লগ ইননিবন্ধন
পরীক্ষা আর্কাইভ

৪৯তম বিসিএস ⎯ ফলিত গণিত [৫৬১]

পরীক্ষা৪৯তম বিসিএস ⎯ ফলিত গণিত [৫৬১]তারিখতারিখ অনির্ধারিতসময়40 minutes৩৫ বৈধ · ৪ অসম্পূর্ণ
মোট প্রশ্ন৩৯
সিলেবাস
Exam - 06 Topics: Mathematical Method (a) Bessel’s Equation: Solution, Generating function, Recurrence relation, Orthogonality. (b) Legendre’s Equation: Solution, Generating function, Recurrence relation, Rodrigui’s formula and orthogonality of Legndre Polynomial. (c) Fortier Series: Fortier Coefficients, Sine and Cosine series, Dirichlet’s theorem, Properties and applications. [Source: Class - 05 and Relevant Books]
প্রশ্ন দেখুনঅনুশীলন
ঘনত্ব
উত্তর
উত্তরিতবর্তমানপুনরায় দেখুনঅসম্পূর্ণ

৪৯তম বিসিএস ⎯ ফলিত গণিত [৫৬১]

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

১.
Which of the following is true about Legendre polynomials?
  1. ক
    They are orthogonal over [−1,1] with weight w(x)=1.
  2. খ
    They satisfy a Sturm–Liouville problem.
  3. গ
    They appear in solutions of Laplace’s equation in spherical coordinates.
  4. ঘ
    All of the above
২.
Bessel functions arise as eigenfunctions of:
  1. ক
    Laplace’s equation in cylindrical coordinates
  2. খ
    Heat equation in Cartesian coordinates
  3. গ
    Wave equation in Cartesian coordinates
  4. ঘ
    All of these
ব্যাখ্যা

Separation of variables in cylindrical systems leads to Bessel’s equation for the radial part.

৩.
Which of the following best explains why Bessel functions are important in wave propagation?
  1. ক
    They approximate sinusoidal motion
  2. খ
    They represent standing waves in circular/elliptical geometries
  3. গ
    They only apply to 1D vibrations
  4. ঘ
    They remove damping effects
অনির্ধারিত
ব্যাখ্যা
Bessel functions describe radial oscillations in 2D/3D geometries (circular membranes, cylindrical pipes, antennas, waveguides).
৪.
Which of the following is NOT an application of Fourier series?
  1. ক
    Solving the heat equation
  2. খ
    Signal processing and communication
  3. গ
    Image compression (JPEG)
  4. ঘ
    Finding eigenvalues of Legendre equation
ব্যাখ্যা
Fourier series is widely used in PDEs (heat, wave, Laplace equations), signal & image processing, and data compression.
But Legendre’s equation is solved by Legendre polynomials, not Fourier series.
৫.
If f(x) is an odd function, then its Fourier series contains:
  1. ক
    Only cosine terms
  2. খ
    Only sine terms
  3. গ
    Both sine and cosine terms
  4. ঘ
    Only the constant term
৬.
Which of the following is a direct consequence of Riemann’s theorem?
  1. ক
    Parseval’s identity always holds
  2. খ
    Fourier series converges uniformly for all functions
  3. গ
    an​,bn ​→ 0 as n → ∞
  4. ঘ
    f(x) must be even
ব্যাখ্যা

৭.
Which of the following functions does NOT satisfy the condition for Riemann’s theorem?
  1. ক
    A piecewise continuous function
  2. খ
    A bounded function
  3. গ
    A function with infinite discontinuities in [- π, π]
  4. ঘ
    A function defined on [- π, π]
ব্যাখ্যা


The theorem requires the function to be piecewise continuous (finite discontinuities allowed). If there are infinitely many discontinuities, the condition fails.

৮.
Using Rodrigue's Formula P1(x) = ?
  1. ক
    1
  2. খ
    x
  3. ঘ
    All of these
অনির্ধারিত
ব্যাখ্যা



Using All kind of Formula, P1(x) is always x.
৯.
  1. ঘ
    None of these
ব্যাখ্যা

১০.
A function f(x) with period 2π can be expressed as:
  1. ঘ
    Both B and C
ব্যাখ্যা



১১.
The Fourier series was originally developed by Joseph Fourier to solve problems related to:
  1. ক
    Vibrations of strings
  2. খ
    Heat conduction
  3. গ
    Electrical circuits
  4. ঘ
    Quantum mechanics
ব্যাখ্যা

Fourier first introduced the Fourier series in his study of the heat equation in 1822. Later, it was applied in many other fields like vibrations, circuits, and physics.

১২.
Express J1(x) in terms of J0​(x) and J2(x).
  1. ক
    J1​(x) = [(J0​(x) - J2​(x)]/2
  2. খ
    J1​(x) = [(J0​(x) - J2​(x)]
  3. গ
    J1​(x) = [(J0​(x) + J2​(x)]/2
  4. ঘ
    J1​(x) = [(J0​(x) + J2​(x)]
ব্যাখ্যা


​

১৩.
  1. ঘ
    None of these
ব্যাখ্যা


​

১৪.
The periodic square wave function f(x) defined in (- π, π) as  has Fourier expansion involving:
  1. ক
    Only cosine term
  2. খ
    Only sine terms
  3. গ
    Both sine and cosine terms
  4. ঘ
    Constant term only
ব্যাখ্যা

১৫.
  1. ক
    2π
  2. খ
    0
  3. গ
    Divergent
  4. ঘ
    π2
ব্যাখ্যা


its integral tends to zero by Riemann’s theorem

১৬.
Which one in true?
  1. ঘ
    All of these
ব্যাখ্যা


​

১৭.
Which of the following derivative relations is correct for Legendre polynomials?
    ব্যাখ্যা



    ১৮.
    Riemann’s theorem is most useful in proving which property of Fourier coefficients?
    1. ক
       a0=0 for odd functions
    2. খ
      Fourier coefficients are bounded by O(1/n)
    3. গ
      Fourier coefficients vanish at infinity
    4. ঘ
      Fourier series converges absolutely for every function
    ব্যাখ্যা

    The theorem ensures that as n→∞, an→0and bn→0, which is essential for Fourier series representation.

