III. Arithmetic. 1. Multiply 127. 178. 81⁄2d. by 137. Divide 426 tons 8 cwt. I qr. by 84. 2. Find the G.C.M. of 615, 861, 1435; and the L.C.M. of 84, 126, 294, 630. 3. Find by Practice or otherwise (1) the cost of 1501 articles at 17. 16s. 91⁄2d. each ; (2) the value of 37 cwt. 2 qrs. 7 lbs. at 37. 78. 10d. per cwt. 5 321 7 6. Express as decimal fractions 3, ; and as vulgar fractions in their lowest terms 0512, 0409. 7. Reduce 3 acres 3 roods 3 perches to the decimal of 100 acres; and find the number of minutes and seconds in 06375 of an hour. 8. Extract the square root of (1) 1400; (2) 7.513081; (3) .0081. 9. Find the cost of carpeting a room 15 ft. 6 in. by 12 ft. 9 in., the price of the carpet being 7s. 8d. per yard, and the width of the carpet 24 inches. 10. The yearly rent of a house being 130l., what should be the price that the purchaser may get 6 per cent. on his investment ? 11. Find the compound interest of 6257. in 3 years at 4 per cent. 12. A person holding 750%. of a 3-per cent. stock and 650l. of a 31-per cent. stock sells the former at 96 and the latter at 971, investing the proceeds of the sale in a 5-per cent. stock at 125. Find his income before and after the transfer. IV. Euclid. [N.B. At least two propositions from the Second Book are expected.] 1. Define-figure, diameter of a circle, parallelogram, plane superficies. Write out Euclid's three postulates. 2. If two triangles have two angles of the one equal to two angles of the other, each to each, and the sides adjacent to the equal angles also equal, then shall the other sides be equal, each to each; and also the third angle of the one equal to the third angle of the other. 3. If a straight line be divided into any two parts the square on the whole line is equal to the squares on the two parts, together with twice the rectangle contained by the parts. 4. Draw a straight line perpendicular to a given straight line of unlimited length from a given point without it. 5. Describe a square equal to a given rectilineal figure. 6. Parallelograms upon the same base and between the same parallels are equal to one another. 7. Divide a straight line into two parts, so that the rectangle contained by the whole and one part may be equal to the square on the other part. 8. The three interior angles of every triangle are together equal to two right angles. Prove this, together with one of the two corollaries. 9. If a straight line be bisected and produced to any point, the rectangle contained by the whole line thus pro duced and the part of it produced, together with the square on half the line bisected, is equal to the square on the straight line which is made up of the half and the part produced. 10. Describe a parallelogram that shall be equal to a given triangle and have one of its angles equal to a given rectilineal angle. V. Algebra. 1. State the rules for (1) multiplying, (2) dividing one power of a number by another power of the same number. 2. Add together 3x-5y+4%, 37-5x-4%, 22-3y+8x. From a (4a+3)2 take (a+1)(4a+1)2. 3. Multiply m2+2mn+2n2 by m2 — 2 mn+2n2. 9. Solve the equations : (1) }(x−1)+}(x − 3) = (x − 5) + ž (x − 7) ; (4) 5(x-1)= y; 3(y−2) = x. 10. Solve one or more of the following problems: (1) A lecture is attended by a certain number of undergraduates, there being three times as many commoners as scholars. One scholar resigns his scholarship, and there are then four times as many commoners as scholars. Find the original number of scholars. (2) The denominator of a vulgar fraction exceeds the numerator by 2. When 3 is added to the numerator and 3 to the denominator the value of the fraction is . Find the original fraction. (3) A goods train, running at the rate of 15 miles an hour, starts from London for Reading at the same time that a passenger train, running at the rate of 39 miles an hour, leaves Reading for London. Find how far from London the trains will pass each other, the distance between London and Reading being 36 miles. |