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mathematics
linear algebra

Questions and Answers of Linear Algebra

Use mathematical induction to prove each of the following.a.b. c. The sum of n terms of an arithmetic sequence: a1 + (a1 + d) + (a1 + 2d) + ....... + [a1 + (n - 1) d] = n / 2 [2a1 + (n - 1)d)
Martin received $104 in simple interest one year from three investments. Part is invested at 1.5%, part at 2%, and part at 3%. The amount invested at 2% is twice the amount invested at 1.5%. There is
Use mathematical induction to prove each of the following.a. The sum of n terms of a geometric sequence:b. x + y is a factor of x2n - y2n.
Prove each of the following for any complex numbers z1, z2,........ , zn, where i2 = -1 and z is the conjugate of z.
There are three pegs on a board. On one peg are n disks, each smaller than the one on which it rests. The problem is to move this pile of disks to another peg. The final order must be the same, but
Use mathematical induction to prove each of the following. a. 2 + 4 + 6 + g + 2n = n(n + 1) b. 4 + 8 + 12 + g + 4n = 2n(n + 1) c. 1 + 5 + 9 +....... + (4n - 3) = n (2n - 1)
How many permutations are there of the letters in each of the following words, if all the letters are used without repetition? a. CREDIT b. FRUIT c. EDUCATION
A program is planned to have 5 musical numbers and 4 speeches. In how many ways can this be done if a musical numberand a speech are to alternate and a musical number is to come first?
A professor is going to grade her 24 students on a curve. She will give 3 A's, 5 B's, 9 C's, 4 D's, and 3 F's. In how many ways can she do this?
How many 7-digit phone numbers can be formed with the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9, assuming that the first number cannot be 0 or 1? Accordingly, how many telephone numbers can there be
How many distinguishable code symbols can be formed from the letters of the word BUSINESS? BIOLOGY? MATHEMATICS?
Suppose the expression a2b3c 4 is rewritten without exponents. In how many distinguishable ways can this be done?
A penny, a nickel, a dime, and a quarter are arranged in a straight line.a) Considering just the coins, in how many ways can they be lined up? b) Considering the coins and heads and tails, in how
How many code symbols can be formed using 5 of the 6 letters A, B, C, D, E, F if the letters: a) Are not repeated? b) Can be repeated? c) Are not repeated but must begin with D? d) Are not repeated
A state forms its license plates by first listing a number that corresponds to the county in which the owner of the car resides. (The names of the counties are alphabetized and the number is its
A U.S. postal zip code is a five-digit number. a) How many zip codes are possible if any of the digits 0 to 9 can be used? b) If each post office has its own zip code, how many possible post offices
A social security number is a 9-digit number like 243-47-0825. a) How many different social security numbers can there be? b) There are about 310 million people in the United States. Can each person
Find the zero(s) of the function. a. f(x) = 4x - 9 b. f(x) = x2 + x - 6 c. f(x) = 2x2 - 3x - 1
Show that n! = n(n - 1)(n - 2)(n - 3)!.
In a singleelimination sports tournament consisting of n teams, a team is eliminated when it loses one game. How many games are required to complete the tournament?
In a double elimination softball tournament consisting of n teams, a team is eliminated when it loses two games. At most, how many games are required to complete the tournament?
In each of the following exercises, give an expression for the answer using permutation notation, combination notation, factorial notation, or other operations. Then evaluate. There are 23 students
In each of the following exercises, give an expression for the answer using permutation notation, combination notation, factorial notation, or other operations. Then evaluate. League Games. How many
In each of the following exercises, give an expression for the answer using permutation notation, combination notation, factorial notation, or other operations. Then evaluate. Test Options. On a
In each of the following exercises, give an expression for the answer using permutation notation, combination notation, factorial notation, or other operations. Then evaluate. Of the first 10
In each of the following exercises, give an expression for the answer using permutation notation, combination notation, factorial notation, or other operations. Then evaluate.
Burt Baskin and Irv Robbins began making ice cream in 1945. Initially they developed 31 flavors-one for each day of the month. (Source: Baskin- Robbins) a) How many 2-dip cones are possible using the
A full house in poker consists of three of a kind and a pair (two of a kind). How many full houses are there that consist of 3 aces and 2 queens? (See Section 8.8 for a description of a 52-card deck.)
Find the indicated term of the binomial expansion. a. 3rd; (a + b)7 b. 6th; (x + y)8 c. 6th; (x - y)10
Determine the number of subsets of each of the following. a. A set of 7 elements b. A set of 6 members c. The set of letters of the Greek alphabet, which contains 24 letters
Expand each of the following, where i2 = -1. a. (3 + i2)5 b. (1 + i) 6 c. √2 - i)4
Given that f1x2 = x2 + 1 and g1x2 = 2x - 3, find each of the following. 1. (f + g)(x) 2. (fg) (x) 3. (f °g) (x)
In a survey conducted by the authors, 100 people were polled and asked to select anumber from 1 to 5. The results are shown in the table below.a) What is the probability that the number chosen is 1?
