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)
(2)
(b) Find the H. C. F. and the L. C. M. of
2. (a) Simplify
(b) If 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 shall have equal roots.
(b) Solve the equations
(c) Plot the following two equations, and find from the graphs the approximate values of their common solutions:
5. Two integers are in the ratio 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?
7. (a) Find the seventh term of
(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?