Deck 10: Parametric Equations; Polar Equations

Question 1

The parametric equations of the line segment from (1,0) to (0,1) with t $\in$ [0,1] are

A) $x ( t ) = t , y ( t ) = 1 - t$
B) $x ( t ) = t , y ( t ) = t - 1$
C) $x ( t ) = 1 - t , y ( t ) = t$
D) $x ( t ) = t - 1 , y ( t ) = t$
E) $x ( t ) = t - 1 , y ( t ) = 1 - t$

Answer 1

$x ( t ) = 1 - t , y ( t ) = t$

Question 2

The parametric equations of the line segment from (-5,0) to (0,5) with t $\in$ [0,5] are

A) $x ( t ) = 5 - t , y ( t ) = t$
B) $x ( t ) = t , y ( t ) = 5 - t$
C) $x ( t ) = t , y ( t ) = t - 5$
D) $x ( t ) = t - 5 , y ( t ) = t$
E) $x ( t ) = t - 5 , y ( t ) = 5 - t$

Answer 2

$x ( t ) = t - 5 , y ( t ) = t$

Question 3

The parametric equations of the line segment from (-3,0) to (0,-3) with t $\in$ [0,3] are

A) $x ( t ) = t - 3 , y ( t ) = t - 3$
B) $x ( t ) = 3 - t , y ( t ) = t$
C) $x ( t ) = 3 - t , y ( t ) = - t$
D) $x ( t ) = t - 3 , y ( t ) = - t$
E) $x ( t ) = t - 3 , y ( t ) = 3 - t$

Answer 3

$x ( t ) = t - 3 , y ( t ) = - t$

Question 4

The parametric equations of the line segment from (0,-4) to (4,0) with t $\in$ [0,4] are

A) $x ( t ) = t - 4 , y ( t ) = t$
B) $x ( t ) = t , y ( t ) = 4 - t$
C) $x ( t ) = t , y ( t ) = t - 4$
D) $x ( t ) = 4 - t , y ( t ) = - t$
E) $x ( t ) = t - 4 , y ( t ) = 4 - t$

Question 5

The graph of the parametric equations $x ( t ) = \sin t , y ( t ) = \cos t$ with $$t \in \left[ \frac { \pi } { 2 } , 2 \pi \right] t \in \left[ \frac { \pi } { 2 } , 2 \pi \right]$$ is an arc of the unit circle from

A)(1,0) to (-1,0)
B)(0,1) to (1,0)
C)(1,0) to (0,-1)
D)(1,0) to (0,1)
E)(0,1) to (-1,0)

Question 6

The graph of the parametric equations $x ( t ) = 5 \cos t , y ( t ) = 5 \sin t$ with $t \in [ \pi , 2 \pi ]$ is an arc of a circle centered at the origin with radius 5 from

A)(0,-5) to (5,0)
B)(-5,0) to (0,5)
C)(-5,0) to (5,0)
D)(5,0) to (0,-5)
E)(0,5) to (-5,0)

Question 7

The graph of the parametric equations $x ( t ) = 3 \cos t , y ( t ) = 3$ with $t \in \left[ \frac { 3 \pi } { 2 } , 2 \pi \right]$ is an arc of a circle centered at the origin with radius 3 from

A) $( 0 , - 3 ) \text { to } ( 3,0 )$
B) $( 0,3 ) \text { to } ( 0 , - 3 )$
C) $( - 3,0 ) \text { to } ( 3,0 )$
D) $( 3,0 ) \text { to } ( 0 , - 3 )$
E) $( 0,3 ) \text { to } ( - 3,0 )$

Question 8

The graph of the parametric equations $x ( t ) = 4 \sin t , y ( t ) = 4 \cos t,$ with $t \in [ \pi , 2 \pi ]$ is an arc of a circle centered at the origin with radius 4 from

A) $( 0,4 ) \text { to } ( 0 , - 4 )$
B) $( 0 , - 4 ) \text { to } ( 0,4 )$
C) $( 0 , - 4 ) \text { to } ( 4,0 )$
D) $( 4,0 ) \text { to } ( 0 , - 4 )$
E) $( 0,4 ) \text { to } ( - 4,0 )$

Question 9

The rectangular equation of the plane curve, with parametric equations $x ( t ) = 2 t + 5 , y ( t ) = 4 t - 7,$ is

A) $y = 4 x - 3$
B) $y = 2 x - 7$
C) $y = 2 x - 17$
D) $y = 4 x + 3$
E) $y = 2 x - 2$

