Part 4
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 coördinates 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) + [12]^(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
Six questions are required; two from Group _A_, two from Group _B_, and both questions of Group _C_. No extra credit will be given for more than six questions.
_Group A_
1. (_a_) Resolve the following into their prime factors: (1) (x^2 - y^2)^2 - y^4. (2) 10x^2 - 7x - 6.
(_b_) Find the H. C. F. and the L. C. M. of x^3 - 3x^2 + x - 3, x^3 - 3x^2 - x + 3.
2. (_a_) Simplify [x/y + y/x - 2]/[1/x + 1/y] + [x/y + y/x + 2]/[1/x - 1/y].
(_b_) If x : y = (x - z)^2 : (y - z)^2, prove that z is a mean proportional between x and y.
3. A crew can row 10 miles in 50 minutes downstream, and 12 miles in an hour and a half upstream. Find the rate of the current and of the crew in still water.
_Group B_
4. (_a_) Determine the values of k so that the equation (2 + k)x^2 + 2kx + 1 = 0 shall have equal roots.
(_b_) Solve the equations x^2 - xy + y^2 = 7, 2x - 3y = 0.
(_c_) Plot the following two equations, and find from the graphs the approximate values of their common solutions: x^2 + y^2 = 25, 4x^2 + 9y^2 = 144.
5. Two integers are in the ratio 4 : 5. Increase each by 15, and the difference of their squares is 999. What are the integers?
6. A man has $539 to spend for sheep. He wishes to keep 14 of the flock that he buys, but to sell the remainder at a gain of $2 per head. This he does and gains $28. How many sheep did he buy, and at what price each?
_Group C_
7. (_a_) Find the seventh term of [a + 1/a]^(13).
(_b_) Derive the formula for the sum of n terms of an arithmetic progression.
8. A ball falling from a height of 60 feet rebounds after each fall one third of its last descent. What distance has it passed over when it strikes the ground for the eighth time?
~CORNELL UNIVERSITY~
ELEMENTARY ALGEBRA
1. Find the H. C. F.: x^4 - y^4, x^3 - xy^2 + x^2y - y^3, x^4 + 2x^2y^2 - 3y^4.
2. Solve the following set of equations: x + y = -1, x + 3y + 2z = -4, x - y + 4z = 5.
3. Expand and simplify: [2x^3 - 1/x]^7.
4. An automobile goes 80 miles and back in 9 hours. The rate of speed returning was 4 miles per hour faster than the rate going. Find the rate each way.
5. Simplify: {[(x + 1)/(x - 1)]^2 - 2 + [(x - 1)/(x + 1)]^2} /{[(x + 1)/(x - 1)]^2 - [(x - 1)/(x + 1)]^2}.
6. Solve for x: (2x + 3)/(x - 1) - 6 = 5/(x^2 + 2x - 3).
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.
2. Solve the equation x^2 - 1.6x - 0.23 = 0, obtaining the values of the roots correct to three significant figures.
3. Write out the first four terms of (a - b)^7. Find the fourth term of this expansion when a = [x^(-1) y^(1/2)]^(1/3), b = [9xy^(-4)]^(1/6), expressing the result in terms of a single radical, and without fractional or negative exponents.
4. Reduce the following expression to a polynomial in a and b: (6a^3 + 7ab^2 + 12b^3)/(3a^2 - 5ab - 4b^2) - 1/[3/19b - (5a + 4b)/(19a^2)].
5. The cost of publishing a book consists of two main items: first, the fixed expense of setting up the type; and, second, the running expenses of presswork, binding, etc., which may be assumed to be proportional to the number of copies. A certain book costs 35 cents a copy if 1000 copies are published at one time, but only 19 cents a copy if 5000 copies are published at one time. Find (_a_) the cost of setting up the type for the book, and (_b_) the cost of presswork, binding, etc., per thousand copies.
