TRB BT Assistant - Evolutes, Envelopes and Asymptotes Online Test (MCQ Quiz)

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

A) Envelope B) Evolute C) Asymptote D) Tangent

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

A) Normals B) Tangents C) Secants D) Chords

3. 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.

4. 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''

5. 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)

6. The evolute of a parabola is a:

A) Circle B) Ellipse C) Parabola D) Hyperbola

7. The evolute of an ellipse is a:

A) Ellipse B) Circle C) Astroid D) Cardioid

8. The evolute of a cycloid is a:

A) Cycloid B) Epicycloid C) Hypocycloid D) Parabola

9. 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

10. 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

11. 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

12. 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

13. 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)

14. 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)

15. 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

16. 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

17. 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

18. 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)

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

A) Parabola B) Ellipse C) Circle D) Hyperbola

20. 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.

21. 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

22. 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

23. 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

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

A) x = 0 B) y = 0 C) x = 1 D) y = 1

25. 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

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

A) Another cardioid B) A circle C) An epicycloid D) A nephroid

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

28. 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'')

29. Which curve has the evolute as itself?

A) Circle B) Ellipse C) Parabola D) Straight line

30. 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

31. 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

32. 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

33. 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

34. 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

35. 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.

36. 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

37. 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)

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

A) 1/2 B) 1 C) 2 D) 0

39. 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

40. 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

41. 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

42. 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

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

A) Tangents B) Normals C) Secants D) Chords

44. 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

45. 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

46. 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.

47. 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

48. 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

49. 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.

50. 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

Evolutes, Envelopes and Asymptotes - Online Test

Test Results