Conformal and Bilinear Transformations - Question Bank

1. 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.
2. 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
3. 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.
4. 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.
5. What is the fixed point of the transformation w = 1/z?
A) 1 and -1
B) 0
C) ∞
D) 1
6. Consider the transformation w = z̅. This transformation is:
A) Anti-conformal.
B) Conformal.
C) Bilinear.
D) Analytic.
7. 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.
8. 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.
9. 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.
10. 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
11. 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.
12. 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.
13. The transformation w = z is:
A) Conformal and bilinear.
B) Conformal but not bilinear.
C) Bilinear but not conformal.
D) Neither conformal nor bilinear.
14. 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.
15. 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)
16. 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.
17. 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.
18. 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.
19. 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.
20. What is the cross-ratio of the points 0, 1, i, ∞?
A) -1
B) 1
C) 0
D) ∞
21. 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.
22. 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.
23. 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
24. 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.
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. 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.
27. Which transformation is an example of an anti-conformal map?
A) w = conjugate(z)
B) w = z
C) w = z + c
D) w = az
28. 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.
29. 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.
30. 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))
31. 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.
32. The transformation w = z² is conformal everywhere except at:
A) z = 0
B) z = 1
C) z = ∞
D) z = -1
33. 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.
34. 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.
35. 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.
36. 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.
37. What is the image of 0 under the transformation w = 1/z?
A) ∞
B) 1
C) 0
D) Undefined
38. 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
39. A bilinear transformation maps circles and lines to:
A) Circles or straight lines
B) Only circles
C) Only straight lines
D) Ellipses or parabolas
40. 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.
41. 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)
42. 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
43. 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.
44. 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
45. What is a bilinear transformation also known as?
A) Möbius transformation
B) Cauchy transformation
C) Laplace transformation
D) Fourier transformation
46. 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
47. 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
48. 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.
49. 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.
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