Question 10

The rectangular equation of the plane curve, with parametric equations $x ( t ) = \tan t , y ( t ) = \sec ^ { 2 } t,$ is

A) $y = \sqrt { x }$
B) $y = x ^ { 2 } - 1$
C) $y = x ^ { 2 } + 1$
D) $y = \sqrt { x ^ { 2 } + 1 }$
E) $y = \sqrt { x ^ { 2 } - 1 }$

Question 11

The rectangular equation of the plane curve, with parametric equations $x ( t ) = \sin 3 t , y ( t ) = \cos ^ { 2 } 3 t$ with $t \in \left[ 0 , \frac { \pi } { 6 } \right],$ is

A) $y = 1 - x ^ { 2 }$
B) $y = x ^ { 2 } - 1$
C) $y = x ^ { 2 } + 1$
D) $y = - \sqrt { 1 - x ^ { 2 } }$
E) $y = \sqrt { 1 - x ^ { 2 } }$

Question 12

The rectangular equation of the plane curve, with parametric equations $x ( t ) = e ^ { - t } + 1 , y ( t ) = e ^ { 2 t }$ with $t \in ( - \infty , \infty ),$ is

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

Question 13

The rectangular equation of the plane curve, with parametric equations $x ( t ) = \csc t , y ( t ) = \cot t,$ is

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

Question 14

The rectangular equation of the plane curve, with parametric equations $x ( t ) = 3 - t , y ( t ) = t + 3$ with $t \in ( - \infty , \infty ),$ is

A) $x - y = 6$
B) $x + y = - 6$
C) $x - y = - 6$
D) $x + y = 6$
E) $x + y = - 3$

Question 15

The rectangular equation of the plane curve, with parametric equations $x ( t ) = \sqrt [ 3 ] { 3 - t } , y ( t ) = \sqrt [ 3 ] { t }$ with $t \in ( - \infty , \infty ),$ is

A) $x = \sqrt [ 3 ] { y ^ { 3 } - 3 }$
B) $y = \sqrt [ 3 ] { x ^ { 3 } - 3 }$
C) $x = \sqrt [ 3 ] { y ^ { 3 } } - 3$
D) $y = \sqrt [ 3 ] { x ^ { 3 } } - 3$
E) $x = \sqrt [ 3 ] { 3 - y ^ { 3 } }$

Question 16

The rectangular equation of the plane curve, with parametric equations $x ( t ) = 4 \sin t , y ( t ) = 3 \cos t,$ is

A) $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 16 } = 1$
B) $\frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 9 } = 1$
C) $\frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 9 } = 1$
D) $- \frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 9 } = 1$
E) $16 x ^ { 2 } + 9 y ^ { 2 } = 1$

Question 17

The rectangular equation of the plane curve, with parametric equations $x ( t ) = \sqrt { t } , y ( t ) = 3 t ^ { \frac { 5 } { 2 } }$ is

A) $x = 3 y ^ { 3 }$
B) $y = 3 y ^ { 3 }$
C) $x = 3 y ^ { 5 }$
D) $y = 3 x ^ { 5 }$
E) $y = - 3 x ^ { 5 }$

Question 18

The rectangular equation of the plane curve, with parametric equations $x ( t ) = e ^ { - t } , y ( t ) = 3 e ^ { t }$ with $t \in ( - \infty , \infty )$ , is

A) $y = \frac { 3 } { x }$
B) $y = \frac { x } { 3 }$
C) $y = - \frac { 3 } { x }$
D) $y = 3 x$
E) $y = - \frac { x } { 3 }$

Question 19

Which one of the following sets of parametric equations does not correspond to the rectangular equation $y = 4 x - 2 ?$

A) $x ( t ) = t , y ( t ) = 4 t - 2$
B) $x ( t ) = \frac { t } { 4 } + \frac { 1 } { 2 } , y ( t ) = t$
C) $x ( t ) = \frac { t + 1 } { 2 } , y ( t ) = 2 t$
D) $x ( t ) = 2 t , y ( t ) = 8 t - 2$
E) $x ( t ) = 2 t , y ( t ) = 8 t - 4$

Question 20

Which one of the following sets of parametric equations does not correspond to the rectangular equation $y = 6 x ^ { 3 } ?$

