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Download SGBAU BSc 2019 Summer 5th Sem Mathematics Mathematical Methods Question Paper

Download SGBAU (Sant Gadge Baba Amravati university) BSc 2019 Summer (Bachelor of Science) 5th Sem Mathematics Mathematical Methods Previous Question Paper

This post was last modified on 10 February 2020

SGBAU BSc Last 10 Years 2010-2020 Question Papers || Sant Gadge Baba Amravati university


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B.Sc. (Part—III) Semester—V Examination

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5S : MATHEMATICS (New)

(Mathematical Methods)

Paper—X

Time : Three Hours] [Maximum Marks : 60

Note :— (1) Question No.1 is compulsory and attempt it once.

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(2) Solve ONE question from each Unit.

  1. Choose the correct alternative (1 mark each) :
    1. If p (x) is the solution of Legendre’s D.E., then p (-1) is :
      1. -1
      2. 1
      3. 0
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    3. The value of integral ?x p1(x) dx , where p1(x) is Legendre’s polynomial of degree 1, equals :
      1. 5/3
      2. 2/3
      3. 1/3
      4. 0
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    5. The value of J½(x) equals :
      1. cos x
      2. v(2/px) sin x
      3. v(2/px) cos x
      4. sin x
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    7. Eigen functions corresponding to different Eigen values are :
      1. Linearly dependent
      2. Linearly independent
      3. Real
      4. None
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    9. The coefficient in a half range sine series for the function f(x) = sin x defined on [0, p] is given by :
      1. (2/p) ? sin x cos(nx/p) dx
      2. (2/p) ? cos x cos(nx/p) dx
      3. (2/p) ? sin x sin(nx/p) dx
      4. (2/p) ? sin x sin(nx/p) dx
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    11. The function f(x) = |x| is
      1. Odd
      2. Even
      3. Even and Odd
      4. None of these
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    13. If L[f(t)] = F(s). then L[f(at)] is :
      1. F(s-a)
      2. (1/a)F(s/a)
      3. F(s/a)
      4. aF(s/a)
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    15. The value of L-1[1/(s-a)] is
      1. 1
      2. t
      3. eat
      4. e-at
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    17. The Fourier sine transform of f(x) = e-ax, x = 0 is :
      1. a/(a²+n²)
      2. n/(a²+n²)
      3. 1/(a²+n²)
      4. 1/(a²-n²)
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UNIT—I

    1. Show that pn(x) is the coefficient of hn in the ascending power series expansion of (1 - 2xh + h²)-1/2. 5
    2. Prove that np'n = xp'n - p'n-1. 5
    3. Prove that x³ = (2/5)p5(x) - (3/5)p1(x). 5
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    1. Prove that ?x pn(x)p'n(x) dx = 2n(n+1)/(2n+1). 5
    2. Prove that ?p?(x) dx = 0. 5
    3. Prove that p?(x) = (1/2nn!) dn/dxn (x²-1)n. 5
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UNIT—II

    1. Prove that J½(x) = v(2/px) sin x. 4
    2. Prove that xJ'n = nJn - xJn+1. 4
    3. Evaluate ?[J1(x) - J3(x)]dx. 2
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    1. Prove that Eigen values of the S-L problem are real. 4
    2. Prove that d/dx (xnJn) = xnJn-1. 3
    3. Prove that J(x) = v(2/px) cos x. 3

UNIT—III

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    1. If the trigonometric series S (a? cos nx + b? sin nx) converges uniformly to f(x) in -c < x < c + 2p, then find the Fourier coefficient of f(x). 5
    2. Obtain Fourier Series in [0, 2p] for the function f(x) = x². 5
    1. Obtain Fourier Series in [-p, p] for the function : f(x) = -1, -p < x < 0 x, 0 < x < p 5
    2. Obtain Fourier cosine series in [0, p] for the function f(x) = sin x. 5
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UNIT—IV

    1. Prove that L[tnf(t)]=(-1)n dn/dsn F(s), n=1,2,3..... 4
    2. Find L[sin t - cos 2t - cos 3t]. 3
    3. Show that L(tn) = n!/sn+1, s>0. 3
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    1. Solve the D.E. y'' - 4y' + 4y = e2t, y(0) = 0, y'(0) = -1 by using Laplace transform. 4
    2. Find the inverse Laplace transform of 1/((s-2)(s+1)) by using Convolution theorem. 3
    3. Prove that L(?u/?t) = sL(u(x, t)) - u(x, 0). 3
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UNIT—V

    1. Find the finite Fourier sine and cosine transform of f(x) = sin kx in (0, p). 4
    2. Find the Fourier transform of the function : f(x) = 1, |x|<1 0, |x|>1 4
    3. Prove that ?f'(x) sin(nx) dx = -nFc(n). 2
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    1. Find the Fourier sine and cosine transform of the function f(x) = xn-1 n> 0. 5
    2. Find finite Fourier cosine transform of ?u/?t and ?²u/?t²; where u = u(x, t). 5

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