid. c. Each of three lines meeting in a point O is perpendicular to the plane containing the other two; two of them are inclined at 30°, 45°; show them by a plan and an elevation when the point O is 3" above the paper and in the plane of the elevation. d. A pyramid 3" high has a pentagon ABCDE of 2" side for its base. Show this solid by one elevation and two plans. a. When the two edges BV, CV are horizontal. b. When the edge AV is vertical. e. A prism 3′′ long has an equilateral triangle of 3" side for its base. This solid can be cut into three pyramids of equal volume. Show by a plan and elevation of each two of these three pyramids. (The best mode of proceeding is to draw the lines of section on the plan and elevation of the prism first drawn.) |