Part 41

1. Find two numbers whose ratio is 3 and such that two sevenths of the larger is 15 more than one half the smaller.

2. Determine the factors of the lowest common multiple of 3x^4 (x^3 - y^3), 15 (x^4 - 2x^2y^2 + y^4), and 10y (x^4 + x^2y^2 + y^4).

3. Find to two decimal places the value of [4a^(-2/5) + b^0[ab^(-1)]^(1/2)]^(1/2), when a = -32 and b = -8.

4. Solve the equations: 2x + 5y = 85, 2y + 5z = 103, 2z + 5x = 57.

5. Solve any 3 of these equations:

(a) x^2 + 44 - 15x = 0.

(b) 2/x - x/5 = x/20 - 223/30.

(c) x^2 + 8x - [4x^2 + 32x + 12]^(1/2) = 21.

(d) 5/(x + 1) + 8/(x - 2) = 12/(40 - 2x).

6. The sum of two numbers is 13, and the sum of their cubes is 910. Find the smaller number, correct to the second decimal place.

7. The sum of 9 terms of an arithmetical progression is 46; the sum of the first 5 terms is 25. Find the common difference.

8. Explain the terms, and prove that if four numbers are in proportion, they are in proportion by alternation, by inversion, and by composition. Find x when (3 + x)/(3 - x) = (40 + x^3)/(40 - x^3).

9. Find the value of x in each of these equations:

(a) 7x^(1/4) - 3x^(1/2) = 2.

(b) (x^2 + 2)^(5/2) + 3/{[x^2 + 2]^(1/2)} = 4x^2 + 8.

~YALE UNIVERSITY~

ALGEBRA A

TIME: ONE HOUR

Omit one question in Group II and one in Group III. Credit will be given for six questions only.

Group I

1. Resolve into prime factors: (a) 6x^2 - 7x - 20; (b) (x^2 - 5x)^2 - 2(x^2 - 5x) - 24; (c) a^4 + 4a^2 + 16.

2. Simplify [5 - (a^2 - 19x^2)/(a^2 - 4x^2)] / [3 - (a - 5x)/(a - 2x)].

3. Solve [2(x - 7)]/(x^2 + 3x - 28) + (2 - x)/(4 - x) - (x + 3)/(x + 7) = 0.

Group II

4. Simplify [2^(1/2) + 2[3^(1/2)]]/[2^(1/2) - 12^(1/2)], and compute the value of the fraction to two decimal places.

5. Solve the simultaneous equations { x^(-1/2) + 2y^(-1/2) = 7/6, { 2x^(-1/2) - y^(-1/2) = 2/3.

Group III

6. Two numbers are in the ratio of c : d. If a be added to the first and subtracted from the second, the results will be in the ratio of 3 : 2. Find the numbers.