4. Divide 14 into two parts such that the quotient of the greater divided by the less shall be to the quotient of the less divided by the greater as 16 to 9. 5. Solve the equation \Jx + 8 + + 3 = ^x. 6. The sum of two numbers is 17; and twice the square of the first, increased by 30, is equal to 3 times the square of the second. Find the numbers. 7. Explain the method of inserting a given number of arithmetical means between two given terms. 8. Find the sum of an infinite number of terms of the series 4, ||, &c. 9. What is the seventh term in the expansion of (a — x)10? 10. A and B have the same number of horses. A can make up twice as many teams, taking 3 horses at a time, as B can make up, taking 2 at a time. Find the number of horses. IL 1. Find the least common multiple of a8 — x, a? — 1, and a? + L Obtain the result, if possible, by factoring. 2. Simplify (a* X a*)&. 3. Add together ^40, fL35, ^625. 4. Find both roots of the equation 2x + \]5x + 10 = 11. 5. What two numbers are those whose difference is to the less as 4 to 3, and whose product multiplied by the less is 504? 6. What is the 4th term in the expansion of (c — ^ 7. The difference of two numbers is 3, and the difference of their cubes is 63. What are the numbers? 8. Obtain the formula for the sum of the terms of an Arithmetical Progression. 2 2 9. Find the sum of the series 2, 10. How many arrangements can be made of the letters in the word m. 1. Reduce the following expression to its simplest form: a2— [2ab— {be— (a+b — c) (a — (b — 2. State and prove the rule for the sign of a power and of a root. How do imaginary quantities arise? 2 3. What is denoted by a0? by a-s? by a?? x — 1 4. Eeduce 1 to its simplest form. 1 1_ *+ 1 5. Solve the equation axt-{-2hx-\-b = 0; and prove that the product of the roots = -. 6. There are seven numbers in Arithmetical Progression such that the sum of the 1st and 5th is 16, and the product of the 4th and 7th is 160. Find the numbers. (This question admits two solutions. Both are required.) 7. Multiply 1 — 5 \/7 by — 2 — 3 y7. Divide a yc Y d y cd^c 8. Find the sixth term of — ^ b \/a ^. 9. Find the greatest common divisor and the least common multiple of 6 a£ — 6a? — 72x and 4x* — 16a? — 84 a?. IV. 1. Extract the cube root of 64 — 96 a; — as6 + 40 x3 — 6 a* 2. Solve the equation _ \ — —- — 3 J = 0. 3. Multiply together 2 + 3 sj — 1, 3 — 2 \J — 1, and 12 — 5\T^"1. 4. Three times the product of two numbers, diminished by the square of the first, equals the square of the second plus one. Also the first number is greater by one than twice the second. Find the numbers. (Give both solutions.) 5. Solve the equation ax2 -f- bx -f- c = 0, and state what relative values of a, b, and c will make the roots equal, and what values will make them imaginary. 6. In an Arithmetical Progression, given the number of terms, the common difference, and the sum of the terms; — obtain formulas for the first term and the last. 7. In a Geometrical Progression the first term is 2\, and the fifth term is Find the sum of the series to infinity. 9. How many whole numbers of four figures each can you form, each number either beginning or ending with 5, and no number containing the same figure twice? V. 1. What are eggs a dozen when two more in a shilling's worth lowers the price one penny per dozen? 2. Solve the equations a? — y3 = 63, ofiy — ocy* = 12. 3. Multiply | + fVJ ^ J — 7V£. Divide , ,v .. ,„ by Y 4. Solve the equation ^(21 + Ax) + \fcx + 3) — \/(x+ 8) «= 0. 5. From the letters 6. Prove that the sum of any number of antecedents of a continued proportion is to the sum of the corresponding consequents as any one antecedent is to its consequent. 7. Find the greatest common divisor of 27a6 + 3a3 — lOx2 and 162a6 — 32a. 8. For what values of a, b, and c is ^ positive, and , o — c for what values negative? For what values is it 0? oo? indeterminate? 9. Find r and n in an arithmetical progression when a, I, and S are known. VI. 1. A certain sum of money at simple interest will amount to a dollars in m months, and to b dollars in n months. Find the principal and the rate of interest. Find the answers when a = 1837.50, b = 1890.00, m = 10, n = 16. 2. There are three numbers in geometric progression of which the continued product is 64 and the sum of their cubes 584. Find the numbers. 3. Simplify -1 (a6 — 66). a 4. Find the greatest common divisor of 24x7 + 6*s — 30x and 4z10 — 4x*. 5. Find the square root of 25xe — 20x5y — 6^2 + 34zy _ lla!y _ Qxyz _j_ 9yB 6. Solve the equation 2\Jx — sl^x + \/7x -f- 2 = 1. 7. To find two numbers when their sum and product are given. In what case are the answers imaginary? How must a given number be divided in order that the product of its parts shall be as great as possible? 8. State and prove the Rule of Three. |