~HARVARD UNIVERSITY~
ELEMENTARY ALGEBRA
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~
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?
7. Find a quantity such that by adding it to each of the quantities a, b, c, d, we obtain four quantities in proportion.
8. What values must be given to a and b, so that (3a + 2b + 17)/2, (2a - 3b + 25)/3, 4 - 5a - 13b may be equal?
~MOUNT HOLYOKE COLLEGE~
ELEMENTARY ALGEBRA
TIME: TWO HOURS
1. Factor the following expressions:
(_a_) a^(3/4) - b^(3/4).
(_b_) x^2 y^2 z^2 - x^2 z - y^2 z + 1.
(_c_) 16(x + y)^4 - (2x - y)^4.
2. (_a_) Simplify
(a^2 + b^2){(b^4)/(b^2 - a^2) - a^2}/{a/(a + b) + b/(a - b)}}.
(_b_) Extract the square root of x^4 - 2x^3 + 5x^2 - 4x + 4.
3. Solve the following equations:
(_a_) 1/x + 1/y = 5, 1/(x^2) + 1/(y^2) = 13.
(_b_) x^2 - 5x + 2 = 0.
(_c_) [27x + 1]^(1/2) = 2 - 3[3x^(1/2)].
4. Simplify:
(_a_) 7[54]^(1/3) + 256^(1/6) + [432/(-250)]^(1/3).
(_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 7¢. He sells 1/4 of them at 5 for 12¢, and the rest at 6 for 11¢, making 27¢ 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?
~PRINCETON UNIVERSITY~
ALGEBRA A
TIME: TWO HOURS
Candidates who are at this time taking _both_ Algebra A and Algebra B may omit from Algebra A questions 4, 5, and 6, and from Algebra B questions 1 (_a_), 3, and 4.
1. Simplify (a^3 + a^2b + ab^2)/(a^2 - 3ab - 4b^2) ÷ {(a^2 + 6ab - 7b^2)/(a^2 + 8ab - 9b^2) · (a^3 - b^3)/(a^2 - 7ab + 12b^2)}.
2. (_a_) Divide a^(5/2) + ab^(3/2) + b^(5/2) - 2a^(1/2)b^2 - a^(3/2)b by a^(3/2) - b^(3/2) + a^(1/2)b - ab^(1/2).
(_b_) Simplify (1)/(x^(-1) + y^(-1)} · (x^(1/4)y^(1/2))^3 + 1.
3. Factor: (_a_) (x^2 + 3x)^2 - (2x - 6)^2.
(_b_) a^2 + ac - 4b^2 - 2bc.
4. Solve 1/(x + 1) - (1)/(x - 1) - (1)/(x - 3) + (1)/(x - 5) = 0.
5. Solve for x and y: mx + ax = my - by, x - y = a + b.
6. The road from A to B is uphill for 5 mi., level for 4 mi., and then downhill for 6 mi. A man walks from B to A in 4 hr.; later he walks halfway from A to B and back again to A in 3 hr. and 55 min.; and later he walks from A to B in 3 hr. and 52 min. What are his rates of walking uphill, downhill, and on the level, if these do not vary?
ALGEBRA B
1. Solve
(_a_) (x + 1)/(x - 2) + (2x + 1)/(x + 1) + (3x + 3)/(1 - x) = 0.
(_b_) [2x + 7]^(1/2) + [3x - 18]^(1/2) - [7x + 1]^(1/2) = 0.
(_c_) 6/(x^2 + 2x) = 5 - 2x - x^2.
2. Solve for x and y, checking one solution in each problem:
(_a_) 2x + 3y = 1, 6/x + 1/y = 2.
(_b_) x^2 = x + y, y^2 = 3y - x.
3. A man arranges to pay a debt of $3600 in 40 monthly payments which form an A. P. After paying 30 of them he still owes 1/3 of his debt. What was his first payment?
4. If 4 quantities are in proportion and the second is a mean proportional between the third and fourth, prove that the third will be a mean prop. between the first and second.
