Intermediate Grade. TOTS (to accompany Arithmetic paper). Tuesday, June 20th, 1911.- Morning, 9 to 9.15. Add these up, placing the totals in the spaces indicated :— Add these across, placing the totals in the spaces indicated :— = 279 +1623 +84 +9 + 718+ 2059 + 315 +79 = $50.2+13.25 +8.84+117.34 +6.29+11.3+2.40+6.07= TOTALS. N.B.-The totals are not to be copied into the Answer book. (15) ALGEBRA (Intermediate Grade). Monday, June 19th, 1911.- Morning, 11.30 to 1.15. Care should be taken to write out the work neatly and to omit no steps of the proofs. Squared paper may be obtained from the presiding examiner. 2. Prove that 5x-1 is a factor of 15x3-38x2-23x+6, and find its (iii) 2x2+3x=10, correct to two decimal places. 4. Extract the square root of 4a6+12a5-7a4-44a3-14a2+40a +25. (10) (14) (14) 5. What number must be added to 9x2-24x in order to form a perfect square? If y = 9x2-24x+30, show that y always exceeds 14, whatever be the numerical value of x. Find the values of x which make y equal to 39. (16) 6. A man buys (a+b) tons of coal at (p-q) shillings per ton, and (a-b) tons at (p+q) shillings per ton. If he sells the whole at p shillings per ton, what profit does he make? Verify your result when a 30, b = 11, p = 22, q = 3. (16) Either 7A. Taking inch as unit, draw the graph of (x+2)(x−3), from x=-2 to x = 3. Find, from your graph, the values of x which correspond to the following values of (x+2) (x-3): (i) -2, (ii) -61. Verify your results by algebraical solutions. (20) GEOMETRY (Intermediate Grade). Wednesday, June 21st, 1911.-Morning, 9 to 11. Figures must be drawn neatly. Each candidate may take ONE paper ONLY, A or B or C. Paper A.-Euclid, Books I and II, with Riders. [Any generally recognized symbols or abbreviations may be used, but the proofs must be geometrical.] A 1. Show, with proof, how at a given point in a straight line to make an angle equal to a given angle. (6) A 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 a 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? Prove your statement. (16) A 3. Calculate the magnitude of the angle of a regular polygon of thirteen sides. (9) A 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) A 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) A 6. Prove that, if a straight line is divided into any two parts, the square on the whole line is equal to the sum of the squares on the two parts together with twice the rectangle contained by the two parts, (9) 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) B 3. 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 4; 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) |