COUNCIL OF HIGHER EDUCATION, NEWFOUNDLAND. Preliminary Grade. TOTS (to accompany Arithmetic paper). Wednesday, June 19th, 1912. Morning, 9 to 9.15. Add these up, placing the totals in the spaces indicated:- N.B.-The totals are not to be copied into the Answer book. (15) COUNCIL OF HIGHER EDUCATION, NEWFOUNDLAND. ALGEBRA (Preliminary Grade). Tuesday, June 18th, 1912. Afternoon, 2.30 to 4.30. All the working must be shown. Work neatly. 1. What do you mean by the terms-power, numerical coefficient, factor? Show, by writing out the factors in full, that 3. Find the quotient and remainder when 5-9x+27 is divided by x-3. 4. Factorize- (i) 12x2+x-1, (ii) 15x2+8x+1; (12) and thence write down in factors their Least Common Multiple. 7. One-fifth of the books of a library are out on a certain day. Threefourths of those out are fiction, one-sixth history, and the rest of them, 200 in number, are of other kinds. How many books remained in the library? (18) GEOMETRY (Preliminary Grade). Friday, June 21st, 1912. Morning, 9 to 11. The figures must be drawn neatly and accurately. All construction lines must be shown. Each Candidate may take one paper only, A or B. Paper A.-Euclid, Book I, Props. 1-26, with Riders. [Any generally recognized symbols or abbreviations may be used, but the proofs must be strictly geometrical.] A 1. AB is a given finite straight line. Find the middle point. On a given base describe an isosceles triangle, the sum of the equal sides of which shall be equal to the given straight line PQ. (10) A 2. Let ABC be any triangle. Show that any two of the sides together are greater than the third (i) by producing one of the sides; (ii) by bisecting one of the angles. (10) A 3. Draw a straight line perpendicular to a given straight line of unlimited length from a given point without it. (10) A 4. If, at a point in a straight line, two other straight lines, on opposite sides of it, make the adjacent angles together equal to two right angles, then these two straight lines shall be in one and the same straight line. (10) A 5. Prove that any two angles of a triangle are together less than two right angles. (10) A 6. If, from the ends of a side of a triangle, there be drawn two straight lines to a point within the triangle, demonstrate that these straight lines together shall be less than the other two sides of the triangle. Prove that the sum of the distances of any point within a triangle from its angular points is less than the sum of the sides of the triangle. (15) A 7. On a given base AB describe a triangle with sides respectively equal to the lines PQ and RS. Point out how the construction may fail if any one of these three straight lines is greater than the sum of the other two. If the bisector of the vertical angle of a triangle is also perpendicular to the base, show that the triangle is isosceles. A 8. Enumerate the various cases in which two triangles may be equal to each other in all respects. In a triangle ABC the vertex A is joined to X, the middle point of the base BC. Show that the angle AXB is obtuse or acute according as the side AB is greater or less than AC. (20) Paper B.-Theoretical and Practical Geometry. [Any generally recognized symbols and 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. Candidates are expected to answer questions in both Parts of the Paper.] PART I. THEORETICAL GEOMETRY. B1. Define circle, circumference, diameter, radius, centre; equilateral, equiangular. Name the different kinds of angles dealt with in geometry. (15) B2. By bisecting one of the angles of a triangle, show that any two of the sides together are greater than the third. (10) B 3. If, from the ends of a side of a triangle, there be drawn two straight lines to a point within the triangle, prove that these straight lines together shall be less than the other two sides of the triangle. Prove that the sum of the distances of any point within the triangle from its angular points is less than the sum of the sides of the triangle. (15) B 4. State the various cases in which two triangles may be equal to each other. In a triangle ABC the vertex A is joined to X, the middle point of the base BC. Show that the angle AXB is obtuse or acute according as the side AB is greater or less than AC. (15) PART II. PRACTICAL GEOMETRY. B 5. A gardener made a square bed in the middle of a square grass plot, the side of the plot being 60 yards. The bed had an area equal to half that of the grass plot. Draw the plot and bed to scale, and write down the width of the margin round the bed, (10) In the above triangle, divide AB in the points D and E, in the proportion 1: 2: 3. Draw DF, EG parallel to BC, cutting AC in F and G. Write down the lengths of AF, FG, GC. (10) B7. AB and CD are two straight lines inclined to each other at an angle of 60°, but not necessarily intersecting on your paper. Find the locus of a point from which perpendiculars of equal length could be drawn to AB and CD. (10) B 8. Take a straight line PS = 8 centimetres. Divide it into three equal parts at Q and R. Construct the parallelogram ABCD, with AB=PR, AD = RS, angle DAB = 120°. Take the four points A', B', C', D' in order, so that AA' BB' = CC' = DD'. Join A', B', C', D'. of the figure A'B'C'D'. Write down the lengths of the diagonals |