Question 1

Solve the problem.
-How many different three-digit numbers can be written using digits from the set $\{ 1,2,3,4,5 \}$ without any repeating digits?

Accepted Answer

A)  120 
B)  60 
C)  10 
D)  20 
A)  120 
B)  60 
C)  10 
D)  20

Question 2

Write the first n terms of the given arithmetic sequence (the value of n is indicated in the question).
-$$a _ { 13 } = - 17 x - 9 , a _ { 15 } = - 15 x - 1$$

Accepted Answer

A)  $a _ { n } = ( - 29 + n ) x - 57 + 4 n , a _ { 6 } = - 23 x - 33$ 
B)  $a _ { n } = ( - 29 - n ) x - 57 - 4 n , a _ { 6 } = - 35 x - 81$ 
C)  $a _ { n } = ( - 30 + n ) x - 61 + 4 n , a _ { 6 } = - 24 x - 37$ 
D)  $a _ { n } = ( - 29 + n ) x - 53 + 4 n , a _ { 6 } = - 23 x - 77$ 
A)  $a _ { n } = ( - 29 + n ) x - 57 + 4 n , a _ { 6 } = - 23 x - 33$ 
B)  $a _ { n } = ( - 29 - n ) x - 57 - 4 n , a _ { 6 } = - 35 x - 81$ 
C)  $a _ { n } = ( - 30 + n ) x - 61 + 4 n , a _ { 6 } = - 24 x - 37$ 
D)  $a _ { n } = ( - 29 + n ) x - 53 + 4 n , a _ { 6 } = - 23 x - 77$

Question 3

Find the common ratio r for the given infinite geometric sequence.
-A pendulum bob swings 6.0 cm on its first oscillation. On each subsequent oscillation the bob travels $\frac { 1 } { 3 }$ of the previous distance. Find the total distance the bob travels before coming to rest.

Accepted Answer

A)  13.0 cm 
B)  9.0 cm 
C)  13.3 cm 
D)  12.0 cm 
A)  13.0 cm 
B)  9.0 cm 
C)  13.3 cm 
D)  12.0 cm

Question 4

List the elements in the sample space of the experiment.
-A lottery uses balls numbered 1 through 35 . List the sample space of the event picking one even-numbered ball.

Accepted Answer

A)  $\{ 1,2,3 , \ldots , 35 \}$ 
B)  $\{ 2 \}$ 
C)  $\{ 2,4,6 , \ldots , 34 \}$ 
D)  $\{ 34 \}$ 
A)  $\{ 1,2,3 , \ldots , 35 \}$ 
B)  $\{ 2 \}$ 
C)  $\{ 2,4,6 , \ldots , 34 \}$ 
D)  $\{ 34 \}$

Question 5

Graph the function corresponding to the sequence defined. Use the graph to decide whether the sequence converges or
diverges.
-$$a _ { n } = \left( \frac { 1 } { 20 } + n \right) ^ { 1 / n }$$

Accepted Answer

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

Question 6

Write the binomial expansion of the expression.
-$$\left( \frac { 1 } { 3 } x + 4 \right) ^ { 3 }$$

Accepted Answer

A)  $\frac { 1 } { 27 } x ^ { 6 } + \frac { 8 } { 9 } x ^ { 3 } + 64$ 
B)  $\frac { 1 } { 9 } x ^ { 3 } + \frac { 4 } { 3 } x ^ { 2 } + 64$ 
C)  $\frac { 1 } { 9 } x ^ { 3 } + \frac { 4 } { 3 } x ^ { 2 } + 16 x + 64$ 
D)  $\frac { 1 } { 27 } x ^ { 3 } + \frac { 4 } { 3 } x ^ { 2 } + 16 x + 64$ 
A)  $\frac { 1 } { 27 } x ^ { 6 } + \frac { 8 } { 9 } x ^ { 3 } + 64$ 
B)  $\frac { 1 } { 9 } x ^ { 3 } + \frac { 4 } { 3 } x ^ { 2 } + 64$ 
C)  $\frac { 1 } { 9 } x ^ { 3 } + \frac { 4 } { 3 } x ^ { 2 } + 16 x + 64$ 
D)  $\frac { 1 } { 27 } x ^ { 3 } + \frac { 4 } { 3 } x ^ { 2 } + 16 x + 64$

Question 7

Solve the problem.
-An ordinary die is tossed. What are the odds in favor of the die showing a number less than 3 ?

Accepted Answer

A)  1 to 3 
B)  1 to 2 
C)  1 to 1 
D)  2 to 1 
A)  1 to 3 
B)  1 to 2 
C)  1 to 1 
D)  2 to 1

Question 8

Solve the problem.
-Three noncollinear points determine a triangle. How many triangles can be formed with 8 noncollinear points?

