TRB PG Assistant - Zeros and poles, local mapping properties, maximum modulus principle Online Test (MCQ Quiz)

1. What is the order of a zero at z = 0 for the function f(z) = sin(z) - z + z^3/6?

A) 1 B) 2 C) 3 D) 4

2. If f(z) has a zero of order m at z0, then what is the behavior of f(z) near z0?

A) f(z) is analytic and non-zero at z0 B) f(z) has a pole of order m at z0 C) f(z) behaves like c(z-z0)^m for some non-zero constant c near z0 D) f(z) is constant near z0

3. What is the order of the pole at z = 0 for the function g(z) = 1/sin(z)?

A) 0 B) 1 C) 2 D) 3

4. If f(z) has a pole of order m at z0, then what is the behavior of 1/f(z) near z0?

A) 1/f(z) has a zero of order m at z0 B) 1/f(z) has a pole of order m at z0 C) 1/f(z) is analytic and non-zero at z0 D) 1/f(z) is constant near z0

5. Consider the function f(z) = z^2 * sin(z). What is the order of the zero at z = 0?

A) 1 B) 2 C) 3 D) 4

6. What is the order of the pole at z = 0 for the function h(z) = cot(z)?

A) 1 B) 2 C) 3 D) 4

7. If f(z) is analytic in a region D and has a zero of order m at z0 in D, what can be said about f'(z) at z0?

A) f'(z0) = 0 B) f'(z0) is non-zero C) f'(z0) is infinite D) f'(z0) is undefined

8. Let f(z) = (e^z - 1 - z) / z^2. What is the order of the zero at z = 0?

A) 1 B) 2 C) 3 D) 4

9. What is the order of the pole at z = 1 for the function f(z) = 1 / (z-1)^3?

A) 1 B) 2 C) 3 D) 4

10. If f(z) has a zero of order m at z0, then f'(z0) is:

A) Always zero B) Never zero C) Zero if m > 1 D) Zero if m = 1

11. What is the principal part of the Laurent series expansion of f(z) = 1 / (z * (z-1)) around z = 0?

A) 1/z B) -1/z C) 1/(z-1) D) -1/(z-1)

12. If f(z) has a simple pole at z0, what is the residue of f(z) at z0?

A) The coefficient of (z-z0)^-1 in the Laurent series B) The coefficient of (z-z0)^0 in the Laurent series C) The coefficient of (z-z0)^1 in the Laurent series D) Zero

13. What is the order of the zero at z = 0 for f(z) = z * sin(z)?

A) 1 B) 2 C) 3 D) 4

14. If f(z) = 1/z^2, what is the order of the pole at z = 0?

A) 1 B) 2 C) 3 D) 4

15. Consider the function f(z) = z^3 / (sin(z) - z + z^3/6). What is the order of the zero at z = 0?

A) 1 B) 2 C) 3 D) 4

16. What is the nature of the singularity at z = 0 for f(z) = sin(1/z)?

A) Removable singularity B) Pole C) Essential singularity D) None of the above

17. If f(z) is analytic and non-constant in a domain D, which principle states that |f(z)| cannot attain a maximum value in D?

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

18. According to the Maximum Modulus Principle, if f(z) is analytic and non-constant in a bounded domain D and continuous on its boundary ∂D, where does the maximum value of |f(z)| occur?

A) At an interior point of D B) At a point where f(z) = 0 C) On the boundary ∂D D) It does not attain a maximum

19. What does the Maximum Modulus Principle imply for a non-constant analytic function f(z) inside a disk |z| <= R?

A) The maximum of |f(z)| occurs at the center B) The maximum of |f(z)| occurs on the boundary |z| = R C) The maximum of |f(z)| occurs at a removable singularity D) The maximum of |f(z)| is attained at an interior point

20. If f(z) is analytic in a domain D and |f(z)| is constant on D, what can be concluded about f(z)?

A) f(z) is identically zero B) f(z) is a non-zero constant C) f(z) has zeros in D D) f(z) has poles in D

21. Consider the function f(z) = z^2. What is the maximum value of |f(z)| on the disk |z| <= 2?

A) 0 B) 2 C) 4 D) 8

22. If f(z) is analytic on the closed unit disk |z| <= 1 and f(0) = 0, what can be said about |f(z)| inside the disk?

A) |f(z)| = 0 for all z B) |f(z)| > 0 for z != 0 C) |f(z)| < |f'(0)| for z != 0 D) |f(z)| attains its maximum at z=0

23. What is the minimum value of |f(z)| for f(z) = z - 1 on the closed disk |z| <= 1?

A) 0 B) 1 C) 2 D) The minimum is not attained

24. The Minimum Modulus Principle states that if f(z) is analytic and non-constant in a bounded domain D and continuous on its boundary ∂D, and if f(z) is never zero in D, then the minimum value of |f(z)| occurs:

A) At an interior point of D B) At a point where f(z) = 0 C) On the boundary ∂D D) At a pole

25. If f(z) is analytic in the punctured disk 0 < |z| < 1 and has a pole at z = 0, can |f(z)| attain a minimum value in this domain?

