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~ ALGEBRA A TIME: TWO HOURS 1. Simplify [m + 1/m]^2 + [n + 1/n]^2 + [mn + 1/mn]^2 - [m + 1/m][n + 1/n][mn + 1/mn]. 2. Find the prime factors of (_a_) (x - x^2)^3 + (x^2 - 1)^3 + (1 - x)^3. (_b_) (2x + a - b)^4 - (x - a + b)^4. 3. (_a_) Simplify [(x^q)/(x^r)]^(q + r) [(x^r)/(x^p)]^(r + p)[{x^p/{x^q}]^(p + q). (_b_) Show that ([[x]^[1/(n+1)]]^(1/n))/([[x]^[1/(n+2)]]^[1/(n+1)]) = {x^(1/n) . [x]^[1/(n+2)]}/{[x^2]^[1/(n+1)]}. 4. Define _homogeneous terms_. For what value of n is x^n y^(5 - n/2) + x^(n + 1) y^(2n - 6) a homogeneous binomial? 5. Extract the square root of x(x - 2^(1/2))(x - 8^(1/2))(x - 18^(1/2)) + 4. 6. Two vessels contain each a mixture of wine and water. In the first vessel the quantity of wine is to the quantity of water as 1 : 3, and in the second as 3 : 5. What quantity must be taken from each, so as to form a third mixture which shall contain 5 gallons of wine and 9 gallons of water?