Part 37

(b) 1/[(a - b)(b - c)] + 1/[(c - a)(b - a)].

(c) Find [19 - 8[3^(1/2)]]^(1/2).

5. Plot the graphs of the following system, and determine the solution from the point of intersection: { x - 2y = 0, { 2x - 3y = 4.

6. (a) Derive the formula for the solution of ax^2 + bx + c = 0.

(b) Determine the value of m for which the roots of 2x^2 + 4x + m = 0 are (i) equal, (ii) real, (iii) imaginary.

(c) Form the quadratic equation whose roots are 2 + 3^(1/2) and 2 - 3^(1/2).

7. A page is to have a margin of 1 inch, and is to contain 35 square inches of printing. How large must the page be, if the length is to exceed the width by 2 inches?

8. (a) In an arithmetical progression the sum of the first six terms is 261, and the sum of the first nine terms is 297. Find the common difference.

(b) Three numbers whose sum is 27 are in arithmetical progression. If 1 is added to the first, 3 to the second, and 11 to the third, the sums will be in geometrical progression. Find the numbers.

(c) Derive the formula for the sum of n terms of a geometrical progression.

9. (a) Expand and simplify (2a^2 - 3x^3)^7.

(b) For what value of x will the ratio 7 + x : 12 + x be equal to the ratio 5 : 6?

~UNIVERSITY OF PENNSYLVANIA~

ELEMENTARY ALGEBRA

TIME: THREE HOURS

1. Simplify: [(a + x)/(a - x) - (a - x)/(a + x)] / (4ax)/(a^2 - x^2).

2. Find the H. C. F. and L. C. M. of 10ab^2(x^2 - 2ax), 15a^3b(x^2 - ax - 2a^2), 25b^3(x^2 - a^2)^2.

3. A grocer buys eggs at 4 for 7c. He sells 1/4 of them at 5 for 12c, and the rest at 6 for 11c, making 27c by the transaction. How many eggs does he buy?

4. Solve for t: (t + 4a + b)/(t + a + b) - (4t - a - 2b)/(t + a - b) = -3.

5. Find the square root of a^2 - (3/2)a^(3/2) - (3/2)a^(1/2) + (41/16)a + 1.

6. (a) For what values of m will the roots of 2x^2 + 3mx = -2 be equal?

(b) If 2a + 3b is a root of x^2 - 6bx - 4a^2 + 9b^2 = 0, find the other root without solving the equation.

7. (a) Solve for x: [2x - 3a]^(1/2) + [3x - 2a]^(1/2) = 3[a^(1/2)].

(b) Solve for m: 1 - (1)/(2 - m) = 1/(m + 2) + (m - 6)/(4 - m^2).

8. Solve the system: x^2 + 2y^2 = 17; xy - y^2 = 2.

9. Two boats leave simultaneously opposite shores of a river 2-1/4 mi. wide and pass each other in 15 min. The faster boat completes the trip 6-3/4 min. before the other reaches the opposite shore. Find the rates of the boats in miles per hour.

10. Write the sixth term of [x/(2[y^2]^(1/3)) - (y^(1/2))/x]^9 without writing the preceding terms.

11. The sum of the 2d and 20th terms of an A. P. is 10, and their product is 23-47/64. What is the sum of sixteen terms?