Question 1

Give the probability that the roll of a die will show 3 or 5.

Accepted Answer

A)  $\frac { 1 } { 3 }$ 
B)  $\frac { 1 } { 2 }$ 
C)  2 
D)  $\frac { 5 } { 6 }$ 
A)  $\frac { 1 } { 3 }$ 
B)  $\frac { 1 } { 2 }$ 
C)  2 
D)  $\frac { 5 } { 6 }$

Question 2

Write an equation for the ellipse.
-foci at $( - 2,1 ) , ( 4,1 )$; major axis length of 10

Accepted Answer

A)  $\frac { ( y - 1 ) ^ { 2 } } { 25 } + \frac { ( x - 1 ) ^ { 2 } } { 16 } = 1$ 
B)  $\frac { ( x - 1 ) ^ { 2 } } { 25 } + \frac { ( y - 1 ) ^ { 2 } } { 16 } = 1$ 
C)  $\frac { ( x - 1 ) ^ { 2 } } { 25 } + \frac { ( y + 1 ) ^ { 2 } } { 16 } = 1$ 
D)  $\frac { ( x + 1 ) ^ { 2 } } { 25 } + \frac { ( x + 1 ) ^ { 2 } } { 16 } = 1$ 
A)  $\frac { ( y - 1 ) ^ { 2 } } { 25 } + \frac { ( x - 1 ) ^ { 2 } } { 16 } = 1$ 
B)  $\frac { ( x - 1 ) ^ { 2 } } { 25 } + \frac { ( y - 1 ) ^ { 2 } } { 16 } = 1$ 
C)  $\frac { ( x - 1 ) ^ { 2 } } { 25 } + \frac { ( y + 1 ) ^ { 2 } } { 16 } = 1$ 
D)  $\frac { ( x + 1 ) ^ { 2 } } { 25 } + \frac { ( x + 1 ) ^ { 2 } } { 16 } = 1$

Question 3

Evaluate the series, if it converges.
-$$- 11 + \frac { 55 } { 2 } - \frac { 275 } { 4 } + \frac { 1375 } { 8 } + \ldots$$

Accepted Answer

A)  $\frac { 55 } { 7 }$ 
B)  $- \frac { 55 } { 7 }$ 
C)  Does not converge 
D)  $\frac { 22 } { 7 }$ 
A)  $\frac { 55 } { 7 }$ 
B)  $- \frac { 55 } { 7 }$ 
C)  Does not converge 
D)  $\frac { 22 } { 7 }$

Question 4

A pendulum bob swings 6.0 cm on its first oscillation. On each subsequent oscillation the bob travels $\frac { 2 } { 3 }$ of the previous distance. Find the total distance the bob travels before coming to rest.

Accepted Answer

A)  33.3 cm 
B)  31.0 cm 
C)  18.0 cm 
D)  30.0 cm 
A)  33.3 cm 
B)  31.0 cm 
C)  18.0 cm 
D)  30.0 cm

Question 5

Provide an appropriate response.
-Why is it not possible to choose h and k, neither zero, so the parabola $f ( x ) = a ( x - h ) ^ { 2 } +$ k has its vertex at the
origin?

Accepted Answer

The only way a parabola can have its ver

Question 6

Write an equation for the parabola with vertex at the origin.
-Focus $\left( 0 , - \frac { 1 } { 17 } \right)$

Accepted Answer

A)  $x ^ { 2 } = - \frac { 4 } { 17 } y$ 
B)  $y ^ { 2 } = \frac { 1 } { 68 } x$ 
C)  $y ^ { 2 } = - \frac { 4 } { 17 } x$ 
D)  $x ^ { 2 } = - 68 y$ 
A)  $x ^ { 2 } = - \frac { 4 } { 17 } y$ 
B)  $y ^ { 2 } = \frac { 1 } { 68 } x$ 
C)  $y ^ { 2 } = - \frac { 4 } { 17 } x$ 
D)  $x ^ { 2 } = - 68 y$

Question 7

Identify the equation as a parabola, circle, ellipse, or hyperbola.
-$$4 x ^ { 2 } - y = 13$$

