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Q. 73. Explain how you would find the magnitude and line of action of

the force corresponding to the acceleration of the connecting rod of a steam engine for any specified position of the crank. You may assume that the necessary dimensions and speed of the

engine, and the kinetic elements of the rod are known. Question 73 might easily be answered from mere text-book information, yet the answers were unsatisfactory.

In spite of these unsatisfactory details, the Examiners are fairly well satisfied with the results of the examination. Even when a candidate gets only half marks for an answer he often shows evidence of individuality, and proves that he has been thinking things out for himself. The Examiners are confident that in many cases where the reasoning is not altogether good and the real point seems to be evaded, the candidate shows signs that he will in time see things more clearly.

Report on the Examination in Practical Mathematics.

The number of candidates in this subject is increasing rapidly, as the following table for three successive years will show :

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'This shows a total increase of 133 per cent. in two years, and the rate of increase is seen to be greater in the higher stages. It is satisfactory to be able to record that along with this increase in number there is no lowering of the average quality of the work. In fact, after making due allowance for the somewhat easier questions set this year, there is a distinct improvement noticed in the work of all the stages. There were many classes in the lower stages in which nearly all the candidates passed in the first class. There were some in which the work was almost wholly bad, but it is believed that these unprepared classes are fewer than formerly. Logarithms are now used with facility and with much more certainty than was the case a few years ago. There is still some looseness in writing down expressions, and especially in the use of the sign of equality “ =." Thus it is not uncommon to find such a statement as 200 (104))2 = 200 (0:0170 x 12) = 200 (0-2010).

200 X 1'600 = 320'0. The candidate as a rule knows quite well what he is doing, but when dealing with more complicated expressions such slackness leads to confusion and mistakes, and should never be permitted.

A persistent feature of the examination is that although candidates can use contracted methods, as shown in their answers to Questions 1, 21 and 41, they rarely employ these methods in their arithmetical computations when working any of the other questions of the paper.

With regard to plotting, scales are, as a rule, well selected and plainly figured. There is, however, a tendency in such questions as 5, 11, 12, 13 and 27, after having marked the points, to join them by straight lines or chords, instead of by a fair continuous curve, and to measure results from this figure.

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The significance of the slope of a curve is slowly being acquired, but, in Stage 1 especially, candidates have great difficulty in expressing themselves intelligently. The mere plotting of a curve bas little value apart from the lessons the curve teaches, and students should be induced to think more about their work, and not be content with mere mechanical graphing on squared paper.

STAGE 1.
Results : 1st Class, 930; 2nd Class, 595 ; Failed, 574 ; Total, 2,099.
Q. 1. The four parts (a), (6), (c), and (d) must all be answered to get full

marks:
(a) Compute by contracted methods, to four significant figures
only, and without using logarithms

3 214 x 0*7423 = 7.912.
(6) Write down the logarithms of 32170, 32-17, 0.3217,

0'003217,
(c) Compute, using logarithms

V 84'05 0:1357 - 1.163. (dl) Express £0 178. 9d. as the decimal of a pound. (a) Contracted methods well known. Mistakes as a rule only in the fourth figure. Decimal point almost always correctly placed.

(()). Few mistakes; these generally in the characteristics of the logarithms.

(c) The symbol of the square root instead of the index · gave some trouble in applying logarithms.

(1) Answer generally correct. The usual method was to divide the number of pence in 178. 9d. by 240. Very few used the florin as the basis of their calculation. Q. 2. The three parts (a), (b), and (c) must all be answered to get full

marks :-
(a) If A = P (1+

200, p = 4,
n = 12.
(6) On board a ship there were 1,312 men, 514 women, and 132

children. State these as percentages of the total

number of persons.
(c) When x and y are small we may take
1 + x

as being very nearly equal to 1 + x - y.

What is the error in this when x == 0·02 and y = 0·03 ? (a) Common mistakes were (1 + '04)12 12:48, 1 + Top 1.25 or 1.4,

200 (1'04)12 = 20812. One candidate multiplied the latter out in

full, obtaining an answer with thirty figures. (%) Fairly well done ; a few good graphical solutions. (c) Not well answered as a rule. Q. 3. The four parts (ru), (l), (c), and (d) must all be answered to get full

marks :-
(a) Write down algebraically :--The principal P multiplied
by 1 + twelve times.

100
(6) Divide 17-24 into two parts such that one quarter of the

first added to one third of the second make 5.06. (c) What are the factors of -2--10? (d) A wheel is 3.45 feet in diameter; it makes 1020 revolu

tions rolling along a road; what is the distance passed over ?

)" find A when P

100

1 + y

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(2) A common mistake was 12 P (1 +

(1+0) (6) Pretty well done, but some, after stating the equations in x and y,

could not work them out correctly. (c) The factors of x2 9 would evidently have been correctly given by

almost all, but the 10 proved a stumbling block. Some said,
“There are
no factors.

Others gave such
22

answers

as

- (2-), 10 (-1).

Common mistakes were (x — 3:3) (r + 3-3), (x — 5), (x -- 2). (d) Fairly well done. Some think app is the circumference of a circle others use 2nd, and a not uncommon answer was 3:45 x 1020. Q. 4. If y r— 34x + 2.73, calculate y when a has the values 1, 1-2, 1:4, 1:6, 18, 2, and 2.2. Plot these values of x and y and d.aw

What values of x cause y to be 0 ? Attempted by about 44 per cent. of the candidates, with very satisfactory results. Q. 5. x and t are the distance in miles and the time in hours of a train

from a railway station, Plot on squared paper. Describe why it is that the slope of the curve shows the speed; where approximately is the speed greatest and where is it least ?

a curve.

