+ [n(n - 1)(n - 2)]/(1.2.3) a^(n - 3)b^3
+ [n(n - 1)(n - 2)(n - 3)]/(1.2.3.4) a^(n - 4) b^4 + ....
Show by observation that the formula for the
(r + 1)th term
= [n(n - 1)(n - 2)...(n - r + 1)]/[1.2.3.4 ... r] a^(n - r)b^r.
9. Indicate what the 97th term of (a + b)^n would be.
10. Using the expansion of (a + b)^n in (8), derive a formula for the
rth term by observing how each term is made up, then generalizing.
Using either the formula in (8) or (10), whichever you are familiar
with, find:
11. The 4th term of [a + 1/a]^(30).
12. The 8th term of (1 + x[y^(1/2)])^(13).
13. The middle term of (2a^(3/4) - y[a^(1/3)])^(10).
14. The term not containing x in [x^3 - 2/x]^(12).
15. The term containing x^(18) in [x^2 - a/x]^(15).
~Reference:~ The chapter on The Binomial Theorem in any algebra.
~MISCELLANEOUS EXAMPLES, QUADRATICS AND BEYOND~
1. Solve the equation x^2 - 1.6x - .23 = 0, obtaining the values of
the roots correct to three significant figures. (_Harvard._)
2. Write the roots of (x^2 + 2x)(x^2 - 2x - 3)(x^2 - x + 1) = 0.
(_Sheffield Scientific School._)
3. Solve
2[2x + 2]^(1/2) + [2x + 1]^(1/2) = (12x + 4)/([8x + 8]^{1/2}).
(_Yale._)
4. Solve the equation V = (H/3)(B + x + [Bx]^(1/2)) for x, taking
H = 6, B = 8, and V = 28; and verify your result. (_Harvard._)
5. Solve { x : y = 2 : 3,
{ x^2 + y^2 = 5(x + y) + 2.
6. Solve 2x^2 - 4x + 3[x^2 - 2x + 6]^(1/2) = 15. (_Coll. Ent. Board._)
7. Find all values of x and y which satisfy the equations:
{ x^(1/2) + y^(1/2) = 4,
{ 1/[[x + 1]^(1/2) - x^(1/2)] - 1/[[x + 1]^(1/2) + x^(1/2)] = y.
(_Mass. Inst. of Technology._)
8. If [alpha] and [beta] represent the roots of px^2 + qx + r = 0,
find [alpha] + [beta], [alpha] - [beta], and [alpha][beta] in terms
of p, q, and r. (_Princeton._)
9. Form the equation whose roots are 2 + [3]^(1/2) and 2 - [-3]^(1/2).
10. Determine, without solving, the character of the roots of
9x^2 - 24x + 16 = 0. (_College Entrance Board._)
11. If a : b = c : d, prove that
a + b : c + d = [a^2 + b^2]^(1/2) : [c^2 + d^2]^(1/2).
(_College Entrance Board._)
12. Given a : b = c : d. Prove that
a^2 + b^2 : (a^3)/(a + b) = c^2 + d^2 : (c^3)/(c + d).
(_Sheffield._)
13. The 9th term of an arithmetical progression is 1/6; the 16th term
is 5/2. Find the first term. (_Regents._)
Solve graphically:
1. x^2 - x - 6 = 0.
2. x^2 + 3x - 10 = 0.
3. Find four numbers in arithmetical progression, such that the sum of
the first two is 1, and the sum of the last two is -19.
4. What number added to 2, 20, 9, 34, will make the results
proportional?
5. Find the middle term of [3a^5 + (b^(3/4))/(2)]^8.
6. Solve (x + 1)/(3x + 2) = (2x - 3)/(3x - 2) - 1 - 36/(4 - 9x^2).
(_Princeton._)
7. A strip of carpet one half inch thick and 29-6/7 feet long is
rolled on a roller four inches in diameter. Find how many turns
there will be, remembering that each turn increases the diameter
by one inch, and that the circumference of a circle equals
(approximately) 22/7 times the diameter. (_Harvard._)
8. The sum of the first three terms of a geometrical progression is
21, and the sum of their squares is 189. What is the first term?
(_Yale._)
9. Find the geometrical progression whose sum to infinity is 4, and
whose second term is 3/4.
10. Solve 4x + 4[3x^2 - 7x + 3]^(1/2) = 3x^2 - 3x + 6.
11. Solve { 2x^2 + 3xy - 5y^2 = 4,
{ 2xy + 3y^2 = -3.
12. Two hundred stones are placed on the ground 3 feet apart, the
first being 3 feet from a basket. If the basket and all the stones
are in a straight line, how far does a person travel who starts
from the basket and brings the stones to it one by one?
Solve graphically; and check by solving algebraically:
1. { x^2 + y^2 = 25,
{ x + y = 1.