Accepted Answer

A)  Parabola 
B)  Circle 
C)  Ellipse 
D)  Hyperbola 
A)  Parabola 
B)  Circle 
C)  Ellipse 
D)  Hyperbola

Question 8

Find the first term and the common difference for the arithmetic sequence. Round approximations to the nearest
hundredth.
-$$\mathrm { S } _ { 8 } = - 100 , \mathrm { a } _ { 8 } = - 16$$

Accepted Answer

A)  a $1 = - 9 , \mathrm {~d} = - 1$ 
B)  $\mathrm { a } _ { 1 } = 103 , \mathrm {~d} = - 4$ 
C)  $\mathrm { a } _ { 1 } = - 9 , \mathrm {~d} = - 4$ 
D)  $\mathrm { a } _ { 1 } = 103 , \mathrm {~d} = - 1$ 
A)  a $1 = - 9 , \mathrm {~d} = - 1$ 
B)  $\mathrm { a } _ { 1 } = 103 , \mathrm {~d} = - 4$ 
C)  $\mathrm { a } _ { 1 } = - 9 , \mathrm {~d} = - 4$ 
D)  $\mathrm { a } _ { 1 } = 103 , \mathrm {~d} = - 1$

Question 9

Provide an appropriate response.
-Let f( $f ( x ) = 9 \left( 2 ^ { x } \right)$ ), and consider the sequence f(1), f(2), f(3), . . . . Is this a geometric sequence, and if so, what is the common ratio?

Accepted Answer

A)  Yes; x 
B)  No 
C)  Yes; 2 
D)  Yes; 9 
A)  Yes; x 
B)  No 
C)  Yes; 2 
D)  Yes; 9

Question 10

Decide whether the given sequence is finite or infinite.
-3, 4, 5, 6, . . .

Accepted Answer

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

Question 11

The x-intercepts of the hyperbola with equation $$\frac { x ^ { 2 } } { 81 } - \frac { y ^ { 2 } } { 64 } = 1 \text { are } ( 9,0 ) \text { and } ( - 9,0 )$$

Accepted Answer

A) True 
 B)False

Question 12

Use mathematical induction to prove that the statement is true for every positive integer n.
-$$1 ^ { 2 } + 4 ^ { 2 } + 7 ^ { 2 } + \ldots + ( 3 n - 2 ) ^ { 2 } = \frac { n \left( 6 n ^ { 2 } - 3 n - 1 \right) } { 2 }$$

Accepted Answer

Answers will vary. One possible proof fo

Question 13

It can be shown that an ellipse with foci $( c , 0 )$ and $( - c , 0 )$ where the sum of the distances from any point ( $x , y )$ of the ellipse to the two foci is $2 a$ has equation $\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { a ^ { 2 } - c ^ { 2 } } = 1$. Use this result to find an equation of an ellipse with foci $( 3,0 )$ and $( - 3,0 )$, where the sum of the distances from any point of the ellipse to the two foci is 10 .

Accepted Answer

A)  $\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 16 } = 1$ 
B)  $\frac { x ^ { 2 } } { - 3 } + \frac { y ^ { 2 } } { 100 } = 1$ 
C)  $x ^ { 2 } + y ^ { 2 } = 100$ 
D)  $\frac { x ^ { 2 } } { - 3 } + \frac { y ^ { 2 } } { 22 } = 1$ 
A)  $\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 16 } = 1$ 
B)  $\frac { x ^ { 2 } } { - 3 } + \frac { y ^ { 2 } } { 100 } = 1$ 
C)  $x ^ { 2 } + y ^ { 2 } = 100$ 
D)  $\frac { x ^ { 2 } } { - 3 } + \frac { y ^ { 2 } } { 22 } = 1$

Question 14

Write the first n terms of the given arithmetic sequence (the value of n is indicated in the question).
-The first term is $- 1 + \sqrt { 13 }$, and the common difference is $5 ; \mathrm { n } = 3$

