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

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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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   2. 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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   3. The value of J_½(x) equals :
      1. cos x
      2. √(2/πx) sin x
      3. √(2/πx) cos x
      4. sin x

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   4. Eigen functions corresponding to different Eigen values are :
      1. Linearly dependent
      2. Linearly independent
      3. Real
      4. None

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   5. The coefficient in a half range sine series for the function f(x) = sin x defined on [0, π] is given by :
      1. (2/π) ∫ sin x cos(nx/π) dx
      2. (2/π) ∫ cos x cos(nx/π) dx
      3. (2/π) ∫ sin x sin(nx/π) dx
      4. (2/π) ∫ sin x sin(nx/π) dx

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   6. The function f(x) = |x| is
      1. Odd
      2. Even
      3. Even and Odd
      4. None of these

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   7. 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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   8. The value of L^-1[1/(s-a)] is
      1. 1
      2. t
      3. e^at
      4. e^-at

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   9. 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

2. 1. Show that p_n(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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3. 1. Prove that ∫x p_n(x)p'_n(x) dx = 2n(n+1)/(2n+1). 5
   2. Prove that ∫pₙ(x) dx = 0. 5
   3. Prove that pₙ(x) = (1/2ⁿn!) dⁿ/dxⁿ (x²-1)ⁿ. 5

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UNIT—II

4. 1. Prove that J_½(x) = √(2/πx) sin x. 4
   2. Prove that xJ'_n = nJ_n - xJ_n+1. 4
   3. Evaluate ∫[J₁(x) - J₃(x)]dx. 2

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5. 1. Prove that Eigen values of the S-L problem are real. 4
   2. Prove that d/dx (xⁿJ_n) = xⁿJ_n-1. 3
   3. Prove that J_-½(x) = √(2/πx) cos x. 3

UNIT—III

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6. 1. If the trigonometric series Σ (aₙ cos nx + bₙ sin nx) converges uniformly to f(x) in -c < x < c + 2π, then find the Fourier coefficient of f(x). 5
   2. Obtain Fourier Series in [0, 2π] for the function f(x) = x². 5

7. 1. Obtain Fourier Series in [-π, π] for the function : f(x) = -1, -π < x < 0 x, 0 < x < π 5
   2. Obtain Fourier cosine series in [0, π] for the function f(x) = sin x. 5

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UNIT—IV

8. 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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9. 1. Solve the D.E. y'' - 4y' + 4y = e^2t, 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

10. 1. Find the finite Fourier sine and cosine transform of f(x) = sin kx in (0, π). 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 = -nF_c(n). 2

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11. 1. Find the Fourier sine and cosine transform of the function f(x) = x^n-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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