6. Find the greatest common measure of the following quantities (1) x3-3x2+4x-2 and x3-2x+4. (2) x-x-m and x+x 7. Solve the equations c-m+2. 8. Give an algebraical solution of the proposition Euclid II. 11, and interpret the two answers. 9. If ==, prove that each of these fractions is equal to α b px+qy+rz 10. Show how to insert n harmonic means between two given quantities a and b. An arithmetical progression and an harmonical progression have each the first term a and the last term l, and the same number of terms n; prove that the product of the (r+1)th term of one series and the (n-r)th of the other is independent of r. 11. Investigate the expression for (1+x)", n being a positive integer. Find the sum of the squares of the coefficients. 12. Define the characteristic of a logarithm, and from your definition show that the mantissa must always be positive. What is the characteristic of 05 to the base 2? If log10'02=2·30103 and log1030=1.47712, log10 (0020736)*, find 13. Express sin A in terms of tan A, and tan A in terms of sin A. Find the cosine, sine, and tangent of one-third of a right angle. 14. Prove geometrically that cos (A+B)=cos A cos B-sin A sin B when A+B<90°, and also when A and B are each >90°, and A+B<270°. 15. If a+ß=w, then cos 2a+cos 23-2 cos a cos ß cos w=sin 2w. 16. In any triangle prove that b cos C+c cos B-a and sin A sin B sin C 1 that where R is the radius of с 2R' the circumscribing circle. 17. Show how to solve a triangle, when two of the sides and the included angle are given. Ex.: A=63°, b=13 ft., c=11 ft.; find the number of degrees, minutes, and seconds in the angles B and C. Given L cotan 31° 30' L tan 7° 44′ L tan 7° 45' 10.21268. log 2=30103 18. Express the distances between the centres of the inscribed and escribed circles of a triangle in terms of the sides. If a new triangle be formed by joining the centres of the three escribed circles of a triangle ABC, prove that the distances of the centres of its escribed circles from the centre of its inscribed circle are B+C 8R sin 8R sin C+1, 8R sin 4 A+B where R is the radius of the circle circumscribing the triangle ABC. WOOLWICH PAPERS. LX. 1. Add together 1, 22%, and 21. 2. Subtract 917 from 103. 3. Multiply together 13, 31, 2, and 118. 4. Divide 127 by 34 451 5. Add together 123.478, 103, 12, 1·0001, and 437. 6. Subtract 83.0007 from 334. 7. Multiply 86.35 by 1.2203. 8. Divide 43555·952 by 324·56. 9. Express 12 oz. 8 drams as the decimal fraction of 2 cwt. 26 lbs. 10. Add together of 13, 1 of 6%, 24 of 2%, and 3 49 12. Multiply together 4 of 91, 131, and of 19. 13. Divide 47 by 19%. 14. Add together 13 of a ton, '001 of a cwt., 93 of a quarter, and 25 of a pound, and express the answer in pounds and the decimal fraction of a pound. 15. Multiply 101-037 by 0173. 16. Divide 00378 by 210. 17. Reduce 4 bushels 1 peck 1 gallon to the decimal fraction of 7 bushels. 18. Divide 26814 acres 1 rood 32 perches 0 yds. 1 ft. 3 in. by 72. 19. The construction of a railway 12 miles 6 furlongs 20 poles in length cost £328,000. What was the average expense per mile? 20. Find (by Practice) the cost of 12,480 things at £3 7s. 51d. each. 21. Find the simple interest on £3,121 6s. for 5 years at 33 per cent per annum. 22. If 45 men can excavate 2,790 tons of earth in 28 days of 10 hours, in how many days will 35 men working 12 hours per diem do twice that amount of work? 23. Find the present value of £2,315 5s. at 5 per cent. (compound interest) due in 3 years. 24. Extract the cube root of 30.664297; and the square root of 7 to four places of decimals. 25. Compute by means of the tables√52 × (?) ̄ ̄. 511.31 26. On the birth of an infant, 1,500 is invested so that it may accumulate at compound interest (3 per cent. per annum payable half-yearly) during the child's minority, calculate by logarithms the amount at the end of the 21 years. LXI. 1. Find the number of inches in the length 1 mile 1 furlong 3 poles. 2. Find the value of 17 quarters 1 bushel of corn at the price of 4s. 8d. the bushel. 3. What is the least weight which can be expressed either by a number of Troy pennyweights or by a number of Avoirdupois ounces? Give the answer in Troy weight. 4. A vessel holds 23 pints. How many times can it be filled from a cask of 56 gallons, and will there be any remainder? 5. Find the value of ⚫06812 of £2 Os. 14d. 6. A school of boys and girls contains 453 children, and the boys are 525252.... of the girls. How many boys are there? 7. If a metric system of area were adopted wherein 1 acre 1 rood 3 perches is represented by 5.12, express the unit of measurement in square yards and decimal parts of a square yard. 8. At what rate per cent. is the deduction made when 19s. 10d. is taken from an account of £39 15s. in consideration of immediate payment? 9. Prove that if all the logarithms in a table were multiplied by the same quantity, the results would be the logarithms of the same numbers as before, but to a different base. If, for instance, all the logarithms in the table supplied to you were doubled, what would the new base be to which they would continue to be logarithms of the same numbers? Give its value to four places of decimals. Could we rely on the results of calculation made by a system of logarithms where the value of the base itself cannot be exactly determined? 10. Compute by logarithms the fraction. five places of decimals. 3 0073288 05482637 to 11. What is the compound interest to the nearest penny on £83 14s. 7d. in 7 years at the rate of 3 per cent. per annum, interest being due at the end of each year? How much does this compound interest exceed the simple interest in the same time on this principal at the same rate per cent.? LXII. 1. Add together 13, 2%, and 31. 2. Subtract 914 from 113. 3. Multiply together 11, 14, 2, and 3}. 4. Divide 21 by 319. 5. Add together 11.234, 007, 8, 9-0008, and 7582. 6. Subtract 25.789 from 100.081. 7. Multiply 1.2345 by 9.807. |