Ex. 67. Find, by logarithms, 1. The square Root of 3.621409. 2. The cube Root of 3852. 3. The cube Rt. of 199586.251. 5. The cube Rt. of 00052653. 7. The 5th Rt. of '0856329. 9. The 19th Rt. of 00123456. 11. The 365th Rt. of 1·045. 13. The 8th Rt. of '08. Ex. 68. Find a fourth proportional1. To 8352, 3'69, 30'57. 197 64 3. TO, 상품, '099. 4. The 5th Rt. of 24871.53. 2. To 357 109, 5000·8, ·031. 5. To (*00058309)§, (•2839)3, (·018÷25)†. Find a third proportional 6. To 00709, 1508. 8. To (948), ('00052653). Find a mean proportional 9. To ('03)3, ('529807)3. 11. To (387-908), ('0187). Ex. 69. Find the value 7. To 5'241, 9'5308. 10. To (·01)3, (·20)*. Ex. 70. 1. Given log 2='301030, log 3='477121, log 7='845098; find the logs of 6, 15, 5°4, 17.5, 875 and 6860. 2. Given log 18=1255272, log 25=1*397940; find the logs of 2, 3, 16, 450, 075 and 375. 3. Find log 256 to the base 2 √2. 1. Find the amount of £660 at 5 per cent. per annum, compound interest, payable quarterly for 6 years. 2. What is the amount of £2639 16s. 31d. in 5 years, at 4 per cent. per annum, compound interest, payable monthly? Ex. 72. 3. What sum will amount to £1000 in 10 years at 5 per cent. per annum, compound interest? 4. What sum will amount to £839 78. 14d. in 15 years at 31 per cent. per annum, compound interest, payable quarterly? 5. At what rate per cent. will £300 amount to £500 in 4 years, compound interest? 6. At what rate per cent. will a given sum double itself in 6 years, compound interest? 7. In how many years will £600 amount to £6000 at 5 per cent., compound interest? 8. In what time will a given sum treble itself, at 3 per cent. per annum, compound interest, payable half-yearly? 9. In how many years will £2653 78. 6d. invested at 3 per cent. per annum, compound interest, payable quarterly, amount to £3327 18s. 1104d.? 10. What is the discount on £100 due 3 years hence, at 4 per cent. per annum, compound interest? 11. A person puts out £25 at 4 per cent. per annum, compound interest, and adds to his capital at the end of every year a sum equal to the third part of the interest for that year; find the amount at the end of 20 years. 12. A sum of £3500 is left for three children, A, B and C, in such a manner that at the end of 7, 9, and 12 years, when they respectively will come of age, they are to receive equal sums; find the present values of each share at 4 per cent. per annum, compound interest. 13. A debt of £500 accumulating at 4 per cent. per annum, compound interest, is discharged in n years by annual payments of £41 138. 4d.; find the value of n. 14. A banker borrows money at 3 per cent. per annum, and pays the interest at the end of the year; he lends it out at the rate of 5 per cent. per annum, but receives the interest quarterly, and by this means gains £200 a-year; how much does he borrow? 15. A person spends in the 1st year n times the interest of his property; in the 2nd year 2n times that of the remainder; in the 3rd year 3n times that at the end of the 2nd year; and so on: at the end of 2t years he has nothing left; show that in the tth year he spends as much as he has left at the end of that year. Ex. 72. 16. What is the present worth of an annuity of £425 at 5 per cent. per annum, payable quarterly, for 12 years, at compound interest? 17. A Freehold Estate yielding £330 a-year is sold for £6000; required the rate of interest allowed to the purchaser. 18. If a lease of 65 years be purchased for £250, what rent ought to be received that the purchaser may make 7 per cent. per annum on his money? 19. A person purchases the reversion of an estate after 12 years for £1000; what rent ought he to receive that he may realise 6 per cent. per annum on his money ? 20. The reversion of an estate in fee simple producing £85 a-year is made over for the discharge of a debt of £946 128. 73d.; how soon ought the creditor to take possession, if he be allowed 5 per cent. per annum interest for his debt? 21. Find the present value of an annuity of £25, to commence in 8 years, and then to continue for 15 years, at 3 per cent. compound interest. 22. An annuity of £50 is to commence at the end of 12 years and to continue for 25 years; find the equivalent annuity to commence immediately and to continue 25 years, at 34 per cent. per annum in both cases. 23. An annuity of £50 which is to continue for 36 years is left equally between A and B; A receives the whole for the first 12 years, and B the whole for the remainder of the time: what is the present worth of the annuity, and the rate of compound interest per annum? 9. What is the 4th term of the equation whose roots are -2, I, I, 3, 4? 10. Find the middle term of the equation whose roots are 5, 3, I, −1, −2, −4. 11. Determine which of the Nos. 7, 6, 5, 4, 2, are roots of the equation 24-19x3 +128x2 −356x+336=0. 12. Investigate the 1st, 2nd, &c. derived polynomials of x3 — 10x1+29x3 — 10x2−62x+60; and x3—px3—qx2 + s. Ex. 2. Find the equation containing the other roots of— 1. xa—19x3+132x2 - 302x+56=0, one root being 4. 2. x2-16x3+86x2-176x+105=0, two rts. being 1, 5. 3. x5-2x-67x3 +200x2+588x=1440, two rts. being 2, 6. 4. +33x3+148x +240=x5+2124+40x2, two rts. being 3, -1. 5. x9+x3 —9x7+ 3x6 −8x5 +8x+− 3x3 +9x2 —x=1, one rt. being 1. Ex. 3. Find all the roots 1. Of x3-112+37x-35=0, one root being 3+√2. 2. Of xa— 3x2 — 42x-40=0, one rt. being —(3+ √−31). 3. Of 5-10x+29x3-10x2-62x+60=0, two rts. being 3, V2. 4. Of x3-7x2+16x-12=0, two rts. being equal. 5. Of x3-5x+8x-4=0, two rts. being equal. 6. Of x3-x2-8x+12=0, two rts. being equal. 7. Of x2-12x+3=0, two rts. being equal. 8. Of x++13x3 +33x2+31x+10=0, three rts. being equal. |