Question 1

Graph the complex number.
-$$6 - 3 i$$

Accepted Answer

A) 
  
B) 
 . 
C) 
  
D) 
  
A) 
  
B) 
 . 
C) 
  
D)

Question 2

For the given rectangular equation, give its equivalent polar equation.
-$$2 x + 3 y = 6$$

Accepted Answer

A)  $r = \frac { 6 } { 2 \sin \theta + 3 \cos \theta }$ 
B)  $r = \frac { 2 \cos \theta + 3 \sin \theta } { 6 }$ 
C)  $r = \frac { 6 } { 2 \cos \theta + 3 \sin \theta }$ 
D)  $r = \frac { 2 \sin \theta + 3 \cos \theta } { 6 }$ 
A)  $r = \frac { 6 } { 2 \sin \theta + 3 \cos \theta }$ 
B)  $r = \frac { 2 \cos \theta + 3 \sin \theta } { 6 }$ 
C)  $r = \frac { 6 } { 2 \cos \theta + 3 \sin \theta }$ 
D)  $r = \frac { 2 \sin \theta + 3 \cos \theta } { 6 }$

Question 3

Find the quotient and write in rectangular form. First convert the numerator and denominator to trigonometric form.
-$$\frac { 9 \operatorname { cis } 154 ^ { \circ } } { 3 \operatorname { cis } 34 ^ { \circ } }$$

Accepted Answer

A)  $- \frac { 3 } { 2 } \sqrt { 3 } + \frac { 3 } { 2 } \mathrm { i }$ 
B)  $\frac { 3 } { 2 } + \frac { 3 } { 2 } i \sqrt { 3 }$ 
C)  $- \frac { 3 } { 2 } \sqrt { 3 } - \frac { 3 } { 2 } i$ 
D)  $- \frac { 3 } { 2 } + \frac { 3 } { 2 } i \sqrt { 3 }$ 
A)  $- \frac { 3 } { 2 } \sqrt { 3 } + \frac { 3 } { 2 } \mathrm { i }$ 
B)  $\frac { 3 } { 2 } + \frac { 3 } { 2 } i \sqrt { 3 }$ 
C)  $- \frac { 3 } { 2 } \sqrt { 3 } - \frac { 3 } { 2 } i$ 
D)  $- \frac { 3 } { 2 } + \frac { 3 } { 2 } i \sqrt { 3 }$

Question 4

Solve.
-A complex number $z$ does not belong to the Mandelbrot set if any of the complex numbers in the sequence $z , z ^ { 2 } + z , \left( z ^ { 2 } + z \right) ^ { 2 } + z , \left[ \left( z ^ { 2 } + z \right) ^ { 2 } + z \right] ^ { 2 } + z , \ldots$ has modulus exceeding 2. Does $z = 0.3 + 0.2 i$ belong to the Mandelbrot set?

Accepted Answer

A)  Yes 
B)  No 
A)  Yes 
B)  No

Question 5

Find the quotient and write in rectangular form. First convert the numerator and denominator to trigonometric form.
-$$\frac { 4 \mathrm { i } } { 1 - 10 \mathrm { i } }$$

Accepted Answer

A)  $- \frac { 40 } { 101 } + \frac { 4 } { 101 } \mathrm { i }$ 
B)  $\frac { 40 } { 99 } - \frac { 4 } { 99 } \mathrm { i }$ 
C)  $- \frac { 40 } { 99 } - \frac { 4 } { 99 } \mathrm { i }$ 
D)  $\frac { 40 } { 101 } + \frac { 4 } { 101 } \mathrm { i }$ 
A)  $- \frac { 40 } { 101 } + \frac { 4 } { 101 } \mathrm { i }$ 
B)  $\frac { 40 } { 99 } - \frac { 4 } { 99 } \mathrm { i }$ 
C)  $- \frac { 40 } { 99 } - \frac { 4 } { 99 } \mathrm { i }$ 
D)  $\frac { 40 } { 101 } + \frac { 4 } { 101 } \mathrm { i }$

Question 6

Answer the question.
-In which quadrants do the fourth roots of a complex number lie?

