Chapter 11

Show that \(B\) is the multiplicative inverse of \(A,\) where $$ A=\left[\begin{array}{ll} {2} & {3} \\ {1} & {2} \end{array}\right] \text { and } B=\left[\begin{array}{rr} {2} & {-3} \\ {-1} & {2} \end{array}\right] $$

Use this information to solve Exercises \(47-48 .\) The mathematics department of a college has 8 male professors, 11 female professors, 14 male teaching assistants, and 7 female teaching assistants. If a person is selected at random from the group, find the probability that the selected person is a professor or a female.

Express each repeating decimal as a fraction in lowest terms. $$ 0 . \overline{83}=\frac{83}{100}+\frac{83}{10,000}+\frac{83}{1,000,000}+\cdots $$

Find the term indicated in each expansion. $$(x+2 y)^{10} ; the\quad term\quad containing\quad y^{6}$$

express each sum using summation notation. Use 1 as the lower limit of summation and i for the index of summation. $$ 1+2+3+\dots+40 $$

How many arrangements can be made using four of the letters of the word COMBINE if no letter is to be used more than once?

Write out the first three terms and the last term. Then use the formula for the sum of the first n terms of an arithmetic sequence to find the indicated sum. $$ \sum_{i=1}^{40}(-2 i+6) $$

Graph the hyperbola whose equation is $$ 25 x^{2}-16 y^{2}-100 x-96 y-444=0 $$ Where are the foci located? What are the equations of the asymptotes?

A single die is rolled twice. Find the probability of rolling a 2 the first time and a 3 the second time.

Express each repeating decimal as a fraction in lowest terms. $$ 0 . \overline{257} $$

Use the Binomial Theorem to expand each expression and write the result in simplified form. $$ \left(x^{3}+x^{-2}\right)^{4} $$

express each sum using summation notation. Use 1 as the lower limit of summation and i for the index of summation. $$ \frac{1}{2}+\frac{2}{3}+\frac{3}{4}+\dots+\frac{14}{14+1} $$

An election ballot asks voters to select three city commissioners from a group of six candidates. In how many ways can this be done?

Write out the first three terms and the last term. Then use the formula for the sum of the first n terms of an arithmetic sequence to find the indicated sum. $$ \sum_{i=1}^{100} 4 i $$

Verify the identity: $$\cot x+\tan x=\csc x \sec x$$

A single die is rolled twice. Find the probability of rolling a 5 the first time and a 1 the second time.

Express each repeating decimal as a fraction in lowest terms. $$ 0 . \overline{529} $$

Use the Binomial Theorem to expand each expression and write the result in simplified form. $$ \left(x^{2}+x^{-3}\right)^{4} $$

express each sum using summation notation. Use 1 as the lower limit of summation and i for the index of summation. $$ \frac{1}{3}+\frac{2}{4}+\frac{3}{5}+\dots+\frac{16}{16+2} $$

A four-person committee is to be elected from an organization’s membership of 11 people. How many different committees are possible?

Write out the first three terms and the last term. Then use the formula for the sum of the first n terms of an arithmetic sequence to find the indicated sum. $$ \sum_{i=1}^{50}(-4 i) $$

Will help you prepare for the material covered in the next section. Each exercise involves observing a pattern in the expanded form of the binomial expression \((a+b)^{n}\). \((a+b)^{1}=a+b\) \((a+b)^{2}=a^{2}+2 a b+b^{2}\) \((a+b)^{3}=a^{3}+3 a^{2} b+3 a b^{2}+b^{3}\) \((a+b)^{4}=a^{4}+4 a^{3} b+6 a^{2} b^{2}+4 a b^{3}+b^{4}\) \((a+b)^{5}=a^{5}+5 a^{4} b+10 a^{3} b^{2}+10 a^{2} b^{3}+5 a b^{4}+b^{5}\) Describe the pattern for the exponents on a.

A single die is rolled twice. Find the probability of rolling an even number the first time and a number greater than 2 the second time.

The general term of a sequence is given. Determine whether the sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference; if it is geometric, find the common ratio. $$ a_{n}=n+5 $$

Use the Binomial Theorem to expand each expression and write the result in simplified form. $$ \left(x^{\frac{1}{3}}-x^{-\frac{1}{3}}\right)^{3} $$

express each sum using summation notation. Use 1 as the lower limit of summation and i for the index of summation. $$ 4+\frac{4^{2}}{2}+\frac{4^{3}}{3}+\dots+\frac{4^{n}}{n} $$

Of 12 possible books, you plan to take 4 with you on vacation. How many different collections of 4 books can you take?