Accepted Answer

A)  24 
B)  6720 
C)  56 
D)  336 
A)  24 
B)  6720 
C)  56 
D)  336

Question 9

Use mathematical induction to prove that the statement is true for every positive integer n.
-$5 ^ { n } > 1$

Accepted Answer

Answers will vary. One possible proof fo

Question 10

Write out the first five terms of the sequence.
-$$a _ { n } = n ^ { 2 } - n$$

Accepted Answer

A)  $2,6,12,20,30$ 
B)  $1,4,9,16,25$ 
C)  $0,2,6,12,20$ 
D)  $0,3,8,15,24$ 
A)  $2,6,12,20,30$ 
B)  $1,4,9,16,25$ 
C)  $0,2,6,12,20$ 
D)  $0,3,8,15,24$

Question 11

Find the sum of the geometric series.
-$\sum _ { \mathrm { i } = 1 } ^ { 5 } 2 ( 4 ) ^ { \mathrm { i } }$

Accepted Answer

A)  40 
B)  5140 
C)  264 
D)  2728 
A)  40 
B)  5140 
C)  264 
D)  2728

Question 12

Solve the problem.
-A musician plans to perform 9 selections. In how many ways can she arrange the musical selections?

Accepted Answer

A)  162 
B)  81 
C)  181,440 
D)  362,880 
A)  162 
B)  81 
C)  181,440 
D)  362,880

Question 13

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

Accepted Answer

A)  $16 x ^ { 8 } + 8 x ^ { 6 } - 24 x ^ { 4 } + 24 x ^ { 2 } - 8$ 
B)  $8 x ^ { 6 } - 24 x ^ { 4 } + 24 x ^ { 2 } - 8$ 
C)  $8 x ^ { 3 } - 24 x ^ { 2 } + 24 x - 8$ 
D)  $\left( 4 x ^ { 4 } - 8 x ^ { 2 } + 4 \right) ^ { 3 }$ 
A)  $16 x ^ { 8 } + 8 x ^ { 6 } - 24 x ^ { 4 } + 24 x ^ { 2 } - 8$ 
B)  $8 x ^ { 6 } - 24 x ^ { 4 } + 24 x ^ { 2 } - 8$ 
C)  $8 x ^ { 3 } - 24 x ^ { 2 } + 24 x - 8$ 
D)  $\left( 4 x ^ { 4 } - 8 x ^ { 2 } + 4 \right) ^ { 3 }$

Question 14

Solve the problem.
-Four boys and three girls are seated, in a row, at random, to watch a play. What is the probability th
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) $\frac { 1 } { 7 }$ b) $\frac { 4 } { 7 }$ c) $\frac { 1 } { 35 }$ 
B)  a) $\frac { 3 } { 7 }$ b) $\frac { 4 } { 7 }$ c) $\frac { 1 } { 7 }$ 
C)  a) $\frac { 2 } { 7 }$ b) $\frac { 1 } { 7 }$ c) $\frac { 1 } { 35 }$ 
D)  a) $\frac { 1 } { 7 }$ b) $\frac { 3 } { 7 }$ c) $\frac { 1 } { 35 }$ 
A)  a) $\frac { 1 } { 7 }$ b) $\frac { 4 } { 7 }$ c) $\frac { 1 } { 35 }$ 
B)  a) $\frac { 3 } { 7 }$ b) $\frac { 4 } { 7 }$ c) $\frac { 1 } { 7 }$ 
C)  a) $\frac { 2 } { 7 }$ b) $\frac { 1 } { 7 }$ c) $\frac { 1 } { 35 }$ 
D)  a) $\frac { 1 } { 7 }$ b) $\frac { 3 } { 7 }$ c) $\frac { 1 } { 35 }$

Question 15

Graph the function corresponding to the sequence defined. Use the graph to decide whether the sequence converges or
diverges.
-$$a _ { n } = \frac { n + 92 } { 2 n }$$

Accepted Answer

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

Question 16

Find the first term and the common ratio for the geometric sequence. Round approximations to the nearest hundredth.
-$$a _ { 2 } = 14 , a _ { 6 } = 224$$

Accepted Answer

A)  $a _ { 1 } = 7 , r = 2$ 
B)  $\mathrm { a } _ { 1 } = 224 , \mathrm { r } = 0.5$ 
C)  $a _ { 1 } = 14 , r = 2$ 
D)  $a _ { 1 } = 7 , r = 0.5$ 
A)  $a _ { 1 } = 7 , r = 2$ 
B)  $\mathrm { a } _ { 1 } = 224 , \mathrm { r } = 0.5$ 
C)  $a _ { 1 } = 14 , r = 2$ 
D)  $a _ { 1 } = 7 , r = 0.5$

Question 17

Solve the problem.
-In how many ways can 7 people line up for play tickets?

Accepted Answer

A)  823,543 
B)  7 
C)  1 
D)  5040 
A)  823,543 
B)  7 
C)  1 
D)  5040

Question 18

Find the common ratio r for the given infinite geometric sequence.
-The beginning population of a small town was 13,000 people. Due to decline in industrial growth the population has since been decreasing at a rate of 5% every year. What was the population of
This town 10 years later?

