Q. 9. when x = y = ax2 + 6x3 ; 1, y is 43, 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 4 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? Not a favourite question, nor well done, most going wrong in finding the "increased displacement," giving the difference in the areas at 175 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 : 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 4 (14′11+13′50), or 13·82 as read from the curve, was Not very often attempted. Fairly well answered. Recurring mistakes:- (pu) 1.0648 ; pu 1.0046 table Q. 13. The following corresponding values of x and y are given in the What is the probable value of x 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 × 0.7423 ÷ 7′912. (b) Using logarithms compute (1.342 × 001731÷00274)0-317. (c) Explain why we multiply a logarithm by 3 when we wish to find the cube of a number. (d) Express £18 17s. 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. (b) 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 ul 1.0646 (b) y = a x2 + bx. When x is 1, y is 43, 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 (c) a good many gave themselves needless trouble by working out the volumes by uncontracted multiplication. Q. 23. A body has moved through the distance 8 feet in the time t seconds and it is known that s = bt when b is a constant. Find the distance when t is 4. Find the distance when the time is 4+ t. What is the average speed during the interval St As St 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 marks: -y. What is the 0'03 ? as being very nearly equal to 1 + ∞ — (c) ABC is a triangle, the angle C is a right angle. The side (a) Well answered by a very considerable number of candidates. (b) A good many candidates used logarithms, and a few the slide rule methods, not well adapted for finding the difference of two nearly equal quantities. In connection with this answer, a point may be noticed, viz., looseness 1.02 1'03 in stating approximations. Here we have =099029. . . This was much oftener stated to be 0.9902 instead of 0.9903, the candidate stopping at the 2 in dividing out without considering the value of the succeeding figure. (c) A point to note was the large number of candidates who were apparently unable to find an angle directly from the tangent. Many first found the hypothenuse of the given triangle, and were then able to find the angle from the sine. Q. 25. A man is 100 feet above the earth which is assumed to be a sphere of 8,000 miles diameter; what is his distance in miles from the furthest point he can see on the surface? Do not give more than three figures in the answer. Not a favourite question. Generally the formula A=√(CB) was employed. A fair number expressed this in the form A= √(C+B) (C-B) and evaluated by logs., but the majority laboriously calculated the differences of the two squares and extracted the square root of the result by the ordinary arithmetical process. The point to which attention has already been drawn in Q. 24 (6) was often conspicuous here, e.g., 100 feet 00189.... miles. This was often stated to be approximately 0.018 miles. Of course if only two significant figures were to be retained, 0019 should have been taken. = 10 528 miles = x2 - 42 x Q. 26. If y 2'93 calculate y for various values of x and plot on squared paper. What values of x cause y to be 0? A favourite question and fairly well answered, but the diagram was often drawn on too small a scale. This generally arose in part from plotting all the values of y worked out by the candidate-the higher ones being obviously of no use for the determination of the points in question. But in nearly all cases the scale for a was too small. Q. 27. x and t are the distance in miles and the time in hours of a train from a railway terminus. Plot on squared paper. Describe why it is that the slope of the curve shows the speed. What is the greatest speed in this case and where approximately does it occur? What is the average speed during the whole time of observation? 0 1.5 6.0 14.0 19.0 21.0 21.5 21.8 23.0 24.7 26.8 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Often taken and, except as regards one point, usually answered well. The point which was missed was the relation between the speed and the slope of the curve. Only a few candidates showed evidence that they clearly understood how either of these at any point on the curve is to be estimated. There is an objectionable practice on the part of some candidates, previously mentioned, of joining plotted points of a curve by ruled straight lines. This is particularly inappropriate here. ρ Q. 28. A disc rotating with angular velocity a, its density being 8, has an outer radius ro 50. There is a hole in the middle whose radius r is 10. Then at any place whose distance from the centre is r, there is a hoop tensile stress Q where Taking a === 122.5, and arranging the formula for systematic calculation, find for the values of 7, 10, 15, 20, 30, 40, and 50. Plot Q and on squared paper. Only a few candidates took this question. Good answers were somewhat scarce. Q. 29. In the following table 4 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? Not often answered. The remarks under Q. 11 apply here. Q. 30. In the following table x and y are the co-ordinates of points in a curve, which you need not draw. Tabulate the values of the average slope of the curve in each interval. Also tabulate the area between the two ordinates in each interval. You had better write in columns rather than in rows. Well answered by a fair number of candidates. Q. 31. x being distance in feet across a river measuring from one side and y the depth of water in feet, the following measurements were made : Find the area of the cross section. If the average speed of the water normal to the section is 3'2 feet per second, what is the quantity flowing in cubic feet per second. A few cases occurred of the inaccurate method of estimating the mean height as the mean of equi-distant ordinates including the first and last. A certain number of candidates went wrong by taking an even number of ordinates in attempting to use Simpson's rule. Q. 32. In a price list I find the following prices of a certain type of steam electric generator of different powers. According to what rule has this price list been made up? What is the list price of a generator of 400 kilowatts ? The price was usually correctly estimated, but only a moderate number made out the rule. STAGE 3. Results 1st Class, 51; 2nd Class, 104; Failed, 190; Total, 345. : Q. 41. 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 × 0.74237912. (b) Using logarithms, compute, (1.342 × 0·01731÷0'00274)−0·317. (c) Explain why we multiply the logarithm of a by 35 when we wish to find a35. Start your explanation from the fact that a3 means a xa xa, (d) Write down the values of sin 203°, cos 140°, tan 278°, (0'4226) The parts (a) and (d) were well answered. In (b) candidates often went wrong in subtracting the negative characteristic. The answers to (c) were not sufficiently complete, and were generally only valid for positive integral indices. Q. 42. Define the scalar and vector product of two vectors. Give an illustration of each. Some schools were totally unacquainted with the meaning to be attached to the products of vectors, and their crude guesses often served to display their ignorance of vectors in general. On the other hand, there were quite a large proportion of schools in which the answers were full and satisfactory. One candidate gave a good example of a vector product by explaining the precession of a gyroscope under the action of a couple. The favourite illustration, however, was the force action of an electric current which traverses a magnetic field. يع y Q. 43. The following values of y and x being given tabulate dy/dx and òA in each interval, dA being the area in the interval between two ordinates. Tabulate the values of A if A=0 when x=3. 3 4 5 6 7 8 9 10 11 12 13 1.75 10:45 19:08 27.56 35.84 43.84 51.50 58.78 65 61 71.93 77·71 Well answered, a large proportion of candidates obtaining full or nearly full marks. Let the curve rotate about the axis of x, forming a surface of revolution. Find the volume of the slice between the sections at x and x+dx. What is the volume between the two sections at x=2 and x=5? The answers were satisfactory and the question was a favourite one. A few plotted a curve and adopted an approximate method, but the great majority proceeded by integration. There were some cases in which candidates went wrong only in the very last step, viz. the evaluation of 55·75 — 25·75, the mistake being to take log (55.75 25.75) as equivalent to 5'75 log (5 — 2), was made. |