Question 1

Convert the equation to polar form.  x2−y2=4x ^ { 2 } - y ^ { 2 } = 4x2−y2=4

Accepted Answer

A)    r=4csc⁡2θr= 4 \csc 2 	hetar=4csc2θ 
B)    r=2csc⁡θr = 2 \csc 	hetar=2cscθ 
C)    r=2sec⁡2θr= 2 \sec 2 	hetar=2sec2θ 
D)    r2=4sec⁡2θr ^ { 2 } = 4 \sec 2 	hetar2=4sec2θ 
E)  none of these F) A) and B)
G) A) and C)

Question 2

Sketch a graph of the rectangular equation. [Hint: First convert the equation to polar coordinates.]  (x2+y2+3y) 2=9(x2+y2) \left( x ^ { 2 } + y ^ { 2 } + 3 y \right)  ^ { 2 } = 9 \left( x ^ { 2 } + y ^ { 2 } \right) (x2+y2+3y) 2=9(x2+y2)

Accepted Answer

A)   
B)   
C)   
D)    E) B) and C)
F) C) and D)

Question 3

Convert the rectangular coordinates to polar coordinates with $r > 0 \text { and } 0 \leq \theta < 2 \pi$ $( - \sqrt { 3 } , - 1 )$

Accepted Answer

The answer of Convert the rectangular coordinates to polar coordinates...

Question 4

Find the modulus and the argument for the complex number.  z=−iz = - iz=−i

Accepted Answer

A)    r=−1,θ=π/2r = - 1 , 	heta = \pi / 2r=−1,θ=π/2 
B)    r=1,θ=πr = 1 , 	heta = \pir=1,θ=π 
C)    r=i,θ=0r= i , 	heta = 0r=i,θ=0 
D)    r=2,θ=3π/2r= \sqrt { 2 } , 	heta = 3 \pi / 2r=2c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/>​,θ=3π/2 
E)    r=1,θ=3π/2r = 1 , 	heta = 3 \pi / 2r=1,θ=3π/2F) None of the above
G) A) and B)

Question 5

Sketch a graph of the polar equation.  r=3−cos⁡θr=\sqrt{3}-\cos 	hetar=3​−cosθ

Accepted Answer

A)   
B)   
C)   
D)    E) B) and D)
F) None of the above

Question 6

Sketch a graph of the polar equation. $y = \sqrt { 3 } + 2 \sin \theta$

Accepted Answer

The answer of Sketch a graph of the polar equation....

Question 7

Find a rectangular-coordinate equation for the curve by eliminating the parameter.  x=t+2,y=tt+2x = t + 2 , y = \frac { t } { t + 2 }x=t+2,y=t+2t​

Accepted Answer

A)    y=x−4xy = \frac { x - 4 } { x }y=xx−4​ 
B)    y=x−2xy = \frac { x - 2 } { x }y=xx−2​ 
C)    y=x−24xy = \frac { x - 2 } { 4 x }y=4xx−2​ 
D)    y=2x−12xy = \frac { 2 x - 1 } { 2 x }y=2x2x−1​ 
E)   none of these F) C) and D)
G) A) and C)

Question 8

Convert the equation to polar form. $x ^ { 2 } + y ^ { 2 } = 25$

Accepted Answer

The answer of Convert the equation to polar form. \(x...

Question 9

Find a rectangular-coordinate equation for the curve by eliminating the parameter.  x=t+3,y=tt+3x = t + 3 , y = \frac { t } { t + 3 }x=t+3,y=t+3t​

Accepted Answer

A)    y=x−4xy = \frac { x - 4 } { x }y=xx−4​ 
B)    y=x−2xy = \frac { x - 2 } { x }y=xx−2​ 
C)    y=x−23xy = \frac { x - 2 } { 3 x }y=3xx−2​ 
D)    y=2x−12xy = \frac { 2 x - 1 } { 2 x }y=2x2x−1​ 
E)   none of these F) D) and E)
G) A) and E)

Question 10

Write the complex conjugate of z in polar form with argument $\theta$ between 0 and $2 \pi$  $z = - 5 - 5 i \sqrt { 3 }$

Accepted Answer

The answer of Write the complex conjugate of z in...

