4.09a-c Polar curves, sketching and area enclosed Quiz: OCR Further Maths, Unit 1

The 20 questions

1. How is the Cartesian x-coordinate expressed in polar coordinates?

   • r sec θ
   • r sin θ
   • r cos θ
   • r tan θ

2. What is the relation between polar radius r and Cartesian coordinates x and y?

   • r² = x² - y²
   • r² = x² + y²
   • r² = x - y
   • r = x + y

3. What is the standard formula for the area enclosed by a polar curve?

   • ½ ∫ r² dθ
   • ∫ r² dθ
   • ½ ∫ r dθ
   • ∫ r dθ

4. What shape does the polar equation r = a(1 + cos θ) represent?

   • Circle
   • Lemniscate
   • Cardioid
   • Spiral

5. Which of these polar equations represents the figure-of-eight curve known as a lemniscate?

   • r = a² cos θ
   • r = a cos(2θ)
   • r² = a cos θ
   • r² = a² cos(2θ)

6. What curve is defined by the polar equation r = aθ for θ ≥ 0?

   • Cardioid
   • Circle
   • Spiral
   • Lemniscate

7. What geometric shape is described by the simple polar equation r = 3?

   • Line length 3
   • Cardioid radius 3
   • Circle radius 3
   • Circle diameter 3

8. A polar curve r = f(θ) is symmetric about the initial line if f(-θ) equals what?

   • f(θ)
   • 1/f(θ)
   • -f(θ)
   • f(π - θ)

9. What are the Cartesian coordinates of the polar point with r = 4 and θ = π/3?

   • (2√3, 2)
   • (4, 2)
   • (2, 2√3)
   • (2, 4)

10. What is the maximum distance from the pole for the cardioid r = 5(1 + cos θ)?

    • 10
    • 5
    • 2.5
    • 0

11. At what angle θ in the interval [0, π] does r = 2 sin θ reach maximum distance?

    • π
    • π/2
    • π/4
    • 0

12. How many petals does the rose curve given by r = a sin(3θ) have in total?

    • 2
    • 3
    • 6
    • 4

13. How many petals does the rose curve given by r = a cos(2θ) have in total?

    • 8
    • 4
    • 3
    • 2

14. To find tangents parallel to the initial line, which derivative with respect to θ must be zero?

    • dx/dθ
    • dy/dθ
    • dr/dθ
    • dy/dx

15. To find tangents perpendicular to the initial line, which derivative with respect to θ must be zero?

    • dy/dθ
    • dr/dθ
    • dθ/dr
    • dx/dθ

16. Using polar integration of ½ ∫₀²π a² dθ, what is the enclosed total area?

    • ½πa²
    • 2πa²
    • 4πa²
    • πa²

17. What is the area bounded by the spiral r = θ from θ = 0 to θ = π?

    • π³/2
    • π²/6
    • π³/6
    • π³/3

18. What is the total area enclosed by the polar curve r = 2 cos θ?

    • 4π
    • π
    • 2π
    • π/2

19. At what non-zero angle θ in [0, π) does the curve r = 2 sin(2θ) pass through pole?

    • π/2
    • 3π/4
    • π/3
    • π/6

20. What is the expansion of r² for the cardioid given by r = 3(1 + cos θ)?

    • 9(1 + 2 cos θ + cos² θ)
    • 9(1 + cos² θ)
    • 3(1 + 2 cos θ + cos² θ)
    • 9(1 + cos θ)

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