TRB PG Assistant - Analytic mappings - conformality, arcs and closed curves, conformal mapping, linear fractional transformations, cross ratio and symmetry Online Test (MCQ Quiz)

1. What is the primary condition for a function f(z) to be conformal at a point z₀?

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

2. If a function f(z) is analytic in a region R, and f'(z) ≠ 0 for all z in R, then f(z) is conformal in R. What property does conformality preserve?

A) Distances between points B) Angles between intersecting curves C) Areas of regions D) The magnitude of complex numbers

3. What does it mean for a mapping to be conformal at a point?

A) It preserves distances and angles. B) It preserves angles but not necessarily distances. C) It preserves distances but not necessarily angles. D) It distorts angles and distances uniformly.

4. Which of the following transformations is NOT a linear fractional transformation?

A) f(z) = az + b B) f(z) = 1/z C) f(z) = (az + b) / (cz + d) D) f(z) = z²

5. A linear fractional transformation is also known as a:

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

6. What is the general form of a linear fractional transformation?

A) f(z) = az² + bz + c B) f(z) = az + b C) f(z) = a/(cz+d) D) f(z) = (az + b) / (cz + d), where ad - bc ≠ 0

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

A) It ensures the transformation is conformal. B) It ensures the transformation maps the extended complex plane to itself. C) It ensures the transformation is linear. D) It ensures the transformation is invertible.

8. The cross-ratio of four distinct complex numbers z₁, z₂, z₃, z₄ is denoted by:

A) (z₁, z₂, z₃, z₄) B) [z₁, z₂, z₃, z₄] C) {z₁, z₂, z₃, z₄} D) <z₁, z₂, z₃, z₄>

9. What is the formula for the cross-ratio (z₁, z₂, z₃, z₄)?

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

10. A key property of the cross-ratio is that it is invariant under:

A) Translation B) Rotation C) Linear Fractional Transformations D) Differentiation

11. What is the value of the cross-ratio (z₁, z₂, z₃, z₄) if z₁, z₂, z₃ are collinear?

A) 0 B) 1 C) ∞ D) Undefined

12. The cross-ratio of four points on a circle or a line is always:

A) A purely imaginary number B) A real number C) A complex number with magnitude 1 D) Zero

13. What is meant by an 'arc' in the context of complex analysis?

A) A closed curve. B) A continuous path that is not necessarily closed. C) A straight line segment. D) A curve with infinite length.

14. What is a 'closed curve' in complex analysis?

A) A curve that does not intersect itself. B) A curve where the starting point and ending point are the same. C) A curve that is always differentiable. D) A curve that lies entirely within a bounded region.

15. Consider a conformal mapping f(z). If two curves C₁ and C₂ intersect at a point z₀ at an angle θ, what is the angle between their images f(C₁) and f(C₂) under the mapping?

A) θ/2 B) 2θ C) θ D) π - θ

16. A mapping f(z) is conformal at z₀ if it is analytic at z₀ and:

A) f'(z₀) = 0 B) f'(z₀) ≠ 0 C) f''(z₀) = 0 D) f''(z₀) ≠ 0

17. Which property is NOT preserved by a conformal mapping?

A) The angle between two intersecting curves. B) The orientation of the angle between two intersecting curves. C) The ratio of infinitesimal distances. D) The area of a region.

18. The transformation f(z) = 1/z maps circles and lines to:

A) Circles only B) Lines only C) Circles or lines D) Ellipses

19. What is the image of the unit circle |z| = 1 under the mapping f(z) = 1/z?

A) The unit circle |w| = 1 B) The real axis C) The imaginary axis D) The line Re(w) = 1

20. What is the image of the real axis under the mapping f(z) = 1/z?

A) The real axis B) The imaginary axis C) The unit circle |w| = 1 D) The line Im(w) = 1

21. A linear fractional transformation maps the extended complex plane (C ∪ {∞}) to:

A) The complex plane C B) The extended complex plane (C ∪ {∞}) C) The unit disk D) The upper half-plane

22. How many points uniquely determine a linear fractional transformation?

A) Two B) Three C) Four D) Infinitely many

23. If f(z) is a linear fractional transformation, then f(∞) is defined as:

A) 0 B) 1 C) a/c (if c ≠ 0) D) ∞

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

A) 1 B) -1 C) i D) -i

25. If the cross-ratio (z₁, z₂, z₃, z₄) = 1, what can be said about the points z₁, z₂, z₃, z₄?

A) They are collinear. B) They are concyclic (lie on a circle). C) They are not necessarily related in any specific geometric way. D) They form a square.

