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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.
- Choose the correct alternative (1 mark each) :
- If p (x) is the solution of Legendre’s D.E., then p (-1) is :
- -1
- 1
- 0
- The value of integral ?x p1(x) dx , where p1(x) is Legendre’s polynomial of degree 1, equals :
- 5/3
- 2/3
- 1/3
- 0
- The value of J½(x) equals :
- cos x
- v(2/px) sin x
- v(2/px) cos x
- sin x
- Eigen functions corresponding to different Eigen values are :
- Linearly dependent
- Linearly independent
- Real
- None
- The coefficient in a half range sine series for the function f(x) = sin x defined on [0, p] is given by :
- (2/p) ? sin x cos(nx/p) dx
- (2/p) ? cos x cos(nx/p) dx
- (2/p) ? sin x sin(nx/p) dx
- (2/p) ? sin x sin(nx/p) dx
- The function f(x) = |x| is
- Odd
- Even
- Even and Odd
- None of these
- If L[f(t)] = F(s). then L[f(at)] is :
- F(s-a)
- (1/a)F(s/a)
- F(s/a)
- aF(s/a)
- The value of L-1[1/(s-a)] is
- 1
- t
- eat
- e-at
- The Fourier sine transform of f(x) = e-ax, x = 0 is :
- a/(a²+n²)
- n/(a²+n²)
- 1/(a²+n²)
- 1/(a²-n²)
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- If p (x) is the solution of Legendre’s D.E., then p (-1) is :
UNIT—I
-
- Show that pn(x) is the coefficient of hn in the ascending power series expansion of (1 - 2xh + h²)-1/2. 5
- Prove that np'n = xp'n - p'n-1. 5
- Prove that x³ = (2/5)p5(x) - (3/5)p1(x). 5
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-
- Prove that ?x pn(x)p'n(x) dx = 2n(n+1)/(2n+1). 5
- Prove that ?p?(x) dx = 0. 5
- Prove that p?(x) = (1/2nn!) dn/dxn (x²-1)n. 5
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UNIT—II
-
- Prove that J½(x) = v(2/px) sin x. 4
- Prove that xJ'n = nJn - xJn+1. 4
- Evaluate ?[J1(x) - J3(x)]dx. 2
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-
- Prove that Eigen values of the S-L problem are real. 4
- Prove that d/dx (xnJn) = xnJn-1. 3
- Prove that J-½(x) = v(2/px) cos x. 3
UNIT—III
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-
- 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
- Obtain Fourier Series in [0, 2p] for the function f(x) = x². 5
-
- Obtain Fourier Series in [-p, p] for the function : f(x) = -1, -p < x < 0 x, 0 < x < p 5
- Obtain Fourier cosine series in [0, p] for the function f(x) = sin x. 5
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UNIT—IV
-
- Prove that L[tnf(t)]=(-1)n dn/dsn F(s), n=1,2,3..... 4
- Find L[sin t - cos 2t - cos 3t]. 3
- Show that L(tn) = n!/sn+1, s>0. 3
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-
- Solve the D.E. y'' - 4y' + 4y = e2t, y(0) = 0, y'(0) = -1 by using Laplace transform. 4
- Find the inverse Laplace transform of 1/((s-2)(s+1)) by using Convolution theorem. 3
- Prove that L(?u/?t) = sL(u(x, t)) - u(x, 0). 3
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UNIT—V
-
- Find the finite Fourier sine and cosine transform of f(x) = sin kx in (0, p). 4
- Find the Fourier transform of the function : f(x) = 1, |x|<1 0, |x|>1 4
- Prove that ?f'(x) sin(nx) dx = -nFc(n). 2
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-
- Find the Fourier sine and cosine transform of the function f(x) = xn-1 n> 0. 5
- Find finite Fourier cosine transform of ?u/?t and ?²u/?t²; where u = u(x, t). 5
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