TRB BT Assistant - Evolutes, Envelopes and Asymptotes Question Bank with Options

1. The evolute of a curve is also known as the locus of the:

A) Poles of the tangents

B) Centers of the osculating circles

C) Intersections of tangents

D) Points of inflection

2. What is the relationship between the radius of curvature and the distance from the center of curvature to the curve?

A) They are equal.

B) The radius of curvature is half the distance.

C) The radius of curvature is twice the distance.

D) There is no direct relationship.

3. The envelope of the family of circles with radius r and center on the x-axis is:

A) The x-axis

B) The y-axis

C) The lines y = r and y = -r

D) The origin

4. The evolute of the curve x = a cos^3 t, y = a sin^3 t (astroid) is:

A) The astroid itself

B) A circle

C) Another astroid (scaled and rotated)

D) A pair of lines

5. Which of the following is NOT a property of asymptotes?

A) A curve can intersect its asymptote multiple times.

B) Asymptotes help describe the behavior of a curve at infinity.

C) Vertical asymptotes occur where the function approaches infinity.

D) A curve can have at most two oblique asymptotes.

6. The curve y = x + 1/x has an oblique asymptote. What is it?

A) y = x

B) y = 1

C) y = x + 1

D) y = 0

7. For the curve y = x^3 - x, what is the limit of y/x as x approaches infinity?

A) 0

B) 1

C) ∞

D) Does not exist

8. The evolute of a curve is the locus of the intersection of two adjacent:

A) Tangents

B) Normals

C) Secants

D) Chords

9. If the family of curves depends on two parameters, say F(x, y, a, b) = 0, the envelope is found by solving F=0 and ∂F/∂a=0 (or ∂F/∂b=0) and eliminating the parameter. This is a method for finding the:

A) Evolute

B) Asymptote

C) Envelope

D) Center of curvature

10. The envelope of the family of chords of an ellipse that are normal to the ellipse is:

A) The ellipse itself

B) The evolute of the ellipse

C) A circle

D) A point

11. An asymptote is a tangent to the curve at infinity. This is a conceptual definition of:

A) Evolute

B) Envelope

C) Asymptote

D) Pedal curve

12. The evolute of the circle x^2 + y^2 = a^2 is:

A) The circle itself

B) The origin (0,0)

C) A point circle at the origin

D) A straight line

13. The radius of curvature of y = x^2 at (0,0) is:

A) 1/2

B) 1

C) 2

D) 0

14. For the curve y = x^3, the center of curvature at x=0 is:

A) (0, 0)

B) (0, 1)

C) (1, 0)

D) (1, 1)

15. The envelope of the family of lines x cos α + y sin α = p (constant) is:

A) A circle

B) An ellipse

C) A parabola

D) A hyperbola

16. What is the equation of the tangent to the evolute of a curve at a point?

A) It is the normal to the original curve at the corresponding point.

B) It is the tangent to the original curve at the corresponding point.

C) It is perpendicular to the normal of the original curve.

D) It is parallel to the tangent of the original curve.

17. The envelope of the family of circles passing through the origin and having their centers on the line y = x is:

A) The x-axis

B) The y-axis

C) The line y = x

D) The origin

18. The evolute of the curve r = e^(aθ) (logarithmic spiral) is:

A) The same logarithmic spiral

B) A circle

C) An epicycloid

D) A cardioid

19. The asymptotes of the hyperbola x^2/a^2 - y^2/b^2 = 1 are given by:

A) x^2/a^2 + y^2/b^2 = 0

B) x^2/a^2 - y^2/b^2 = 1

C) y = ±(b/a)x

D) x = ±a

20. The asymptote of the curve y = sqrt(x^2 + 1) as x approaches infinity is:

A) y = x

B) y = -x

C) y = x + 1

D) y = 1

21. Consider the family of lines tangent to the parabola y^2 = 4ax. The envelope of this family is:

A) The parabola itself

B) The directrix

C) The axis of the parabola

D) A point

22. Which curve has the evolute as itself?

A) Circle

B) Ellipse

C) Parabola

D) Straight line

23. The center of curvature (h, k) for a curve defined parametrically x = x(t), y = y(t) is:

A) h = x - y'(x'^2 + y'^2)/(x'y'' - y'x''), k = y + x'(x'^2 + y'^2)/(x'y'' - y'x'')

B) h = x + y'(x'^2 + y'^2)/(x'y'' - y'x''), k = y - x'(x'^2 + y'^2)/(x'y'' - y'x'')

C) h = x - y''(x'^2 + y'^2)/(x'y' - y'x''), k = y + x''(x'^2 + y'^2)/(x'y' - y'x'')

D) h = x + y''(x'^2 + y'^2)/(x'y' - y'x''), k = y - x''(x'^2 + y'^2)/(x'y' - y'x'')

24. If a curve is represented parametrically by x = x(t) and y = y(t), the radius of curvature is given by:

A) ρ = (x'^2 + y'^2)^3/2 / |x'y'' - y'x''|

B) ρ = |x'y'' - y'x''| / (x'^2 + y'^2)^3/2

C) ρ = (x'^2 + y'^2) / |x'y'' - y'x''|

D) ρ = |x'y'' - y'x''| / (x'^2 + y'^2)

