7.1-7.2 Polar coordinates and sketching polar curves Quiz: Pearson Edexcel Further Maths, Unit 7

The 20 questions

1. Which formulae convert polar coordinates (r, theta) to Cartesian coordinates (x, y)?

   • x = r^2, y = theta
   • x = r cos theta, y = r sin theta
   • x = r + theta, y = r - theta
   • x = r sin theta, y = r cos theta

2. Which expression gives r^2 in terms of Cartesian coordinates?

   • x + y
   • x^2 - y^2
   • x^2 + y^2
   • 2xy

3. What is the polar equation r = a (a > 0) on an Argand or polar diagram?

   • A circle with centre (a, 0) and radius a
   • An Archimedean spiral
   • A straight line through the origin
   • A circle with centre at the origin and radius a

4. What is the shape of the polar curve r = 2a cos theta?

   • A circle with centre (0, a) and radius 2a
   • A cardioid
   • A circle with centre (a, 0) and radius a
   • A straight line x = 2a

5. Which curve is given by r = a(1 + cos theta) with a > 0?

   • A cardioid
   • A lemniscate
   • A four-petal rose
   • An Archimedean spiral

6. Convert the polar point (r, theta) = (2, pi/3) to Cartesian coordinates.

   • (2, sqrt(3))
   • (1, sqrt(3))
   • (sqrt(3), 1)
   • (-1, sqrt(3))

7. Convert the Cartesian point (-1, 1) to polar coordinates with r > 0 and -pi < theta <= pi.

   • (2, 3pi/4)
   • (sqrt(2), pi/4)
   • (sqrt(2), -3pi/4)
   • (sqrt(2), 3pi/4)

8. What Cartesian equation corresponds to the polar equation r cos theta = 2?

   • x = -2
   • y = 2
   • x^2 + y^2 = 4
   • x = 2

9. What Cartesian equation corresponds to the polar curve r = 2 cos theta?

   • x^2 + y^2 = 1
   • x^2 + y^2 = 2x
   • x^2 + y^2 = 2y
   • x^2 + y^2 = 4x

10. What is the area enclosed by the polar curve r = 2 for 0 <= theta <= 2pi?

    • 4pi
    • pi
    • 8pi
    • 2pi

11. What is the area enclosed by the cardioid r = 1 + cos theta?

    • 2pi
    • 3pi/2
    • 3pi/4
    • pi

12. What is the area enclosed by the circle r = 2 cos theta?

    • 4pi
    • pi
    • 2pi
    • pi/2

13. For the cardioid r = a(1 + cos theta), what is the point on the curve at theta = pi?

    • (2a, 0)
    • (-2a, 0)
    • The origin, a cusp of the curve
    • (0, 2a)

14. For the curve r = 3 + 2 cos theta, at which value of theta is r greatest, and what is that value?

    • theta = pi/2, r = 3
    • theta = pi/2, r = 5
    • theta = pi, r = 5
    • theta = 0, r = 5

15. How many petals does the rose r = a cos 2theta have?

    • 8
    • 4
    • 2
    • 3

16. What kind of curve does the polar equation r = 3 sec(theta - pi/4) describe?

    • A circle of radius 3 centred at the origin
    • A straight line at distance 3 from the origin, perpendicular to the direction at angle pi/4
    • A straight line at distance 3 from the origin in the direction pi/3
    • A line through the origin at angle pi/4

17. What is the area enclosed by the polar curve r = 2 for 0 <= theta <= pi/2?

    • 4pi
    • pi
    • pi/2
    • 2pi

18. Where do the polar curves r = 1 and r = 1 + cos theta intersect for 0 <= theta < 2pi?

    • theta = pi/2 and 3pi/2
    • theta = pi/4 and 7pi/4
    • theta = pi/3 and 5pi/3
    • theta = 0 and pi

19. What is the total area enclosed by the lemniscate r^2 = cos 2theta?

    • pi
    • 2
    • 1
    • pi/2

20. How many petals does the rose r = 4 cos 3theta have?

    • 4
    • 6
    • 2
    • 3

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