Accepted Answer

A)  7784 people 
B)  6292 people 
C)  7002 people 
D)  8643 people 
A)  7784 people 
B)  6292 people 
C)  7002 people 
D)  8643 people

Question 19

Find the sum of the first n terms of the following arithmetic sequence.
-$a _ { 2 } = 21 , a _ { 5 } = - 42 ; \quad \mathrm { n } = 10$

Accepted Answer

A)  735 
B)  1365 
C)  -735 
D)  -525 
A)  735 
B)  1365 
C)  -735 
D)  -525

Question 20

Graph the function corresponding to the sequence defined. Use the graph to decide whether the sequence converges or
diverges.
-The series $\ln ( x ) = ( x - 1 ) - \frac { ( x - 1 ) ^ { 2 } } { 2 } + \frac { ( x - 1 ) ^ { 3 } } { 3 } - \ldots + ( - 1 ) ^ { n - 1 } \frac { ( x - 1 ) ^ { n } } { n }$ can be used to estimate the value of $\ln ( \mathrm { a } )$ for any positive real number a such that $0 < \mathrm { a } \leq 2$. Use the first five terms of this series to estimate $\ln ( 0.86 )$. Round to the nearest thousandth.

Accepted Answer

A)  $- 1.081$ 
B)  $0.739$ 
C)  $- 0.151$ 
D)  $1.409$ 
A)  $- 1.081$ 
B)  $0.739$ 
C)  $- 0.151$ 
D)  $1.409$

Solve the problem. -How many different three-digit numbers can be written using digits from the set $\{ 1,2,3,4,5 \}$ without any repeating digits?

Write the first n terms of the given arithmetic sequence (the value of n is indicated in the question). - $a _ { 13 } = - 17 x - 9 , a _ { 15 } = - 15 x - 1$

Find the common ratio r for the given infinite geometric sequence. -A pendulum bob swings 6.0 cm on its first oscillation. On each subsequent oscillation the bob travels $\frac { 1 } { 3 }$ of the previous distance. Find the total distance the bob travels before coming to rest.

List the elements in the sample space of the experiment. -A lottery uses balls numbered 1 through 35 . List the sample space of the event picking one even-numbered ball.

Graph the function corresponding to the sequence defined. Use the graph to decide whether the sequence converges or diverges. - $a _ { n } = \left( \frac { 1 } { 20 } + n \right) ^ { 1 / n }$

Write the binomial expansion of the expression. - $\left( \frac { 1 } { 3 } x + 4 \right) ^ { 3 }$

Solve the problem. -An ordinary die is tossed. What are the odds in favor of the die showing a number less than 3 ?

Solve the problem. -Three noncollinear points determine a triangle. How many triangles can be formed with 8 noncollinear points?

Use mathematical induction to prove that the statement is true for every positive integer n. - $5 ^ { n } > 1$

Write out the first five terms of the sequence. - $a _ { n } = n ^ { 2 } - n$

Find the sum of the geometric series. - $\sum _ { \mathrm { i } = 1 } ^ { 5 } 2 ( 4 ) ^ { \mathrm { i } }$

Solve the problem. -A musician plans to perform 9 selections. In how many ways can she arrange the musical selections?

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

Solve the problem. -Four boys and three girls are seated, in a row, at random, to watch a play. What is the probability th 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?

Graph the function corresponding to the sequence defined. Use the graph to decide whether the sequence converges or diverges. - $a _ { n } = \frac { n + 92 } { 2 n }$

Find the first term and the common ratio for the geometric sequence. Round approximations to the nearest hundredth. - $a _ { 2 } = 14 , a _ { 6 } = 224$

Solve the problem. -In how many ways can 7 people line up for play tickets?

Find the common ratio r for the given infinite geometric sequence. -The beginning population of a small town was 13,000 people. Due to decline in industrial growth the population has since been decreasing at a rate of 5% every year. What was the population of This town 10 years later?

Find the sum of the first n terms of the following arithmetic sequence. - $a _ { 2 } = 21 , a _ { 5 } = - 42 ; \quad \mathrm { n } = 10$

Review of Basic Concepts

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

Filters

Exam 12: Further Topics in Algebra

Solve the problem. -How many different three-digit numbers can be written using digits from the set {1,2,3,4,5}\{ 1,2,3,4,5 \}{1,2,3,4,5} without any repeating digits?

Write the first n terms of the given arithmetic sequence (the value of n is indicated in the question). - a13=−17x−9,a15=−15x−1a _ { 13 } = - 17 x - 9 , a _ { 15 } = - 15 x - 1a13​=−17x−9,a15​=−15x−1

Find the common ratio r for the given infinite geometric sequence. -A pendulum bob swings 6.0 cm on its first oscillation. On each subsequent oscillation the bob travels 13\frac { 1 } { 3 }31​ of the previous distance. Find the total distance the bob travels before coming to rest.