Polar Curves and Tangents - Question Bank

1. What is the equation of the tangent to the curve r = e^(aθ) at θ = 0?
A) r = e^(aθ) cos(θ)
B) r = e^(aθ) sec(θ)
C) r = cos(θ)
D) r = sec(θ)
2. For the curve r = 2 sin θ, find the slope of the tangent at θ = π/6.
A) 1/√3
B) √3
C) 1/2
D) 2
3. The condition r' sin θ + r cos θ = 0 for a polar curve means the tangent is:
A) Parallel to the polar axis
B) Perpendicular to the polar axis
C) Passing through the origin
D) Coincident with the polar axis
4. The condition r' cos θ - r sin θ = 0 for a polar curve means the tangent is:
A) Parallel to the polar axis
B) Perpendicular to the polar axis
C) Passing through the origin
D) Coincident with the polar axis
5. What is the polar equation of the tangent to the cardioid r = 1 - cos θ at θ = π/2?
A) r = 1
B) r = -1
C) r = cos θ
D) r = sin θ
6. What is the polar equation of the tangent to the circle r = a cos θ at the point (a/2, π/3)?
A) r = a cos(θ - π/3)
B) r = a cos(θ + π/3)
C) r = a cos(θ)
D) r = a sin(θ)
7. The tangent to the polar curve r = a sin³θ is given by:
A) r = a sin³θ cos(θ - θ₀)
B) r = a sin³θ sec(θ - θ₀)
C) r = a sin³θ csc(θ - θ₀)
D) r = a sin³θ sin(θ - θ₀)
8. For the curve r = sin(2θ), what is the slope of the tangent at θ = 0?
A) 0
B) 1
C) -1
D) Undefined
9. For the curve r = sin(2θ) (a four-petal rose), what is the slope of the tangent at θ = π/4?
A) 0
B) 1
C) -1
D) Undefined
10. The tangent to the polar curve r = f(θ) at the origin (r=0) is given by the line θ = θ₀, where θ₀ is the smallest positive value for which f(θ₀) = 0.
A) True
B) False
C) Only if f'(θ₀) is not zero
D) Only if f'(θ₀) is zero
11. What is the angle between the tangent and the initial line for r = aθ at θ = π/2?
A) tan⁻¹(2/π)
B) tan⁻¹(π/2)
C) tan⁻¹(a)
D) tan⁻¹(1)
12. For the curve r = 2 + cos θ, find the slope of the tangent at θ = 0.
A) 0
B) 1
C) -1
D) Undefined
13. For the curve r = 2 + cos θ, find the slope of the tangent at θ = π/2.
A) 0
B) 1
C) -1
D) Undefined
14. For the curve r = 2 + cos θ, find dr/dθ.
A) -sin θ
B) sin θ
C) cos θ
D) -cos θ
15. The polar curve r = a csc θ represents:
A) A circle
B) An ellipse
C) A vertical line
D) A horizontal line
16. The polar curve r = a sec θ represents:
A) A circle
B) An ellipse
C) A vertical line
D) A horizontal line
17. What is the equation of the tangent line to the polar curve r = f(θ) at the point (r₀, θ₀)?
A) r = r₀ cos(θ - θ₀)
B) r = r₀ sin(θ - θ₀)
C) r = r₀ sec(θ - θ₀)
D) r = r₀ csc(θ - θ₀)
18. Consider the lemniscate r² = a² cos(2θ). What is the condition for the tangent to be parallel to the polar axis?
A) cos(2θ) = 0
B) sin(2θ) = 0
C) tan(2θ) = 0
D) cot(2θ) = 0
19. Consider the lemniscate r² = a² cos(2θ). What is the condition for the tangent to be perpendicular to the polar axis?
A) cos(2θ) = 0
B) sin(2θ) = 0
C) tan(2θ) = 0
D) cot(2θ) = 0
20. At what point on the cardioid r = 1 + cos θ is the tangent parallel to the polar axis?
A) θ = 0
B) θ = π/2
C) θ = π
D) θ = 3π/2
21. At what point on the cardioid r = 1 + cos θ is the tangent perpendicular to the polar axis?
A) θ = 0
B) θ = π/2
C) θ = π
D) θ = 3π/2
22. For the polar curve r = aθ (spiral of Archimedes), find dr/dθ.
A) a
B) aθ
C) θ
D) 1
23. For the polar curve r = e^(aθ) (logarithmic spiral), find dr/dθ.
A) a e^(aθ)
B) e^(aθ)
C) a θ e^(aθ)
D) θ e^(aθ)
24. If tan φ is undefined for a polar curve, what is the relationship between the radius vector and the tangent?
A) The tangent is perpendicular to the radius vector.
B) The tangent is parallel to the radius vector.
C) The tangent is coincident with the radius vector.
D) The tangent is parallel to the initial line.
25. If tan φ = 0 for a polar curve, what is the relationship between the radius vector and the tangent?
A) The tangent is perpendicular to the radius vector.
B) The tangent is parallel to the radius vector.
