Part 32

8. A boat's crew, rowing at half their usual speed, row 3 miles downstream and back again in 2 hours and 40 minutes. At full speed they can go over the same course in 1 hour and 4 minutes. Find the rate of the crew, and the rate of the current in miles per hour. (College Entrance Board.)

9. Find the sides of a rectangle whose area is unchanged if its length is increased by 4 feet and its breadth decreased by 3 feet, but which loses one third of its area if the length is increased by 16 feet and the breadth decreased by 10 feet. (M. I. T.)

COLLEGE ENTRANCE EXAMINATIONS

~UNIVERSITY OF CALIFORNIA~

ELEMENTARY ALGEBRA

1. If a = 4, b = -3, c = 2, and d = -4, find the value of: (a) ab^3 - 3cd^2 + 2(3a - b)(c - 2d). (b) 2a^3 - 3b^4 + (4c^3 + d^3)(4c^2 + d^2).

2. Reduce to a mixed number: (3a^4 - 4a^3 - 10a^2 + 41a - 28)/(a^2 - 3a + 4).

Simplify:

3. (a + 2)/(a^2 + 3a - 40) - (b - 2)/(ab - 5b + 3a - 15).

4. [1 - (2 - 3b - 2c)/(a + 2)] / (a^2 - 4c^2 + 9b^2 + 6ab)/(2a^2 + a - 6).

5. A's age 10 years hence will be 4 times what B's age was 11 years ago, and the amount that A's age exceeds B's age is one third of the sum of their ages 8 years ago. Find their present ages.

6. Draw the lines represented by the equations 3x - 2y = 13 and 2x + 5y = -4, and find by algebra the cooerdinates of the point where they intersect.

7. Solve the equations { bx - ay = b^2 - ab, { y - b = 2(x - 2a).

8. Solve (2x + 1)(3x - 2) - (5x - 7)(x - 2) = 41.

~COLORADO SCHOOL OF MINES~

ELEMENTARY ALGEBRA

1. Solve by factoring: x^3 + 30x = 11x^2.

2. Show that 1 - [(a^2 + b^2 - c^2)/(2ab)]^2 = (a + b + c)(a + b - c)(a - b + c)(b + c - a) / 4a^2b^2.

3. How many pairs of numbers will satisfy simultaneously the two equations { 3x + 2y = 7, { x + y = 3?

Show by means of a graph that your answer is correct.

What is meant by eliminating x in the above equations by substitution? by comparison? by subtraction?

4. Find the square root of 223,728.

5. Simplify: (a) [1/3]^(1/2) + ^(1/2) - [3/4]^(1/2). (b) (-[-3[-4]^(1/2)]^(1/2))^4.

6. Solve the equation .03x^2 - 2.23x + 1.1075 = 0.

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