Give examples of converse propositions from the first book of Euclid. (These form one question.) SECTION II.-1. If the equal sides of an isosceles triangle be produced, the angles made by these lines produced with the base will be equal. Show briefly that this property may be proved by a method similar to that employed in the 4th proposition of the first book. 2. The greater angle of every triangle is subtended by the greater side, or has the greater side opposite to it. Write out the converse of this proposition, and show that one of the two propositions is proved directly and the other indirectly. 3. If two triangles have two angles of the one equal to two angles of the other, each to each, and one side opposite to one of the equal angles of the one equal to the side opposite to the corresponding equal angle of the other, the third angle of the one shall be equal to the third angle of the other. Show that this property would follow as a direct corollary from the 32nd proposition of the first book. SECTION III.-1. Straight lines which are parallel to the same straight line, are parallel to one another. If the straight line, to which two others are parallel, lie between them, show that this property follows from Euclid's definition of parallel straight lines. 2. Triangles upon equal bases and between the same parallels, are equal. Given the middle points of the sides of a triangle, construct the triangle. 3. If the square described on one of the sides of a triangle be equal to the squares described on the other two sides of it, the angle contained by those two sides is a right angle. Write out the corresponding proposition for obtuseangled triangles. SECTION IV.-1. If a straight line be divided into two equal parts and also into two unequal parts, the rectangle contained by the unequal parts, together with the Euclid, Algebra, and Mensuration. 209 square on the line between the points of section, is equal to the square on half the line. At what point must a given line be divided, so that the rectangle contained by the two parts shall be the greatest possible? 2. To draw a straight line which shall touch a given circle, from a given point without it. Two of the common tangents to two equal circles which do not cut or touch one another intersect in a point equi-distant from their centres. 3. The angle in the segment of a circle less than a semicircle is greater than a right angle. Show that only four equal triangles can be described on the same base having equal vertical angles. ALGEBRA. SECTION V.-Simplify 7a-4b-{5a-3 [b-2 (a−b)]} Resolve into factors 256b-81a1, 4a3-9a2b-16ab2+ 3663. If x2-ax-g is divisible by x-2, what is the value of a? (These form one question.) SECTION VI.-1. If x2+ax+b, and x2+cx+d have a b-d common measure of the form x+e, show that e=" a-c 2. Show that a quadratic equation cannot have more than two roots. Form the equation whose roots are 2 and -3. 3. If a+b+c=0, show that a1+b1+c1=2 (ab+be+ca)2. SECTION VII.-Solve the equations: x+3 x-3 3x+7 5x+6 1. = 7 2. (x+2) (x-2) (x+8)=x (x−3) (a+16). SECTION VIII.-1. A square field contains 1 acre 2 roods 27 poles 234 square yards: find the length and breadth of an oblong field, one of whose sides is as much longer than the side of the square, as the other is shorter than it, and whose area contains 6,400 square yards. 2. A and B can do a piece of work in m days; A works n days alone, when B joins him, and both together finish the work in p days more: how long would either require to do it singly ? MENSURATION. SECTION IX.-1. Find the area in acres, roods and poles, of a triangular field, whose perimeter is 30 chains 32 links, the triangle being right-angled and isosceles. 2. The floor of a rectangular space, one of whose sides is twice as long as the other, contains 3,200 square feet, and is surrounded by a wall 22 inches thick: find the space occupied by the base of the wall. Candidates are not permitted to answer more than one question in each Section. The solution must in every instance be given at such length as to be intelligible to the Examiner, otherwise the answer will be considered of no value. SECTION 1.-1. Multiply £70,396 17s. 41d. by 378, and prove your result by division. 2. Divide 1,290 tons 12 cwt. 1 qr. 24 lb. 5 oz. by 73, and prove your result by multiplication. SECTION II.-What is the value of 2 tons 13 cwt. 3 qrs. 11 lb. of rice at 34d. per lb. ? and what would be the difference if the price were reduced d. per lb ? SECTION III.-Make a bill of the following articles, and deduct 2d. in the shilling for ready money:17 reams of paper at 94d. for 5 quires. 13 dozen copy books at 3d. each copy book. Arithmetic. Or, 67 yards of long cloth at 54d. per yard, 29 yards of silk at 3s. 7åd. per yard, and deduct 5 per cent. for ready money. 211 SECTION IV.-1. Find by Practice the value of 9 acres 1 rood 16 perches of land, at £1 1s. 8d. per acre. 2. Find by Practice the value of 37 lb 4 oz. 16 dwts. of silver at £3 4s. 91d. per lb. SECTION V.-1. If the railway fare for 113 miles be 18s. 1d., what ought the fare to be for 69 miles ? 2. A rate of 1d. in the £ is required to raise a sum of £525 12s. 6d. what is the rateable value of the town? SECTION VI.-If I give £1 10s. 11d. for 4 yards of velvet, what quantity could I purchase for £14 18. 10d. P SECTION VII.-1. Explain what you understand by multiplication by a fraction, and multiply 7 tons 4 cwt. 1 qr. 15 lb. by 23. gr 2. Add together,,, and, both as vulgar and decimal fractions, and show that the two results coincide. SECTION VIII.-1. What decimal is 3s. 7d. of 188. 23d. P Divide 299 by 13 and 3525 by 7110. 2. Find the value of 3275 of £10; multiply 3.275 by 12.8; and divide 0625 by 00005. SECTION IX.-1. A tradesman commenced business with a capital of £3,200; he increased his capital at the rate of 15 per cent. for 5 years, simple interest; what is its present amount ? 2. At what rate per cent., simple interest, will £2,700 amount to £3,219 15s. in 7 years? SCHOLARSHIP QUESTIONS, 1881. SCHOOL MANAGEMENT. Three hours allowed for this Paper with that on Music. Males and Females. Those who are or have been Pupil Teachers are not to answer more than one question in any Section. Candidates who have not been Pupil Teachers may answer any seven questions they think fit, except in Section I., from which only one subject may be selected for notes of a lesson. No Candidate is to answer more than seven questions. SECTION I.-Write full notes of a lesson on one of the following subjects: (1) Animals of the cat kind. (2) Cardinal Wolsey. (3) The river Mississippi. The manufacture of a cup, or a needle, or a boot. SECTION II.-1. Why should young teachers be restricted from the use of corporal punishment (a) for the sake of their scholars? (b) for their own sake? 2. What bad habits are produced by careless correction of exercises, and by want of attention to home lessons? 3. Point out some of the ways in which school discipline may be useful in producing habits of ready obedience, and name some characteristic features of good discipline. SECTION III.-1. Enumerate Froebel's seven gifts, and show the progressive nature of their lessons. |