2. x^2 - 3x - 18 = 0.
3. x^2 + 3x - 10 = 0.
Determine the value of m for which the roots of the equation will be
equal: (HINT: See page 40. To have the roots equal, b^2 - 4ac must
equal 0.)
4. 2x^2 - mx + 12-1/2 = 0.
5. (m - 1)x^2 + mx + 2m - 3 = 0.
6. If 2a + 3b is a root of x^2 - 6bx - 4a^2 + 9b^2 = 0, find the other
root without solving the equation. (_Univ. of Penn._)
7. How many times does a common clock strike in 12 hours?
8. Find the sum to infinity of
2/(2^(1/2)), 1/(2^(1/2)), 1/(2[2]^(1/2)), ....
9. Solve [x/2 + 6/x]^2 - 6[x/2 + 6/x] + 8 = 0.
10. Find the value of the recurring decimal 2.214214....
11. A man purchases a $500 piano by paying monthly installments of $10
and interest on the debt. If the yearly rate is 6%, what is the
total amount of interest?
12. The arithmetical mean between two numbers is 42-1/2, and their
geometrical mean is 42. Find the numbers.
(_College Entrance Exam. Board._)
13. If the middle term of [3x - (1)/(2[x^(1/2)])]^4 is equal to the
fourth term of [2[x^(1/2)] + 1/2x]^7, find the value of x.
(_M. I. T._)
~PROBLEMS~
~Linear Equations, One Unknown~
1. A train running 30 miles an hour requires 21 minutes longer to go
a certain distance than does a train running 36 miles an hour. How
great is the distance? (_Cornell._)
2. A man can walk 2-1/2 miles an hour up hill and 3-1/2 miles an hour
down hill. He walks 56 miles in 20 hours on a road no part of which
is level. How much of it is up hill? (_Yale._)
3. A physician having 100 cubic centimeters of a 6% solution of a
certain medicine wishes to dilute it to a 3-1/2% solution. How much
water must he add? (A 6% solution contains 6% of medicine and 94%
of water.) (_Case._)
4. A clerk earned $504 in a certain number of months. His salary was
increased 25%, and he then earned $450 in two months less time than
it had previously taken him to earn $504. What was his original
salary per month? (_College Entrance Board._)
5. A person who possesses $15,000 employs a part of the money in
building a house. He invests one third of the money which remains
at 6%, and the other two thirds at 9%, and from these investments
he obtains an annual income of $500. What was the cost of the
house? (_M. I. T._)
6. Two travelers have together 400 pounds of baggage. One pays $1.20
and the other $1.80 for excess above the weight carried free. If
all had belonged to one person, he would have had to pay $4.50. How
much baggage is allowed to go free? (_Yale._)
7. A man who can row 4-1/3 miles an hour in still water rows
downstream and returns. The rate of the current is 2-1/4 miles per
hour, and the time required for the trip is 13 hours. How many
hours does he require to return?
~Simultaneous Equations, Two and Three Unknowns~
1. A manual training student in making a bookcase finds that the
distance from the top of the lowest shelf to the under side of
the top shelf is 4 ft. 6 in. He desires to put between these four
other shelves of inch boards in such a way that the book space
will diminish one inch for each shelf from the bottom to the top.
What will be the several spaces between the shelves?
2. A quantity of water, sufficient to fill three jars of different
sizes, will fill the smallest jar 4 times, or the largest jar twice
with 4 gallons to spare, or the second jar three times with 2
gallons to spare. What is the capacity of each jar? (_Case._)
3. A policeman is chasing a pickpocket. When the policeman is 80 yards
behind him, the pickpocket turns up an alley; but coming to the
end, he finds there is no outlet, turns back, and is caught just as
he comes out of the alley. If he had discovered that the alley had
no outlet when he had run halfway up and had then turned back, the
policeman would have had to pursue the thief 120 yards beyond the
alley before catching him. How long is the alley? (_Harvard._)
4. A and B together can do a piece of work in 14 days. After they have
worked 6 days on it, they are joined by C who works twice as fast
as A. The three finish the work in 4 days. How long would it take
each man alone to do it? (_Columbia._)
5. In a certain mill some of the workmen receive $1.50 a day, others
more. The total paid in wages each day is $350. An assessment made
by a labor union to raise $200 requires $1.00 from each man
receiving $1.50 a day, and half of one day's pay from every man
receiving more. How many men receive $1.50 a day? (_Harvard._)
6. There are two alloys of silver and copper, of which one contains
twice as much copper as silver, and the other three times as much
silver as copper. How much must be taken from each to obtain a
kilogram of an alloy to contain equal quantities of silver and
copper? (_M. I. T._)
7. Two automobiles travel toward each other over a distance of 120
miles. A leaves at 9 A.M., 1 hour before B starts to meet him, and
they meet at 12:00 M. If each had started at 9:15 A.M., they would
have met at 12:00 M. also. Find the rate at which each traveled.
