MISCELLANEOUS EXAMPLES VI. 336
6. Solve 2(x + a) x+b + 3(x + b) x + a =5. 7. The sum of a certain number of terms of an A. P. is 45, and the first and last of these terms are 1 and 17 respectively. Find the number of terms and the common difference of the series. 8. Solve (i.) 2x-3 x-2 -1 6 + 2x-1 1-x = 0. (ii.) 12x-5 + 3x-1 = 27x-2. 9. Find the value of the seventh term in the expansion of (a + x)^n when a = 12, x= 13, n = 9. 10. A man starting from A at 11 o'clock passed the fourth mile-stone at 11.30 and met another man (who started from B at 12) at 12.48; the second man passed the fourth milestone from A at 1.40: find the distance between A and B, and the second man's rate. 11. Show that x^3 + 13 a^2x > 5ax^2 + 9a^3, if x>a. 13. Extract the cube root of 44x^3 + 63x^2 + x^6 + 27 + 6x^5 + 21 x^4 + 54 x. 13. Solve (i.) x - y = 3, x^2 + xy + y^2 = 93. (ii.) 2x^2 - 9xy + 9y^2 = 5, 4x2 - 10xy + 11y^2 = 35. 14. Find a mean proportional between v 13 - 8 - 3 (a^0+b^0)^-2 ab and the reciprocal of 4- -3 16 a^3b. 15. Two vessels, one of which sails 2 miles an hour faster than the other, start together upon voyages of 1680 and 1152 miles respectively; the slower vessel reaches its destination one day before the other: how many miles per hour did the faster vessel sail? 16. Solve (i.) x^6 = 8 + 7x^3. (ii.) x^2n + b^2 = c^2 - 2b^n 17. Two numbers are in the ratio 2:7; the numbers obtained by adding 6 to each of the given numbers are in the duplicate ratio of 2:3. Find the numbers. 18. Solve (i.) 2bx^2 - 2b = 4x + b^2x. (ii.) x+4 2x+3 + 3x+10 2x = 2x+3 x-1. (iii.) x+3 + x+8 = 4x + 21. (iv.) x^2 + xy + y = 137, y^2 + xy + x = 205.
