Question 1

Solve the problem.
-A die is rolled 10 times. Find the probability of rolling no more than 4 ones.

Accepted Answer

A)  Approximately 0.155 
B)  Approximately 0.930 
C)  Approximately 0.054 
D)  Approximately 0.985 
A)  Approximately 0.155 
B)  Approximately 0.930 
C)  Approximately 0.054 
D)  Approximately 0.985

Question 2

Graph the function corresponding to the sequence defined. Use the graph to decide whether the sequence converges or
diverges.
-$$a _ { n } = \frac { 1 } { 44 } e ^ { n }$$

Accepted Answer

A)  Diverges 
B)  Converges 
A)  Diverges 
B)  Converges

Question 3

List the elements in the sample space of the experiment.
-A box contains 10 red cards numbered 1 through 10. List the sample space of picking one card from the box.

Accepted Answer

A)  {100} 
B)  None of these 
C)  {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} 
D)  {10} 
A)  {100} 
B)  None of these 
C)  {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} 
D)  {10}

Question 4

Evaluate the expression.
-$\frac { 10 ! } { 8 ! 2 ! }$

Accepted Answer

A)  1 
B)  45 
C)  10 
D)  0 ! 
A)  1 
B)  45 
C)  10 
D)  0 !

Question 5

Solve the problem.
-During the first day of an epidemic 9000 people get sick, during the second day 18,000 people get sick, and so on with the number of people getting sick increasing by 9000 each day. The epidemic peaks after 30 days with 9000 fewer people getting sick on the 31st day than on the 30th day, and so forth. What is the total number of people who get sick?

Accepted Answer

A)  $8,370,000$ 
B)  $8,235,000$ 
C)  $7,830,000$ 
D)  $8,100,000$ 
A)  $8,370,000$ 
B)  $8,235,000$ 
C)  $7,830,000$ 
D)  $8,100,000$

Question 6

Write the series using summation notation.
-$$1 ^ { 9 } + 2 ^ { 9 } + 3 ^ { 9 } + 4 ^ { 9 } + \ldots$$

Accepted Answer

A)  $\sum _ { \mathrm { k } = 1 } ^ { \infty } \mathrm { k } ^ { 8 }$ 
B)  $\sum _ { k = 1 } ^ { 4 } k ^ { 9 }$ 
C)  $\sum _ { k = 1 } ^ { \infty } \mathrm { k } ^ { 9 }$ 
D)  $\sum _ { k = 0 } ^ { \infty } 9 \mathrm { k }$ 
A)  $\sum _ { \mathrm { k } = 1 } ^ { \infty } \mathrm { k } ^ { 8 }$ 
B)  $\sum _ { k = 1 } ^ { 4 } k ^ { 9 }$ 
C)  $\sum _ { k = 1 } ^ { \infty } \mathrm { k } ^ { 9 }$ 
D)  $\sum _ { k = 0 } ^ { \infty } 9 \mathrm { k }$

Question 7

Write the binomial expansion of the expression.
-$$\left( 2 x ^ { 2 } - 1 \right) ^ { 3 }$$

Accepted Answer

A)  $8 x ^ { 6 } - 12 x ^ { 4 } + 6 x ^ { 2 } - 1$ 
B)  $8 x ^ { 3 } - 12 x ^ { 2 } + 6 x - 1$ 
C)  $\left( 4 x ^ { 4 } - 4 x ^ { 2 } + 1 \right) ^ { 3 }$ 
D)  $16 x ^ { 8 } + 8 x ^ { 6 } - 12 x ^ { 4 } + 6 x ^ { 2 } - 1$ 
A)  $8 x ^ { 6 } - 12 x ^ { 4 } + 6 x ^ { 2 } - 1$ 
B)  $8 x ^ { 3 } - 12 x ^ { 2 } + 6 x - 1$ 
C)  $\left( 4 x ^ { 4 } - 4 x ^ { 2 } + 1 \right) ^ { 3 }$ 
D)  $16 x ^ { 8 } + 8 x ^ { 6 } - 12 x ^ { 4 } + 6 x ^ { 2 } - 1$

