TRB BT Assistant - Conformal and Bilinear Transformations Online Test (MCQ Quiz)

1. What is the primary characteristic of a conformal transformation?

A) It preserves angles between intersecting curves, including orientation. B) It only preserves the magnitude of angles, not their orientation. C) It distorts angles but preserves distances. D) It maps circles to lines and lines to circles.

2. Under what condition is a transformation w = f(z) conformal at a point z₀?

A) If f(z) is analytic at z₀ and f'(z₀) ≠ 0. B) If f(z) is analytic at z₀ and f'(z₀) = 0. C) If f(z) is continuous at z₀. D) If f(z) is differentiable at z₀.

3. What geometric property is preserved by a conformal mapping?

A) Angles between curves B) Areas of regions C) Lengths of curves D) Distances between points

4. Which of the following is NOT a property of conformal transformations?

A) Preservation of angles B) Preservation of local distances (up to a scaling factor) C) Mapping of straight lines to straight lines D) Mapping of circles to circles or straight lines

5. What is a bilinear transformation also known as?

A) Möbius transformation B) Cauchy transformation C) Laplace transformation D) Fourier transformation

6. What is the general form of a bilinear transformation?

A) w = (az + b) / (cz + d), where ad - bc ≠ 0 B) w = az + b, where a ≠ 0 C) w = z² D) w = e^z

7. What is the condition ad - bc ≠ 0 crucial for in a bilinear transformation?

A) To ensure it is a non-degenerate transformation and invertible. B) To ensure it is conformal at all points. C) To ensure it maps circles to circles. D) To ensure it maps lines to lines.

8. How many points does a bilinear transformation uniquely determine?

A) Three distinct points B) Two distinct points C) Four distinct points D) An infinite number of points

9. What is the cross-ratio of four points z₁, z₂, z₃, z₄?

A) (z₁ - z₃)(z₂ - z₄) / (z₁ - z₄)(z₂ - z₃) B) (z₁ - z₂)(z₃ - z₄) / (z₁ - z₃)(z₂ - z₄) C) (z₁ + z₂)(z₃ + z₄) / (z₁ + z₃)(z₂ + z₄) D) (z₁ - z₂)(z₃ - z₄)

10. What property of bilinear transformations is related to the cross-ratio?

A) The cross-ratio is invariant under bilinear transformations. B) The cross-ratio is preserved only for specific points. C) The cross-ratio is multiplied by a constant factor. D) The cross-ratio is not preserved at all.

11. A bilinear transformation maps circles and lines to:

A) Circles or straight lines B) Only circles C) Only straight lines D) Ellipses or parabolas

12. What is the image of the point at infinity under the transformation w = 1/z?

A) 0 B) 1 C) ∞ D) The point at infinity is undefined

13. What is the image of 0 under the transformation w = 1/z?

A) ∞ B) 1 C) 0 D) Undefined

14. Consider the transformation w = z + c. Is this transformation conformal?

A) Yes, it is a translation and preserves angles. B) No, it distorts angles. C) Yes, but only for specific values of c. D) No, it maps points to lines.

15. Consider the transformation w = az, where a is a non-zero complex number. Is this transformation conformal?

A) Yes, it is a rotation and scaling, preserving angles. B) No, it distorts angles. C) Yes, but only if a is real. D) No, it maps circles to ellipses.

16. What is the effect of the transformation w = e^(iθ)z on the complex plane, where θ is real?

A) Rotation by an angle θ about the origin. B) Translation by θ units. C) Scaling by a factor of θ. D) Reflection across the real axis.

17. What is the effect of the transformation w = rz on the complex plane, where r > 0?

A) Scaling by a factor of r from the origin. B) Rotation by r radians. C) Translation by r units. D) Shearing.

18. The transformation w = z² is conformal everywhere except at:

A) z = 0 B) z = 1 C) z = ∞ D) z = -1

19. Why is w = z² not conformal at z = 0?

A) Because f'(0) = 0. B) Because f'(0) ≠ 0. C) Because it maps circles to circles. D) Because it maps lines to lines.

20. What is the Jacobian of the transformation w = f(z) = u(x, y) + iv(x, y) at a point z where f'(z) exists?

A) |f'(z)|² B) |f'(z)| C) Re(f'(z)) D) Im(f'(z))

21. The Jacobian of a conformal map represents:

A) The local area scaling factor. B) The local angle scaling factor. C) The local distance scaling factor. D) The local shear factor.

22. What is the condition for a transformation to be anti-conformal?

A) It preserves angles but reverses their orientation. B) It preserves angles and their orientation. C) It distorts angles. D) It maps circles to lines.

23. Which transformation is an example of an anti-conformal map?

A) w = conjugate(z) B) w = z C) w = z + c D) w = az

24. If w = f(z) is conformal, what can be said about w = conjugate(f(z))?

A) It is anti-conformal. B) It is also conformal. C) It is a translation. D) It is a rotation.