Accepted Answer

A)  $- 1 + \sqrt { 13 } , 5 + \sqrt { 13 } , 10 + \sqrt { 13 }$ 
B)  $1 + \sqrt { 13 } , 4 + \sqrt { 13 } , 9 + \sqrt { 13 }$ 
C)  $- 1 + \sqrt { 13 } , 4 + \sqrt { 13 } , 9 + \sqrt { 13 }$ 
D)  $- 1 - \sqrt { 13 } , 4 + \sqrt { 13 } , 9 + \sqrt { 13 }$ 
A)  $- 1 + \sqrt { 13 } , 5 + \sqrt { 13 } , 10 + \sqrt { 13 }$ 
B)  $1 + \sqrt { 13 } , 4 + \sqrt { 13 } , 9 + \sqrt { 13 }$ 
C)  $- 1 + \sqrt { 13 } , 4 + \sqrt { 13 } , 9 + \sqrt { 13 }$ 
D)  $- 1 - \sqrt { 13 } , 4 + \sqrt { 13 } , 9 + \sqrt { 13 }$

Question 15

Write out the first five terms of the sequence.
-$$a _ { n } = 3 ( 2 n - 3 )$$

Accepted Answer

A)  $- 1,1,3,5,7$ 
B)  $- 9 , - 3,3,9,15$ 
C)  $- 3,3,9,15,21$ 
D)  $- 3 , - 6 , - 9 , - 12 , - 15$ 
A)  $- 1,1,3,5,7$ 
B)  $- 9 , - 3,3,9,15$ 
C)  $- 3,3,9,15,21$ 
D)  $- 3 , - 6 , - 9 , - 12 , - 15$

Question 16

Write the binomial expansion of the expression.
-$( a - b ) ^ { 6 }$

Accepted Answer

A)  $a ^ { 6 } - 6 a ^ { 5 } b - 15 a ^ { 4 } b ^ { 2 } - 20 a ^ { 3 } b ^ { 3 } - 15 a ^ { 2 } b ^ { 4 } - 6 a b ^ { 5 } - b ^ { 6 }$ 
B)  $a^{ 6} - 6 a ^ { 5 } b + 15 a ^ { 4 } b ^ { 2 } - 20 a ^ { 3 } b ^ { 3 } + 15 a ^ { 2 } b ^ { 4 } - 6 a b ^ { 5 } + b ^ { 6 }$ 
C)  $a ^ { 6 } + 6 a ^ { 5 } b + 15 a ^ { 4 } b ^ { 2 } + 20 a ^ { 3 } b ^ { 3 } + 15 a ^ { 2 } b ^ { 4 } + 6 a b ^ { 5 } + b ^ { 6 }$ 
D)  $- a ^ { 6 } + 6 a ^ { 5 } b - 15 a ^ { 4 } b ^ { 2 } + 20 a ^ { 3 } b ^ { 3 } - 15 a ^ { 2 } b ^ { 4 } + 6 a b ^ { 5 } - b ^ { 6 }$ 
A)  $a ^ { 6 } - 6 a ^ { 5 } b - 15 a ^ { 4 } b ^ { 2 } - 20 a ^ { 3 } b ^ { 3 } - 15 a ^ { 2 } b ^ { 4 } - 6 a b ^ { 5 } - b ^ { 6 }$ 
B)  $a^{ 6} - 6 a ^ { 5 } b + 15 a ^ { 4 } b ^ { 2 } - 20 a ^ { 3 } b ^ { 3 } + 15 a ^ { 2 } b ^ { 4 } - 6 a b ^ { 5 } + b ^ { 6 }$ 
C)  $a ^ { 6 } + 6 a ^ { 5 } b + 15 a ^ { 4 } b ^ { 2 } + 20 a ^ { 3 } b ^ { 3 } + 15 a ^ { 2 } b ^ { 4 } + 6 a b ^ { 5 } + b ^ { 6 }$ 
D)  $- a ^ { 6 } + 6 a ^ { 5 } b - 15 a ^ { 4 } b ^ { 2 } + 20 a ^ { 3 } b ^ { 3 } - 15 a ^ { 2 } b ^ { 4 } + 6 a b ^ { 5 } - b ^ { 6 }$

Question 17

An ordinary die is tossed. What are the odds in favor of the die showing a 4?