Question 8

Provide an appropriate response.
-For $13 , x , 2$ to be an arithmetic sequence, $x$ must be:

Accepted Answer

A)  $\frac { 11 } { 2 }$ 
B)  $\frac { 15 } { 2 }$ 
C)  $\frac { 28 } { 3 }$ 
D)  $\frac { 17 } { 3 }$ 
A)  $\frac { 11 } { 2 }$ 
B)  $\frac { 15 } { 2 }$ 
C)  $\frac { 28 } { 3 }$ 
D)  $\frac { 17 } { 3 }$

Question 9

Evaluate the sum.
-$$\sum _ { \mathrm { i } = 4 } ^ { 14 } ( \mathrm { i } + 8 )$$

Accepted Answer

A)  169 
B)  187 
C)  173 
D)  183 
A)  169 
B)  187 
C)  173 
D)  183

Question 10

Use mathematical induction to prove that the statement is true for every positive integer n.
-If $0 < a < 1$, then $a ^ { n } < 1$.
(Assume that a is a constant.)

Accepted Answer

Answers will vary. One possible proof fo

Question 11

Evaluate the expression.
-$\left( \begin{array} { c } 102 \ 2 \end{array} \right)$

Accepted Answer

A)  100 
B)  10,302 
C)  $\frac { 102 ! } { 100 ! }$ 
D)  5151 
A)  100 
B)  10,302 
C)  $\frac { 102 ! } { 100 ! }$ 
D)  5151

Question 12

Use the summation properties to evaluate the series. The following rules may be needed: $$\begin{array} { l l l } 
\mathrm { n } & \mathrm { n } & \mathrm { n } \
\sum \mathrm { i } = \frac { \mathrm { n } ( \mathrm { n } + 1 ) } { 2 } ; & \sum \mathrm { i } ^ { 2 } = \frac { \mathrm { n } ( \mathrm { n } + 1 ) ( 2 \mathrm { n } + 1 ) } { 6 } ; & \sum \mathrm { i } ^ { 3 } = \frac { \mathrm { n } ^ { 2 } ( \mathrm { n } + 1 ) ^ { 2 } } { 4 } \
\mathrm { i } = 1 & \mathrm { i } = 1 & \mathrm { i } = 1
\end{array}$$
-$$\sum _ { i = 1 } ^ { 6 } 4 i$$

Accepted Answer

A)  21 
B)  300 
C)  84 
D)  42 
A)  21 
B)  300 
C)  84 
D)  42

Question 13

Find the nth term of the geometric sequence.
-$$\frac { 1 } { 2 } , \frac { 1 } { 4 } , \frac { 1 } { 8 } , \ldots ; \mathrm { n } = 8$$

Accepted Answer

A)  $\frac { 1 } { 256 }$ 
B)  $\frac { 1 } { 512 }$ 
C)  $\frac { 1 } { 128 }$ 
D)  $\frac { 1 } { 32 }$ 
A)  $\frac { 1 } { 256 }$ 
B)  $\frac { 1 } { 512 }$ 
C)  $\frac { 1 } { 128 }$ 
D)  $\frac { 1 } { 32 }$

Question 14

Provide an appropriate response.
-Consider determining how many possible phone numbers are in an area code (repeated numbers allowed). Is this a combination, a permutation, or neither?

Accepted Answer

A)  Neither 
B)  Permutation 
C)  Combination 
A)  Neither 
B)  Permutation 
C)  Combination

Question 15

Solve the problem.
-Four boys and three girls are seated, in a row, at random, to watch a play. What is the probability that
a) a girl is seated at each end of the row?
b) one of the boys is seated in the middle seat?
c) boys and girls are seated alternately?