Question 11

Let  z1=8(cos⁡11π6+isin⁡11π6) z _ { 1 } = 8 \left( \cos \frac { 11 \pi } { 6 } + i \sin \frac { 11 \pi } { 6 } \right) z1​=8(cos611π​+isin611π​)   and  z2=23(cos⁡π3+isin⁡π3) z _ { 2 } = 2 \sqrt { 3 } \left( \cos \frac { \pi } { 3 } + i \sin \frac { \pi } { 3 } \right) z2​=23​(cos3π​+isin3π​)  )  Find  z1/z2z _ { 1 } / z _ { 2 }z1​/z2​

Accepted Answer

A)    3−4i3 - 4 i3−4i 
B)    4−3i4 - \sqrt { 3 } i4−3c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/>​i 
C)    4+3i4 + \sqrt { 3 } i4+3c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/>​i 
D)    43\frac { 4 } { 3 }34​ 
E)    −433i- \frac { 4 \sqrt { 3 } } { 3 } i−343c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/>​​iF) A) and E)
G) None of the above

Question 12

Write  z1=2+2iz _ { 1 } = 2 + 2 iz1​=2+2i  in polar form then find  1/z11 / z _ { 1 }1/z1​

Accepted Answer

A)    14−14i\frac { 1 } { 4 } - \frac { 1 } { 4 } i41​−41​i 
B)    34−14i\frac { 3 } { 4 } - \frac { 1 } { 4 } i43​−41​i 
C)    14+14i\frac { 1 } { 4 } + \frac { 1 } { 4 } i41​+41​i 
D)    −14−14i- \frac { 1 } { 4 } - \frac { 1 } { 4 } i−41​−41​i 
E)    14−34i\frac { 1 } { 4 } - \frac { \sqrt { 3 } } { 4 } i41​−43c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/>​​iF) None of the above
G) D) and E)

Question 13

Graph the polar equation $r= 4 \sin \theta$

Accepted Answer

The answer of Graph the polar equation \(r= 4 \sin...

Question 14

Find two polar coordinate representations for the point $( 3 , \pi / 3 )$ , one with $r > 0$ , and both with $0 \leq \theta < 2 \pi$

Accepted Answer

The answer of Find two polar coordinate representations for the...

Question 15

Write the complex conjugate of z in polar form with argument $\theta$ between 0 and $2 \pi$ $z = 5 - 5 i \sqrt { 3 }$

Accepted Answer

The answer of Write the complex conjugate of z in...

Question 16

Use DeMoivre's Theorem to find the indicated power. $(1+i)^{2}$

Accepted Answer

The answer of Use DeMoivre's Theorem to find the indicated...

Question 17

Find a rectangular-coordinate equation for the curve by eliminating the parameter. $x = 4 - t ^ { 2 } , y = 2 + t$

Accepted Answer

The answer of Find a rectangular-coordinate equation for the curve...

Question 18

Find two polar coordinate representations for the point $( 3 , \pi / 3 )$ , one with $r> 0$ , and the other with $r < 0$

Accepted Answer

The answer of Find two polar coordinate representations for the...

Question 19

Convert the rectangular coordinates to polar coordinates with $r > 0 \text { and } 0 \leq \theta < 2 \pi$  $( - 2 \sqrt { 3 } , - 2 )$

Accepted Answer

The answer of Convert the rectangular coordinates to polar coordinates...

Question 20

Convert the equation to polar form. $x ^ { 2 } + y ^ { 2 } = 16$

Accepted Answer

The answer of Convert the equation to polar form. \(x...

Exam 9: Vectors in Two and Three Dimensions

Sketch a graph of the polar equation. y=3+2sin⁡θy = \sqrt { 3 } + 2 \sin \thetay=3​+2sinθ

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Graph the polar equation r=4sin⁡θr= 4 \sin \thetar=4sinθ

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Convert the rectangular coordinates to polar coordinates with r>0 and 0≤θ<2πr > 0 \text { and } 0 \leq \theta < 2 \pir>0 and 0≤θ<2π (−23,−2)( - 2 \sqrt { 3 } , - 2 )(−23​,−2)

Correct AnswerverifiedShow Answer

Find the modulus and the argument for the complex number. z=−iz = - iz=−i

Correct AnswerverifiedShow Answer

Write z1=2+2iz _ { 1 } = 2 + 2 iz1​=2+2i in polar form then find 1/z11 / z _ { 1 }1/z1​

Correct AnswerverifiedShow Answer

Find a rectangular-coordinate equation for the curve by eliminating the parameter. x=t+2,y=tt+2x = t + 2 , y = \frac { t } { t + 2 }x=t+2,y=t+2t​

Correct AnswerverifiedShow Answer

Correct AnswerverifiedShow Answer

Convert the rectangular coordinates to polar coordinates with r>0 and 0≤θ<2πr > 0 \text { and } 0 \leq \theta < 2 \pir>0 and 0≤θ<2π (−3,−1)( - \sqrt { 3 } , - 1 )(−3​,−1)

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Convert the equation to polar form. x2+y2=25x ^ { 2 } + y ^ { 2 } = 25x2+y2=25

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Convert the equation to polar form. x2+y2=16x ^ { 2 } + y ^ { 2 } = 16x2+y2=16

