MISCELLANEOUS EXAMPLES VI. 337
men in each outer side of the square as there were in the depth of the column; how many men were there at first in the regiment ? 32. Solve (i.) 2x^2 + xy + y^2 = 37, 8x^2 + 4xy + y^2 = 73. (ii.) 27x^3 + y^3 = 152, 3x^2y + xy^2 = 40. 33. Simplify 8^{4}{3} + 3(2 x 4^-5) - 7 2 4^{3 7} - (32)^{- 3 5} 34. A man bought a field the length of which was to its breadth as 8 to 5. The number of dollars that he paid for 1 acre was equal to the number of rods in the length of the field; and 13 times the number of rods round the field equalled the number of dollars that it cost. Find the length and breadth of the field. 35. Solve (i.) x^2 + xy + 3y^2 = 14 + 2 2, 2x^2 + xy + 5y^2 = 24 + 2 2. (ii.) 2x + 3y = 10, 5x^2 + x + y = 4 3 4. 36. Find two numbers whose sum added to their product is 34, and the sum of whose squares diminished by their sum is 42. 37. Find the sixth term in the expansion of each of the following expressions: (i.) (a + 3b^-2)^7. (ii.) 2a-b^{1 2} c^{-2}. (iii.) (x 2 - y 3)^9 38. A varies directly as B and inversely as C; A = 2 3 when B = 3 7 and C = 9 14: find B when A = 48 and C = 75. 39. Solve (i.) x+12 + x+12 = 6. (ii.) x^2 - y xy = 9, y^2 + x xy = 18. 40. Form an equation whose roots shall be the arithmetic and harmonic means between the roots of x^2 - px + q = 0.
