Question 1

The graph of a polar equation is given. Select the polar equation for the graph.
-

Accepted Answer

A)  $r = 4 \sin \theta$ 
B)  $r = 2 + \sin \theta$ 
C)  $r = 2 + \cos \theta$ 
D)  $r = 4 \cos \theta$

Question 2

Perform the indicated operation. Write the result in standard form.
-$$( - 4 + 6 i ) - 5$$

Accepted Answer

A)  $- 9 + 6 i$ 
B)  $+ 6 \mathrm { i }$ 
C)  $- 6 \mathrm { i }$ 
D)  $9 - 6 i$

Question 3

The rectangular coordinates of a point are given. Express the point in polar coordinates with $$\mathrm { r } \geq 0 \text { and } 0 ^ { \circ } \leq \theta < 360 ^ { \circ }$$
-$( 3,3 )$

Accepted Answer

A)  $\left( 3,90 ^ { \circ } \right)$ 
B)  $\left( 3 \sqrt { 2 } , 45 ^ { \circ } \right)$ 
C)  $\left( 3 \sqrt { 2 } , 135 ^ { \circ } \right)$ 
D)  $\left( 3,45 ^ { \circ } \right)$

Question 4

Find the product. Write the answer in standard form.
-$$\mathrm { i } ( 6 - 9 \mathrm { i } ) ( 6 - 5 \mathrm { i } )$$

Accepted Answer

A)  $45 \mathrm { i } ^ { 3 } + 84 \mathrm { i } ^ { 2 } + 36 \mathrm { i }$ 
B)  $- 84 + 9 \mathrm { i }$ 
C)  $81 - 24 i$ 
D)  $84 - 9 \mathrm { i }$

Question 5

Graph the cycloid for t in the indicated interval.
-$$x = 4 ( t - \sin t ) , y = 4 ( 1 - \cos t ) , 0 \leq t \leq 4 \pi$$

Accepted Answer

A) 
 
B) 
 
C) 
 
D)

Question 6

For the given rectangular equation, give its equivalent polar equation.
-$$2 x - 9 y = - 10$$

Accepted Answer

A)  $r = \frac { - 10 } { 2 \cos \theta + 9 \sin \theta }$ 
B)  $r = \frac { - 10 } { 2 \sin \theta - 9 \cos \theta }$ 
C)  $r = \frac { - 10 } { 2 \cos \theta - 9 \sin \theta }$ 
D)  $r = \frac { - 10 } { 9 \cos \theta - 2 \sin \theta }$

Question 7

Find the polar coordinates of the point(s)of intersection of the given curves for 0 $$0 \leq \theta < 2 \pi$$
-$$\mathrm { r } = 7 , \mathrm { r } = 7 + \sin \theta$$

Accepted Answer

A)  $( 0,0 ) , ( 0,7 ) , ( \pi , 7 )$ 
B)  $( 0,7 ) , ( \pi , 7 )$ 
C)  $( 7,0 ) , ( 7 , \pi )$ 
D)  $( 0,0 ) , ( 7,0 ) , ( 7 , \pi )$

Question 8

Find all specified roots.
-Fifth roots of i.

Accepted Answer

A)  $\operatorname { cis } \frac { 2 \pi } { 5 } , \operatorname { cis } \frac { 4 \pi } { 5 } , \operatorname { cis } \frac { 6 \pi } { 5 } , \operatorname { cis } \frac { 8 \pi } { 10 } , 1$ 
B)  $\operatorname { cis } \frac { \pi } { 10 } , i , \operatorname { cis } \frac { 9 \pi } { 10 } , \operatorname { cis } \frac { 13 \pi } { 10 } , \operatorname { cis } \frac { 17 \pi } { 10 }$ 
C)  $\operatorname { cis } \frac { \pi } { 5 } , \operatorname { cis } \frac { 3 \pi } { 5 } , - 1 , \operatorname { cis } \frac { 7 \pi } { 10 } , \operatorname { cis } \frac { 9 \pi } { 10 }$ 
D)  $\operatorname { cis } \frac { \pi } { 10 } , \operatorname { cis } \frac { 7 \pi } { 10 } , \operatorname { cis } \frac { 9 \pi } { 10 } , \operatorname { cis } \frac { 13 \pi } { 10 } , \operatorname { cis } \frac { 17 \pi } { 10 }$

Question 9

Find the product. Write the answer in standard form.
-$( 3 - 7 \mathrm { i } ) ( 2 - 5 \mathrm { i } )$

Accepted Answer

A)  $- 29 + 29 i$ 
B)  $41 + 1 \mathrm { i }$ 
C)  $35 i ^ { 2 } - 29 i + 6$ 
D)  $- 29 - 29 \mathrm { i }$

Question 10

Find the given power. Write answer in rectangular form.
-$$\left( - \frac { 1 } { 2 } - \frac { \sqrt { 3 } } { 2 } i \right) ^ { 10 }$$

