Rucete ✏ AP Precalculus In a Nutshell
9. Polar Functions — Practice Questions 2
This chapter explores how polar equations describe curves and how to convert between rectangular and polar forms.
(Multiple Choice — Click to Reveal Answer)
1. Which of the following defines the relationship between r, x, and y?
(A) r = x² + y²
(B) r² = x² + y²
(C) r² = x + y
(D) r = √(x + y)
Answer
(B) — Derived from the Pythagorean theorem, r² = x² + y².
2. Which of the following formulas converts from polar to rectangular coordinates?
(A) x = r sinθ, y = r cosθ
(B) x = r cosθ, y = r sinθ
(C) x = r tanθ, y = r cotθ
(D) x = r cos²θ, y = r sin²θ
Answer
(B) — The conversion is x = r cosθ and y = r sinθ.
3. What is the rectangular coordinate of (r, θ) = (2, π/3)?
(A) (−1, √3)
(B) (1, √3)
(C) (−√3, 1)
(D) (√3, 1)
Answer
(D) — x = 2cos(π/3) = 1, y = 2sin(π/3) = √3.
4. Which equation represents a circle in polar coordinates?
(A) r = 3
(B) r = 3sinθ
(C) r = 3 + 3cosθ
(D) r = 3 + sinθ
Answer
(A) — r = constant defines a circle centered at the origin.
5. Which of the following is the correct polar form of the complex number 1 + i?
(A) √2(cos π/4 + i sin π/4)
(B) 2(cos π/3 + i sin π/3)
(C) 1(cos π/4 + i sin π/4)
(D) √2(cos π/2 + i sin π/2)
Answer
(A) — r = √2 and θ = π/4 → √2(cos π/4 + i sin π/4).
6. Which complex number corresponds to 3(cos π + i sin π)?
(A) 3
(B) −3
(C) 3i
(D) −3i
Answer
(B) — cosπ = −1, sinπ = 0 → −3.
7. The polar coordinate (r, θ) = (−2, π/3) corresponds to which rectangular point?
(A) (−1, √3)
(B) (1, −√3)
(C) (−1, −√3)
(D) (1, √3)
Answer
(B) — Negative r reverses direction: (x, y) = (2cos(π + π/3), 2sin(π + π/3)) = (1, −√3).
8. Which curve is represented by r = 4sinθ?
(A) Circle centered on x-axis
(B) Circle centered on y-axis
(C) Cardioid
(D) Spiral
Answer
(B) — r = a sinθ → circle centered at (0, a/2).
9. What happens when r < 0 in polar coordinates?
(A) The point is reflected across the origin.
(B) The point disappears.
(C) The point lies on the positive x-axis.
(D) The radius becomes undefined.
Answer
(A) — Negative r reflects the point through the origin.
10. Which of the following represents a spiral in polar form?
(A) r = θ
(B) r = 1 + cosθ
(C) r = 4sinθ
(D) r = 2
Answer
(A) — r = θ forms an Archimedean spiral.
11. The polar equation r = 3 + 2cosθ is what type of curve?
(A) Circle
(B) Limaçon
(C) Cardioid
(D) Ellipse
Answer
(B) — r = a + bcosθ with unequal a, b → limaçon.
12. Which point is on the polar curve r = 2 + 2sinθ?
(A) θ = 0, r = 2
(B) θ = π/2, r = 4
(C) θ = π, r = 4
(D) θ = 3π/2, r = 2
Answer
(B) — Substitute θ = π/2 → r = 2 + 2(1) = 4.
13. Which represents the polar form of −i?
(A) 1(cos π/2 + i sin π/2)
(B) 1(cos π + i sin π)
(C) 1(cos 3π/2 + i sin 3π/2)
(D) 1(cos 0 + i sin 0)
Answer
(C) — −i lies on the negative imaginary axis, θ = 3π/2.
14. Which of the following is symmetric about the y-axis?
(A) r = a cosθ
(B) r = a sinθ
(C) r = a + cosθ
(D) r = θ
Answer
(B) — Sine-based polar equations are symmetric about θ = π/2.
15. What is the polar form of z = −2 − 2i?
(A) 2√2(cos π/4 + i sin π/4)
(B) 2√2(cos 5π/4 + i sin 5π/4)
(C) 2√2(cos 3π/4 + i sin 3π/4)
(D) 2(cos π + i sin π)
Answer
(B) — Located in QIII, r = 2√2, θ = 5π/4.