25. The evolute of the cardioid r = a(1 - cos θ) is:

A) Another cardioid

B) A circle

C) An epicycloid

D) A nephroid

26. What is the envelope of the family of ellipses x^2/a^2 + y^2/b^2 = 1, where a + b = k (a constant)?

A) An ellipse

B) A circle

C) A pair of lines

D) A parabola

27. The curve y = ln(x) has an asymptote at:

A) x = 0

B) y = 0

C) x = 1

D) y = 1

28. For the curve y = e^x, what is the nature of its asymptotes?

A) One horizontal asymptote

B) One vertical asymptote

C) One oblique asymptote

D) No asymptotes

29. The evolute of the rectangular hyperbola xy = c^2 is:

A) x^2 + y^2 = 4c^2

B) (x^2 + y^2)^2 = 4c^2(x^2 - y^2)

C) (x^2 + y^2)^2 = 4c^2(x^2 + y^2)

D) x^2 - y^2 = c^2

30. The equation of the evolute of the curve x = a(cos t + t sin t), y = a(sin t - t cos t) is:

A) x^2 + y^2 = a^2

B) x^2 + y^2 = 2a^2

C) x^2 + y^2 = a^2 t^2

D) x = a cos t, y = a sin t

31. What is the relationship between the evolute and the involute of a curve?

A) The evolute of the involute is the original curve.

B) The involute of the evolute is the original curve.

C) They are the same curve.

D) They are always perpendicular.

32. The envelope of the family of lines y = mx + c, where c = -m^2, is a:

A) Parabola

B) Ellipse

C) Circle

D) Hyperbola

33. Consider the family of circles (x-a)^2 + y^2 = r^2. The envelope of this family (for fixed r) is:

A) The line y = r

B) The line y = -r

C) The lines y = r and y = -r

D) The point (0, 0)

34. The envelope of a family of curves F(x, y, a) = 0 is found by solving F = 0 and ∂F/∂a = 0 for x and y in terms of a, and then eliminating a. This is the method for finding the:

A) Evolute

B) Asymptote

C) Envelope

D) Center of curvature

35. An envelope of a family of curves is a curve that is tangent to each member of the family. This statement defines:

A) Evolute

B) Asymptote

C) Envelope

D) Tangent

36. The curve xy = c^2 has asymptotes at:

A) x = 0 and y = 0

B) x = c and y = c

C) x = -c and y = -c

D) y = x

37. After finding m, c for the oblique asymptote y = mx + c is calculated as:

A) lim (x->inf) (y/x - m)

B) lim (x->inf) (y - mx)

C) lim (x->inf) (y + mx)

D) lim (x->inf) (y - x/m)

38. To find the oblique asymptote y = mx + c for a curve y = f(x), m is calculated as:

A) lim (x->inf) y/x

B) lim (x->inf) (y-c)/x

C) lim (x->inf) x/y

D) lim (x->inf) (y - x)

39. For a rational function f(x) = P(x)/Q(x), if the degree of P(x) is one greater than the degree of Q(x), the curve has a:

A) Horizontal asymptote

B) Vertical asymptote

C) Oblique (slant) asymptote

D) No asymptote

40. If a curve has a horizontal asymptote at y = b, it means that as x tends to infinity (positive or negative), the value of y tends to:

A) A finite value

B) Zero

C) Infinity (positive or negative)

D) The value of b

41. If a curve has a vertical asymptote at x = a, it means that as x approaches a, the value of y tends to:

A) A finite value

B) Zero

C) Infinity (positive or negative)

D) The value of a

42. An asymptote is a line such that the distance between the curve and the line approaches zero as they tend to infinity. This statement defines:

A) Evolute

B) Envelope

C) Asymptote

D) Tangent

43. The evolute of a cycloid is a:

A) Cycloid

B) Epicycloid

C) Hypocycloid

D) Parabola

44. The evolute of an ellipse is a:

A) Ellipse

B) Circle

C) Astroid

D) Cardioid

45. The evolute of a parabola is a:

A) Circle

B) Ellipse

C) Parabola

D) Hyperbola

46. The radius of curvature (ρ) for a curve y = f(x) is given by:

A) ρ = (1 + y'^2)^3/2 / |y''|

B) ρ = |y''| / (1 + y'^2)^3/2

C) ρ = (1 + y'^2) / |y''|

D) ρ = |y''| / (1 + y'^2)

47. For a curve y = f(x), the center of curvature (h, k) is given by:

A) h = x - y'(1 + y'^2)/y'' , k = y + (1 + y'^2)/y''

B) h = x + y'(1 + y'^2)/y'' , k = y - (1 + y'^2)/y''

C) h = y - x'(1 + x'^2)/x'' , k = x + (1 + x'^2)/x''

D) h = y + x'(1 + x'^2)/x'' , k = x - (1 + x'^2)/x''

48. Which of the following is a property of the evolute?

A) It is always a straight line.

B) It is the locus of the centers of curvature.

C) It intersects the original curve at all points.

D) It is parallel to the original curve.

49. The evolute of a curve is the envelope of its:

A) Normals

B) Tangents

C) Secants

D) Chords

50. What is the locus of the centers of curvature of a given curve called?

A) Envelope

B) Evolute

C) Asymptote

D) Tangent

Evolutes, Envelopes and Asymptotes - Question Bank