A) $x ( t ) = t , y ( t ) = 6 t ^ { 3 }$
B) $x ( t ) = 2 t , y ( t ) = 48 t ^ { 3 }$
C) $x ( t ) = \sqrt [ 3 ] { t } , y ( t ) = 6 t$
D) $x ( t ) = t ^ { 4 } , y ( t ) = 6 t ^ { 6 }$
E) $x ( t ) = \frac { \sqrt [ 3 ] { t } } { 2 } , y ( t ) = \frac { 3 t } { 4 }$

Question 21

Let $x ( t ) = 1 - e ^ { t } , y ( t ) = 1 + e ^ { t }$ be the parametric equations of a curve. Then $\frac { d y } { d x }$ is

A) $e ^ { t }$
B) $e ^ { - t }$
C)-1
D)1
E) $1 - e ^ { t }$

Question 22

Let $x ( t ) = 1 - e ^ { t } , y ( t ) = e ^ { 2 t }$ be the parametric equations of a curve. Then $\frac { d y } { d x }$ is

A) $e ^ { t }$
B) $e ^ { - t }$
C)-1
D) $e ^ { - 2 t }$
E) $- 2 e ^ { t }$

Question 23

Let $x ( t ) = \frac { t ^ { 3 } } { 1 + t } , y ( t ) = \frac { 1 } { 1 + t }$ be the parametric equations of a curve. Then $\frac { d y } { d x }$ is

A) $- \frac { t ^ { 2 } } { 2 t + 3 }$
B) $- \frac { 1 } { t ^ { 2 } ( 2 t + 3 ) }$
C)-1
D) $\frac { 1 } { t ^ { 2 } ( 2 \mathrm { t } + 3 ) }$
E) $- \frac { t ^ { 2 } } { 2 \mathrm { t } + 3 }$

Question 24

Let $x ( t ) = 4 t , y ( t ) = t ^ { 2 } + 3$ be the parametric equations of a curve. Then $\frac { d y } { d x }$ is

A) $- \frac { 2 } { t }$
B) $\frac { 2 } { t }$
C) $\frac { t ^ { 2 } } { 2 }$
D) $\frac { t } { 2 }$
E) $- \frac { t } { 2 }$

Question 25

Let $x ( t ) = 4 \cos t , y ( t ) = 3 \sin t$ be the parametric equations of a curve. Then $\frac { d y } { d x }$ is

A) $- \frac { 3 \cot t } { 4 }$
B) $\frac { 3 \cot t } { 4 }$
C) $- \frac { 3 \tan t } { 4 }$
D) $\frac { 3 \tan t } { 4 }$
E) $- \frac { 4 \cot t } { 3 }$

Question 26

Let $x ( t ) = 2 t ^ { 3 } , y ( t ) = 3 t ^ { 2 }$ be the parametric equations of a curve. Then $\frac { d y } { d x }$ is

A) $- \frac { 1 } { t }$
B) $\frac { 2 } { t }$
C) $\frac { t ^ { 2 } } { 2 }$
D) $\frac { 1 } { t }$
E) $- \frac { t } { 2 }$

Question 27

Let $x ( t ) = \sin t , y ( t ) = \cos t - t$ be the parametric equations of a curve. Then $\frac { d y } { d x }$ is

A) $\tan t - \sec t$
B) $\cot t - \csc t$
C) $- ( \tan t + \sec t )$
D) $\tan t + \sec t$
E) $\cot t + \csc t$

Question 28

Let $x ( t ) = t + \sin t , y ( t ) = \cos t$ be the parametric equations of a curve. Then $\frac { d y } { d x }$ is

A) $\frac { \sin t } { 1 + \cos t }$
B) $- \frac { \sin t } { 1 - \cos t }$
C) $- \frac { \cos t } { 1 + \sin t }$
D) $\frac { \sin t } { 1 - \cos t }$
E) $- \frac { \sin t } { 1 + \cos t }$

Question 29

Let $x ( t ) = e ^ { t } \sin t , y ( t ) = e ^ { t } \cos t$ be the parametric equations of a curve. Then $\frac { d y } { d x }$ is

A) $- \frac { \cos t - \sin t } { \cos t + \sin t }$
B) $\frac { \cos t + \sin t } { \cos t - \sin t }$
C) $- \frac { \cos t + \sin t } { \cos t - \sin t }$
D) $\frac { \cos t - \sin t } { \cos t + \sin t }$
E) $\frac { \cos t - \sin t } { 2 ( \cos t + \sin t ) }$

Question 30

Let $x ( t ) = \ln t , y ( t ) = \frac { 1 } { t }$ be the parametric equations of a curve. Then $\frac { d y } { d x }$ is

A) $- \frac { 2 } { t ^ { 2 } }$
B) $\frac { 1 } { t ^ { 2 } }$
C) $- \frac { 1 } { t }$
D) $\frac { 1 } { t }$
E) $- \frac { 1 } { t ^ { 2 } }$

Question 31

Find the points on the curve $x ( t ) = \frac { t ^ { 3 } } { 1 + t } , y ( t ) = \frac { 1 } { 1 + t }$ where the tangent line is vertical.