16. Which of the following shows an inner loop?
(A) r = 1 + 2cosθ
(B) r = 2 + 2cosθ
(C) r = 3 + 2sinθ
(D) r = 4sinθ
Answer
(A) — |b| > |a| → inner loop.
17. Which polar equation forms a rose curve with 5 petals?
(A) r = 5sin(5θ)
(B) r = 5cos(2θ)
(C) r = 3sin(2θ)
(D) r = 3 + cosθ
Answer
(A) — For odd n, r = a sin(nθ) → n petals.
18. For r = 4cosθ, what is the maximum radius?
(A) 0
(B) 2
(C) 4
(D) 8
Answer
(C) — Maximum when cosθ = 1 → r = 4.
19. The equation x² + y² = 8x can be expressed in polar form as:
(A) r = 8cosθ
(B) r = 4cosθ
(C) r = 2cosθ
(D) r = 8sinθ
Answer
(B) — Substitute x = r cosθ: r² = 8r cosθ → r = 8cosθ/2 = 4cosθ.
20. Which of the following is a cardioid?
(A) r = 2 + 3cosθ
(B) r = 2 + 2cosθ
(C) r = 2cosθ
(D) r = 4sin2θ
Answer
(B) — a = b → cardioid.
21. Which graph represents r = 1 + 3cosθ?
(A) Circle
(B) Limaçon with inner loop
(C) Cardioid
(D) Line
Answer
(B) — Since |b| > |a|, it forms a limaçon with inner loop.
22. Which curve passes through the origin?
(A) r = 3
(B) r = 3sinθ
(C) r = 2 + 2cosθ
(D) r = 1 + cosθ
Answer
(B) — At θ = 0, r = 0 for sine-type curves.
23. Which of the following statements is true about r = cos(2θ)?
(A) It is a limaçon.
(B) It is a rose curve with 4 petals.
(C) It is a cardioid.
(D) It is a spiral.
Answer
(B) — Even n → 2n petals; n = 2 → 4 petals.
24. Which of the following represents a conic section?
(A) r = 3sinθ
(B) r = 3/(1 − cosθ)
(C) r = 3 + 3cosθ
(D) r = θ
Answer
(B) — r = eℓ/(1 ± e cosθ) or (1 ± e sinθ) represents a conic.
25. The equation r = 6/(1 + sinθ) represents which conic?
(A) Circle
(B) Parabola
(C) Ellipse
(D) Hyperbola
Answer
(B) — Denominator of form 1 + e sinθ with e = 1 → parabola.
26. Convert the complex number 1 − i to polar form.
(A) √2(cos 7π/4 + i sin 7π/4)
(B) √2(cos π/4 + i sin π/4)
(C) √2(cos 3π/4 + i sin 3π/4)
(D) 2(cos π/2 + i sin π/2)
Answer
(A) — r = √2, θ = −π/4 ≡ 7π/4.
27. Which rectangular equation corresponds to r = 2/(1 + cosθ)?
(A) (x + 1)² + y² = 1
(B) y² = 4x
(C) y² = 8(x − 1)
(D) (x − 2)² + y² = 4
Answer
(B) — r = 2/(1 + cosθ) ⇒ r(1 + cosθ) = 2 ⇒ r + x = 2 ⇒ √(x² + y²) = 2 − x ⇒ squaring gives y² = 4x.
28. Which conic has eccentricity e > 1?
(A) Circle
(B) Parabola
(C) Ellipse
(D) Hyperbola
Answer
(D) — e > 1 → hyperbola.
29. For r = 1 + sinθ, find the coordinates where r = 0.
(A) θ = π/2
(B) θ = 3π/2
(C) θ = 0
(D) θ = π
Answer
(B) — r = 0 when sinθ = −1 → θ = 3π/2.
30. Which of the following graphs shows a line through the pole?
(A) r = 2
(B) θ = π/4
(C) r = 2 + 3cosθ
(D) r = sinθ
Answer
(B) — θ = constant → line through the pole.
31. Convert z = 3√3 − 3i into polar form.
(A) 6(cos π/6 + i sin π/6)
(B) 6(cos 11π/6 + i sin 11π/6)
(C) 3(cos π/3 + i sin π/3)
(D) 3√2(cos π/4 + i sin π/4)
Answer
(B) — r = √[(3√3)² + (−3)²] = 6, θ = −π/6 → 11π/6.
32. Find the average rate of change for r = 4 − 2sinθ on [0, π/2].
(A) −2/π
(B) −4/π
(C) 4/π
(D) 2/π
Answer
(B) — r(0) = 4, r(π/2) = 2 ⇒ (2 − 4)/(π/2) = −4/π.