26. The property that a linear fractional transformation preserves the cross-ratio is fundamental to:

A) Proving conformality B) Mapping circles and lines to circles and lines C) Calculating derivatives D) Determining singularities

27. The set of all linear fractional transformations forms a group under:

A) Addition B) Multiplication C) Composition D) Differentiation

28. What is the image of the upper half-plane Im(z) > 0 under the mapping f(z) = iz?

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

29. If f(z) is analytic and non-constant in a region R, then the image of any open set in R under f is:

A) A closed set B) An open set C) A connected set D) A discrete set

30. What is the Jacobian of the transformation w = f(z) = u(x,y) + iv(x,y) at a point z₀ where f'(z₀) ≠ 0?

A) |f'(z₀)|² B) Re(f'(z₀)) C) Im(f'(z₀)) D) 0

31. A mapping is called 'sense-preserving' if it preserves:

A) Distances B) Angles C) The orientation of angles D) Areas

32. A linear fractional transformation is conformal everywhere except possibly at:

A) z = 0 and z = ∞ B) The poles of the transformation C) The fixed points of the transformation D) Points where ad - bc = 0

33. What are the fixed points of the transformation f(z) = 1/z?

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

34. Consider the cross-ratio (z₁, z₂, z₃, z₄). If we permute z₃ and z₄, how does the cross-ratio change?

A) It remains the same. B) It is inverted (1/value). C) It is negated (-value). D) It becomes its complex conjugate.

35. Which of the following is NOT a property of the cross-ratio (z₁, z₂, z₃, z₄)?

A) It is invariant under linear fractional transformations. B) It is real if and only if the four points are concyclic or collinear. C) It is zero if any two of the points coincide. D) It is equal to 1 if the points form a harmonic range.

36. The condition for four points z₁, z₂, z₃, z₄ to be concyclic or collinear is that their cross-ratio is:

A) Purely imaginary B) Real C) Equal to 1 D) Equal to 0

37. What is the image of the unit circle |z|=1 under the mapping f(z) = z + 1/z?

A) The unit circle B) An ellipse C) A line segment D) The real axis

38. The transformation f(z) = z² is conformal at z₀ if:

A) z₀ ≠ 0 B) z₀ = 0 C) z₀ = 1 D) z₀ = i

39. At z₀ = 0, the mapping f(z) = z² is:

A) Conformal B) Not conformal C) Conformal only for real z D) Conformal only for imaginary z

40. What is the angle of magnification at a point z₀ where f(z) is conformal?

A) Re(f'(z₀)) B) Im(f'(z₀)) C) |f'(z₀)| D) arg(f'(z₀))

41. What is the argument of the angle of magnification at a point z₀ where f(z) is conformal?

A) Re(f'(z₀)) B) Im(f'(z₀)) C) |f'(z₀)| D) arg(f'(z₀))

42. A mapping f(z) = u + iv is conformal at z₀ if and only if u and v satisfy the:

A) Laplace equations B) Cauchy-Riemann equations C) Heat equations D) Wave equations

43. What is the condition for a linear fractional transformation f(z) = (az+b)/(cz+d) to map the real axis to itself?

A) a, b, c, d are real B) a, b, c, d are purely imaginary C) a, c are real and b, d are imaginary D) a, d are real and b, c are imaginary

44. The inverse of a linear fractional transformation f(z) = (az+b)/(cz+d) is given by:

A) f⁻¹(w) = (dw-b)/(-cw+a) B) f⁻¹(w) = (aw-b)/(cw+d) C) f⁻¹(w) = (-dw+b)/(cw-a) D) f⁻¹(w) = (dw+b)/(cw+a)

45. Which theorem states that if f(z) is analytic and non-constant in a region R, then f maps open sets to open sets?

A) Cauchy's Integral Theorem B) Maximum Modulus Principle C) Open Mapping Theorem D) Liouville's Theorem

46. What is the image of the unit circle |z|=1 under the mapping f(z) = z + 1?

A) The unit circle |w|=1 B) The circle |w-1|=1 C) The circle |w|=2 D) The line Re(w)=1

47. The cross-ratio of four points is invariant under the group of:

A) Isometries B) Conformal mappings C) Linear fractional transformations D) Analytic functions

48. If f(z) is conformal at z₀, then the angle between any two smooth curves passing through z₀ is preserved in magnitude and:

A) Orientation B) Direction C) Sense D) Magnitude

49. Symmetry with respect to a circle or a line in the complex plane is preserved under:

A) All conformal mappings B) Only translations C) Only rotations D) Linear fractional transformations

Analytic mappings - conformality, arcs and closed curves, conformal mapping, linear fractional transformations, cross ratio and symmetry - Online Test

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