Will help you prepare for the material covered in the next section. Each exercise involves observing a pattern in the expanded form of the binomial expression \((a+b)^{n}\). \((a+b)^{1}=a+b\) \((a+b)^{2}=a^{2}+2 a b+b^{2}\) \((a+b)^{3}=a^{3}+3 a^{2} b+3 a b^{2}+b^{3}\) \((a+b)^{4}=a^{4}+4 a^{3} b+6 a^{2} b^{2}+4 a b^{3}+b^{4}\) \((a+b)^{5}=a^{5}+5 a^{4} b+10 a^{3} b^{2}+10 a^{2} b^{3}+5 a b^{4}+b^{5}\) Describe the pattern for the exponents on b.

A single die is rolled twice. Find the probability of rolling an odd number the first time and a number less than 3 the second time.

The general term of a sequence is given. Determine whether the sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference; if it is geometric, find the common ratio. $$ a_{n}=n-3 $$

Use the Binomial Theorem to expand each expression and write the result in simplified form. $$ \left(x^{\frac{2}{3}}-\frac{1}{\sqrt[3]{x}}\right)^{3} $$

express each sum using summation notation. Use 1 as the lower limit of summation and i for the index of summation. $$ \frac{1}{9}+\frac{2}{9^{2}}+\frac{3}{9^{3}}+\dots+\frac{n}{9^{n}} $$

There are 14 standbys who hope to get seats on a flight, but only 6 seats are available on the plane. How many different ways can the 6 people be selected?

Will help you prepare for the material covered in the next section. Each exercise involves observing a pattern in the expanded form of the binomial expression \((a+b)^{n}\). \((a+b)^{1}=a+b\) \((a+b)^{2}=a^{2}+2 a b+b^{2}\) \((a+b)^{3}=a^{3}+3 a^{2} b+3 a b^{2}+b^{3}\) \((a+b)^{4}=a^{4}+4 a^{3} b+6 a^{2} b^{2}+4 a b^{3}+b^{4}\) \((a+b)^{5}=a^{5}+5 a^{4} b+10 a^{3} b^{2}+10 a^{2} b^{3}+5 a b^{4}+b^{5}\) Describe the pattern for the sum of the exponents on the variables in each term.

If you toss a fair coin six times, what is the probability of getting all heads?

The general term of a sequence is given. Determine whether the sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference; if it is geometric, find the common ratio. $$ a_{n}=2^{n} $$

Find\(\frac{f(x+h)-f(x)}{h}\) and simplify. $$ f(x)=x^{4}+7 $$

express each sum using summation notation. Use 1 as the lower limit of summation and i for the index of summation. $$ 1+3+5+\dots+(2 n-1) $$

You volunteer to help drive children at a charity event to the zoo, but you can fit only 8 of the 17 children present in your van. How many different groups of 8 children can you drive?

If you toss a fair coin seven times, what is the probability of getting all tails?

The general term of a sequence is given. Determine whether the sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference; if it is geometric, find the common ratio. $$ a_{n}=\left(\frac{1}{2}\right)^{n} $$

Find\(\frac{f(x+h)-f(x)}{h}\) and simplify. $$ f(x)=x^{5}+8 $$

express each sum using summation notation. Use 1 as the lower limit of summation and i for the index of summation. $$ a+a r+a r^{2}+\dots+a r^{n-1} $$

The probability that South Florida will be hit by a major hurricane (category 4 or 5 ) in any single year is \(\frac{1}{16}\) (Source: National Hurricane Center) a. What is the probability that South Florida will be hit by a major hurricane two years in a row? b. What is the probability that South Florida will be hit by a major hurricane in three consecutive years? c. What is the probability that South Florida will not be hit by a major hurricane in the next ten years? d. What is the probability that South Florida will be hit by a major hurricane at least once in the next ten years?

The general term of a sequence is given. Determine whether the sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference; if it is geometric, find the common ratio. $$ a_{n}=n^{2}+5 $$

Find the middle term in the expansion of \(\left(\frac{3}{x}+\frac{x}{3}\right)^{10}\)

express each sum using summation notation. Use a lower limit of summation of your choice and k for the index of summation. $$ 5+7+9+11+\dots+31 $$

To win at LOTTO in the state of Florida, one must correctly select 6 numbers from a collection of 53 numbers (1 through 53). The order in which the selection is made does not matter. How many different selections are possible?

Explaining the Concepts Describe the difference between theoretical probability and empirical probability.

The general term of a sequence is given. Determine whether the sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference; if it is geometric, find the common ratio. $$ a_{n}=n^{2}-3 $$