Accepted Answer

A)  The first and third quadrants if the number is real, and the second and fourth quadrants if the number is not real 
B)  All four quadrants 
C)  The first and second quadrants if the number is real, and the third and fourth quadrants if the number is not real 
D)  The first and third quadrants if the number is real and positive, and the second and fourth quadrants otherwise 
A)  The first and third quadrants if the number is real, and the second and fourth quadrants if the number is not real 
B)  All four quadrants 
C)  The first and second quadrants if the number is real, and the third and fourth quadrants if the number is not real 
D)  The first and third quadrants if the number is real and positive, and the second and fourth quadrants otherwise

Question 7

For the given rectangular equation, give its equivalent polar equation.
-$6 x - y = 14$

Accepted Answer

A)  $r = \frac { 14 } { 6 \sin \theta - \cos \theta }$ 
B)  $r = \frac { 14 } { 6 \cos \theta - \sin \theta }$ 
C)  $r = \frac { 6 } { 14 \cos \theta - \sin \theta }$ 
D)  $r = \frac { 14 } { 6 \cos \theta + \sin \theta }$ 
A)  $r = \frac { 14 } { 6 \sin \theta - \cos \theta }$ 
B)  $r = \frac { 14 } { 6 \cos \theta - \sin \theta }$ 
C)  $r = \frac { 6 } { 14 \cos \theta - \sin \theta }$ 
D)  $r = \frac { 14 } { 6 \cos \theta + \sin \theta }$

Question 8

Give the rectangular coordinates for the point.
-(2, 270°)

Accepted Answer

A)  (0, 2) 
B)  (-2, 0) 
C)  (0, -2) 
D)  (2, 0) 
A)  (0, 2) 
B)  (-2, 0) 
C)  (0, -2) 
D)  (2, 0)

Question 9

Find all specified roots.
-Cube roots of $\mathrm { i }$.

Accepted Answer

A)  $\frac { \sqrt { 3 } } { 2 } - \frac { 1 } { 2 } \mathrm { i } , - \frac { \sqrt { 3 } } { 2 } - \frac { 1 } { 2 } \mathrm { i } , - \mathrm { i }$ 
B)  $\frac { \sqrt { 3 } } { 2 } + \frac { 1 } { 2 } \mathrm { i } , - \frac { \sqrt { 3 } } { 2 } + \frac { 1 } { 2 } \mathrm { i }$, i 
C)  $\frac { \sqrt { 3 } } { 2 } + \frac { 1 } { 2 } \mathrm { i } , - \frac { \sqrt { 3 } } { 2 } + \frac { 1 } { 2 } \mathrm { i } , - \mathrm { i }$ 
D)  $\frac { \sqrt { 3 } } { 2 } - \frac { 1 } { 2 } \mathrm { i } , - \frac { \sqrt { 3 } } { 2 } - \frac { 1 } { 2 } \mathrm { i }$, i 
A)  $\frac { \sqrt { 3 } } { 2 } - \frac { 1 } { 2 } \mathrm { i } , - \frac { \sqrt { 3 } } { 2 } - \frac { 1 } { 2 } \mathrm { i } , - \mathrm { i }$ 
B)  $\frac { \sqrt { 3 } } { 2 } + \frac { 1 } { 2 } \mathrm { i } , - \frac { \sqrt { 3 } } { 2 } + \frac { 1 } { 2 } \mathrm { i }$, i 
C)  $\frac { \sqrt { 3 } } { 2 } + \frac { 1 } { 2 } \mathrm { i } , - \frac { \sqrt { 3 } } { 2 } + \frac { 1 } { 2 } \mathrm { i } , - \mathrm { i }$ 
D)  $\frac { \sqrt { 3 } } { 2 } - \frac { 1 } { 2 } \mathrm { i } , - \frac { \sqrt { 3 } } { 2 } - \frac { 1 } { 2 } \mathrm { i }$, i

Question 10

Find the polar coordinates of the point(s) of intersection of the given curves for 0 $$0 \leq \theta < 2 \pi$$
-$r = 4 + \cos \theta , r = 4 - \sin \theta$

