Part 38

~PRINCETON UNIVERSITY~

ALGEBRA A

TIME: TWO HOURS

Candidates who are at this time taking both Algebra A and Algebra B may omit from Algebra A questions 4, 5, and 6, and from Algebra B questions 1 (a), 3, and 4.

1. Simplify (a^3 + a^2b + ab^2)/(a^2 - 3ab - 4b^2) / {(a^2 + 6ab - 7b^2)/(a^2 + 8ab - 9b^2) . (a^3 - b^3)/(a^2 - 7ab + 12b^2)}.

2. (a) Divide a^(5/2) + ab^(3/2) + b^(5/2) - 2a^(1/2)b^2 - a^(3/2)b by a^(3/2) - b^(3/2) + a^(1/2)b - ab^(1/2).

(b) Simplify (1)/(x^(-1) + y^(-1)} . (x^(1/4)y^(1/2))^3 + 1.

3. Factor: (a) (x^2 + 3x)^2 - (2x - 6)^2.

(b) a^2 + ac - 4b^2 - 2bc.

4. Solve 1/(x + 1) - (1)/(x - 1) - (1)/(x - 3) + (1)/(x - 5) = 0.

5. Solve for x and y: mx + ax = my - by, x - y = a + b.

6. The road from A to B is uphill for 5 mi., level for 4 mi., and then downhill for 6 mi. A man walks from B to A in 4 hr.; later he walks halfway from A to B and back again to A in 3 hr. and 55 min.; and later he walks from A to B in 3 hr. and 52 min. What are his rates of walking uphill, downhill, and on the level, if these do not vary?

ALGEBRA B

1. Solve

(a) (x + 1)/(x - 2) + (2x + 1)/(x + 1) + (3x + 3)/(1 - x) = 0.

(b) [2x + 7]^(1/2) + [3x - 18]^(1/2) - [7x + 1]^(1/2) = 0.

(c) 6/(x^2 + 2x) = 5 - 2x - x^2.

2. Solve for x and y, checking one solution in each problem:

(a) 2x + 3y = 1, 6/x + 1/y = 2.

(b) x^2 = x + y, y^2 = 3y - x.

3. A man arranges to pay a debt of $3600 in 40 monthly payments which form an A. P. After paying 30 of them he still owes 1/3 of his debt. What was his first payment?

4. If 4 quantities are in proportion and the second is a mean proportional between the third and fourth, prove that the third will be a mean prop. between the first and second.

5. In the expansion of [2x + 1/3x]^6 the ratio of the fourth term to the fifth is 2 : 1. Find x.

6. Two men A and B can together do a piece of work in 12 days; B would need 10 days more than A to do the whole work. How many days would it take A alone to do the work?

ALGEBRA TO QUADRATICS

1. Simplify (ab^(-2)c^2)^(1/2) . (a^3b^2c^(-3))^(1/3) + [(a^6)/(b)]^(1/3).

2. Simplify a/[(a - b)(a - c)] + b/[(b - c)(b - a)] + c/[(c - a)(c - b)].

3. Factor (a) x^4 - 10x^2 + 9.