7. A, B, and C, all working together, can do a piece of work in 2-2/3 days. A works twice as fast as C, and A and C together could do the work in 4 days. How long would it take each one of the three to do the work alone? ~CORNELL UNIVERSITY~ INTERMEDIATE ALGEBRA 1. Solve the following set of equations: x + y = -1, 2z + 5w = 1, x + 3y + 2z = -4, x - y + 4z + 4w = 5. 2. Simplify: (_a_) [6 - 20^(1/2)]^(1/2). (_b_) [1 + [x^2 + 1]^(1/2)]/[1 + [x^2 + 1]^(1/2) + x^2]. 3. Find, and simplify, the 23d term in the expansion of [(2x^2)/(3) - 3/4]^(28). 4. The weight of an object varies directly as its distance from the center of the earth when it is below the earth's surface, and inversely as the square of its distance from the center when it is above the surface. If an object weighs 10 pounds at the surface, how far above, and how far below the surface will it weigh 9 pounds? (The radius of the earth may be taken as 4000 miles.) 5. Solve the following pair of equations for x and y: x^2 + y^2 = 4, x = (1 + 2^(1/2))y - 2. 6. Find the value of [1 + 8^(-x/3)]/[(8x)^(1/2) + 10^(x - 2)], when x = 2. 7. From a square of pasteboard, 12 inches on a side, square corners are cut, and the sides are turned up to form a rectangular box. If the squares cut out from the corners had been 1 inch larger on a side, the volume of the box would have been increased 28 cubic inches. What is the size of the square corners cut out? (See the figure on the blackboard.) ~HARVARD UNIVERSITY~ ELEMENTARY ALGEBRA TIME: ONE HOUR AND A HALF Arrange your work neatly and clearly, beginning each question on a separate page. 1. Simplify the following expression: [[1/a + 1/(b + c)]/[1/a - 1/(b + c)] [1 + (b^2 + c^2 - a^2)/(2bc)]. 2. (_a_) Write the middle term of the expansion of (a - b)^14 by the binomial theorem. (_b_) Find the value of a^7b^7, if a = x^(2/7)y^(-3/2) and b = (1/2) x^(-1/7)y^(1/2), and reduce the result to a form having only positive exponents. 3. Find correct to three significant figures the negative root of the equation 1 - 2/(x + 1) + 4x/{(x + 1)^2} = 0. 4. Prove the rule for finding the sum of n terms of a geometrical progression of which the first term is a and the constant ratio is r. Find the sum of 8 terms of the progression 5 + 3-1/3 + 2-2/9 + .... 5. A goldsmith has two alloys of gold, the first being 3/4 pure gold, the second 5/12 pure gold. How much of each must he take to produce 100 ounces of an alloy which shall be 2/3 pure gold? ~HARVARD UNIVERSITY~ ELEMENTARY ALGEBRA TIME: ONE HOUR AND A HALF 1. Solve the simultaneous equations x + y = a + b, (y + b)/(x + a) = a/b, and verify your results.