Accepted Answer

A)  a) $\frac { 3 } { 7 }$ b) $\frac { 4 } { 7 }$ c) $\frac { 1 } { 7 }$ 
B)  a) $\frac { 1 } { 7 }$ b) $\frac { 4 } { 7 }$ c) $\frac { 1 } { 35 }$ 
C)  a) $\frac { 2 } { 7 } \quad$ b) $\frac { 1 } { 7 } \quad$ c) $\frac { 1 } { 35 }$ 
D)  a) $\frac { 1 } { 7 }$ b) $\frac { 3 } { 7 }$ c) $\frac { 1 } { 35 }$ 
A)  a) $\frac { 3 } { 7 }$ b) $\frac { 4 } { 7 }$ c) $\frac { 1 } { 7 }$ 
B)  a) $\frac { 1 } { 7 }$ b) $\frac { 4 } { 7 }$ c) $\frac { 1 } { 35 }$ 
C)  a) $\frac { 2 } { 7 } \quad$ b) $\frac { 1 } { 7 } \quad$ c) $\frac { 1 } { 35 }$ 
D)  a) $\frac { 1 } { 7 }$ b) $\frac { 3 } { 7 }$ c) $\frac { 1 } { 35 }$

Question 16

It can be shown that $( 1 + x ) ^ { n } = 1 + n x + \frac { n ( n - 1 ) } { 2 ! } x ^ { 2 } + \frac { n ( n - 1 ) ( n - 2 ) } { 3 ! } x ^ { 3 } \ldots$ is true for any real number n (not just positive
integer values) and any real number x, where $| x | < 1$ Use this series to approximate the given number to the nearest
thousandth.
-$$5 + 2 \cdot 5 + 3 \cdot 5 + \ldots + 5 n = \frac { 5 n ( n + 1 ) } { 2 }$$

Accepted Answer

Answers will vary. One possible proof fo

Question 17

Use mathematical induction to prove that the statement is true for every positive integer n.
-$0.16 n < 0.16 n - 1$

Accepted Answer

Answers will vary. One possible proof fo

Question 18

Decide whether the given sequence is finite or infinite.
-$$\mathrm { a } _ { 1 } = 8 ; \text { for } 2 \leq \mathrm { n } \leq 45 , \mathrm { a } _ { \mathrm { n } } = 4 \cdot \mathrm { a } _ { \mathrm { n } - 1 }$$

Accepted Answer

A)  Infinite 
B)  Finite 
A)  Infinite 
B)  Finite

Question 19

Evaluate the expression.
-$\left( \begin{array} { l } 9 \ 9 \end{array} \right)$

Accepted Answer

A)  9 
B)  181,440 
C)  362,880 
D)  1 
A)  9 
B)  181,440 
C)  362,880 
D)  1

Question 20

Solve the problem.
-One digit from the number 6,212,552 is written on each of seven cards. What is the probability of drawing a card that shows 6, 2, or 1?

Accepted Answer

A)  $\frac { 3 } { 7 }$ 
B)  $\frac { 2 } { 7 }$ 
C)  $\frac { 4 } { 7 }$ 
D)  $\frac { 5 } { 7 }$ 
A)  $\frac { 3 } { 7 }$ 
B)  $\frac { 2 } { 7 }$ 
C)  $\frac { 4 } { 7 }$ 
D)  $\frac { 5 } { 7 }$

Solve the problem. -A die is rolled 10 times. Find the probability of rolling no more than 4 ones.

Graph the function corresponding to the sequence defined. Use the graph to decide whether the sequence converges or diverges. - $a _ { n } = \frac { 1 } { 44 } e ^ { n }$

List the elements in the sample space of the experiment. -A box contains 10 red cards numbered 1 through 10. List the sample space of picking one card from the box.