5. In the expansion of [2x + 1/3x]^6 the ratio of the fourth term to the fifth is 2 : 1. Find x.
6. Two men A and B can together do a piece of work in 12 days; B would need 10 days more than A to do the whole work. How many days would it take A alone to do the work?
ALGEBRA TO QUADRATICS
1. Simplify (ab^(-2)c^2)^(1/2) · (a^3b^2c^(-3))^(1/3) + [(a^6)/(b)]^(1/3).
2. Simplify a/[(a - b)(a - c)] + b/[(b - c)(b - a)] + c/[(c - a)(c - b)].
3. Factor (_a_) x^4 - 10x^2 + 9.
(_b_) x^2 + 2xy - a^2 - 2ay.
(_c_) (a + b)^2 + (a + c)^2 - (c + d)^2 - (b + d)^2.
4. Find H. C. F. of x^4 - x^3 + 2x^2 + x + 3 and (x + 2)(x^3 - 1).
5. Solve x/(x - 2) + (x - 9)/(x - 7) = (x + 1)/(x - 1) + (x - 8)/(x - 6).
6. The sum of three numbers is 51; if the first number be divided by the second, the quotient is 2 and the remainder 5; if the second number be divided by the third, the quotient is 3 and the remainder 2. What are the numbers?
~SMITH COLLEGE~
ELEMENTARY ALGEBRA
1. Factor e^(2x) - 2 + e^(-2x), x^(12) - 8, x^2 - x - y^2 - y, 18a^2x^2 -24axy - 10y^2.
2. Solve [7 + 4x + 3[2x^2 + 5x + 7]^(1/2)]^(1/2) - 3 = 0.
3. The second term of a geometrical progression is 3[2^(1/2)], and the fifth term is 3/16. Find the first term and the ratio.
4. Solve the following equations and check your results by plotting:
{ x^2 + y^2 - xy = 7, { x + y = 4.
5. Solve
1/(x^3) + 1/(y^3) = 243/8, 1/x + 1/y = 9/2.
6. In an arithmetical progression d = -11, n = 13, s = 0. Find a and l.
7. Expand by the binomial theorem and simplify:
[(2x)/(y^3) - (y^4)/(x^5 [-6]^(1/2))]^5.
8. The diagonal of a rectangle is 13 ft. long. If each side were longer by 2 ft., the area would be increased by 38 sq. ft. Find the lengths of the sides.
~SMITH COLLEGE~
ELEMENTARY ALGEBRA
1. Find the H. C. F. of 8x^3 - 27, 32x^5 - 243, and 6x^3 - 9x^2 + 4x - 6.
2. Solve:
(_a_) (2x + 5)^(-5) + 31(2x + 5)^(-5/2) = 32.
(_b_) (x - 1)^(1/2) + (3x + 1)^(1/2) = 4.
3. A farmer sold a horse at $75 for which he had paid x dollars. He realized x per cent profit by his sale. Find x.
4. Find the 13th term and the sum of 13 terms of the arithmetical progression
(2^(1/2) - 1)/2, (2^(1/2))/2, (1)/[2([2]^(1/2) - 1)], ···.
5. The difference between two numbers is 48. Their arithmetical mean exceeds their geometrical mean by 18. Find the numbers.
6. Expand by the binomial theorem and simplify
[3a^(-2) - a/[-2]^(1/2)]^5.
7. Solve:
1/x + 1/y = 3/2, 1/(x^2) + 1/(y^2) = 5/4.
8. Solve the following equations and check the results by finding the intersections of the graphs of the two equations:
{ x^2 = 4y, { x + 2y = 4.
~VASSAR COLLEGE~
ELEMENTARY AND INTERMEDIATE ALGEBRA
Answer any six questions.
1. Find the product of
[1 + 2a/3 - (5a^2)/(6)] and [2 - 3a/4 + (a^2)/(3)].
