COLUMBIA UNIVERSITY
Reading edition
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)
1. (b) Find the H. C. F. and the L. C. M. of
2. (a) Simplify
2. (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
4. (a) shall have equal roots.
4. (b) Solve the equations
4. (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?
Group C
7. (a) Find the seventh term of
7. (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?