Evaluate the expression. - $\frac { 10 ! } { 8 ! 2 ! }$

Write the series using summation notation. - $1 ^ { 9 } + 2 ^ { 9 } + 3 ^ { 9 } + 4 ^ { 9 } + \ldots$

Write the binomial expansion of the expression. - $\left( 2 x ^ { 2 } - 1 \right) ^ { 3 }$

Provide an appropriate response. -For $13 , x , 2$ to be an arithmetic sequence, $x$ must be:

Evaluate the sum. - $\sum _ { \mathrm { i } = 4 } ^ { 14 } ( \mathrm { i } + 8 )$

Use mathematical induction to prove that the statement is true for every positive integer n. -If $0 < a < 1$ , then $a ^ { n } < 1$ . (Assume that a is a constant.)

Evaluate the expression. - $\left( \begin{array} { c } 102 \\ 2 \end{array} \right)$

Use the summation properties to evaluate the series. The following rules may be needed: =; =; = =1 =1 =1 - $\sum _ { i = 1 } ^ { 6 } 4 i$

Find the nth term of the geometric sequence. - $\frac { 1 } { 2 } , \frac { 1 } { 4 } , \frac { 1 } { 8 } , \ldots ; \mathrm { n } = 8$

Provide an appropriate response. -Consider determining how many possible phone numbers are in an area code (repeated numbers allowed). Is this a combination, a permutation, or neither?

Solve the problem. -Four boys and three girls are seated, in a row, at random, to watch a play. What is the probability that a) a girl is seated at each end of the row? b) one of the boys is seated in the middle seat? c) boys and girls are seated alternately?

Use mathematical induction to prove that the statement is true for every positive integer n. - $0.16 n < 0.16 n - 1$

Decide whether the given sequence is finite or infinite. - $\mathrm { a } _ { 1 } = 8 ; \text { for } 2 \leq \mathrm { n } \leq 45 , \mathrm { a } _ { \mathrm { n } } = 4 \cdot \mathrm { a } _ { \mathrm { n } - 1 }$

Evaluate the expression. - $\left( \begin{array} { l } 9 \\ 9 \end{array} \right)$

Solve the problem. -One digit from the number 6,212,552 is written on each of seven cards. What is the probability of drawing a card that shows 6, 2, or 1?

Equations and Inequalities

Graphs and Functions

Polynomial and Rational Functions

Inverse, Exponential, and Logarithmic Functions

Trigonometric Functions

The Circular Functions and Their Graphs

Trigonometric Identities and Equations

Applications of Trigonometry

Systems and Matrices

Analytic Geometry

Review of Basic Concepts

Filters

Exam 11: Further Topics in Algebra

Solve the problem. -A die is rolled 10 times. Find the probability of rolling no more than 4 ones.

Graph the function corresponding to the sequence defined. Use the graph to decide whether the sequence converges or diverges. - an=144ena _ { n } = \frac { 1 } { 44 } e ^ { n }an​=441​en

List the elements in the sample space of the experiment. -A box contains 10 red cards numbered 1 through 10. List the sample space of picking one card from the box.

Evaluate the expression. - 10!8!2!\frac { 10 ! } { 8 ! 2 ! }8!2!10!​

Write the series using summation notation. - 19+29+39+49+…1 ^ { 9 } + 2 ^ { 9 } + 3 ^ { 9 } + 4 ^ { 9 } + \ldots19+29+39+49+…

Write the binomial expansion of the expression. - (2x2−1)3\left( 2 x ^ { 2 } - 1 \right) ^ { 3 }(2x2−1)3

Provide an appropriate response. -For 13,x,213 , x , 213,x,2 to be an arithmetic sequence, xxx must be:

Evaluate the sum. - ∑i=414(i+8)\sum _ { \mathrm { i } = 4 } ^ { 14 } ( \mathrm { i } + 8 )∑i=414​(i+8)

Use mathematical induction to prove that the statement is true for every positive integer n. -If 0<a<10 < a < 10<a<1 , then an<1a ^ { n } < 1an<1 . (Assume that a is a constant.)