Accepted Answer

A)  $\left( 4 , \frac { 3 \pi } { 4 } \right) , \left( 4 , \frac { 7 \pi } { 4 } \right)$ 
B)  $\left( 4 - \frac { 1 } { \sqrt { 2 } } , \frac { 3 \pi } { 4 } \right) , \left( 4 + \frac { 1 } { \sqrt { 2 } } , \frac { 7 \pi } { 4 } \right)$ 
C)  $\left( 4 + \frac { 1 } { \sqrt { 2 } } , \frac { 3 \pi } { 4 } \right) , \left( 4 - \frac { 1 } { \sqrt { 2 } } , \frac { 7 \pi } { 4 } \right)$ 
D)  $\left( \frac { 7 } { 2 } , \frac { 3 \pi } { 4 } \right) , \left( \frac { 9 } { 2 } , \frac { 7 \pi } { 4 } \right)$ 
A)  $\left( 4 , \frac { 3 \pi } { 4 } \right) , \left( 4 , \frac { 7 \pi } { 4 } \right)$ 
B)  $\left( 4 - \frac { 1 } { \sqrt { 2 } } , \frac { 3 \pi } { 4 } \right) , \left( 4 + \frac { 1 } { \sqrt { 2 } } , \frac { 7 \pi } { 4 } \right)$ 
C)  $\left( 4 + \frac { 1 } { \sqrt { 2 } } , \frac { 3 \pi } { 4 } \right) , \left( 4 - \frac { 1 } { \sqrt { 2 } } , \frac { 7 \pi } { 4 } \right)$ 
D)  $\left( \frac { 7 } { 2 } , \frac { 3 \pi } { 4 } \right) , \left( \frac { 9 } { 2 } , \frac { 7 \pi } { 4 } \right)$

Question 11

Determine whether the pair of vectors is orthogonal.
--1, -1 , -64, 32

Accepted Answer

A)  Yes 
B)  No 
A)  Yes 
B)  No

Question 12

Graph the complex number.
-$$3 + 2 i \sqrt { 7 }$$

Accepted Answer

A) 
  
B) 
  
C) 
  
D) 
  
A) 
  
B) 
  
C) 
  
D)

Question 13

Give the rectangular coordinates for the point.
-$$\left( - 3 , \frac { 3 \pi } { 2 } \right)$$

Accepted Answer

A)  $( 0 , - 3 )$ 
B)  $( 3,0 )$ 
C)  $( 0,3 )$ 
D)  $( - 3,0 )$ 
A)  $( 0 , - 3 )$ 
B)  $( 3,0 )$ 
C)  $( 0,3 )$ 
D)  $( - 3,0 )$

Question 14

Determine whether the pair of vectors is orthogonal.
--2, 6 , -8, -5

Accepted Answer

A)  Yes 
B)  No 
A)  Yes 
B)  No

Question 15

Considering the given value of t, choose the ordered pair that lies on the graph of the given pair of parametric equations.
-$$x = t , y = t ^ { 2 } ; t = - 2$$

Accepted Answer

A)  $( - 2,4 )$ 
B)  $( - 2 , - 2 )$ 
C)  $( 2 , - 4 )$ 
D)  $( 4 , - 2 )$ 
A)  $( - 2,4 )$ 
B)  $( - 2 , - 2 )$ 
C)  $( 2 , - 4 )$ 
D)  $( 4 , - 2 )$

Question 16

Solve.
-A complex number $z$ does not belong to the Mandelbrot set if any of the complex numbers in the sequence $z , z ^ { 2 } + z , \left( z ^ { 2 } + z \right) ^ { 2 } + z , \left[ \left( z ^ { 2 } + z \right) ^ { 2 } + z \right] ^ { 2 } + z , \ldots$ has modulus exceeding 2. Does $z = - 1.11$ belong to the Mandelbrot set?

