A 7. Prove that the rectangle contained by the sum and the difference of two straight lines is equal to the difference of the squares on them. The sides of a triangle are 6, 7, 9 feet; a perpendicular is drawn on the longest side from the opposite angle. Calculate the distance of the foot of this perpendicular from the middle point of the longest side. (18) A 8. Show how to divide a finite straight line into two parts so that the rectangle under the whole line and one part shall be equal to the square on the other part. (9) Paper B.-Euclid, Book I and Book III, Props. 1-19, with Riders. [Any generally recognized symbols or abbreviations may be used, but the proofs must be geometrical.] B1. Show, with proof, how at a given point in a straight line to make an angle equal to a given angle. (6) B 2. Prove that, if two triangles have two angles of the one equal to two angles of the other, each to each, and also have a side of one equal to a side of the other, these sides being adjacent to the equal angles, the triangles are equal in all respects. A, B, C are three points on the bank of the river, P the point on the other bank directly opposite to A, and AB, BC are equal; on walking 100 feet from C, in a direction perpendicular to BC, I find myself in a line with P, B. What is the breadth of the river? (16) B3. Calculate the magnitude of the angle of a regular polygon of thirteen sides. (9) B 4. Prove that equal triangles on equal bases, in the same straight line and on the same side of it, are between the same parallels. Two equal triangles stand on the same base and on opposite sides of it. Prove that the base bisects the straight line which joins their vertices. (18) B 5. The sides of a triangle are 24, 25, 7 feet long. Prove that one angle is a right angle, and calculate the length of the perpendicular drawn from this angle to the opposite side. (15) B 6. Prove that, if two circles cut one another, they cannot have the same centre. (8) B 7. Show how to describe a circle of given radius which shall touch each of two given circles externally. (10) B8 Prove that the diameter is the greatest chord in a circle; and, of others, that which is nearer the centre is greater than that which is more remote. Show how, through a given point within a circle, to draw the least possible chord. (18) Paper C.-Theoretical and Practical Geometry. [Any generally recognized symbols or abbreviations may be used. Figures should be drawn accurately. In the Practical Geometry, candidates are not required to prove the validity of the constructions, but all the lines required in the constructions must be clearly shown.] C1. Two sides of a triangular field are 315 and 260 yards, and the included angle is 39°. Draw a plan (1 inch to 100 yards), and find, by measurement, the length of the remaining side of the field. (12) C2. Construct the quadrilateral ABCD, given the lengths of AB, BC, CD, DA to be 5.6, 2.5, 4.0, and 3.3 centimetres, and the angle A to be 60°. Measure the diagonals of the quadrilateral. (12) C 3. Find, by measurement, the radius of the circumscribed circle of the triangle whose sides are 6·3, 3·0, 5·1 centimetres. (12) C4. Draw two circles of radii 1·7′′ and 1·0", having their centres 2.1′′ apart. Draw their common tangents, and measure their lengths. (14) C5. Show how to find the locus of a point which moves so that its perpendicular distances from two given straight lines are equal. Prove that the bisectors of the angles of a triangle are con current. (14) C6. Prove that the square described on the hypotenuse of a right-angled triangle is equal to the sum of the squares described on the other two sides. ABC is a triangle right-angled at A; the sides AB, AC are (18) C7. Prove that, of any two chords in a circle, that which is nearer the centre is greater than that which is more remote. AB is a fixed chord of a circle, and XY any other chord having its middle point Z on AB. What is the greatest value XY can have? Prove your statement. (18) MECHANICS (Intermediate Grade). Friday, June 23rd, 1911.- Morning, 9 to 11. Work neatly. 1. In what respects may forces be represented by straight lines, and what are the advantages of so representing them? (12) A weight of 20 lb. is supported by two cords attached to two hooks in a horizontal beam and to a point in the weight. The cords are 5 ft. and 2 ft. long respectively. Draw a diagram which will enable you by measurement to find the tensions of the two strings. (14) 2. Show how to find the resultant of two parallel forces acting in the same direction. A uniform rod, 4 ft. long and weighing 12 lb., rests horizontally with its ends on two props 4 ft. apart, and a body weighing 20 lb. is suspended from the rod at a point 1 foot from one end. Find the pressures on the props. (28) 3. Describe, and sketch, some simple machine by which a force of 50 lb. will sustain a load of 5 cwt. (24) 4. How much work is required to lift 5 bags of grain, each weighing 56 lb., to a floor 30 ft. high? What difference would it make in the amount of work if the grain was lifted by means of a single movable pulley? (26) 5. A stone is let fall from a tower, and falls 16 ft. in the first second. How far will it fall in the next second, and how far in the third second? (24) 6. Write out Newton's First Law of Motion, and give evidence of its truth. (24) 7. Describe an Air Pump, and also some experiments which may be made with it. (24) 8. How would you find the specific gravity of a piece of metal? Give a numerical example. (24) MENSURATION (Intermediate Grade). Friday, June 23rd, 1911.-Morning, 9 to 11. Work neatly. Diagrams and formula should accompany the solutions, which should be given in full. 1. The top of a ladder rests against the wall of a house at a point 48 ft. above the ground. The length of the ladder is 52 ft. How far is the foot of the ladder from the foot of the wall? (8) 2. A square pyramid has a height of 50 ft., and its volume is 60000 cubic ft. Find the length of a side of the base. (8) 3. How many cubic yards of earth must be removed to make a ditch 4 ft. deep, 6 ft. broad at the top, 3 ft. broad at the bottom, and 174 ft. long? (10) 4. A spherical iron shot is 6 inches in diameter. Find its weight to the nearest ounce. [1 cubic ft. of iron weighs 480 lb.] (10) 5. A sector of a circle has a radius of 16 inches and an angle of 70°. Find its area and the length of the arc. (10) 6. The conical spire of a church is 60 ft. high and 10 ft. in diameter at the base. Find the cost of covering it with sheet copper at 80 cents per square foot. Answer to the nearest dollar. (10) 7. A rectilinear field ABCD has AB = 613 links, AD 581 links, BC 672 links, and the angles at B and D are right angles. Find the area to the nearest acre. = (14) 8. A quadrilateral plot of land has two adjacent sides, each 17 chains, and the other two sides 20 chains, and one diagonal is 35 chains. Find its area to the nearest acre, and the length of the other diagonal to the nearest link. (14) 9. Draw neatly a Plan of the field ABCDEFG, and find its area in acres, roods, perches, from the following Field-Book entries in links: BOOK-KEEPING-Double Entry (Intermediate Grade). Saturday, June 24th, 1911.- Morning, 9 to 11. N.B.-Extra marks are awarded for neat writing, with legible figures. Dates and Names of Persons must be given as stated. 1. Journalize the following Balances and Transactions. 2. Post the Journal entries into the Ledger. 3. Extract a Trial Balance. 4. Close the Accounts and compile a Balance Sheet. On May 1st, 1911, the Books of Mr. W. Welsh showed the following balances-Capital a/c to be calculated therefrom :— and paid Carriage thereon, chargeable to Scot 25.50 24.75 Credit Note for same 150.00 18. Sold returned Goods for Cash (paid into Bank).. 145.00 400.50 |