    ১৯.
    Which of the following functions does NOT satisfy Dirichlet’s conditions in (- π, π)?
    1. ক
      f(x) = sin(x)
    2. খ
      f(x) =।x।
    3. গ
      f(x) = tan(x)
    4. ঘ
      f(x) = x2
    ব্যাখ্যা


    ২০.
    Which of the following is the differential equation satisfied by the modified Bessel function In​(x) ?
    1. ঘ
      All of these
    ব্যাখ্যা

    ২১.
    which one is correct?
      অনির্ধারিত
      ব্যাখ্যা

      ২২.
      Which of the following problems is most suitable for half-range Fourier expansion?
      1. ক
         A vibrating string fixed at both ends, displacement given in (0,L).
      2. খ
        Motion in free space without boundaries
      3. গ
        Probability distribution problems
      4. ঘ
        All of these
      ব্যাখ্যা

      Half range Fourier expansions are widely used in vibrating string and heat conduction problems where functions are defined only on (0,L).

      ২৩.
      If a function is not periodic, can its Fourier series representation exist?
      1. ক
        Yes, always
      2. খ
        No, never
      3. গ
        Yes, if it is extended periodically
      4. ঘ
        Only if it is discontinuous
      ব্যাখ্যা

      Fourier series require periodicity. A non-periodic function can still have a Fourier series if it is periodically extended outside the given interval.

      ২৪.
      Which differential equation is satisfied by Legendre polynomials?
        ব্যাখ্যা

        ২৫.
        J2(- x) = ?
        1. ক
          J- 2​(x)
        2. খ
          - J- 2​(x)
        3. গ
          - J2​(x)
        4. ঘ
          J2​(x)
        ব্যাখ্যা

        ২৬.
        Dirichlet’s conditions ensure that:
        1. ক
          The Fourier series always converges uniformly.
        2. খ
           The Fourier series always converges to the original function everywhere.
        3. গ
          The Fourier series converges pointwise to the function or to the average at discontinuities.
        4. ঘ
          The Fourier series does not converge.
        ব্যাখ্যা

        Dirichlet’s theorem guarantees pointwise convergence. At discontinuities, it converges to the average of left and right-hand limits. It does not guarantee uniform convergence.

        ২৭.
        The normalization constant for J2(αmx) is:
        1. ঘ
          0
        ব্যাখ্যা


        ২৮.
        If f(x) = x in [-π, π], then its Fourier series:
        1. ক
          Converges uniformly.
        2. খ
           Does not converge uniformly but converges pointwise.
        3. গ
          Diverges everywhere.
        4. ঘ
          Converges only at x=0.
        ব্যাখ্যা


        Here, f(−π)=−π and f(π)=π, which are not equal. Thus, the periodic extension introduces a jump discontinuity. The Fourier series converges pointwise everywhere except at the discontinuity, but not uniformly

        ২৯.
        Which of the following waves satisfies Dirichlet’s conditions?
        1. ক
          Square wave
        2. খ
          Triangular wave
        3. গ
          Sawtooth wave
        4. ঘ
          All of the above
        ব্যাখ্যা

        ৩০.
        For m > p 
        1. ক
          0
        2. খ
          1
        3. গ
          ∞
        4. ঘ
          None of These
        ব্যাখ্যা


        ​m≠p
        ​

        ৩১.
        Which is the differential equation for modified Bessel functions In(x)?
        1. ঘ
          All of these
        অনির্ধারিত
        ব্যাখ্যা


        ৩২.
        1. গ
          sinx
        2. ঘ
          cosx
        ৩৩.
        1. ক
          2/7
        2. খ
          0
        3. গ
          7/2
        4. ঘ
          2/9
        ব্যাখ্যা


        ৩৪.
        The standard form of Bessel’s differential equation of order n is:
          ব্যাখ্যা

          ৩৫.
          The Bessel equation arises in which type of physical problems?
          1. ক
            Vibrations of a circular membrane
          2. খ
            Heat conduction in a rod
          3. গ
            Projectile motion
          4. ঘ
            Simple harmonic oscillator
          ব্যাখ্যা

          Bessel equations appear in problems with cylindrical symmetry, e.g., circular membranes, cylindrical waves, and electromagnetic waves in cylinders.

          ৩৬.
          1. ক
            2/3
          2. খ
            1/3
          3. গ
            3/2
          4. ঘ
            0
          ব্যাখ্যা

           

          ৩৭.
          J0(0) and J1(0)=?
          1. ক
            0 and 1 respectively
          2. খ
            1 and 0 respectively
          3. গ
            Both are 0
          4. ঘ
            Both are 1
          ব্যাখ্যা

          okay give

          ৩৮.
          Which of the following integrals is nonzero?
            ব্যাখ্যা

            ৩৯.
            P2(-1)=?
            1. ক
              0
            2. খ
              1
            3. গ
              -1
            4. ঘ
              2
            ব্যাখ্যা

            Pn(-1)=(-1)n
            ​Putting n=2 we get P2(-1)=1