Accepted Answer

A)  1 to 6 
B)  1 to 4 
C)  1 to 5 
D)  2 to 3 
A)  1 to 6 
B)  1 to 4 
C)  1 to 5 
D)  2 to 3

Question 18

Find the first term and the common ratio for the geometric sequence. Round approximations to the nearest hundredth.
-$$a _ { 2 } = 9 , a _ { 6 } = 729$$

Accepted Answer

A)  $\mathrm { a } _ { 1 } = 3 , \mathrm { r } = 0.33$ 
B)  $\mathrm { a } _ { 1 } = 3 , \mathrm { r } = 3$ 
C)  $\mathrm { a } _ { 1 } = 729 , \mathrm { r } = 0.33$ 
D)  $a _ { 1 } = 9 , r = 3$ 
A)  $\mathrm { a } _ { 1 } = 3 , \mathrm { r } = 0.33$ 
B)  $\mathrm { a } _ { 1 } = 3 , \mathrm { r } = 3$ 
C)  $\mathrm { a } _ { 1 } = 729 , \mathrm { r } = 0.33$ 
D)  $a _ { 1 } = 9 , r = 3$

Question 19

Find the nth term of the geometric sequence.
-$$a _ { 1 } = 2 , r = - 3 , n = 11$$

Accepted Answer

A)  $a _ { 11 } = 531,441$ 
B)  $a _ { 11 } = - 39,366$ 
C)  $a _ { 11 } = 118,098$ 
D)  $a _ { 11 } = - 66$ 
A)  $a _ { 11 } = 531,441$ 
B)  $a _ { 11 } = - 39,366$ 
C)  $a _ { 11 } = 118,098$ 
D)  $a _ { 11 } = - 66$

Question 20

Find the eccentricity $ e $ of the hyperbola.
-$$x ^ { 2 } - 25 y ^ { 2 } = 25$$

Accepted Answer

A)  $\frac { 2 \sqrt { 6 } } { 5 }$ 
B)  $2 \sqrt { 6 }$ 
C)  $\frac { \sqrt { 26 } } { 5 }$ 
D)  $\sqrt { 26 }$ 
A)  $\frac { 2 \sqrt { 6 } } { 5 }$ 
B)  $2 \sqrt { 6 }$ 
C)  $\frac { \sqrt { 26 } } { 5 }$ 
D)  $\sqrt { 26 }$

Give the probability that the roll of a die will show 3 or 5.

Write an equation for the ellipse. -foci at $( - 2,1 ) , ( 4,1 )$ ; major axis length of 10

Evaluate the series, if it converges. - $- 11 + \frac { 55 } { 2 } - \frac { 275 } { 4 } + \frac { 1375 } { 8 } + \ldots$

A pendulum bob swings 6.0 cm on its first oscillation. On each subsequent oscillation the bob travels $\frac { 2 } { 3 }$ of the previous distance. Find the total distance the bob travels before coming to rest.

Provide an appropriate response. -Why is it not possible to choose h and k, neither zero, so the parabola $f ( x ) = a ( x - h ) ^ { 2 } +$ k has its vertex at the origin?

Write an equation for the parabola with vertex at the origin. -Focus $\left( 0 , - \frac { 1 } { 17 } \right)$

Identify the equation as a parabola, circle, ellipse, or hyperbola. - $4 x ^ { 2 } - y = 13$

Find the first term and the common difference for the arithmetic sequence. Round approximations to the nearest hundredth. - $\mathrm { S } _ { 8 } = - 100 , \mathrm { a } _ { 8 } = - 16$

Provide an appropriate response. -Let f( $f ( x ) = 9 \left( 2 ^ { x } \right)$ ), and consider the sequence f(1), f(2), f(3), . . . . Is this a geometric sequence, and if so, what is the common ratio?

Decide whether the given sequence is finite or infinite. -3, 4, 5, 6, . . .