A) $\left( \frac { 27 } { 4 } , - 2 \right)$
B) $( 0,1 ) , \left( \frac { 27 } { 4 } , - 2 \right)$
C) $( 0,1 )$
D) $( 0,1 ) , \left( \frac { 27 } { 4 } , 2 \right)$
E) $( 0,1 ) , \left( - \frac { 27 } { 4 } , 2 \right)$

Question 32

Find the points on the curve $x ( t ) = 4 t , y ( t ) = t ^ { 2 } + 3$ where the tangent line is horizontal.

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

Question 33

Find the points on the curve $x ( t ) = t ^ { 3 } - 9 t , y ( t ) = t ^ { 2 }$ where the tangent line is vertical.

A) $( 6 \sqrt { 3 } , 3 )$
B) $( - 6 \sqrt { 3 } , 3 )$
C) $( - 6 \sqrt { 3 } , 3 \sqrt { 3 } )$
D) $( 6 \sqrt { 3 } , 3 \sqrt { 3 } )$
E) $( - 6 \sqrt { 3 } , 3 ) , ( 6 \sqrt { 3 } , 3 )$

Question 34

Find the points on the curve $x ( t ) = 3 - t ^ { 2 } , y ( t ) = t ^ { 2 } + 4 t$ where the tangent line is horizontal.

A)(-4,-1)
B)(-1,4)
C)(-1,-4)
D)(1,-4)
E)(1,4)

Question 35

The length of the curve $x ( t ) = t ^ { 3 } , y ( t ) = t ^ { 2 }$ with $t \in [ 0,2 ]$ is

A) $\frac { 8 ( 10 \sqrt { 10 } - 1 ) } { 3 }$
B) $\frac { 8 ( 10 \sqrt { 10 } - 1 ) } { 18 }$
C) $\frac { 8 ( 10 \sqrt { 10 } - 1 ) } { 36 }$
D) $\frac { 8 ( 10 \sqrt { 10 } - 1 ) } { 9 }$
E) $\frac { 8 ( 10 \sqrt { 10 } - 1 ) } { 27 }$

Question 36

The length of the curve $x ( t ) = 9 - t ^ { 3 } , y ( t ) = t ^ { 2 }$ with $t \in [ 0,2 ]$ is

A) $\frac { 8 ( 10 \sqrt { 10 } - 1 ) } { 3 }$
B) $\frac { 8 ( 10 \sqrt { 10 } - 1 ) } { 18 }$
C) $\frac { 8 ( 10 \sqrt { 10 } - 1 ) } { 36 }$
D) $\frac { 8 ( 10 \sqrt { 10 } - 1 ) } { 9 }$
E) $\frac { 8 ( 10 \sqrt { 10 } - 1 ) } { 27 }$

Question 37

The length of the curve $x ( t ) = t ^ { 2 } - 2 , y ( t ) = \frac { t ^ { 3 } } { 2 } + 3$ with $t \in [ 0,2 ]$ is

A) $\frac { 4 ( 13 \sqrt { 13 } - 8 ) } { 3 }$
B) $\frac { 4 ( 13 \sqrt { 13 } - 8 ) } { 9 }$
C) $\frac { 4 ( 13 \sqrt { 13 } - 8 ) } { 27 }$
D) $\frac { 4 ( 13 \sqrt { 13 } - 8 ) } { 36 }$
E) $\frac { 4 ( 13 \sqrt { 13 } - 8 ) } { 18 }$

Question 38

The length of the curve $x ( t ) = 1 - \cos t , y ( t ) = 1 + \sin t$ with $t \in \left[ - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right]$ is

A) $\frac { \pi } { 2 }$
B) $\pi$
C) $\frac { 3 \pi } { 8 }$
D) $\frac { 3 \pi } { 4 }$
E) $2 \pi$

Question 39

The length of the curve $x ( t ) = 4 - 3 \cos t , y ( t ) = 3 + 3 \sin t$ with $t \in \left[ - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right]$ is

