Jawaharlal Nehru Technological University Hyderabad
Jawaharlal Nehru Technological University Hyderabad B.Tech I Year MATHEMATICS - II Previous Question Paper (June - 2022 - R16)
Answer any five questions. All questions carry equal marks.
Main
Find L-1{ (s+5)/(s2+2s+5) + (1+s)/(s2+3s+6) + 4/(s-2)3 + 5/(s+3)6 }.
Answer
To find the inverse Laplace transform of the given function, we use the method of partial fraction decomposition. Let the function be:
We express F(s) as:
Equating the numerators:
Setting s = 4:
Comparing coefficients of s²:
Comparing constants:
Substituting back:
Rewriting the second term to match standard forms:
Applying the inverse Laplace transform:
Find L-1{ 1/(s(s-1)) }.
Answer
To find the inverse Laplace transform of 1/(s(s-1)), we use partial fraction decomposition:
Equating numerators:
Setting s = 0 gives A = -1. Setting s = 1 gives B = 1. Thus:
Taking the inverse Laplace transform:
Use Laplace transform to prove that ∫0∞ (sin t)/t dt = π/4.
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Find the inverse Laplace transform of e-πs/(s2+1).
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Show that ∫01 xm-1(1-x)n-1 dx = β(m, n).
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Evaluate ∫ sin5 x dx and hence find ∫0π/2 sin5 x dx.
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Change the order of integration and solve ∫0a ∫x2/a√(2ax-x2) xy dy dx.
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Evaluate ∫∫ dx dy / (x4+y4), over the region bounded by x ≥ y ≥ 1.
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Prove that ∇(f/g) = 1/g2 (g ∇f - f ∇g).
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Given the velocity potential f of a flow, find the velocity field ∇f = v and its value v(P) at P where f = cos x cosh y, P: (π/2, ln 2).
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Prove that ∇(fn) = nfn-1∇f.
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For what points P(x, y, z) does ∇f with f = 25x2 + 9y2 + 16z2 have the direction from P to the origin?
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Evaluate ∫C F ⋅ dr around the boundary C of the region R by Green’s Theorem, where F = grad(x3 cos2(xy)) and R: 1 ≤ y ≤ 2 - x2.
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Evaluate ∮C ez dx + 2y dy - dz by Stoke’s theorem when c is the curve x2 + y2 = 4, z = 2.
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