The x-intercepts of the hyperbola with equation $\frac { x ^ { 2 } } { 81 } - \frac { y ^ { 2 } } { 64 } = 1 \text { are } ( 9,0 ) \text { and } ( - 9,0 )$

Use mathematical induction to prove that the statement is true for every positive integer n. - $1 ^ { 2 } + 4 ^ { 2 } + 7 ^ { 2 } + \ldots + ( 3 n - 2 ) ^ { 2 } = \frac { n \left( 6 n ^ { 2 } - 3 n - 1 \right) } { 2 }$

Write the first n terms of the given arithmetic sequence (the value of n is indicated in the question). -The first term is $- 1 + \sqrt { 13 }$ , and the common difference is $5 ; \mathrm { n } = 3$

Write out the first five terms of the sequence. - $a _ { n } = 3 ( 2 n - 3 )$

Write the binomial expansion of the expression. - $( a - b ) ^ { 6 }$

An ordinary die is tossed. What are the odds in favor of the die showing a 4?

Find the first term and the common ratio for the geometric sequence. Round approximations to the nearest hundredth. - $a _ { 2 } = 9 , a _ { 6 } = 729$

Find the nth term of the geometric sequence. - $a _ { 1 } = 2 , r = - 3 , n = 11$

Find the eccentricity $e$ of the hyperbola. - $x ^ { 2 } - 25 y ^ { 2 } = 25$

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Exam 7: Arithmetic Sequence: Common Difference and First n Terms

Give the probability that the roll of a die will show 3 or 5.

Write an equation for the ellipse. -foci at (−2,1),(4,1)( - 2,1 ) , ( 4,1 )(−2,1),(4,1) ; major axis length of 10

Evaluate the series, if it converges. - −11+552−2754+13758+…- 11 + \frac { 55 } { 2 } - \frac { 275 } { 4 } + \frac { 1375 } { 8 } + \ldots−11+255​−4275​+81375​+…

A pendulum bob swings 6.0 cm on its first oscillation. On each subsequent oscillation the bob travels 23\frac { 2 } { 3 }32​ of the previous distance. Find the total distance the bob travels before coming to rest.

Provide an appropriate response. -Why is it not possible to choose h and k, neither zero, so the parabola f(x)=a(x−h)2+f ( x ) = a ( x - h ) ^ { 2 } +f(x)=a(x−h)2+ k has its vertex at the origin?

Write an equation for the parabola with vertex at the origin. -Focus (0,−117)\left( 0 , - \frac { 1 } { 17 } \right)(0,−171​)

Identify the equation as a parabola, circle, ellipse, or hyperbola. - 4x2−y=134 x ^ { 2 } - y = 134x2−y=13

Find the first term and the common difference for the arithmetic sequence. Round approximations to the nearest hundredth. - S8=−100,a8=−16\mathrm { S } _ { 8 } = - 100 , \mathrm { a } _ { 8 } = - 16S8​=−100,a8​=−16

Provide an appropriate response. -Let f( f(x)=9(2x)f ( x ) = 9 \left( 2 ^ { x } \right)f(x)=9(2x) ), and consider the sequence f(1), f(2), f(3), . . . . Is this a geometric sequence, and if so, what is the common ratio?

Decide whether the given sequence is finite or infinite. -3, 4, 5, 6, . . .

The x-intercepts of the hyperbola with equation x281−y264=1 are (9,0) and (−9,0)\frac { x ^ { 2 } } { 81 } - \frac { y ^ { 2 } } { 64 } = 1 \text { are } ( 9,0 ) \text { and } ( - 9,0 )81x2​−64y2​=1 are (9,0) and (−9,0)

Use mathematical induction to prove that the statement is true for every positive integer n. - 12+42+72+…+(3n−2)2=n(6n2−3n−1)21 ^ { 2 } + 4 ^ { 2 } + 7 ^ { 2 } + \ldots + ( 3 n - 2 ) ^ { 2 } = \frac { n \left( 6 n ^ { 2 } - 3 n - 1 \right) } { 2 }12+42+72+…+(3n−2)2=2n(6n2−3n−1)​