Accepted Answer

A)  Yes 
B)  No 
A)  Yes 
B)  No

Question 17

Solve the problem.
-Lookout station B is located $12 \mathrm { mi }$ due east of station A. The bearing of a fire from A is $\mathrm { S } 10 ^ { \circ } 10 ^ { \prime } \mathrm { W }$ and the bearing from B is $\mathrm { S } 38 ^ { \circ } 40 ^ { \prime } \mathrm { W }$. Determine the distance from the fire to $\mathrm { B }$ (to the nearest tenth of a mile).

Accepted Answer

A)  $14.4 \mathrm { mi }$ 
B)  $24.8 \mathrm { mi }$ 
C)  $26.8 \mathrm { mi }$ 
D)  $12.4 \mathrm { mi }$ 
A)  $14.4 \mathrm { mi }$ 
B)  $24.8 \mathrm { mi }$ 
C)  $26.8 \mathrm { mi }$ 
D)  $12.4 \mathrm { mi }$

Question 18

Find all specified roots.
-Seventh roots of $i$.

Accepted Answer

A)  $\operatorname { cis } \frac { \pi } { 7 }$, cis $\frac { 5 \pi } { 14 } , \operatorname { cis } \frac { 9 \pi } { 14 } , \operatorname { cis } \frac { 13 \pi } { 14 } , \operatorname { cis } \frac { 17 \pi } { 14 } , - \mathrm { i } , \operatorname { cis } \frac { 25 \pi } { 14 }$ 
B)  $\operatorname { cis } \frac { \pi } { 14 } , \operatorname { cis } \frac { 5 \pi } { 14 } , \operatorname { cis } \frac { 9 \pi } { 14 } , \operatorname { cis } \frac { 13 \pi } { 14 }$, cis $\frac { 17 \pi } { 14 } , - \mathrm { i } , \operatorname { cis } \frac { 25 \pi } { 14 }$ 
C)  cis $\frac { \pi } { 14 } , \operatorname { cis } \frac { 5 \pi } { 14 } , \operatorname { cis } \frac { 9 \pi } { 14 } , \operatorname { cis } \frac { 13 \pi } { 14 }$, cis $\frac { 17 \pi } { 14 }$, i, cis $\frac { 25 \pi } { 14 }$ 
D)  cis $\frac { \pi } { 14 }$, cis $\frac { 5 \pi } { 14 }$, cis $\frac { 9 \pi } { 14 }$, cis $\frac { 13 \pi } { 14 }$, cis $\frac { 17 \pi } { 14 } , - \mathrm { i }$, cis $\frac { 25 \pi } { 14 }$, i 
A)  $\operatorname { cis } \frac { \pi } { 7 }$, cis $\frac { 5 \pi } { 14 } , \operatorname { cis } \frac { 9 \pi } { 14 } , \operatorname { cis } \frac { 13 \pi } { 14 } , \operatorname { cis } \frac { 17 \pi } { 14 } , - \mathrm { i } , \operatorname { cis } \frac { 25 \pi } { 14 }$ 
B)  $\operatorname { cis } \frac { \pi } { 14 } , \operatorname { cis } \frac { 5 \pi } { 14 } , \operatorname { cis } \frac { 9 \pi } { 14 } , \operatorname { cis } \frac { 13 \pi } { 14 }$, cis $\frac { 17 \pi } { 14 } , - \mathrm { i } , \operatorname { cis } \frac { 25 \pi } { 14 }$ 
C)  cis $\frac { \pi } { 14 } , \operatorname { cis } \frac { 5 \pi } { 14 } , \operatorname { cis } \frac { 9 \pi } { 14 } , \operatorname { cis } \frac { 13 \pi } { 14 }$, cis $\frac { 17 \pi } { 14 }$, i, cis $\frac { 25 \pi } { 14 }$ 
D)  cis $\frac { \pi } { 14 }$, cis $\frac { 5 \pi } { 14 }$, cis $\frac { 9 \pi } { 14 }$, cis $\frac { 13 \pi } { 14 }$, cis $\frac { 17 \pi } { 14 } , - \mathrm { i }$, cis $\frac { 25 \pi } { 14 }$, i

Question 19

Write the complex number in rectangular form.
-$\sqrt { 2 } \operatorname { cis } 270 ^ { \circ }$