C) The tangent is coincident with the radius vector.
D) The tangent is perpendicular to the initial line.
26. What is the angle (φ) between the radius vector and the tangent to the polar curve r = f(θ)?
A) tan φ = r / (dr/dθ)
B) tan φ = (dr/dθ) / r
C) tan φ = r * (dr/dθ)
D) tan φ = r + (dr/dθ)
27. For the circle r = a sin θ, find the slope dy/dx at θ = π/2.
A) 0
B) 1
C) -1
D) Undefined
28. For the circle r = a sin θ, find dr/dθ.
A) a cos θ
B) -a cos θ
C) a sin θ
D) -a sin θ
29. For the circle r = a cos θ, find the slope dy/dx at θ = 0.
A) 0
B) 1
C) -1
D) Undefined
30. For the circle r = a cos θ, find dr/dθ.
A) -a sin θ
B) a sin θ
C) -a cos θ
D) a cos θ
31. For the cardioid r = a(1 - cos θ), find the slope dy/dx at θ = π/2.
A) 1
B) -1
C) 0
D) Undefined
32. For the cardioid r = a(1 - cos θ), find dr/dθ.
A) a sin θ
B) -a sin θ
C) a cos θ
D) -a cos θ
33. What is the condition for a tangent to a polar curve r = f(θ) to be parallel to the initial line (polar axis)?
A) r' sin θ + r cos θ = 0
B) r' cos θ - r sin θ = 0
C) r' cos θ + r sin θ = 0
D) r' sin θ - r cos θ = 0
34. What is the condition for a tangent to a polar curve r = f(θ) to be perpendicular to the initial line (polar axis)?
A) r' cos θ - r sin θ = 0
B) r' sin θ + r cos θ = 0
C) r' cos θ + r sin θ = 0
D) r' sin θ - r cos θ = 0
35. In the formula for dy/dx of a polar curve, what does r' represent?
A) The second derivative of r with respect to θ
B) The derivative of r with respect to θ (dr/dθ)
C) The derivative of θ with respect to r (dθ/dr)
D) The radius r itself
36. For a polar curve r = f(θ), what is the formula for the slope dy/dx of the tangent line in terms of r and θ?
A) (r' sin θ + r cos θ) / (r' cos θ - r sin θ)
B) (r' cos θ + r sin θ) / (r' sin θ - r cos θ)
C) (r cos θ + r' sin θ) / (r sin θ - r' cos θ)
D) (r sin θ + r' cos θ) / (r cos θ - r' sin θ)
37. A tangent to a polar curve is a line that intersects the curve at a point and has the same direction as the curve at that point.
A) True
B) False
C) Only if the curve is a straight line
D) Only if the curve is a circle
38. The polar curve r² = a² cos(2θ) is known as:
A) Cardioid
B) Lemniscate of Bernoulli
C) Rose curve
D) Archimedean spiral
39. What does the polar curve r = a sin θ represent?
A) A line
B) A circle passing through the origin
C) An ellipse
D) A cardioid
40. What does the polar curve r = a cos θ represent?
A) A line
B) A circle passing through the origin
C) An ellipse
D) A cardioid
41. What does the polar curve r = a(1 - cos θ) represent?
A) A circle
B) A cardioid
C) A lemniscate
D) A spiral of Archimedes
42. The equation θ = α in polar coordinates represents which shape?
A) A circle
B) An ellipse
C) A straight line through the origin
D) A hyperbola
43. The equation r = a in polar coordinates represents which shape?
A) A straight line through the origin
B) A circle centered at the origin
C) A spiral
D) A parabola
44. What is the relationship between Cartesian coordinates (x, y) and polar coordinates (r, θ) for θ?
A) θ = tan⁻¹(y/x)
B) θ = tan⁻¹(x/y)
C) θ = sin⁻¹(y/x)
D) θ = cos⁻¹(x/y)
45. What is the relationship between Cartesian coordinates (x, y) and polar coordinates (r, θ) for r?
A) r = sqrt(x^2 + y^2)
B) r = sqrt(x^2 - y^2)
C) r = sqrt(y^2 - x^2)
D) r = sqrt(x + y)
46. What is the relationship between Cartesian coordinates (x, y) and polar coordinates (r, θ)?
A) x = r cos θ, y = r sin θ
B) x = r sin θ, y = r cos θ
C) x = r tan θ, y = r cot θ
D) x = r sec θ, y = r csc θ
47. In polar coordinates (r, θ), what does 'θ' represent?
A) The distance from the origin
B) The x-coordinate
C) The angle from the positive x-axis
D) The y-coordinate
48. In polar coordinates (r, θ), what does 'r' represent?
A) The angle from the positive x-axis
B) The distance from the origin
C) The slope of the line segment
D) The y-coordinate
49. What is the polar form of a complex number z = x + iy?
A) r(cos θ - i sin θ)
B) r(sin θ + i cos θ)
C) r(cos θ + i sin θ)
D) r(sin θ - i cos θ)
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