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৪৯তম বিসিএস ⎯ ফলিত গণিত [৫৬১]

পরীক্ষা৪৯তম বিসিএস ⎯ ফলিত গণিত [৫৬১]তারিখতারিখ অনির্ধারিতসময়45 minutes
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সিলেবাস
Exam - 09 Topics: Complex Analysis (a) Complex Functions: Single and many valued functions. Limit, Continuity and differentiability of complex function. (b) Analytic Functions: Necessary and sufficient conditions. Harmonic functions. Mobius transformation and power series. [Source: Class - 07 and Relevant Books]
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৪৯তম বিসিএস ⎯ ফলিত গণিত [৫৬১]

৪৯তম বিসিএস ⎯ ফলিত গণিত [৫৬১] · তারিখ অনির্ধারিত · ৩৯ প্রশ্ন

১.
 Which of the following is a single-valued function in complex analysis?
    ব্যাখ্যা

     

    ২.
    Which is a necessary and sufficient condition for a function f(z) to be analytic in a region?
    1. ক
      It is differentiable at one point
    2. খ
      It satisfies Cauchy–Riemann equations at every point of the region
    3. গ
      It satisfies CR equations and u,v have continuous partial derivatives in the region
    4. ঘ
      It is bounded in the region
    ব্যাখ্যা

    CR equations + continuity of partial derivatives in a neighborhood are the necessary and sufficient conditions for analyticity.

    ৩.
    Let f(z) = ez. Which of the following is true?
    1. ক
      Not analytic anywhere
    2. খ
      Analytic only on real axis
    3. গ
      Analytic everywhere
    4. ঘ
      Analytic only for Re(z) > 0
    ব্যাখ্যা

    ৪.
    Which of the following is a Möbius transformation?
      ব্যাখ্যা

      ৫.
      Which property is always true for Möbius transformations?
      1. ক
        Maps circles and lines to circles or lines
      2. খ
        Maps only circles to circles
      3. গ
        Preserves distance
      4. ঘ
        Maps analytic functions to harmonic functions
      ব্যাখ্যা

      Möbius transformations are conformal and map circles/lines to circles/lines; they do not necessarily preserve distance.

      ৬.
      The principal value of a multi-valued function is:
      1. ক
        The average of all possible values
      2. খ
        One specific branch chosen to make the function single-valued
      3. গ
        The limit of the function as z→0
      4. ঘ
        The sum of all possible values
      ব্যাখ্যা

      ৭.
      Which of the following is a Laurent series?
        ব্যাখ্যা

        ৮.
        Which of the following complex functions is continuous at every point of the complex plane?
          ব্যাখ্যা

          ৯.
          Which of the following functions is harmonic?
          1. ঘ
            None
          ব্যাখ্যা

          ১০.
            ব্যাখ্যা

            These are the Cauchy–Riemann equations, the necessary conditions for differentiability of a complex function.

            ১১.
            Which of the following functions is differentiable everywhere in C?
              ব্যাখ্যা

              ১২.
              Which of the following statements is true?
              1. ক
                Every continuous complex function is differentiable
              2. খ
                Every differentiable complex function is continuous
              3. গ
                If limit exists along one path, it exists everywhere
              4. ঘ
                All real-valued functions of complex variables are differentiable
              ব্যাখ্যা

              ১৩.
              Which of the following statements about the limit of a complex function is true?
              1. ক
                The limit along one path guarantees the general limit exists
              2. খ
                A limit exists only if the function is differentiable
              3. গ
                A limit exists if the function approaches the same value from all directions
              4. ঘ
                Limits are not defined in complex analysis
              ব্যাখ্যা

              ১৪.
              1. ক
                Yes, limit = 0
              2. খ
                Yes, limit = 1
              3. গ
                No, limit does not exist
              4. ঘ
                Yes, limit = ∞
              ব্যাখ্যা

              ১৫.
              The function f(z) = z1/n (where n > 1) is:
              1. ক
                Always single-valued
              2. খ
                Multi-valued with infinitely many values
              3. গ
                Multi-valued with exactly n distinct values
              4. ঘ
                Not a valid complex function
              ব্যাখ্যা

              ১৬.
              1. ক
                1 + i
              2. খ
                1 - i
              3. গ
                1
              4. ঘ
                - 1
              ব্যাখ্যা

              ১৭.
              1. ক
                Differentiable everywhere
              2. খ
                Differentiable only at z = 0
              3. গ
                Not differentiable anywhere
              4. ঘ
                Analytic everywhere
              ব্যাখ্যা

              ১৮.
              The real part of an analytic function f(z) = u + iv is:
              1. ক
                Always zero
              2. খ
                Harmonic
              3. গ
                Constant
              4. ঘ
                Non-differentiable
              ব্যাখ্যা

              ১৯.
              The function f(z) = ∣z∣ is:
              1. ক
                Differentiable everywhere
              2. খ
                Differentiable only at z = 0
              3. গ
                Not differentiable anywhere
              4. ঘ
                Analytic everywhere
              ব্যাখ্যা

              ২০.
              If f(z) = u(x, y) + iv(x, y) satisfies CR equations everywhere but u,v do not have continuous partial derivatives, then:
              1. ক
                f is analytic
              2. খ
                f may fail to be analytic
              3. গ
                f is entire
              4. ঘ
                f is harmonic
              ব্যাখ্যা

              CR equations alone are necessary but not sufficient. Without continuity of partials in a neighborhood, differentiability may fail ⇒ function may not be analytic.