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

Question 40

The length of the curve $x ( t ) = e ^ { t } \cos t , y ( t ) = e ^ { t } \sin t$ with $t \in [ 0 , \pi ]$ is

A) $\sqrt { 3 } \left( e ^ { \pi } - 1 \right)$
B) $\sqrt { 3 } \left( e ^ { \pi } + 1 \right)$
C) $\sqrt { 5 } \left( e ^ { \pi } - 1 \right)$
D) $\sqrt { 2 } \left( e ^ { \pi } + 1 \right)$
E) $\sqrt { 2 } \left( e ^ { \pi } - 1 \right)$

Question 41

The area of the surface generated by revolving the curve with $x ( t ) = 4 - 3 \cos t,$ about the x-axis is

A) $\pi \ln \left( \frac { \pi + 4 } { 4 } \right)$
B) $6 \pi \ln \left( \frac { \pi - 2 } { 4 } \right)$
C) $6 \pi \ln \left( \frac { \pi + 2 } { 4 } \right)$
D) $6 \pi \ln \left( \frac { \pi + 2 } { 2 } \right)$
E) $6 \pi \ln \left( \frac { \pi - 2 } { 2 } \right)$

Question 42

The area of the surface generated by revolving the curve with $x ( t ) = 4 - 3 \cos t,$ about the y-axis is

A) $6 \pi ( 2 \pi - 3 )$
B) $6 \pi ( 2 \pi + 3 )$
C) $8 \pi ( 2 \pi - 3 )$
D) $8 \pi ( 2 \pi + 3 )$
E) $9 \pi ( 2 \pi - 3 )$

Question 43

The area of the surface generated by revolving the curve $x ( t ) = t , \quad y ( t ) = t ^ { 2 }$ with $t \in [ 0,2 ]$ about the y-axis is

A) $\frac { \pi ( 17 \sqrt { 17 } - 1 ) } { 2 }$
B) $\frac { \pi ( 17 \sqrt { 17 } + 1 ) } { 3 }$
C) $\frac { \pi ( 17 \sqrt { 17 } - 1 ) } { 3 }$
D) $\frac { \pi ( 17 \sqrt { 17 } + 1 ) } { 6 }$
E) $\frac { \pi ( 17 \sqrt { 17 } - 1 ) } { 6 }$

Question 44

The area of the surface generated by revolving the curve $x ( t ) = \cos t , \quad y ( t ) = 2 + \sin t$ with $t \in [ 0,2 \pi ]$ about the x-axis is

A) $4 \pi ^ { 2 }$
B) $8 \pi ^ { 2 }$
C) $12 \pi ^ { 2 }$
D) $16 \pi ^ { 2 }$
E) $2 \pi ^ { 2 }$

Question 45

The area of the surface generated by revolving the curve $x ( t ) = t , \quad y = 3 t ^ { 2 }$ with $t \in [ 0,2 ]$ about the y-axis is

A) $\frac { \pi ( 145 \sqrt { 145 } - 1 ) } { 60 }$
B) $\frac { \pi ( 145 \sqrt { 145 } - 1 ) } { 54 }$
C) $\frac { \pi ( 145 \sqrt { 145 } - 1 ) } { 48 }$
D) $\frac { \pi ( 145 \sqrt { 145 } - 1 ) } { 42 }$
E) $\frac { \pi ( 145 \sqrt { 145 } - 1 ) } { 36 }$

Question 46

The area of the surface generated by revolving the curve $x ( t ) = \frac { 2 t ^ { \frac { 3 } { 2 } } } { 3 } , y = 2 \sqrt { t }$ with $t \in [ 0 , \sqrt { 3 } ]$ about the y-axis is

A) $\frac { 28 \pi } { 7 }$
B) $\frac { 29 \pi } { 8 }$
C) $\frac { 28 \pi } { 9 }$
D) $\frac { 27 \pi } { 8 }$
E) $\frac { 29 \pi } { 9 }$

Question 47

The area of the surface generated by revolving the curve $x ( t ) = 4 t , \quad y = t ^ { 2 } - 2 \ln t$ with $t \in [ 0,1 ]$ about the y-axis is

A) $\frac { 2 \pi } { 3 }$
B) $\frac { 4 \pi } { 3 }$
C) $\frac { 8 \pi } { 3 }$
D) $\frac { 16 \pi } { 3 }$
E) $\frac { 64 \pi } { 3 }$