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About half the candidates selected this question. The curve was well plotted, and the places of greatest and least speed successfully indicated. Answers as to why the slope shows speed often amounted, when sifted, simply to the statement it is so. Q. 6. The point B is 4 miles North and 2 miles East of the point 1.

What is the distance from A to B and what angle does the line

AB make with the due East direction ? The favourite optional question. Graphical solutions were common, but in these cases the scale was often too small. Of those who worked by tables, the majority were content with “63°” or “ 64°” for the angle, or “between 63° and 64°," or "about 63}",” Few obtained the correct angle to within 0:1° by interpolation.

Q. 7. The horse-power of the engines of a ship being proportional to the cube of the speed ; if the horse-power is 2,000 at a speed of 10 knots, what is the power when the speed is 15 knots ?

Fairly often attempted. Correct answers were frequent, but many used simple proportion obtaining 3,000 horse-power. Q. 8. There are two maps, one to the scale of 2 inches to the mile, the

other to the scale of half an inch to the mile. The area of an estate on the first map is 1:46 square inches, what is the area of

this estate on the second map ? Pretty frequently attempted but with poor results. Answers like the following occurred 1'46 + ]

1:46 + 2 = 0.98 ;

6.92. 2 1:46 + 22

21.84.

Q. 9.

y

ax2 + 6x3 ; when x = 1, y is 4:3, and when x = 2, y is 30 ; find a and b.

What is y when x is 1:5? Seldom attempted, but the answers were good. Quite a number discovered and pointed out the “misprint" and corrected it, and those who stuck to the equation as written were very leniently dealt with. Q. 10. In the following table A is the area in square feet of the horizontal

section of a ship at the level of the surface of the water when the vertical draught of the ship is h feet. When the draught changes from 175 to 185 feet, what is the increased displacement of the vessel in cubic feet ?

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Not a favourite question, nor well done, most going wrong in finding the “increased displacement,” giving the difference in the areas at 17.5 and 18*5 feet, viz. 6870-6500 instead of } (6870 + 6500) x 1 or 6685 cubic feet. Q. 11. The speed of a ship in knots (nautical miles per hour) has been

noted at the following times :

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Plot on squared paper. What is the distance passed through

during the hour after 6 o'clock. Frequently attempted. The plotting was good and there were many correct answers to the first part. But in the latter part the mistake 14:11 - 13.50 instead of } (14:11+13:50), or 13.82 as read from the curve, was common. Q. 12. If

ри 479, find p when u is 3:25. Not very often attempted. Fairly well answered. Recurring mistakes : ри

; pu

px 1.0646u. Q. 13. The following corresponding values of x and y are given in the

table :

1-0616

1.0646

1.0646

(pu) 1.0046

S

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What is the probable value of when y is 8 ? A favourite question. Answers fairly good. Many took far too small a scale on one of the axes. A straight line was often taken between the points instead of a curve through them.

STAGE 2. Results : 1st Class, 187; 2nd Class, 554 ; Failed, 303 ; Total, 1,044. Q. 21. The four parts (a), (b), (c) and (d) must all be answered to get

full marks :(a) Compute by contracted methods to four significant figures only, and without using logarithms,

3:214 x 0°7423 = 7.912.

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(6) Using logarithms compute

(1-342 (-01731 = 0·0274)0-317.
(c) Explain why we multiply a logarithm by 3 when we wish

to find the cube of a number.

(d) Express £18 178. 3d. in pounds. (a) On the whole this was not well answered. A large number of candidates omitted the first “carrying " figure in multiplying: In many cases the last figure of the required quotient was incorrectly given, owing to this omission of the carrying figure.

(6) This was often wrong. The principal source of error was in the misplacing of the decimal point after multiplying the logarithm by the index 0:317.

(c) Some good answers were given to this, but the great bulk of the candidates are evidently accustomed to use this and similar rules without clearly understanding the theorems on which they rest.

(d) A number went wrong in this very easy question. As noticed in
Q.1, few used the florin as the base of their calculations.
Q. 22. The three parts (a), (b), and (c) must all be answered to get full

marks :

(a) If p ulei 479, find u when p is 120.
(6) y = a x2 + b x? When x is 1, y is 4:3, and when x is 2,

y is 30; find a and b. What is y when x is 1:5?
(c) Two men measure a rectangular box; one finds its length,

breadth and depth in inches to be 8.54, 5:17 and 3:19.
The other finds them to be 8.50, 5:12 and 3:16. Calculate
the volume in each case ; what is the mean of the two ?
What is the percentage difference of either from the

mean?
All three parts were usually done well In (e) a good many gave them-
selves needless trouble by working out the volumes by uncontracted
multiplication.
Q. 23. A body has moved through the distance & feet in the time t

seconds and it is known that 8 = = b t" when b is a constant.

Find the distance when t is 4. Find the distance when the time is 4 + o't. What is the average speed during the interval 8 t? As d t is imagined to be smaller and smaller, what does

the average speed become ? Comparatively few gave a really satisfactory answer to this question. A great many failed to grasp the notion that for average speed the distance described must be divided by the time occupied. A very considerable number seem to have a notion that in finding an average we must divide by two. Q. 24. The three parts (a), (b) and (c) must all be answered to get full

(a) If

marks :

αθ e

If a

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where e 2718.

0'3 and 0 2.85 and if
Y * 550, find x.
(6) When x and y are small we may take

1 + x

1 + y
as being very nearly equal to 1 + & - - y. What is the

error in this when x = 0:02 and y = 0'03 ?
(c) ABC is a triangle, the angle C is a right angle. The side

AC is 21:32 feet, the side BC is 12:56 feet, find the
angles A and B.

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