Part 35

TIME: ONE HOUR AND A HALF

1. Find the highest common factor and the lowest common multiple of the three expressions a^4 - b^4; a^3 + b^3; a^3 + 2a^2 b + 2ab^2 + b^3.

2. Solve the quadratic equation x^2 - 1.6x + 0.3 = 0, computing the value of the larger root correct to three significant figures.

3. In the expression x^2 - 2xy + y^2 - 4[2^(1/2)](x + y) + 8, substitute for x and y the values x = (u + v + 1)/[2^(1/2)], y = (u - v + 1)/[2^(1/2)], and reduce the resulting expression to its simplest form.

4. State and prove the formula for the sum of the first n terms of a geometric progression in which a is the first term and r the constant ratio.

5. A state legislature is to elect a United States senator, a majority of all the votes cast being necessary for a choice. There are three candidates, A, B, and C, and 100 members vote. On the first ballot A has the largest number of votes, receiving 9 more votes than his nearest competitor, B; but he fails of the necessary majority. On the second ballot C's name is withdrawn, and all the members who voted for C now vote for B, whereupon B is elected by a majority of 2. How many votes were cast for each candidate on the first ballot?

~MASSACHUSETTS INSTITUTE OF TECHNOLOGY~

ALGEBRA A

TIME: ONE HOUR AND THREE QUARTERS

1. Factor the expressions: x^3 + x^2 = 2x. x^3 + x^2 - 4x - 4.

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

3. Find the value of x + [1 + x^2]^(1/2), when x = (1/2)[[a/b]^(1/2) - [b/a]^(1/2)].

4. Solve the equations: (7x + 6)/11 + y - 16 = (5x - 13)/2 - (8y - x)/5, 3(3x + 4) = 10y - 15.

5. Solve the equations: A + C = 2, -A + B + C + D = 1, 2A - B + 2C + D = 5, B + D = 1.

6. Two squares are formed with a combined perimeter of 16 inches. One square contains 4 square inches more than the other. Find the area of each.

7. A man walked to a railway station at the rate of 4 miles an hour and traveled by train at the rate of 30 miles an hour, reaching his destination in 20 hours. If he had walked 3 miles an hour and ridden 35 miles an hour, he would have made the journey in 18 hours. Required the total distance traveled.

~MASSACHUSETTS INSTITUTE OF TECHNOLOGY~

ALGEBRA B

TIME: ONE HOUR AND THREE QUARTERS

1. How many terms must be taken in the series 2, 5, 8, 11, ... so that the sum shall be 345?

2. Prove the formula x = [-b +- [b^2 - 4ac]^(1/2)]/(2a) for solving the quadratic equation ax^2 + bx + c = 0.

3. Find all values of a for which [\sq]a is a root of x^2 + x + 20 = 2a, and check your results.

4. Solve {x^2 + 3y^2 = 10, x - y = 2,} and sketch the graphs.

5. The sum of two numbers x and y is 5, and the sum of the two middle terms in the expansion of (x + y)^3 is equal to the sum of the first and last terms. Find the numbers.

6. Solve x^4 - 2x^3 + 3x^2 - 2x + 1 = 0.

(HINT: Divide by x^2 and substitute x + 1/x = z.)

7. In anticipation of a holiday a merchant makes an outlay of $50, which will be a total loss in case of rain, but which will bring him a clear profit of $150 above the outlay if the day is pleasant. To insure against loss he takes out an insurance policy against rain for a certain sum of money for which he has to pay a certain percentage. He then finds that whether the day be rainy or pleasant he will make $80 clear. What is the amount of the policy, and what rate did the company charge him?

~MASSACHUSETTS INSTITUTE OF TECHNOLOGY~