Accepted Answer

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

Question 11

Considering the given value of t, choose the ordered pair that lies on the graph of the given pair of parametric equations.
-

Accepted Answer

A)  $r = 8 \sin \theta$ 
B)  $r = 4$ 
C)  $r = 8 \cos \theta$ 
D)  $r \sin \theta = 4$

Question 12

Perform the indicated operation. Give answers in rectangular form expressing real and imaginary parts to four decimal
places.
-$\left( 12 \operatorname { cis } 13.5 ^ { \circ } \right) \left( 2 \operatorname { cis } 7.5 ^ { \circ } \right)$

Accepted Answer

A)  $22.4059 + 8.6008 \mathrm { i }$ 
B)  $- 6.5728 + 10.0399 \mathrm { i }$ 
C) 
$- 1.0955 \mathrm { i }$ 
D) 
$+ 22.4059 \mathrm { i }$

Question 13

Solve the problem.
-A projectile is fired with an initial velocity of 400 feet per second at an angle of 70° with the horizontal. In how many seconds will the projectile reach its maximum altitude? (Round your
Answer to the nearest tenth of a second.)

Accepted Answer

A) 16.6 sec 
B) 8.4 sec 
C) 11.8 sec 
D) 19.9 sec

Question 14

Solve the quadratic equation and express all nonreal complex solutions in terms of i.
-$$\mathrm { x } ^ { 2 } + 35 = 5 \mathrm { x }$$

Accepted Answer

A)  $\{ 0,5 \}$ 
B)  $\left\{ \frac { 5 } { 2 } \pm \frac { 5 } { 2 } i \right\}$ 
C)  $\{ - 5,5 \}$ 
D)  $\left\{ \frac { 5 } { 2 } \pm \frac { \sqrt { 115 } } { 2 } \mathrm { i } \right\}$

Question 15

Find the quotient and write in rectangular form. First convert the numerator and denominator to trigonometric form.
-$\frac { 2 \sqrt { 3 } + 2 i } { \sqrt { 3 } - i }$

Accepted Answer

A)  $\sqrt { 3 } \mathrm { i }$ 
B)  $\sqrt { 3 }$ 
C)  $\sqrt { 3 } + \mathrm { i }$ 
D)  $1 + \sqrt { 3 } \mathrm { i }$

Question 16

Find the quotient. Write the answer in standard form.
-$$\frac { 5 - 6 i } { 4 + 2 i }$$

Accepted Answer

A)  $\frac { 8 } { 3 } + \frac { 17 } { 12 } \mathrm { i }$ 
B)  $\frac { 2 } { 5 } - \frac { 17 } { 10 } \mathrm { i }$ 
C)  $\frac { 16 } { 5 } + \frac { 7 } { 5 } \mathrm { i }$ 
D)  $\frac { 1 } { 3 } + \frac { 17 } { 12 } \mathrm { i }$

Question 17

Considering the given value of t, choose the ordered pair that lies on the graph of the given pair of parametric equations.
-$$x = t , y = t ^ { 3 } ; t = - 2$$

Accepted Answer

A)  $( - 8 , - 2 )$ 
B)  $( 2,8 )$ 
C)  $( - 2 , - 8 )$ 
D)  $( - 2$,

Question 18

Considering the given value of t, choose the ordered pair that lies on the graph of the given pair of parametric equations.
-$x = - 4 t + 8 , y = 3 t - 3 ; t = - 7$

Accepted Answer

A)  $( - 3 , - 7 )$ 
B)  $( 36 , - 24 )$ 
C)  $( 20 , - 18 )$ 
D)  $( 4,0 )$

Question 19

Find the given power. Write answer in rectangular form.
-$\left( \cos 30 ^ { \circ } + i \sin 30 ^ { \circ }) \right.$$^{18}$

Accepted Answer

A)  $- 1$ 
B)  $\mathrm { i }$ 
C)  $- \mathrm { i }$

Question 20

Write the complex number in rectangular form.
-$9 \left( \cos 180 ^ { \circ } + i \sin 180 ^ { \circ } \right)$

Accepted Answer

A)  $- 9$ 
B)  $9 \mathrm { i }$ 
C)  9 
D)  $- 9 \mathrm { i }$

Exam 8: Complex Numbers, Polar Equations, and Parametric Equations

The graph of a polar equation is given. Select the polar equation for the graph. - $The graph of a polar equation is given. Select the polar equation for the graph. - A) r = 4 \sin \theta B) r = 2 + \sin \theta C) r = 2 + \cos \theta D) r = 4 \cos \theta$

Perform the indicated operation. Write the result in standard form. - $( - 4 + 6 i ) - 5$

The rectangular coordinates of a point are given. Express the point in polar coordinates with $\mathrm { r } \geq 0 \text { and } 0 ^ { \circ } \leq \theta < 360 ^ { \circ }$ - $( 3,3 )$

