APPENDIX I. The Questions proposed to the Battersea Masters in July, 1847; the Papers of the first Government Examination, which was undertaken with a view of certificating Masters. ARITHMETIC. [Only one question in each section is to be answered.]* The method best suited for explanation to learners is to be preferred, and all the work to appear. Section I. 1. If 670 articles cost £67 17s. 10d., what is the price of each? 2. If 2 cwt. 3 qrs. of sugar cost £7 15s., what is the least price per pound, in current coin, at which it must be sold in order to gain 7 per cent. ? 3. If two equally good workmen, and three boys, who also work equally well, by labouring 7 hours a-day perform in 7 days, what the two men alone could do in 10 days by working 81 hours a-day, what ratio does the work done by a boy bear to that done in the same time by a man? * This request was placed after the heading of each of the following papers, except the last. Section II. 1. What part of a guinea is equivalent to ths of 6s. 8d.? 2. Reduce 2 d. to the decimal of a pound, and divide the result by 7500. What is meant by incommensurables ? 3. Find, by cross-multiplication (or duodecimals), the area of a board, whose length is 6 ft. 11 in., and breadth 1 ft. 6 in. State clearly what the answer is. Explain the steps of the process, and the interpretation of the result. Do you see any reason why, in representing quanti ties by means of numbers, incommensurables should occur? Section III. 1. Explain the reason of each process employed in multiplying 607 by 404. 2. Show that a fraction may be regarded as a quantity resulting from the division of the numerator by the denominator; and also that, if the numerator of a fraction be multiplied by any number, the result is equivalent to that obtained by dividing the deno minator of the same fraction by the same number. 3. 'Explain the reason of the rules for extracting the square root of any number. MENSURATION. SECTION I. 1. If the sides of a rectangle be 25 feet, and 13 feet 3 inches, what is the side of a square, whose area is the same as that of the rectangle ? 2. What is the area of a right-angled triangle, whose hypotenuse is 84} yards, and one of the remaining sides 304 yards ? 3. One angle of a four-sided field is a right angle, and the sides that include the right angle are 10:47 chains and 14:6 chains respectively; the remaining sides are 8:57 chains and 13•64 chains respectively. Required the area of the field. : SECTION II. 1. The circumference of a circle is 47 yards and 2 feet 5 inches. What is its area? 2. The length of the slant side of a cone is 15 feet 7 inches, and the diameter of its base 151 inches. Required its solid content. 3. A round hole, 14} yards in diameter at the bottom, is dug in a level field; its sides have a regular slope, whose breadth is 5 yards 1 foot; and the diameter of the upper circular edge is 17 yards 1 foot 9 inches. What quantity of earth has been excavated ? SECTION III. 1. Describe Gunter's chain, point out its peculiar ad vantages, and explain how it is used in the field. 2. What instruments are required, and what observa tions to be made, in order to determine the difference of level between two distant parts of a town? ALGEBRA. SECTION I. 1. Show that if b-c be subtracted from a, the result is a - b + c. 2. Prove the rules for determining the sign of the product, when two quantities are multiplied together; and explain the meaning of the expressions ao, a — m. SECTION II. 1. Multiply 2 a 3 b by 2 a + 3 b, and 9 x — 6 x 1 + 4 by 3 2. Also calculate the value of 2 a 3b + v a2 when a = 6 and b 3. 2. Divide 4 a® aba 12 b3 by 2 a 3 b, and 3 9 (2 a 36) 16 2 a + 3b . 1 3. Reduce and (–a) x + 1 2 i' 4 a® - 12 a 26.+ 17 aba 12 63 -4. Reduce to its and 2 to its -b v -1 a + b ✓ 1 simplest form. mt n Show also that am X an = a for all values of m and n. 5 x a a SECTION III. 3 2 X 1. Find the value of x in the equation. 2 (2x - 5) 2 - 6 + 3 4 6 Find also two numbers, whose sum is 100 and their x + y difference 36. 2. At what time, between 10 and 11 o'clock, will the hour and minute hands of a watch first include between them a portion of the circle, on which the minutes are marked, just equal to 6 of the minute divisions ? Solve the equations (1) ax + Vax-bx +c=f; (2) acxa + bcx = abx + 62. 3. Solve the equations = 29 ✓ x + y = 75 There are 20 boys in a class, each of whom is to have a copy of the same book. Some of the books are to be bound, and the rest unbound; and the binding of each volume will cost 1s. 6d. It is determined to spend £1 16s. on each set. Of how many books will the bound set consist, and what will be the cost of a bound volume ? SECTION IV. a 1. Show that if a : 6:: 0 :d, then ad = bc; and illustrate the result by a geometrical example. 2. If a: 6 c:d::e:f; then : 6 a tc:6+,d, and latcte:b.+ d + f 3. Explain the meaning of the terms “ Arithmetical mean,” and “ Geometrical mean ;" Find the Arithmetical and Geometrical means between a and b, and show that the latter can never be greater than the former. Illustrate the results geome trically. 4. If A a B when C is given ; and A & C when B is given : then, when B and C both vary, A & B x C. Illustrate the proposition by reference to a body moving uniformly with differing velocities and during different times. |