Evaluate the expression. - (1022)\left( \begin{array} { c } 102 \\ 2 \end{array} \right)(1022​)

Use the summation properties to evaluate the series. The following rules may be needed: =; =; = =1 =1 =1 - ∑i=164i\sum _ { i = 1 } ^ { 6 } 4 i∑i=16​4i

Find the nth term of the geometric sequence. - 12,14,18,…;n=8\frac { 1 } { 2 } , \frac { 1 } { 4 } , \frac { 1 } { 8 } , \ldots ; \mathrm { n } = 821​,41​,81​,…;n=8

Provide an appropriate response. -Consider determining how many possible phone numbers are in an area code (repeated numbers allowed). Is this a combination, a permutation, or neither?

Solve the problem. -Four boys and three girls are seated, in a row, at random, to watch a play. What is the probability that a) a girl is seated at each end of the row? b) one of the boys is seated in the middle seat? c) boys and girls are seated alternately?

Use mathematical induction to prove that the statement is true for every positive integer n. - 0.16n<0.16n−10.16 n < 0.16 n - 10.16n<0.16n−1

Decide whether the given sequence is finite or infinite. - a1=8; for 2≤n≤45,an=4⋅an−1\mathrm { a } _ { 1 } = 8 ; \text { for } 2 \leq \mathrm { n } \leq 45 , \mathrm { a } _ { \mathrm { n } } = 4 \cdot \mathrm { a } _ { \mathrm { n } - 1 }a1​=8; for 2≤n≤45,an​=4⋅an−1​

Evaluate the expression. - (99)\left( \begin{array} { l } 9 \\ 9 \end{array} \right)(99​)

Solve the problem. -One digit from the number 6,212,552 is written on each of seven cards. What is the probability of drawing a card that shows 6, 2, or 1?

Equations and Inequalities

Graphs and Functions

Polynomial and Rational Functions

Inverse, Exponential, and Logarithmic Functions

Trigonometric Functions

The Circular Functions and Their Graphs

Trigonometric Identities and Equations

Applications of Trigonometry

Systems and Matrices

Analytic Geometry

Review of Basic Concepts

Filters

Graph the function corresponding to the sequence defined. Use the graph to decide whether the sequence converges or diverges. - $a _ { n } = \frac { 1 } { 44 } e ^ { n }$

Evaluate the expression. - $\frac { 10 ! } { 8 ! 2 ! }$

Write the series using summation notation. - $1 ^ { 9 } + 2 ^ { 9 } + 3 ^ { 9 } + 4 ^ { 9 } + \ldots$

Write the binomial expansion of the expression. - $\left( 2 x ^ { 2 } - 1 \right) ^ { 3 }$

Provide an appropriate response. -For $13 , x , 2$ to be an arithmetic sequence, $x$ must be:

Evaluate the sum. - $\sum _ { \mathrm { i } = 4 } ^ { 14 } ( \mathrm { i } + 8 )$

Use mathematical induction to prove that the statement is true for every positive integer n. -If $0 < a < 1$ , then $a ^ { n } < 1$ . (Assume that a is a constant.)

Evaluate the expression. - $\left( \begin{array} { c } 102 \\ 2 \end{array} \right)$

Use the summation properties to evaluate the series. The following rules may be needed: =; =; = =1 =1 =1 - $\sum _ { i = 1 } ^ { 6 } 4 i$

Find the nth term of the geometric sequence. - $\frac { 1 } { 2 } , \frac { 1 } { 4 } , \frac { 1 } { 8 } , \ldots ; \mathrm { n } = 8$

Use mathematical induction to prove that the statement is true for every positive integer n. - $0.16 n < 0.16 n - 1$

Decide whether the given sequence is finite or infinite. - $\mathrm { a } _ { 1 } = 8 ; \text { for } 2 \leq \mathrm { n } \leq 45 , \mathrm { a } _ { \mathrm { n } } = 4 \cdot \mathrm { a } _ { \mathrm { n } - 1 }$

Evaluate the expression. - $\left( \begin{array} { l } 9 \\ 9 \end{array} \right)$