2. Resolve into linear factors:
(_a_) 4x^2 - 25;
(_b_) 6x^2 - x - 12;
(_c_) a^2b^2 + 1 - a^2 - b^2;
(_d_) y^3 + (x - 3)y^2 - (3x - 2)y + 2x.
3. Reduce to simplest form:
(_a_) z/(1/x - 1/y) + y/(1 - y/x) - x/(1 - x/y).
(_b_) [-(x^3)^(1/2)]^(1/3) × (4y^(-3))^(1/2).
4. (_a_) Divide x^(3/2) - x^(-3/2) by x^(1/2) - x^(-1/2).
(_b_) Find correct to one place of decimals the value of [5^(1/2) + 7^(1/2)]/[2 - 3^(1/2)].
5. (_a_) If a/b = c/d, show that (a^2 + c^2)/(b^2 + d^2) = ac/bd.
(_b_) Two numbers are in the ratio 3 : 4, and if 7 be subtracted from each the remainders are in the ratio 2 : 3. Find the numbers.
6. Solve the equations:
(_a_) (x + 1)/(2) - 3/x = x/3 - (5 - x)/(6).
(_b_) 11x^2 - 11-1/4 = 9x.
(_c_) { x^2 - 2y^2 = 71, { x + y = 20.
7. A field could be made into a square by diminishing the length by 10 feet and increasing the breadth by 5 feet, but its area would then be diminished by 210 square feet. Find the length and the breadth of the field.
~VASSAR COLLEGE~
ELEMENTARY AND INTERMEDIATE ALGEBRA
Answer six questions, including No. 5 and No. 7 or 8. Candidates in Intermediate Algebra will answer Nos. 5-9.
1. Find two numbers whose ratio is 3 and such that two sevenths of the larger is 15 more than one half the smaller.
2. Determine the factors of the lowest common multiple of 3x^4 (x^3 - y^3), 15 (x^4 - 2x^2y^2 + y^4), and 10y (x^4 + x^2y^2 + y^4).
3. Find to two decimal places the value of [4a^(-2/5) + b^0[ab^(-1)]^(1/2)]^(1/2), when a = -32 and b = -8.
4. Solve the equations: 2x + 5y = 85, 2y + 5z = 103, 2z + 5x = 57.
5. Solve any 3 of these equations:
(_a_) x^2 + 44 - 15x = 0.
(_b_) 2/x - x/5 = x/20 - 223/30.
(_c_) x^2 + 8x - [4x^2 + 32x + 12]^(1/2) = 21.
(_d_) 5/(x + 1) + 8/(x - 2) = 12/(40 - 2x).
6. The sum of two numbers is 13, and the sum of their cubes is 910. Find the smaller number, correct to the second decimal place.
7. The sum of 9 terms of an arithmetical progression is 46; the sum of the first 5 terms is 25. Find the common difference.
8. Explain the terms, and prove that if four numbers are in proportion, they are in proportion by _alternation_, by _inversion_, and by _composition_. Find x when (3 + x)/(3 - x) = (40 + x^3)/(40 - x^3).
9. Find the value of x in each of these equations:
(_a_) 7x^(1/4) - 3x^(1/2) = 2.
(_b_) (x^2 + 2)^(5/2) + 3/{[x^2 + 2]^(1/2)} = 4x^2 + 8.
~YALE UNIVERSITY~
ALGEBRA A
TIME: ONE HOUR
Omit one question in Group II and one in Group III. Credit will be given for _six_ questions only.
_Group I_
1. Resolve into prime factors: (_a_) 6x^2 - 7x - 20; (_b_) (x^2 - 5x)^2 - 2(x^2 - 5x) - 24; (_c_) a^4 + 4a^2 + 16.
2. Simplify [5 - (a^2 - 19x^2)/(a^2 - 4x^2)] ÷ [3 - (a - 5x)/(a - 2x)].