Question 48

The area of the surface generated by revolving the curve $x ( t ) = 8 \sin t , y = 2 t - \sin ( 2 t )$ with $t \in \left[ 0 , \frac { \pi } { 2 } \right]$ about the y-axis is

A) $\frac { 32 \pi } { 3 }$
B) $\frac { 64 \pi } { 3 }$
C) $\frac { 128 \pi } { 3 }$
D) $\frac { 256 \pi } { 3 }$
E) $\frac { 512 \pi } { 3 }$

Question 49

The area of the surface generated by revolving the curve $x ( t ) = 3 t ^ { 2 } , y = 2 t ^ { 3 }$ with $t \in [ 0,1 ]$ about the y-axis is

A) $\frac { 24 \pi ( 1 + \sqrt { 2 } ) } { 7 }$
B) $\frac { 24 \pi ( \sqrt { 2 } - 1 ) } { 3 }$
C) $\frac { 24 \pi ( \sqrt { 2 } - 1 ) } { 5 }$
D) $\frac { 24 \pi ( 1 + \sqrt { 2 } ) } { 5 }$
E) $\frac { 24 \pi ( 1 + \sqrt { 2 } ) } { 3 }$

Question 50

The area of the surface generated by revolving the curve $x ( t ) = t + \sqrt { 2 } , \quad y = \frac { t ^ { 2 } } { 2 } + \sqrt { t }$ with $t \in [ - \sqrt { 2 } , \sqrt { 2 } ]$ about the y-axis is

A) $\frac { 52 \pi } { 3 }$
B) $\frac { 26 \pi } { 3 }$
C) $\frac { 13 \pi } { 3 }$
D) $\frac { 11 \pi } { 5 }$
E) $\frac { 8 \pi } { 3 }$

Question 51

The area of the surface generated by revolving the curve $x ( t ) = 3 t , \quad y = \sqrt { 7 } t + 1$ with $t \in [ 0,1 ]$ about the x-axis is

A) $256 \pi \ln 2$
B) $128 \pi \ln 2$
C) $64 \pi \ln 2$
D) $32 \pi \ln 2$
E) $16 \pi \ln 2$

Question 52

The area of the surface generated by revolving the curve $x ( t ) = 1 - t ^ { 2 } , y = 3 + 2 t$ with $t \in [ 0,1 ]$ about the x-axis is

A) $4 \pi ( \sqrt { 2 } + 1 ) \left( 1 + \sinh ^ { - 1 } 1 \right)$
B) $\pi ( \sqrt { 2 } + 1 ) \left( 1 + \sinh ^ { - 1 } 1 \right)$
C) $2 \pi ( \sqrt { 2 } - 1 ) \left( 1 + \sinh ^ { - 1 } 1 \right)$
D) $4 \pi ( \sqrt { 2 } - 1 ) \left( 1 + \sinh ^ { - 1 } 1 \right)$
E) $8 \pi ( \sqrt { 2 } - 1 ) \left( 1 + \sinh ^ { - 1 } 1 \right)$

Question 53

The rectangular equation that corresponds to the polar equation $\theta = \frac { \pi } { 4 }$ is

A) $y = 4 x$
B) $4 y = x$
C) $y = - 4 x$
D) $y = x$
E) $y = - x$

Question 54

The rectangular equation that corresponds to the polar equation $r = 3 \csc \theta$ is

A)y = 3
B)x = 3
C)y = x
D)y = 3x
E)y = -3

Question 55

The rectangular equation that corresponds to the polar equation $r = 5 \sec \theta$ is

A) $y = 5$
B) $y = 5 x$
C) $x = 5$
D) $y = - 5 x$
E) $5 y = x$

Question 56

The rectangular equation that corresponds to the polar equation $r = 4 \cos \theta$ is

A) $x ^ { 2 } + y ^ { 2 } = 4 y$
B) $x ^ { 2 } + y ^ { 2 } = 4 x$
C) $x ^ { 2 } + y ^ { 2 } = - 4 x$
D) $x ^ { 2 } + y ^ { 2 } = - 4 y$
E) $y = - 4 x$

Question 57

The rectangular equation that corresponds to the polar equation $r = - 7 \sin \theta$ is

A) $x ^ { 2 } + y ^ { 2 } = - 7 y$
B) $x ^ { 2 } + y ^ { 2 } = - 7 x$
C) $x ^ { 2 } + y ^ { 2 } = 7 y$
D) $x ^ { 2 } + y ^ { 2 } = 7 x$
E) $x ^ { 2 } + y ^ { 2 } = \frac { y } { 7 }$

