Timedora
Free Practice Quiz Question List

4 Polar Coordinates Online Quiz Questions

Use this free practice quiz with 20 questions to review 4 Polar Coordinates, test your knowledge, and prepare for your next test or exam.

20 questions
01
Choose one
1 point

Convert the rectangular point (−3,33)(-3,3\sqrt{3}) to polar coordinates with r≥0r\ge 0 and 0≤θ<2π0\le\theta<2\pi.

  1. A

    (6,π3)\left(6,\frac{\pi}{3}\right)

  2. B

    (6,2π3)\left(6,\frac{2\pi}{3}\right)

  3. C

    (33,2π3)\left(3\sqrt{3},\frac{2\pi}{3}\right)

  4. D

    (6,4π3)\left(6,\frac{4\pi}{3}\right)

02
Choose one
1 point

What rectangular graph is represented by the polar equation r=4sin⁡θr=4\sin\theta?

  1. A

    A circle centered at (0,0)(0,0) with radius 4

  2. B

    A circle centered at (2,0)(2,0) with radius 2

  3. C

    A circle centered at (0,2)(0,2) with radius 2

  4. D

    A line through the origin with slope 4

03
Choose one
1 point

How many petals does the rose curve r=3sin⁡(2θ)r=3\sin(2\theta) have?

  1. A

    1 petal

  2. B

    2 petals

  3. C

    3 petals

  4. D

    4 petals

04
Choose all
1 point

Select all statements that correctly describe sufficient symmetry tests for a polar equation r=f(θ)r=f(\theta).

  1. A

    Replacing θ\theta by −θ-\theta tests symmetry about the polar axis.

  2. B

    Replacing θ\theta by π−θ\pi-\theta tests symmetry about the polar axis.

  3. C

    Replacing rr by −r-r tests symmetry about the pole.

  4. D

    Replacing θ\theta by θ+π/2\theta+\pi/2 tests symmetry about the pole.

05
Choose all
1 point

Which steps are valid when finding all intersections of two polar curves? Select all that apply.

  1. A

    Set the radial expressions equal and solve for candidate angles.

  2. B

    Check separately whether both curves pass through the pole.

  3. C

    Assume that every equal-radius solution represents a distinct point without further checking.

  4. D

    Check equivalent polar representations such as (r,θ)=(−r,θ+π)(r,\theta)=(-r,\theta+\pi).

06
True or false
1 point

True or false: In polar coordinates, a negative value of rr means that the point is plotted opposite the terminal side of θ\theta.

  1. A

    True

  2. B

    False

07
True or false
1 point

True or false: The polar arc-length formula can give the intended length only when the chosen interval traces the intended portion of the curve exactly once.

  1. A

    True

  2. B

    False

08
Written response
1 point

Enter the polar-coordinate variable that represents the directed distance from the pole.

09
Written response
1 point

What curve family is represented by a polar equation of the form r=a+bθr=a+b\theta?

10
Fill in the blank
1 point

Complete the conversion from polar to rectangular coordinates: x=x= and y=y=.

11
Fill in the blank
1 point

Complete the area formula for a region between two polar curves: A=\frac12\int_\alpha^\beta\left(2−^2-^2\right)d\theta.

12
Open ended
1 point

Explain how to find the slope dydx\frac{dy}{dx} of a polar curve r=f(θ)r=f(\theta). Include the derivative formula and the conditions for horizontal and vertical tangents.

13
Choose one
1 point

Which condition identifies an ordinary vertical tangent to a polar curve r=f(θ)r=f(\theta)?

  1. A

    drdθsin⁡θ+rcos⁡θ=0\frac{dr}{d\theta}\sin\theta+r\cos\theta=0 and drdθcos⁡θ−rsin⁡θ≠0\frac{dr}{d\theta}\cos\theta-r\sin\theta\ne0

  2. B

    drdθcos⁡θ−rsin⁡θ=0\frac{dr}{d\theta}\cos\theta-r\sin\theta=0 and drdθsin⁡θ+rcos⁡θ≠0\frac{dr}{d\theta}\sin\theta+r\cos\theta\ne0

  3. C

    drdθ=0\frac{dr}{d\theta}=0 and r=0r=0

  4. D

    rcos⁡θ=0r\cos\theta=0 and rsin⁡θ=0r\sin\theta=0

14
Choose one
1 point

Convert the rectangular point (−3,33)(-3,3\sqrt{3}) to polar coordinates subject to r≥0r\ge 0 and 0≤θ<2π0\le \theta<2\pi.

  1. A

    (6,π3)\left(6,\frac{\pi}{3}\right)

  2. B

    (6,2π3)\left(6,\frac{2\pi}{3}\right)

  3. C

    (6,4π3)\left(6,\frac{4\pi}{3}\right)

  4. D

    (33,−3)\left(3\sqrt{3},-3\right)

15
Choose one
1 point

Convert the polar coordinates (4,5π6)\left(4,\frac{5\pi}{6}\right) to rectangular coordinates.

  1. A

    (−23,−2)(-2\sqrt{3},-2)

  2. B

    (23,2)(2\sqrt{3},2)

  3. C

    (−23,2)(-2\sqrt{3},2)

  4. D

    (23,−2)(2\sqrt{3},-2)

16
True or false
1 point

True or false: The polar representations (r,θ)(r,\theta) and (−r,θ+π)(-r,\theta+\pi) describe the same point.

  1. A

    True

  2. B

    False

17
Choose one
1 point

Which rectangular equation is equivalent to the polar equation r=6cos⁡θr=6\cos\theta?

  1. A

    x2+y2=6xx^2+y^2=6x

  2. B

    x2+y2=6yx^2+y^2=6y

  3. C

    x+y=6x+y=6

  4. D

    x2+y2=36x^2+y^2=36

18
Written response
1 point

What is the name of the axis about which a polar graph is tested for symmetry by replacing θ\theta with −θ-\theta?

19
Choose one
1 point

How many petals does the rose curve r=5sin⁡(3θ)r=5\sin(3\theta) have?

  1. A

    1

  2. B

    2

  3. C

    3

  4. D

    6

20
Written response
1 point

For the polar curve r=θr=\theta, what is the value of dy/dxdy/dx at θ=0\theta=0? Enter the exact numerical slope.