G1-G2 Polar coordinates and polar curves Quiz: AQA Further Maths, Unit 2

The 20 questions

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

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

2. Which polar expression equals x^2 + y^2?

   • r cos(theta)
   • 2r
   • r^2
   • r

3. Convert the polar point (2, pi/3) to Cartesian coordinates.

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

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

   • (2, -pi/4)
   • (sqrt(2), -3 pi/4)
   • (sqrt(2), 3 pi/4)
   • (1, -3 pi/4)

5. What is the polar curve r = 3?

   • A straight line y = 3
   • A cardioid
   • A straight line through the pole
   • A circle with centre at the pole and radius 3

6. The polar curve r = 2 cos(theta) is which shape?

   • A circle with centre (2, 0) and radius 2
   • A circle with centre (0, 1) and radius 1
   • A circle with centre (1, 0) and radius 1
   • A straight line x = 2

7. What is the name of the polar curve r = a(1 + cos(theta))?

   • Archimedean spiral
   • Cardioid
   • Rose curve
   • Straight line

8. How many petals does the rose curve r = sin(2 theta) have?

   • 3
   • 8
   • 2
   • 4

9. What is the polar distance r of the point (3, 4) from the pole?

   • 5
   • 12
   • 1
   • 7

10. Which curve is r = 2 sin(theta)?

    • The straight line y = 2
    • A circle with centre (0, 2) and radius 2
    • A circle with centre (1, 0) and radius 1
    • A circle with centre (0, 1) and radius 1

11. Convert the Cartesian curve x^2 + y^2 = 4x into a polar equation.

    • r = 2 cos(theta)
    • r = 4 cos(theta)
    • r = 4
    • r = 4 sin(theta)

12. Convert the polar point (-2, pi/6) to Cartesian coordinates.

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

13. For the curve r = 2 + 2 cos(theta), what is the maximum value of r?

    • 2
    • 0
    • 4
    • 1

14. Find the polar coordinates of the points where r = 1 meets r = 2 cos(theta), other than at the pole.

    • (1, pi/6) and (1, -pi/6)
    • (1, pi/3) and (1, -pi/3)
    • (2, pi/3) and (2, -pi/3)
    • (1, pi/2) only

15. A polar curve r = f(theta) is symmetric about the initial line when which condition holds?

    • f(theta + pi) = f(theta)
    • f(theta) = 0
    • f(-theta) = f(theta)
    • f(pi/2 - theta) = f(theta)

16. Where do r = 2 sin(theta) and r = 2 cos(theta) meet, other than at the pole?

    • (1, pi/4)
    • (sqrt(2), pi/4)
    • (sqrt(2), pi/2)
    • (2, pi/4)

17. In polar coordinates, the points (r, theta) and (-r, theta + pi) represent which relationship?

    • They are reflections in the x-axis
    • They are both the origin
    • They differ by a rotation of pi/2
    • They are the same point

18. Express the Cartesian point (0, -3) in polar coordinates with r > 0 and 0 <= theta < 2 pi.

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

19. The polar curve r = theta for theta >= 0 is which curve?

    • Limacon
    • Archimedean spiral
    • Rose curve
    • Cardioid

20. Convert the polar equation r cos(theta) = 2 into Cartesian form.

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

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