Question 58

The rectangular equation that corresponds to the polar equation $r ^ { 2 } = \cos \theta$ is

A) $\left( x ^ { 2 } + y ^ { 2 } \right) ^ { \frac { 3 } { 2 } } = x$
B) $\left( x ^ { 2 } + y ^ { 2 } \right) ^ { \frac { 3 } { 2 } } = y$
C) $\left( x ^ { 2 } - y ^ { 2 } \right) ^ { \frac { 3 } { 2 } } = x$
D) $\left( x ^ { 2 } - y ^ { 2 } \right) ^ { \frac { 3 } { 2 } } = y$
E) $\left( - x ^ { 2 } + y ^ { 2 } \right) ^ { \frac { 3 } { 2 } } = x$

Question 59

The rectangular equation that corresponds to the polar equation $\frac { r } { 2 } = \theta$ is

A) $y = 2 x$
B) $x ^ { 2 } + y ^ { 2 } = 2 x$
C) $\sqrt { x ^ { 2 } + y ^ { 2 } } = 2 \tan ^ { - 1 } \frac { x } { y }$
D) $x ^ { 2 } + y ^ { 2 } = - 2 x$
E) $\sqrt { x ^ { 2 } + y ^ { 2 } } = 2 \tan ^ { - 1 } \frac { y } { x }$

Question 60

The rectangular equation that corresponds to the polar equation $r = \frac { 2 } { 1 - \cos \theta }$ is

A) $x ^ { 2 } = 4 y - 4$
B) $x ^ { 2 } + y ^ { 2 } = 2 x$
C) $y ^ { 2 } = 4 x + 4$
D) $y ^ { 2 } = 4 x - 4$
E) $x ^ { 2 } = 4 y + 4$

Question 61

The rectangular equation that corresponds to the polar equation $\cot \theta = 5$ is

A) $y = \frac { x } { 5 }$
B) $y = - \frac { x } { 5 }$
C) $x = \frac { y } { 5 }$
D) $x = - \frac { y } { 5 }$
E) $x ^ { 2 } + y ^ { 2 } = 25$

Question 62

The rectangular equation that corresponds to the polar equation $r = 1 + \sin \theta$ is

A) $x ^ { 2 } + y ^ { 2 } = \sqrt { x ^ { 2 } + y ^ { 2 } } + x$
B) $x ^ { 2 } + y ^ { 2 } = \sqrt { x ^ { 2 } + y ^ { 2 } } - y$
C) $x ^ { 2 } + y ^ { 2 } = \sqrt { x ^ { 2 } + y ^ { 2 } } - x$
D) $x ^ { 2 } + y ^ { 2 } = \sqrt { x ^ { 2 } + y ^ { 2 } } + y$
E) $x ^ { 2 } + y ^ { 2 } = - \sqrt { x ^ { 2 } + y ^ { 2 } } + y$

Question 63

The polar equation that corresponds to the rectangular equation x + y = 0 is

A) $\theta = \frac { \pi } { 4 }$
B) $\theta = - \frac { \pi } { 4 }$
C) $\theta = - \frac { \pi } { 2 }$
D) $\theta = \frac { \pi } { 3 }$
E) $\theta = - \frac { \pi } { 6 }$

Question 64

The polar equation that corresponds to the rectangular equation $x = - 2$ is

A) $\theta = \frac { \pi } { 4 }$
B) $\theta = - \frac { \pi } { 4 }$
C) $r \cos \theta = - 2$
D) $r \sin \theta = - 2$
E) $\theta = \pi$

Question 65

The polar equation that corresponds to the rectangular equation $x ^ { 2 } + y ^ { 2 } = 7 x$ is

A) $r = 7 \sin \theta$
B) $r = - 7 \sin \theta$
C) $r = \frac { \pi } { 4 }$
D) $r = - 7 \cos \theta$
E) $r = 7 \cos \theta$

Question 66

The polar equation that corresponds to the rectangular equation $2 x + y = 4$ is

A) $r = \frac { 4 } { 2 \cos \theta + \sin \theta }$
B) $r = \frac { 4 } { 2 \cos \theta - \sin \theta }$
C) $r = \frac { 4 } { \cos \theta + 2 \sin \theta }$
D) $r = \frac { 4 } { \cos \theta - 2 \sin \theta }$
E) $r = \frac { - 4 } { 2 \cos \theta + \sin \theta }$