Suppose that a card is drawn from a well-shuffled deck of 52 cards. What is the probability of drawing each of the following? a) A 7 b) A jack or a king c) A black ace d) A red card
Suppose that 3 cards are drawn from a well-shuffled deck of 52 cards. What is the probability that they are all aces?
Suppose that 4 cards are drawn from a well-shuffled deck of 52 cards. What is the probability that they are all red?
The sales force of a business consists of 10 men and 10 women. A production unit of 4 people is set up at random. What is the probability that 2 men and 2 women are chosen?
A sack contains 7 dimes, 5 nickels, and 10 quarters. Eight coins are drawn at random. What is the probability of getting 4 dimes, 3 nickels, and 1 quarter?
Suppose that 5 cards are drawn from a deck of 52 cards. What is the probability of drawing each of the following? 1. 3 sevens and 2 kings 2. 5 aces 3. 5 spades
Three coins are flipped. An outcome might be HTH. a) Find the sample space. What is the probability of getting each of the following? b) Exactly one head c) At most two tails d) At least one head e)
Made by the Tootsie Industries of Chicago, Illinois, Mason Dots® is a gumdrop candy. A box was opened by the authors and was found to contain the following number of gumdrops: Orange 9 Lemon
An American roulette wheel contains 38 slots numbered 00, 0, 1, 2, 3, c, 35, 36. Eighteen of the slots numbered 1-36 are colored red and 18 are colored black. The 00 and 0 slots are considered to be
In each of Exercises 29-36, fill in the blank with the correct term. Some of the given choices will be used more than once. Others will not be used.1. A(n)________________ of a function is an input
In experimental studies, the U.S. Postal Service has found that the probability that a piece of advertising is opened and read is 78%. A business sends out 15,000 pieces of advertising. How many of
Suppose that 5 cards are drawn from a deck of 52 cards. For the following exercises, give both a reasoned expression and an answer. 37. Two Pairs. A hand with two pairs is a hand like Q-Q-3-3-A. a)
An experiment was conducted by the authors to determine the relative occurrence of various letters of the English alphabet. The front page of a newspaper was considered. In all, there were 9136
Suppose that we select, without looking, one marble from a bag containing 4 red marbles and 10 green marbles. What is the probability of selecting each of the following? a) A red marble b) A green
Suppose that we select, without looking, one coin from a bag containing 5 pennies, 3 dimes, and 7 quarters. What is the probability of selecting each of the following? a) A dime b) A quarter c) A
Suppose that a card is drawn from a well-shuffled deck of 52 cards. What is the probability of drawing each of the following? a) A queen b) An ace or a 10 c) A heart d) A black 6
Determine whether the statement is true or false.1. A sequence is a function. [8.1]2. An infinite geometric series with r = -1 has a limit. [8.3]3. Permutations involve order and arrangements of
Find the sum of each infinite geometric series, if it exists. [8.3] 1. 25 + 27.5 + 30.25 + 33.275 + g 2. 0.27 + 0.0027 + 0.000027 + g 3. 1 / 2 - 1 /6 + 1 / 18 - g
A golf ball is dropped from a height of 30 ft to the pavement. It always rebounds three-fourths of the distance that it drops. How far (up and down) will the ball have traveled when it hits the
To create a college fund, a parent makes a sequence of 18 yearly deposits of $2000 each in a savings account on which interest is compounded annually at 2.8%
Suppose you receive 10¢ on the first day of the year, 12¢ on the 2nd day, 14¢ on the 3rd day, and so on. a) How much will you receive on the 365th day? [8.2] b) What is the sum of these 365 gifts?
Suppose the government is making a $24,000,000,000 expenditure for travel to Mars. If 73% of this amount is spent again, and so on, what is the total effect on the economy? [8.3]
Use mathematical induction to prove each of the following.1. For every natural number n,2. For every natural number n, 3. For every natural number n Ãš 2,
How many code symbols can be formed using 5 out of 6 of the letters of G, H, I, J, K, L if the letters: a) Cannot be repeated? [8.5] b) Can be repeated? [8.5] c) Cannot be repeated but must begin
Find the 12th term of (2a - b)18. Do not multiply out the factorials. [8.7]
What is the probability of getting a 10 on a roll of a pair of dice? on a roll of 1 die? [8.8]
From a deck of 52 cards, 1 card is drawn at random. What is the probability that it is a club? [8.8]
From a deck of 52 cards, 3 are drawn at random without replacement. What is the probability that 2 are aces and 1 is a king? [8.8]
Three people were running for mayor in an election campaign. A poll was conducted to see which candidate was favored. During the polling, 86 favored candidate A, 97 favored B, and 23 favored C.
