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NOTE.— The Board of Education publish herein sections of the

Reports of the Examiners, which will be of service
to those engaged in teaching classes under the
Board.

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GROUP I.-PURE AND APPLIED MATHEMATICS.

BOARD OF EXAMINERS.

V.-Pure Mathematics Rev. J. F. Twisden, M.A., Chairman. VI.-Theoretical Mechanics A. R. Willis, M.A., D.Sc. XX.-Navigation

P. T. Wrigley, M.A. XXI.--Spherical and Nautical Major P. X. Macmahon, R.A., D.Sc., F.R.S. Astronomy

H. B. Goodwin, M.A., late R.N.

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Report on the Examinations in Pure Mathematics.

EVENING EXAMINATIONS.

STAGE 1.

Results : 1st Class, 515; 2nd Class, 1,245 ; Failed, 1,113; Total, 2,873.

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A. On the whole the work in ARITHMETIC was well done. The following points may be noticed :Q. 2. Assume that a metre equals 3.281 ft. ; find, as a decimal true to

the nearest second place, how many square metres there are in 10 square yards. The distance between two railway stations is 125 kilometres ; find the distance in miles and a decimal true

to the nearest third place. Work in approximate decimals should, as a rule, be carried a place or two further than what the answer demands.

Q. 3. How many square yards are there in an acre ?

A rectangular piece of ground has an area of one acre, and its length is twice its breadth ; find the length and breadth to the nearest tenth of a yard. Verify your result by multiplying the length by the breadth, and explain any discrepancy that you find. If a border 5 yards wide were trenched round the acre, find the fractional part of the acre which remains

untrenched. Was not taken so often as some of the other questions, but it came in for several good answers.

Q. 5. (a) A man buys 24 per cent. Consols at 863 ; find the rate per cent.

of the interest which accrues. (6) A man invests £1,280 in a 3 per cent. stock, and gets from it

a yearly income of £36 168. Od. ; find the price of a hundred

pounds of the stock. The price of the stock, in (b), was commonly given in a fractional or decimal form, e.g., £104.5, instead of £104 7s. Q. 6. Find at what time between 12 and 1 o'clock the hands of a clock

are exactly opposite each other.

At what times between 1 and 2 o'clock will the hands be at

right angles to each other? A few students answered this question correctly. 9897. 2625—Wt, 23129. 12/07. Wy. & S. 6175r,

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B. The work in GEOMETRY was fairly well done. The questions most often attempted were Q. 7 and Q. 10. Q. 8. ABC and DEF are two triangles, and it is given that the sides

AB, AC are equal to the sides DE, DF respectively; also it is given that the angle A BC equals the angle DE Y Under what circumstances cannot we draw the conclusion that the triangles are equal in all respects ?

Draw a triangle which has two sides 2 in. and 14 in. long, and the angle opposite to the shorter side equal to a third of a right angle. From the same data draw a second triangle not equal in all respects to the former, and show from your diagrams that the third side of one triangle is more than twice as

long as the third side of the other triangle. The answers to this question suggest the following remark : There are two triangles A BC and abc ; the sides AC, CB of the one equal the sides ac, cb, of the other, each to each ; also the angle A equals the angle at a. With these data, most of the students think that they are justified in drawing the conclusion that the triangles are equal in all respects. This calls for the attention of the teachers. The same weakness was shown in the answers to Q. 22. Q. 9. Show that, of all straight lines that can be drawn to a given

straight line from a given point outside it, the perpendicular is the shortest.

Show that the locus of a point which is equidistant from two fixed points is the perpendicular bisector of the straight line

joining the two fixed points. The locus in the second part was seldom completely made out. Q. 10. Define parallel straight lines, parallelogram, rhombus, square.

Show that parallelograms on the same base and between the same parallels have equal areas.

Two four-sided figures are on the same base, and between the same parallels ; also the sides opposite to the base are equal ;

show that the areas are equal. The definitions were often unsatisfactory, e.g., it was seldom stated that two parallel lines must be in one plane.

In the deduction it was almost invariably assumed that the sides opposite to the base are not merely equal, but are equal to the base. Q. 11. AB, CD, EF are three given parallel straight lines such that a

straight line, cutting them in P, Q, R respectively, makes PQ equal to QR. If any other straight line cuts them in X, Y, Ž respectively, show that XY is equal to YZ.

ABC is a triangle, and through D the middle point of AB a parallel is drawn to BC, cutting AC at E ; show that DE is half

as long as BC. Was often attempted, but the answers were seldom well given. Q. 12. Show how to construct a rhombus which has one diagonal of given

length, and whose area equals that of a given square. As might have been expected, this question was seldom attempted, and most of the attempts failed. There were, however, a few good answers by the best students.

C. In ALGEBRA, the questions in elementary rules and in simplifications were often well answered. The answers to the other questions often failed. Q. 13. (a) Find the sum of (x2 – 2x + 3) (x2 + 3x – 5) and (x? – 3.0 — 1) (x + 3)

2 (6) Find the value of the sum,

when x equals The result of the substitution, in (), was often wrong.

3.

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