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Write the complex conjugate of z in polar form with argument θ\thetaθ between 0 and 2π2 \pi2π z=5−5i3z = 5 - 5 i \sqrt { 3 }z=5−5i3​

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Find two polar coordinate representations for the point (3,π/3)( 3 , \pi / 3 )(3,π/3) , one with r>0r> 0r>0 , and the other with r<0r < 0r<0

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Write the complex conjugate of z in polar form with argument θ\thetaθ between 0 and 2π2 \pi2π z=−5−5i3z = - 5 - 5 i \sqrt { 3 }z=−5−5i3​

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Find a rectangular-coordinate equation for the curve by eliminating the parameter. x=4−t2,y=2+tx = 4 - t ^ { 2 } , y = 2 + tx=4−t2,y=2+t

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Sketch a graph of the rectangular equation. [Hint: First convert the equation to polar coordinates.] (x2+y2+3y) 2=9(x2+y2) \left( x ^ { 2 } + y ^ { 2 } + 3 y \right) ^ { 2 } = 9 \left( x ^ { 2 } + y ^ { 2 } \right) (x2+y2+3y) 2=9(x2+y2)

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Find two polar coordinate representations for the point (3,π/3)( 3 , \pi / 3 )(3,π/3) , one with r>0r > 0r>0 , and both with 0≤θ<2π0 \leq \theta < 2 \pi0≤θ<2π

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Sketch a graph of the polar equation. r=3−cos⁡θr=\sqrt{3}-\cos \thetar=3​−cosθ

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Convert the equation to polar form. x2−y2=4x ^ { 2 } - y ^ { 2 } = 4x2−y2=4

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Use DeMoivre's Theorem to find the indicated power. (1+i)2(1+i)^{2}(1+i)2

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Find a rectangular-coordinate equation for the curve by eliminating the parameter. x=t+3,y=tt+3x = t + 3 , y = \frac { t } { t + 3 }x=t+3,y=t+3t​

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Sketch a graph of the polar equation. $y = \sqrt { 3 } + 2 \sin \theta$

Correct Answer
verified

Graph the polar equation $r= 4 \sin \theta$

Correct Answer
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Convert the rectangular coordinates to polar coordinates with $r > 0 \text { and } 0 \leq \theta < 2 \pi$ $( - 2 \sqrt { 3 } , - 2 )$

Correct Answer
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Find the modulus and the argument for the complex number. $z = - i$

Correct Answer
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Write $z _ { 1 } = 2 + 2 i$ in polar form then find $1 / z _ { 1 }$

Correct Answer
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Find a rectangular-coordinate equation for the curve by eliminating the parameter. $x = t + 2 , y = \frac { t } { t + 2 }$

Correct Answer
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Correct Answer
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Convert the rectangular coordinates to polar coordinates with $r > 0 \text { and } 0 \leq \theta < 2 \pi$ $( - \sqrt { 3 } , - 1 )$

Correct Answer
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Convert the equation to polar form. $x ^ { 2 } + y ^ { 2 } = 25$

Correct Answer
verified

Convert the equation to polar form. $x ^ { 2 } + y ^ { 2 } = 16$

Correct Answer
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Write the complex conjugate of z in polar form with argument $\theta$ between 0 and $2 \pi$ $z = 5 - 5 i \sqrt { 3 }$

Correct Answer
verified

Find two polar coordinate representations for the point $( 3 , \pi / 3 )$ , one with $r> 0$ , and the other with $r < 0$

Correct Answer
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Write the complex conjugate of z in polar form with argument $\theta$ between 0 and $2 \pi$ $z = - 5 - 5 i \sqrt { 3 }$

Correct Answer
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Find a rectangular-coordinate equation for the curve by eliminating the parameter. $x = 4 - t ^ { 2 } , y = 2 + t$

Correct Answer
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Sketch a graph of the rectangular equation. [Hint: First convert the equation to polar coordinates.] $\left( x ^ { 2 } + y ^ { 2 } + 3 y \right) ^ { 2 } = 9 \left( x ^ { 2 } + y ^ { 2 } \right)$

Correct Answer
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Find two polar coordinate representations for the point $( 3 , \pi / 3 )$ , one with $r > 0$ , and both with $0 \leq \theta < 2 \pi$

Correct Answer
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Sketch a graph of the polar equation. $r=\sqrt{3}-\cos \theta$

Correct Answer
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Convert the equation to polar form. $x ^ { 2 } - y ^ { 2 } = 4$

Correct Answer
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Use DeMoivre's Theorem to find the indicated power. $(1+i)^{2}$

Correct Answer
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Find a rectangular-coordinate equation for the curve by eliminating the parameter. $x = t + 3 , y = \frac { t } { t + 3 }$

Correct Answer
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