25. What is the fixed point of the bilinear transformation w = (2z + 1) / (z - 1)?

A) z = 1 ± √2 B) z = 1 C) z = 2 D) z = 0

26. How can we find the fixed points of a bilinear transformation w = (az + b) / (cz + d)?

A) By solving z = (az + b) / (cz + d) for z. B) By setting a = 0. C) By setting d = 0. D) By calculating the cross-ratio.

27. The transformation w = z + 1/z is conformal except at:

A) z = 1 and z = -1 B) z = 0 and z = ∞ C) z = i and z = -i D) z = 0

28. What is the image of the unit circle |z| = 1 under the transformation w = z + 1/z?

A) The line segment [-2, 2] on the real axis. B) The unit circle |w| = 1. C) The real axis. D) The imaginary axis.

29. What is the image of the real axis under the transformation w = z + 1/z?

A) The line segment [-2, 2] on the real axis and the rest of the real axis. B) The unit circle |w| = 1. C) The real axis. D) The imaginary axis.

30. What is the cross-ratio of the points 0, 1, i, ∞?

A) -1 B) 1 C) 0 D) ∞

31. If w = f(z) is a conformal map, and w₁, w₂, w₃, w₄ are the images of z₁, z₂, z₃, z₄ respectively, then the cross-ratio of w₁, w₂, w₃, w₄ is:

A) Equal to the cross-ratio of z₁, z₂, z₃, z₄. B) The negative of the cross-ratio of z₁, z₂, z₃, z₄. C) Zero. D) Undefined.

32. Consider the transformation w = 1/z. What is the image of the upper half-plane Im(z) > 0?

A) The lower half-plane Im(w) < 0. B) The upper half-plane Im(w) > 0. C) The right half-plane Re(w) > 0. D) The left half-plane Re(w) < 0.

33. What is the condition for a transformation to be analytic?

A) It must be differentiable in a neighborhood. B) It must be continuous. C) It must be differentiable at a single point. D) Its derivative must be constant.

34. A transformation w = f(z) is conformal if and only if it is:

A) Analytic and its derivative is non-zero. B) Analytic. C) Differentiable. D) Continuous.

35. Which of the following is a bilinear transformation?

A) w = (3z - 1) / (z + 2) B) w = z² C) w = e^z D) w = log(z)

36. What is the image of the circle |z - 1| = 1 under the transformation w = 1/z?

A) A straight line. B) A circle passing through the origin. C) A circle not passing through the origin. D) An ellipse.

37. The transformation w = z is:

A) Conformal and bilinear. B) Conformal but not bilinear. C) Bilinear but not conformal. D) Neither conformal nor bilinear.

38. The transformation w = az + b (where a ≠ 0) is:

A) Conformal and a special case of bilinear transformation. B) Conformal but not a bilinear transformation. C) Bilinear but not conformal. D) Neither conformal nor bilinear.

39. What is the condition for w = f(z) to be anti-conformal at z₀?

A) f(z) is differentiable and f'(z₀) = 0. B) f(z) is differentiable and f'(z₀) ≠ 0. C) f(z) is conjugate analytic and f'(z₀) ≠ 0. D) f(z) is conjugate analytic and f'(z₀) = 0.

40. Which of the following transformations maps the unit disk |z| < 1 to itself and is conformal?

A) Möbius transformations with |a|² - |b|² = 1 - |c|² + |d|² B) w = z C) w = z² D) w = 1/z

41. If a bilinear transformation maps three points on the real axis to three points on the real axis, what can be said about the transformation?

A) It maps the real axis to the real axis. B) It maps the real axis to the imaginary axis. C) It maps the real axis to a circle. D) It maps the real axis to a spiral.

42. The Schwarz-Christoffel transformation is used to map:

A) A polygon to a half-plane. B) A half-plane to a polygon. C) A circle to a circle. D) A disk to a disk.

43. What is the condition for a transformation w = f(z) to preserve angles in magnitude but reverse orientation?

A) f(z) is anti-analytic. B) f(z) is analytic. C) f(z) is differentiable. D) f(z) is continuous.

44. Consider the transformation w = z̅. This transformation is:

A) Anti-conformal. B) Conformal. C) Bilinear. D) Analytic.

45. What is the fixed point of the transformation w = 1/z?

A) 1 and -1 B) 0 C) ∞ D) 1

46. If f(z) is analytic and f'(z₀) ≠ 0, then f(z) is conformal at z₀. This implies that f(z) preserves:

A) Angles between intersecting curves at z₀. B) Distances between points near z₀. C) Areas of regions near z₀. D) The shape of small circles around z₀.

47. What is the fundamental reason why bilinear transformations are important in complex analysis?

A) They map circles and lines to circles and lines, and are conformal except at poles. B) They are conformal everywhere. C) They preserve distances exactly. D) They can map any region to any other region.

48. The transformation w = az + b / cz + d is conformal everywhere except at:

A) z = -d/c (if c ≠ 0) B) z = 0 C) z = ∞ D) z = a/c

49. Which property ensures that a transformation is conformal at a point z₀?

A) The transformation is analytic at z₀ and f'(z₀) ≠ 0. B) The transformation is differentiable at z₀. C) The transformation preserves distances at z₀. D) The transformation maps circles to circles at z₀.

Conformal and Bilinear Transformations - Online Test

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