The table below lists the number of pounds of American cheese consumed per capita for selected years.a) Find a linear sequence function an = an + b that models the data. Let n represent the number of
Which of the following is the 25th term of the arithmetic sequence 12, 10, 8, 6, c? [8.2] A. -38 B. -36 C. 32 D. 60
What is the probability of getting a total of 4 on a roll of a pair of dice? [8.8] A. 1 / 12 B. 1 / 9 C. 1 / 6 D. 5 / 36
The graph of the sequence whose general term is an = n - 1 is which of the following? [8.1]
Suppose that a1, a2........., an and b1, b2.........., bn are geometric sequences. Prove that c1, c2, c, cn........ is a geometric sequence, where cn = anbn. [8.3]
Suppose that a1, a2, c, an is an arithmetic sequence. Is b1, b2, c, bn an arithmetic sequence if: a) bn = |an |? [8.2] b) bn = an + 8? [8.2] c) bn = 7an? [8.2] d) bn = 1 / an ? [8.2] e) bn = log
The zeros of this polynomial function form an arithmetic sequence. Find them. [8.2] f 1x2 = x4 - 4x3 - 4x2 + 16x
Write the first 3 terms of the infinite geometric series with r = -1 / 3 and S∞ = 38. [8.3]
Solve for a:
How "long" is 15? Suppose you own 15 books and decide to make up all the possible arrangements of the books on a shelf. About how long, in years, would it take you if you were to make one arrangement
Give an explanation that you might use with a fellow student to explain that
Discuss the advantages and disadvantages of each method of finding a binomial expansion. Give examples of when you might use one method rather than the other. [8.7]
2/3, 6, √3, - 2.45, 6√26, 18., - 11, 3√27, 5 1/6, 7.151551555 . . . . , - √35, 5√3, - 8/7, 0, √16. a. Which are rational numbers? b. Which are natural numbers? c. Which are irrational
Write interval notation. Then graph the interval. (a) {x| -3 < x ≤ - 5} (b) {x| -2 < x < 6} (c) {x| -3 < x ≤ - 1}
Write interval notation for the graph.(a)(b) (c)
N = the set of natural numbers, W = the set of whole numbers, Z = the set of integers, Q = the set of rational numbers, I = the set of irrational numbers, and R = the set of real numbers. Classify
Name the property illustrated by the sentence. (a) 3 + y = y + 3 (b) 6(xz) = (6x)z (c) -3 ∙ 1 = -3
Find the distance between the given pair of points on the number line. (a) -8, 14 (b) -5.2, 0 (c) -9, -3
Between any two (different) real numbers there are many other real numbers. Find each of the following. Answers may vary.(a) An irrational number between 0.124 and 0.125(b) A rational number between
Write an equivalent expression without negative exponents. (a) 3-7 (b) 1 / (5.9)-4 (c) x-5 / y-4
Francisco wants to have $200,000 accumulated in a retirement account by age 70. If he starts making monthly deposits to the plan at age 30 and can count on an interest rate of 4.5%, compounded
Assume that all exponents are integers, all denominators are nonzero, and zero is not raised to a nonpositive power. (a) (xt ∙ x3t)2 (b) (xy ∙ x-y)3 (c) (ta+x ∙ tx-a)4
Convert to scientific notation. (a) 16,500,000 (b) 359,000 (c) 0.000000437
The mass of a neutron is about 0.00000000000000000000000000167 kg.
It is estimated that $37,800,000,000 will be spent on information technology security systems in 2013 (Source: IDC).
Convert to decimal notation. (a) 7.6 × 105 (b) 3.4 × 10-6 (c) 1.09 × 10-7
The amount of solid waste generated in the United States in a recent year was 2.319 × 108 tons (Source: Franklin Associates, Ltd.).
The mass of a proton is about 1.67 × 10-24 g.
Compute. Write the answer using scientific notation. (a) (4.2 × 107) (3.2 × 10-2) (b) (8.3 × 10-15) (7.7 × 104) (c) (2.6 × 10-18) (8.5 × 107)
Write the answer using scientific notation. It is estimated that there were 51.2 billion pieces of trash on 76 million mi of U.S. roadways in a recent year (Source: Keep America Beautiful). On
A nanometer is 0.000000001 m. Scientists have developed optical nanowires to transmit light wave's short distances. A nanowire with a diameter of 360 nanometers has been used in experiments on the
The tiny country of Monaco has an area of 0.75 mi2. It is estimated that the population of Monaco will be 38,000 in 2050. Find the number of people per square mile in 2050.
The 17.6-mi long Chesapeake Bay Bridge-Tunnel was completed in 1964. Construction costs were $210 million. Find the average cost per mile.
The nearest star, Alpha Centauri C, is about 4.22 light-years from Earth. One light-year is the distance that light travels in 1 year and is about 5.88 × 1012 mi. How many miles is it from Earth to
One parsec is about 3.26 light-years and 1 light-year is about 5.88 × 1012 miles. Find the number of miles in 1 parsec.
One gram of radium produces 37 billion disintegrations per second. How many disintegrations are produced in 1 hour?
The average distance from Earth to the sun is 93 million mi. About how far does Earth travel in a yearly orbit? (Assume a circular orbit.)
Suppose that $3225 is invested at 3.1%, compounded semiannually. How much is in the account at the end of 4 years?

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