Write the first n terms of the given arithmetic sequence (the value of n is indicated in the question). -The first term is −1+13- 1 + \sqrt { 13 }−1+13​ , and the common difference is 5;n=35 ; \mathrm { n } = 35;n=3

Write out the first five terms of the sequence. - an=3(2n−3)a _ { n } = 3 ( 2 n - 3 )an​=3(2n−3)

Write the binomial expansion of the expression. - (a−b)6( a - b ) ^ { 6 }(a−b)6

An ordinary die is tossed. What are the odds in favor of the die showing a 4?

Find the first term and the common ratio for the geometric sequence. Round approximations to the nearest hundredth. - a2=9,a6=729a _ { 2 } = 9 , a _ { 6 } = 729a2​=9,a6​=729

Find the nth term of the geometric sequence. - a1=2,r=−3,n=11a _ { 1 } = 2 , r = - 3 , n = 11a1​=2,r=−3,n=11

Find the eccentricity e e e of the hyperbola. - x2−25y2=25x ^ { 2 } - 25 y ^ { 2 } = 25x2−25y2=25

Review of Basic Concepts

Equations and Inequalities

Graphs and Functions

Polynomials and Rational Functions

Inverse, Exponential, and Logarithmic Functions

Systems and Matrices

Filters

Write an equation for the ellipse. -foci at $( - 2,1 ) , ( 4,1 )$ ; major axis length of 10

Evaluate the series, if it converges. - $- 11 + \frac { 55 } { 2 } - \frac { 275 } { 4 } + \frac { 1375 } { 8 } + \ldots$

A pendulum bob swings 6.0 cm on its first oscillation. On each subsequent oscillation the bob travels $\frac { 2 } { 3 }$ of the previous distance. Find the total distance the bob travels before coming to rest.

Provide an appropriate response. -Why is it not possible to choose h and k, neither zero, so the parabola $f ( x ) = a ( x - h ) ^ { 2 } +$ k has its vertex at the origin?

Write an equation for the parabola with vertex at the origin. -Focus $\left( 0 , - \frac { 1 } { 17 } \right)$

Identify the equation as a parabola, circle, ellipse, or hyperbola. - $4 x ^ { 2 } - y = 13$

Find the first term and the common difference for the arithmetic sequence. Round approximations to the nearest hundredth. - $\mathrm { S } _ { 8 } = - 100 , \mathrm { a } _ { 8 } = - 16$

Provide an appropriate response. -Let f( $f ( x ) = 9 \left( 2 ^ { x } \right)$ ), and consider the sequence f(1), f(2), f(3), . . . . Is this a geometric sequence, and if so, what is the common ratio?

The x-intercepts of the hyperbola with equation $\frac { x ^ { 2 } } { 81 } - \frac { y ^ { 2 } } { 64 } = 1 \text { are } ( 9,0 ) \text { and } ( - 9,0 )$

Use mathematical induction to prove that the statement is true for every positive integer n. - $1 ^ { 2 } + 4 ^ { 2 } + 7 ^ { 2 } + \ldots + ( 3 n - 2 ) ^ { 2 } = \frac { n \left( 6 n ^ { 2 } - 3 n - 1 \right) } { 2 }$

Write the first n terms of the given arithmetic sequence (the value of n is indicated in the question). -The first term is $- 1 + \sqrt { 13 }$ , and the common difference is $5 ; \mathrm { n } = 3$

Write out the first five terms of the sequence. - $a _ { n } = 3 ( 2 n - 3 )$

Write the binomial expansion of the expression. - $( a - b ) ^ { 6 }$

Find the first term and the common ratio for the geometric sequence. Round approximations to the nearest hundredth. - $a _ { 2 } = 9 , a _ { 6 } = 729$

Find the nth term of the geometric sequence. - $a _ { 1 } = 2 , r = - 3 , n = 11$

Find the eccentricity $e$ of the hyperbola. - $x ^ { 2 } - 25 y ^ { 2 } = 25$