Find the product. Write the answer in standard form. - $\mathrm { i } ( 6 - 9 \mathrm { i } ) ( 6 - 5 \mathrm { i } )$

Graph the cycloid for t in the indicated interval. - $x = 4 ( t - \sin t ) , y = 4 ( 1 - \cos t ) , 0 \leq t \leq 4 \pi$ $Graph the cycloid for t in the indicated interval. - x = 4 ( t - \sin t ) , y = 4 ( 1 - \cos t ) , 0 \leq t \leq 4 \pi A) B) C) D)$

For the given rectangular equation, give its equivalent polar equation. - $2 x - 9 y = - 10$

Find the polar coordinates of the point(s) of intersection of the given curves for 0 $0 \leq \theta < 2 \pi$ - $\mathrm { r } = 7 , \mathrm { r } = 7 + \sin \theta$

Find all specified roots. -Fifth roots of i.

Find the product. Write the answer in standard form. - $( 3 - 7 \mathrm { i } ) ( 2 - 5 \mathrm { i } )$

Find the given power. Write answer in rectangular form. - $\left( - \frac { 1 } { 2 } - \frac { \sqrt { 3 } } { 2 } i \right) ^ { 10 }$

Perform the indicated operation. Give answers in rectangular form expressing real and imaginary parts to four decimal places. - $\left( 12 \operatorname { cis } 13.5 ^ { \circ } \right) \left( 2 \operatorname { cis } 7.5 ^ { \circ } \right)$

Solve the problem. -A projectile is fired with an initial velocity of 400 feet per second at an angle of 70° with the horizontal. In how many seconds will the projectile reach its maximum altitude? (Round your Answer to the nearest tenth of a second.)

Solve the quadratic equation and express all nonreal complex solutions in terms of i. - $\mathrm { x } ^ { 2 } + 35 = 5 \mathrm { x }$

Find the quotient and write in rectangular form. First convert the numerator and denominator to trigonometric form. - $\frac { 2 \sqrt { 3 } + 2 i } { \sqrt { 3 } - i }$

Find the quotient. Write the answer in standard form. - $\frac { 5 - 6 i } { 4 + 2 i }$

Considering the given value of t, choose the ordered pair that lies on the graph of the given pair of parametric equations. - $x = t , y = t ^ { 3 } ; t = - 2$

Considering the given value of t, choose the ordered pair that lies on the graph of the given pair of parametric equations. - $x = - 4 t + 8 , y = 3 t - 3 ; t = - 7$

Find the given power. Write answer in rectangular form. - $\left( \cos 30 ^ { \circ } + i \sin 30 ^ { \circ }) \right.$ $^{18}$

Write the complex number in rectangular form. - $9 \left( \cos 180 ^ { \circ } + i \sin 180 ^ { \circ } \right)$

Exam 8: Complex Numbers, Polar Equations, and Parametric Equations

The graph of a polar equation is given. Select the polar equation for the graph. -

Perform the indicated operation. Write the result in standard form. - (−4+6i) −5( - 4 + 6 i ) - 5(−4+6i) −5

The rectangular coordinates of a point are given. Express the point in polar coordinates with r≥0 and 0∘≤θ<360∘\mathrm { r } \geq 0 \text { and } 0 ^ { \circ } \leq \theta < 360 ^ { \circ }r≥0 and 0∘≤θ<360∘ - (3,3) ( 3,3 ) (3,3)

Find the product. Write the answer in standard form. - i(6−9i) (6−5i) \mathrm { i } ( 6 - 9 \mathrm { i } ) ( 6 - 5 \mathrm { i } ) i(6−9i) (6−5i)

Graph the cycloid for t in the indicated interval. - x=4(t−sin⁡t) ,y=4(1−cos⁡t) ,0≤t≤4πx = 4 ( t - \sin t ) , y = 4 ( 1 - \cos t ) , 0 \leq t \leq 4 \pix=4(t−sint) ,y=4(1−cost) ,0≤t≤4π

For the given rectangular equation, give its equivalent polar equation. - 2x−9y=−102 x - 9 y = - 102x−9y=−10

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=7,r=7+sin⁡θ\mathrm { r } = 7 , \mathrm { r } = 7 + \sin \thetar=7,r=7+sinθ

Find all specified roots. -Fifth roots of i.

Find the product. Write the answer in standard form. - (3−7i) (2−5i) ( 3 - 7 \mathrm { i } ) ( 2 - 5 \mathrm { i } ) (3−7i) (2−5i)

Find the given power. Write answer in rectangular form. - (−12−32i) 10\left( - \frac { 1 } { 2 } - \frac { \sqrt { 3 } } { 2 } i \right) ^ { 10 }(−21​−23​​i) 10

Considering the given value of t, choose the ordered pair that lies on the graph of the given pair of parametric equations. -