33. The polar curve r = 3cos(2θ) has how many petals?
(A) 2
(B) 3
(C) 4
(D) 6
Answer
(C) — Even n = 2 → 2n = 4 petals.
34. Which of the following polar forms represents the same point as (r, θ) = (4, 5π/6)?
(A) (−4, 11π/6)
(B) (4, −π/6)
(C) (−4, −π/6)
(D) (−4, 17π/6)
Answer
(A) — Add π to θ when r changes sign: (−4, 5π/6 + π) = (−4, 11π/6).
35. Which of the following is symmetric about the origin?
(A) r = a sinθ
(B) r = a cosθ
(C) r = a sin(θ + π/4)
(D) r = a sin(2θ)
Answer
(D) — r = f(θ) and r = −f(θ + π) symmetry holds for double-angle sine functions.
36. Convert (x, y) = (−2, 2√3) to polar coordinates (r, θ) with r > 0 and 0 ≤ θ < 2π.
Answer
(4, 2π/3) — r = √(4 + 12) = 4; tanθ = (2√3)/(−2) = −√3 with x < 0, y > 0 ⇒ Quadrant II ⇒ θ = 2π/3.
37. Convert (r, θ) = (−5, −π/3) to rectangular coordinates.
Answer
(−5/2, 5√3/2) — x = r cosθ = −5·(1/2) = −5/2; y = r sinθ = −5·(−√3/2) = 5√3/2.
38. For r = 2 + 2cosθ, find rmin and rmax.
Answer
rmin = 0 at θ = π; rmax = 4 at θ = 0 — cosθ ∈ [−1, 1] ⇒ r ∈ [0, 4].
39. Compute the average rate of change of r = 6cosθ on [π/2, π].
Answer
−12/π per radian — r(π/2) = 0, r(π) = −6 ⇒ (−6 − 0)/(π − π/2) = −6/(π/2) = −12/π.
40. Convert r = 4sinθ to a rectangular equation.
Answer
x² + y² = 4y — Multiply by r: r² = 4r sinθ ⇒ x² + y² = 4y.
41. Find all θ in [0, 2π) for which r = 3 + 3sinθ passes through the origin.
Answer
θ = 3π/2 — r = 0 ⇒ 3 + 3sinθ = 0 ⇒ sinθ = −1 ⇒ θ = 3π/2.
42. Find the intersection angles of r = 4 and r = 8cosθ in [0, 2π).
Answer
θ = π/3, 5π/3 — 4 = 8cosθ ⇒ cosθ = 1/2.
43. Write the polar form of the circle x² + y² − 4x = 0.
Answer
r = 4cosθ — r² = 4r cosθ ⇒ r = 4cosθ (for r ≥ 0 representation).
44. For the rose r = 7cos(3θ), list the angles of the petal tips where r is maximal.
Answer
θ = 0, 2π/3, 4π/3 — Max when cos(3θ) = 1 ⇒ 3θ = 2kπ ⇒ θ = 2kπ/3 (k = 0, 1, 2).
45. Express z = −3 + 3i in polar form r(cosθ + i sinθ) with 0 ≤ θ < 2π.
Answer
3√2 [cos(3π/4) + i sin(3π/4)] — r = √(9 + 9) = 3√2; Quadrant II with reference angle π/4 ⇒ θ = 3π/4.
46. Classify the conic r = 5/(1 − (1/2)cosθ) and state its eccentricity.
Answer
Ellipse, e = 1/2 — Matches r = ℓ/(1 − e cosθ) with e = 1/2 < 1.
47. For r = θ, compute the average rate of change of r on [0, 2π].
Answer
1 per radian — (r(2π) − r(0))/(2π − 0) = (2π − 0)/(2π) = 1.
48. Convert (r, θ) = (3, −π/6) to rectangular coordinates.
Answer
(3√3/2, −3/2) — x = 3cos(−π/6) = 3(√3/2); y = 3sin(−π/6) = −3/2.
49. Give a second polar representation (with negative r) of the same point as (5, 2π/5).
Answer
(−5, 7π/5) — Add π to the angle when changing the sign of r: (−5, 2π/5 + π) = (−5, 7π/5).
50. Find the average rate of change of r = 2 + 4cosθ on [0, π/2].
Answer
−8/π per radian — r(0) = 6, r(π/2) = 2 ⇒ (2 − 6)/(π/2 − 0) = −4/(π/2) = −8/π.