Accepted Answer

A)  $- \sqrt { 2 }$ 
B)  $- \mathrm { i } \sqrt { 2 }$ 
C)  $- \mathrm { i }$ 
D)  $- 2 \mathrm { i }$ 
A)  $- \sqrt { 2 }$ 
B)  $- \mathrm { i } \sqrt { 2 }$ 
C)  $- \mathrm { i }$ 
D)  $- 2 \mathrm { i }$

Question 20

Write the complex number in trigonometric form r(c $$r ( \cos \theta + i \sin \theta )$$ ), with $$\theta$$ in the interval [0 $$\left[ 0 ^ { \circ } , 360 ^ { \circ } \right)$$
-$8 \sqrt { 3 } + 8 i$

Accepted Answer

A)  $16 \left( \cos 30 ^ { \circ } + i \sin 30 ^ { \circ } \right)$ 
B)  $16 \left( \cos 300 ^ { \circ } + i \sin 300 ^ { \circ } \right)$ 
C)  $16 \left( \cos 60 ^ { \circ } + i \sin 60 ^ { \circ } \right)$ 
D)  $16 \left( \cos 330 ^ { \circ } + i \sin 330 ^ { \circ } \right)$ 
A)  $16 \left( \cos 30 ^ { \circ } + i \sin 30 ^ { \circ } \right)$ 
B)  $16 \left( \cos 300 ^ { \circ } + i \sin 300 ^ { \circ } \right)$ 
C)  $16 \left( \cos 60 ^ { \circ } + i \sin 60 ^ { \circ } \right)$ 
D)  $16 \left( \cos 330 ^ { \circ } + i \sin 330 ^ { \circ } \right)$

Graph the complex number. - $6 - 3 i$

For the given rectangular equation, give its equivalent polar equation. - $2 x + 3 y = 6$

Find the quotient and write in rectangular form. First convert the numerator and denominator to trigonometric form. - $\frac { 9 \operatorname { cis } 154 ^ { \circ } } { 3 \operatorname { cis } 34 ^ { \circ } }$

Find the quotient and write in rectangular form. First convert the numerator and denominator to trigonometric form. - $\frac { 4 \mathrm { i } } { 1 - 10 \mathrm { i } }$

Answer the question. -In which quadrants do the fourth roots of a complex number lie?

For the given rectangular equation, give its equivalent polar equation. - $6 x - y = 14$

Give the rectangular coordinates for the point. -(2, 270°)

Find all specified roots. -Cube roots of $\mathrm { i }$ .

Find the polar coordinates of the point(s) of intersection of the given curves for 0 $0 \leq \theta < 2 \pi$ - $r = 4 + \cos \theta , r = 4 - \sin \theta$

Determine whether the pair of vectors is orthogonal. --1, -1 , -64, 32

Graph the complex number. - $3 + 2 i \sqrt { 7 }$ $Graph the complex number. - 3 + 2 i \sqrt { 7 }$

Give the rectangular coordinates for the point. - $\left( - 3 , \frac { 3 \pi } { 2 } \right)$

Determine whether the pair of vectors is orthogonal. --2, 6 , -8, -5

Considering the given value of t, choose the ordered pair that lies on the graph of the given pair of parametric equations. - $x = t , y = t ^ { 2 } ; t = - 2$

Find all specified roots. -Seventh roots of $i$ .

Write the complex number in rectangular form. - $\sqrt { 2 } \operatorname { cis } 270 ^ { \circ }$

Write the complex number in trigonometric form r(c $r ( \cos \theta + i \sin \theta )$ ), with $\theta$ in the interval [0 $\left[ 0 ^ { \circ } , 360 ^ { \circ } \right)$ - $8 \sqrt { 3 } + 8 i$

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

Systems and Matrices

Analytic Geometry

Further Topics in Algebra

Filters

Exam 9: Applications of Trigonometry

Graph the complex number. - 6−3i6 - 3 i6−3i

For the given rectangular equation, give its equivalent polar equation. - 2x+3y=62 x + 3 y = 62x+3y=6

Find the quotient and write in rectangular form. First convert the numerator and denominator to trigonometric form. - 9cis⁡154∘3cis⁡34∘\frac { 9 \operatorname { cis } 154 ^ { \circ } } { 3 \operatorname { cis } 34 ^ { \circ } }3cis34∘9cis154∘​

Find the quotient and write in rectangular form. First convert the numerator and denominator to trigonometric form. - 4i1−10i\frac { 4 \mathrm { i } } { 1 - 10 \mathrm { i } }1−10i4i​

Answer the question. -In which quadrants do the fourth roots of a complex number lie?