(_M. I. T._)
~Quadratic Equations~
1. Telegraph poles are set at equal distances apart. In order to have
two less to the mile, it will be necessary to set them 20 feet
farther apart. Find how far apart they are now. (_Yale._)
2. The distance S that a body falls from rest in t seconds is given by
the formula S = 16t^2. A man drops a stone into a well and hears
the splash after 3 seconds. If the velocity of sound in air is 1086
feet a second, what is the depth of the well? (_Yale._)
3. It requires 2000 square tiles of a certain size to pave a hall, or
3125 square tiles whose dimensions are one inch less. Find the area
of the hall. How many solutions has the equation of this problem?
How many has the problem itself? Explain the apparent discrepancy.
(_Cornell._)
4. A rectangular tract of land, 800 feet long by 600 feet broad, is
divided into four rectangular blocks by two streets of equal width
running through it at right angles. Find the width of the streets,
if together they cover an area of 77,500 square feet. (_M. I. T._)
5. (_a_) The height y to which a ball thrown vertically upward with a
velocity of 100 feet per second rises in x seconds is given
by the formula, y = 100x - 16x^2. In how many seconds will
the ball rise to a height of 144 feet?
(_b_) Draw the graph of the equation y = 100x - 16x^2.
(_College Entrance Board._)
6. Two launches race over a course of 12 miles. The first steams 7-1/2
miles an hour. The other has a start of 10 minutes, runs over the
first half of the course with a certain speed, but increases its
speed over the second half of the course by 2 miles per hour,
winning the race by a minute. What is the speed of the second
launch? Explain the meaning of the negative answer.
(_Sheffield Scientific School._)
7. The circumference of a rear wheel of a certain wagon is 3 feet more
than the circumference of a front wheel. The rear wheel performs
100 fewer revolutions than the front wheel in traveling a distance
of 6000 feet. How large are the wheels? (_Harvard._)
8. A man starts from home to catch a train, walking at the rate of 1
yard in 1 second, and arrives 2 minutes late. If he had walked at
the rate of 4 yards in 3 seconds, he would have arrived 2-1/2
minutes early. Find the distance from his home to the station.
(_College Entrance Board._)
~Simultaneous Quadratics~
1. Two cubical coal bins together hold 280 cubic feet of coal, and
the sum of their lengths is 10 feet. Find the length of each bin.
2. The sum of the radii of two circles is 25 inches, and the
difference of their areas is 125[pi] square inches. Find the radii.
3. The area of a right triangle is 150 square feet, and its hypotenuse
is 25 feet. Find the arms of the triangle.
4. The combined capacity of two cubical tanks is 637 cubic feet, and
the sum of an edge of one and an edge of the other is 13 feet.
(_a_) Find the length of a diagonal of any face of each cube.
(_b_) Find the distance from upper left-hand corner to lower
right-hand corner in either cube.
5. A and B run a mile. In the first heat A gives B a start of 20 yards
and beats him by 30 seconds. In the second heat A gives B a start
of 32 seconds and beats him by 9-5/11 yards. Find the rate at which
each runs. (_Sheffield._)
6. After street improvement it is found that a certain corner
rectangular lot has lost 1/10 of its length and 1/15 of its width.
Its perimeter has been decreased by 28 feet, and the new area is
3024 square feet. Find the reduced dimensions of the lot. (_College
Entrance Board._)
7. A man spends $539 for sheep. He keeps 14 of the flock that he buys,
and sells the remainder at an advance of $2 per head, gaining $28
by the transaction. How many sheep did he buy, and what was the
cost of each? (_Yale._)
8. A boat's crew, rowing at half their usual speed, row 3 miles
downstream and back again in 2 hours and 40 minutes. At full speed
they can go over the same course in 1 hour and 4 minutes. Find the
rate of the crew, and the rate of the current in miles per hour.
(_College Entrance Board._)
9. Find the sides of a rectangle whose area is unchanged if its length
is increased by 4 feet and its breadth decreased by 3 feet, but
which loses one third of its area if the length is increased by 16
feet and the breadth decreased by 10 feet. (_M. I. T._)
COLLEGE ENTRANCE EXAMINATIONS
~UNIVERSITY OF CALIFORNIA~
ELEMENTARY ALGEBRA
1. If a = 4, b = -3, c = 2, and d = -4, find the value of:
(_a_) ab^3 - 3cd^2 + 2(3a - b)(c - 2d).
(_b_) 2a^3 - 3b^4 + (4c^3 + d^3)(4c^2 + d^2).
2. Reduce to a mixed number:
(3a^4 - 4a^3 - 10a^2 + 41a - 28)/(a^2 - 3a + 4).
Simplify:
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 cooerdinates 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.