Question 67

The polar equation that corresponds to the rectangular equation $x ^ { 2 } + y ^ { 2 } = 5$ is

A) $r = 5$
B) $r = 25$
C) $r = \sqrt { 5 }$
D) $r = \sqrt { 5 } \cos \theta$
E) $r = 5 \cos \theta$

Question 68

The polar equation that corresponds to the rectangular equation $x ^ { 2 } - y ^ { 2 } = 4$ is

A) $r ^ { 2 } = - \frac { 4 } { 2 \cos ^ { 2 } \theta + \sin ^ { 2 } \theta }$
B) $r ^ { 2 } = \frac { 4 } { \cos ^ { 2 } \theta - 2 \sin ^ { 2 } \theta }$
C) $r ^ { 2 } = \frac { 4 } { 2 \cos ^ { 2 } \theta + \sin ^ { 2 } \theta }$
D) $r ^ { 2 } = \frac { 4 } { \cos ^ { 2 } \theta - \sin ^ { 2 } \theta }$
E) $r ^ { 2 } = \frac { 4 } { 2 \cos ^ { 2 } \theta - \sin ^ { 2 } \theta }$

Question 69

The polar equation that corresponds to the rectangular equation $y = x ^ { 2 }$ is

A) $r = - \tan \theta \sec \theta$
B) $r = \tan \theta \sec \theta$
C) $r = \cot \theta \csc \theta$
D) $r = - \cot \theta \csc \theta$
E) $r ^ { 2 } = \tan \theta \sec \theta$

Question 70

The polar equation that corresponds to the rectangular equation xy = 1 is

A) $r ^ { 2 } = - \tan \theta \sec \theta$
B) $r ^ { 2 } = \tan \theta \sec \theta$
C) $r ^ { 2 } = \csc \theta \sec \theta$
D) $r ^ { 2 } = - \csc \theta \sec \theta$
E) $r ^ { 2 } = \cos \theta$

Question 71

The polar equation that corresponds to the rectangular equation $x - y = 0$ is

A) $r = \frac { 4 } { 2 \cos \theta + \sin \theta }$
B) $r = \tan \theta \sec \theta$
C) $r = \csc \theta \sec \theta$
D) $r ^ { 2 } = \frac { 4 } { 2 \cos ^ { 2 } \theta - \sin ^ { 2 } \theta }$
E) $\theta = \frac { \pi } { 4 }$

Question 72

The polar equation that corresponds to the rectangular equation $y = \sqrt { 3 } x$ is

A) $\theta = \frac { 2 \pi } { 5 }$
B) $\theta = \frac { \pi } { 6 }$
C) $\theta = \frac { \pi } { 2 }$
D) $\theta = \frac { \pi } { 4 }$
E) $\theta = \frac { \pi } { 3 }$

Question 73

For a $\neq$ 0, the polar curve $r = a \cos \theta$ is a

A)Straight line
B)Circle
C)Three-petal rose
D)Cardioid
E)Limaçon

Question 74

For a $\neq$ 0, the polar curve $r = a \sin 3 \theta$ is a

A)Straight line
B)Circle
C)Three-petal rose
D)Cardioid
E)Limaçon

Question 75

For a $\neq$ 0, the polar curve $r = a ( 1 + \cos \theta )$ is a

A)Straight line
B)Circle
C)Three-petal rose
D)Cardioid
E)Limaçon

Question 76

For a $\neq$ 0, the polar curve $r = a ( 1 + 2 \cos \theta )$ is a

A)Straight line
B)Circle
C)Three-petal rose
D)Cardioid
E)Limaçon

Question 77

For a $\neq$ 0, the polar curve $r = a ( 1 - 3 \sin \theta )$ is a

A)Circle
B)Cardioid
C)Limaçon with inner loop
D)Dimpled limaçon
E)Convex limaçon

Question 78

For a $\neq$ 0, the polar curve $r = a ( 3 - 2 \cos \theta )$ is a

A)Circle
B)Cardioid
C)Limaçon with inner loop
D)Dimpled limaçon
E)Convex limaçon

Question 79

For a $\neq$ 0, the polar curve $r = a ( 5 + 2 \sin \theta )$ is a

A)Circle
B)Cardioid
C)Limaçon with inner loop
D)Dimpled limaçon
E)Convex limaçon

Question 80

For a $\neq$ 0, the polar curve $r = a ( 1 - \sin \theta )$ is a

A)Circle
B)Cardioid
C)Limaçon with inner loop
D)Dimpled limaçon
E)Convex limaçon