For the given rectangular equation, give its equivalent polar equation. - 6x−y=146 x - y = 146x−y=14

Give the rectangular coordinates for the point. -(2, 270°)

Find all specified roots. -Cube roots of i\mathrm { i }i .

Find the polar coordinates of the point(s) of intersection of the given curves for 0 0≤θ<2π0 \leq \theta < 2 \pi0≤θ<2π - r=4+cos⁡θ,r=4−sin⁡θr = 4 + \cos \theta , r = 4 - \sin \thetar=4+cosθ,r=4−sinθ

Determine whether the pair of vectors is orthogonal. --1, -1 , -64, 32

Graph the complex number. - 3+2i73 + 2 i \sqrt { 7 }3+2i7​

Give the rectangular coordinates for the point. - (−3,3π2)\left( - 3 , \frac { 3 \pi } { 2 } \right)(−3,23π​)

Determine whether the pair of vectors is orthogonal. --2, 6 , -8, -5

Considering the given value of t, choose the ordered pair that lies on the graph of the given pair of parametric equations. - x=t,y=t2;t=−2x = t , y = t ^ { 2 } ; t = - 2x=t,y=t2;t=−2

Find all specified roots. -Seventh roots of iii .

Write the complex number in rectangular form. - 2cis⁡270∘\sqrt { 2 } \operatorname { cis } 270 ^ { \circ }2​cis270∘

Write the complex number in trigonometric form r(c r(cos⁡θ+isin⁡θ)r ( \cos \theta + i \sin \theta )r(cosθ+isinθ) ), with θ\thetaθ in the interval [0 [0∘,360∘)\left[ 0 ^ { \circ } , 360 ^ { \circ } \right)[0∘,360∘) - 83+8i8 \sqrt { 3 } + 8 i83​+8i

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

Systems and Matrices

Analytic Geometry

Further Topics in Algebra

Filters

Graph the complex number. - $6 - 3 i$

For the given rectangular equation, give its equivalent polar equation. - $2 x + 3 y = 6$

Find the quotient and write in rectangular form. First convert the numerator and denominator to trigonometric form. - $\frac { 9 \operatorname { cis } 154 ^ { \circ } } { 3 \operatorname { cis } 34 ^ { \circ } }$

Find the quotient and write in rectangular form. First convert the numerator and denominator to trigonometric form. - $\frac { 4 \mathrm { i } } { 1 - 10 \mathrm { i } }$

For the given rectangular equation, give its equivalent polar equation. - $6 x - y = 14$

Find all specified roots. -Cube roots of $\mathrm { i }$ .

Find the polar coordinates of the point(s) of intersection of the given curves for 0 $0 \leq \theta < 2 \pi$ - $r = 4 + \cos \theta , r = 4 - \sin \theta$

Graph the complex number. - $3 + 2 i \sqrt { 7 }$ $Graph the complex number. - 3 + 2 i \sqrt { 7 }$

Give the rectangular coordinates for the point. - $\left( - 3 , \frac { 3 \pi } { 2 } \right)$

Considering the given value of t, choose the ordered pair that lies on the graph of the given pair of parametric equations. - $x = t , y = t ^ { 2 } ; t = - 2$

Find all specified roots. -Seventh roots of $i$ .

Write the complex number in rectangular form. - $\sqrt { 2 } \operatorname { cis } 270 ^ { \circ }$

Write the complex number in trigonometric form r(c $r ( \cos \theta + i \sin \theta )$ ), with $\theta$ in the interval [0 $\left[ 0 ^ { \circ } , 360 ^ { \circ } \right)$ - $8 \sqrt { 3 } + 8 i$