A) Yes, at z=0 B) Yes, at some point in the domain C) No, unless f(z) is constant D) No, because of the pole

26. Consider f(z) = e^z. What is the maximum value of |f(z)| on the rectangle defined by 0 <= Re(z) <= 1 and 0 <= Im(z) <= pi?

A) e B) e*pi C) e^pi D) e^(1+pi)

27. If f(z) is analytic in a domain D and has a zero of order m at z0, what is the order of the zero of f'(z) at z0?

A) m-1 B) m C) m+1 D) 0

28. What is the order of the pole at z = 0 for f(z) = 1 / (z^2 * sin(z))?

A) 1 B) 2 C) 3 D) 4

29. Let f(z) = z^n. For what value of n does f(z) have a zero of order 5 at z=0?

A) 0 B) 1 C) 5 D) infinity

30. If f(z) has a pole of order m at z0, then lim (z-z0)^m * f(z) as z->z0 is:

A) 0 B) 1 C) A finite non-zero number D) Infinity

31. What is the order of the zero at z = 0 for the function f(z) = 1 - cos(z)?

A) 1 B) 2 C) 3 D) 4

32. If f(z) is analytic and non-constant in a domain D, and f(z) != 0 in D, then |f(z)| attains its minimum value:

A) Inside D B) On the boundary of D C) At a point where f(z) = 0 D) At a pole

33. Consider f(z) = z^2. What is the order of the zero of f(z) at z=0?

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

34. What is the order of the pole at z = 0 for f(z) = z^-3 * e^z?

A) 1 B) 2 C) 3 D) 4

35. If f(z) is analytic in a region and f(z_0) = f'(z_0) = ... = f^(m-1)(z_0) = 0 and f^(m)(z_0) != 0, then f(z) has a zero of order:

A) m-1 B) m C) m+1 D) 0

36. What is the order of the pole at z = pi for f(z) = cot(z)?

A) 1 B) 2 C) 3 D) 4

37. If f(z) is analytic in the unit disk |z| <= 1 and |f(z)| = 1 on the boundary |z|=1, what can be concluded about f(z)?

A) f(z) is constant B) f(z) has a zero inside the disk C) f(z) has a pole inside the disk D) f(z) is never zero inside the disk

38. Consider the function f(z) = (z-2)^3. What is the order of the zero at z=2?

A) 0 B) 1 C) 2 D) 3

39. What is the order of the pole at z = 0 for f(z) = 1 / (z * (1 - cos(z)))?

A) 1 B) 2 C) 3 D) 4

40. If f(z) is analytic in a punctured disk 0 < |z-z0| < R and has a pole of order m at z0, then |f(z)| tends to ____ as z approaches z0.

A) 0 B) 1 C) infinity D) A finite non-zero value

41. According to the Maximum Modulus Principle, if a non-constant analytic function f(z) has a maximum modulus inside a domain D, then:

A) The maximum modulus must be zero B) The maximum modulus must be 1 C) This is impossible D) The function must be constant

42. What is the order of the zero at z=0 for f(z) = sinh(z) - z?

A) 1 B) 2 C) 3 D) 4

43. If f(z) is analytic in the domain |z| < 1 and |f(z)| <= 1 for all z in the domain, and |f(z_0)| = 1 for some z_0 with |z_0| < 1, what can be concluded?

A) f(z) is constant B) f(z) must be zero at z_0 C) f(z) has a pole at z_0 D) f(z) must be identically 1

44. What is the order of the pole at z=0 for f(z) = z^-2 * cos(z)?

A) 0 B) 1 C) 2 D) 3

45. Consider the function f(z) = 1/(z-a). What is the order of the pole at z=a?

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

46. If f(z) is analytic in a domain D, and z0 is an interior point of D, what is the implication of f(z0) = 0 for the Maximum Modulus Principle?

A) It implies |f(z)| attains its maximum at z0 B) It implies |f(z)| attains its minimum at z0 C) It implies f(z) is constant D) It implies f'(z0) is non-zero

47. What is the order of the zero at z=0 for f(z) = e^z - 1?

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

48. If f(z) is analytic and non-constant in the open disk |z| < R, and f(z) is never zero in this disk, then |f(z)| attains its minimum value:

A) At the center z=0 B) On the boundary |z|=R C) At an interior point where f'(z) = 0 D) This scenario is impossible

49. Let f(z) = (z^2 - 1) / (z - 1). What is the nature of the singularity at z = 1?

A) Pole of order 1 B) Removable singularity C) Essential singularity D) Branch point

50. What is the order of the pole at z = 0 for f(z) = z^-1 * cot(z)?

A) 1 B) 2 C) 3 D) 4

Zeros and poles, local mapping properties, maximum modulus principle - Online Test

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