(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 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?
~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) x (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)].
3. Solve
[2(x - 7)]/(x^2 + 3x - 28) + (2 - x)/(4 - x) - (x + 3)/(x + 7) = 0.
_Group II_
4. Simplify [2^(1/2) + 2[3^(1/2)]]/[2^(1/2) - 12^(1/2)], and compute
the value of the fraction to two decimal places.
5. Solve the simultaneous equations
{ x^(-1/2) + 2y^(-1/2) = 7/6,
{ 2x^(-1/2) - y^(-1/2) = 2/3.
_Group III_
6. Two numbers are in the ratio of c : d. If a be added to the first
and subtracted from the second, the results will be in the ratio
of 3 : 2. Find the numbers.
7. A dealer has two kinds of coffee, worth 30 and 40 cents per pound.
How many pounds of each must be taken to make a mixture of 70
pounds, worth 36 cents per pound?
8. A, B, and C can do a piece of work in 30 hours. A can do half as
much again as B, and B two thirds as much again as C. How long
would each require to do the work alone?
~YALE UNIVERSITY~
ALGEBRA B
TIME: ONE HOUR
Omit one question in Group I and one in Group II. Credit will be given
for _five_ questions only.
_Group I_
1. Solve (x + a)/(x + b) + (x + b)/(x + a) = 5/2.
2. Solve the simultaneous equations
{ x^2y^2 + 28xy - 480 = 0,
{ 2x + y = 11.
Arrange the roots in corresponding pairs.
3. Solve 3x^(-3/2) + 20x^(-3/4) = 32.
_Group II_
4. In going 7500 yd. a front wheel of a wagon makes 1000 more
revolutions than a rear one. If the wheels were each 1 yd. greater
in circumference, a front wheel would make 625 more revolutions
than a rear one. Find the circumference of each.
5. Two cars of equal speed leave A and B, 20 mi. apart, at different
times. Just as the cars pass each other an accident reduces the
power and their speed is decreased 10 mi. per hour. One car makes
the journey from A to B in 56 min., and the other from B to A in 72
min. What is their common speed?
_Group III_
6. Write in the simplest form the last three terms of the expansion
of (4a^(3/2) - a^(1/2) x^(1/3))^8.
7. (_a_) Derive the formula for the sum of an A. P.
(_b_) Find the sum to infinity of the series 1, -1/2, 1/4,
-1/8, .... Also find the sum of the positive terms.
End of Project Gutenberg's A Review of Algebra, by Romeyn Henry Rivenburg