7. How far must a boy run in a potato race if there are n potatoes in a straight line at a distance d feet apart, the first being at a distance a feet from the basket? ~COLUMBIA UNIVERSITY~ ELEMENTARY ALGEBRA COMPLETE TIME: THREE HOURS Six questions are required; two from Group _A_, two from Group _B_, and both questions of Group _C_. No extra credit will be given for more than six questions. _Group A_ 1. (_a_) Resolve the following into their prime factors: (1) (x^2 - y^2)^2 - y^4. (2) 10x^2 - 7x - 6. (_b_) Find the H. C. F. and the L. C. M. of x^3 - 3x^2 + x - 3, x^3 - 3x^2 - x + 3. 2. (_a_) Simplify [x/y + y/x - 2]/[1/x + 1/y] + [x/y + y/x + 2]/[1/x - 1/y]. (_b_) If x : y = (x - z)^2 : (y - z)^2, prove that z is a mean proportional between x and y. 3. A crew can row 10 miles in 50 minutes downstream, and 12 miles in an hour and a half upstream. Find the rate of the current and of the crew in still water. _Group B_ 4. (_a_) Determine the values of k so that the equation (2 + k)x^2 + 2kx + 1 = 0 shall have equal roots. (_b_) Solve the equations x^2 - xy + y^2 = 7, 2x - 3y = 0. (_c_) Plot the following two equations, and find from the graphs the approximate values of their common solutions: x^2 + y^2 = 25, 4x^2 + 9y^2 = 144. 5. Two integers are in the ratio 4 : 5. Increase each by 15, and the difference of their squares is 999. What are the integers? 6. A man has $539 to spend for sheep. He wishes to keep 14 of the flock that he buys, but to sell the remainder at a gain of $2 per head. This he does and gains $28. How many sheep did he buy, and at what price each? _Group C_ 7. (_a_) Find the seventh term of [a + 1/a]^(13). (_b_) Derive the formula for the sum of n terms of an arithmetic progression. 8. A ball falling from a height of 60 feet rebounds after each fall one third of its last descent. What distance has it passed over when it strikes the ground for the eighth time? ~CORNELL UNIVERSITY~ ELEMENTARY ALGEBRA 1. Find the H. C. F.: x^4 - y^4, x^3 - xy^2 + x^2y - y^3, x^4 + 2x^2y^2 - 3y^4. 2. Solve the following set of equations: x + y = -1, x + 3y + 2z = -4, x - y + 4z = 5. 3. Expand and simplify: [2x^3 - 1/x]^7. 4. An automobile goes 80 miles and back in 9 hours. The rate of speed returning was 4 miles per hour faster than the rate going. Find the rate each way. 5. Simplify: {[(x + 1)/(x - 1)]^2 - 2 + [(x - 1)/(x + 1)]^2} /{[(x + 1)/(x - 1)]^2 - [(x - 1)/(x + 1)]^2}. 6. Solve for x: (2x + 3)/(x - 1) - 6 = 5/(x^2 + 2x - 3).