              ২১.
              Which Möbius transformation maps z = ∞?
              1. ক
                f(z) = 1/z
              2. খ
                f(z) = z + 1
              3. গ
                f(z) = 2z
              4. ঘ
                f(z) = z2
              ব্যাখ্যা

              As z→∞z , 1/z→0.

              ২২.
                ব্যাখ্যা

                This is the standard Taylor series for sin⁡z, convergent for all z.

                ২৩.
                1. ক
                  z = 1 to 0, z = −1 to ∞
                2. খ
                  z = -1 to 0, z = 1 to ∞ 
                3. গ
                  z = 1 to 1, z = −1 to -1
                4. ঘ
                  z = ∞ to 0, z = 0 to ∞
                ব্যাখ্যা

                ২৪.
                Which function is harmonic only along certain lines or curves?
                  ব্যাখ্যা

                  ২৫.
                  1. ক
                    Itself
                  2. খ
                    Imaginary axis
                  3. গ
                    Unit circle
                  4. ঘ
                    Infinity
                  ব্যাখ্যা

                  ২৬.
                  1. ক
                    sinh ⁡z
                  2. খ
                    cosz
                  3. গ
                    sinz
                  4. ঘ
                    ez
                  ব্যাখ্যা

                  Standard Taylor expansion of cosine.

                  ২৭.
                  Which function has a Laurent series but no Taylor series expansion at z = 0?
                  1. ক
                    ez
                  2. খ
                    sin⁡z
                  3. গ
                    1/z
                  4. ঘ
                    z2 + 1
                  ব্যাখ্যা

                  ২৮.
                  Which of the following functions is multi-valued?
                  1. ঘ
                    All of These
                  ব্যাখ্যা


                  ২৯.
                  The function f(z)=arg⁡(z) is:
                  1. ক
                    Single-valued
                  2. খ
                    Multi-valued
                  3. গ
                    Analytic
                  4. ঘ
                    Entire
                  ব্যাখ্যা

                  Argument arg⁡(z) can differ by 2πk. Hence it is multi-valued.

                  ৩০.
                  The Laurent series differs from the Taylor series because it:
                  1. ক
                    Has only positive powers
                  2. খ
                    Has both positive and negative powers
                  3. গ
                    Does not converge anywhere
                  4. ঘ
                    Is valid only on real axis
                  ব্যাখ্যা

                  Laurent series generalizes Taylor series by allowing negative powers, useful for singularities.

                  ৩১.
                  Which of the following statements is true for a harmonic function?
                  1. ক
                    It can attain its maximum inside the domain.
                  2. খ
                    It satisfies Laplace’s equation.
                  3. গ
                    Its derivatives are always zero.
                  4. ঘ
                    It cannot have an imaginary part.
                  ব্যাখ্যা

                  By definition, a harmonic function satisfies ∇2u=0. Maximum and minimum occur on boundaries (unless constant). Imaginary part is irrelevant for real harmonic functions. 

                  ৩২.
                  Which of the following is not harmonic?
                  1. ঘ
                    All are Harmonic
                  ব্যাখ্যা

                  All three satisfy Laplace Equation. So All of these are Harmonic

                  ৩৩.
                  Which of the following statements is true?
                  1. ক
                    Analyticity at a point implies differentiability in a neighborhood
                  2. খ
                    Differentiability at a point implies analyticity in a neighborhood
                  3. গ
                    Satisfaction of CR equations at a point is sufficient for analyticity
                  4. ঘ
                    Continuity of f is sufficient for analyticity
                  ব্যাখ্যা

                  Analyticity requires differentiability in a neighborhood.
                  Differentiability at a single point is not enough.
                  CR equations at a point without continuity do not guarantee analyticity

                  ৩৪.
                  The function f(z) = (z - 1)1/2 has how many values?
                  1. ক
                    1
                  2. খ
                    2
                  3. গ
                    Infinitely many
                  4. ঘ
                    None
                  ব্যাখ্যা

                  ৩৫.
                  Which of the following functions is harmonic in R2∖{0}}?
                  1. ঘ
                    All of these
                  ব্যাখ্যা

                  ৩৬.
                  1. ক
                     Yes, everywhere
                  2. খ
                    Only at z = 0
                  3. গ
                    Only along x = 0
                  4. ঘ
                    Not analytic anywhere
                  ব্যাখ্যা

                  ৩৭.
                  If f(z) = u(x, y) + iv(x, y) is analytic at a point, which of the following must hold?
                  1. ঘ
                    None of the above
                  ব্যাখ্যা

                  ৩৮.
                  1. ক
                     Differentiable only at z=0
                  2. খ
                    Differentiable everywhere
                  3. গ
                    Not differentiable anywhere
                  4. ঘ
                    Entire
                  ব্যাখ্যা

                  ৩৯.
                  Which of the following functions is analytic everywhere (entire)?
                  1. ঘ
                    All of these
                  ব্যাখ্যা