JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD
JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD B.Tech I Year MATHEMATICS-II Previous Question Paper (August - 2018 - R16)
Note: This question paper contains two parts A and B. Part A is compulsory which carries 25 marks. Part B consists of 5 Units. Answer any one full question from each unit.
Part A
Answer all questions:
Find the Laplace transform of the function f(t) = t^2.
Answer
- The Laplace transform is defined as L{f(t)} = ∫[0 to ∞] e^(-st) f(t) dt.
- For f(t) = t^n, the formula is L{t^n} = n! / s^(n+1).
- Substituting n = 2, we get L{t^2} = 2! / s^(2+1) = 2 / s^3.
Find Laplace transform of 4 sin t - 3.
Answer
- Using the linearity property: L{4 sin t - 3} = 4 L{sin t} - 3 L{1}.
- Given L{sin at} = a / (s^2 + a^2) and L{1} = 1 / s.
- Therefore, L{4 sin t - 3} = 4(1 / (s^2 + 1)) - 3/s = (4s - 3(s^2 + 1)) / (s(s^2 + 1)) = (4s - 3s^2 - 3) / (s(s^2 + 1)).
Show that Γ(n) = 2 ∫[0 to ∞] e^(-x^2) x^(2n-1) dx.
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Show that β(p, q) = β(p+1, q) + β(p, q+1).
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Find the area bounded by the curves y = x, y = x^2.
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Evaluate ∫[0 to 1] ∫[0 to 1] x^2 y^2 dxdy.
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If φ = x^2 y^2 z^2, then find Grad φ.
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Find a unit normal vector to the surface x^2 + y^2 + 2z^2 = 26 at the point (2,2,3).
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Find curl F when F = 3x^2 i + (2xz - y) j + zk.
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Is the work done by a force in moving a particle from one point to another point in an irrotational field independent of the path of integration? Justify the answer.
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Part B - Unit I
Use Laplace transforms, solve y''(t) + 5y'(t) + 6y(t) = t, y(0)=1, y'(0)=1.
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Solve by using Laplace transforms y'' + 4y' + 3y = e^(-t) with y(0) = y'(0) = 1.
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Part B - Unit II
Prove that ∫[0 to 1] (x^2 / sqrt(1-x^4)) dx * ∫[0 to 1] (dx / sqrt(1+x^4)) = π / (4√2) using β – Γ functions.
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Prove the following:
Prove that Γ(1/2) = √π.
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Prove that β(m, n) = (Γ(m)Γ(n)) / Γ(m+n).
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Part B - Unit III
The plane x/a + y/b + z/c = 1 meets the axes in A, B and C. Find the volume of the tetrahedron OABC.
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Evaluate ∫[0 to 1] ∫[0 to 1-x] ∫[0 to 1-x-y] xyz dz dy dx.
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Part B - Unit IV
Prove that if r is the position vector of any point in space then r^n r is irrotational and is solenoidal if n = -3.
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Answer the following:
Evaluate ∇ ∙ (r ∇(1/r^3)) where r = sqrt(x^2 + y^2 + z^2).
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If R = xi + yj + zk, then find ∇.R and ∇×R.
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Part B - Unit V
Verify Stoke’s theorem for the vector field F = (x^2 - y^2) i + 2xy j integrated round the rectangle in the plane z = 0 and bounded by the lines x = 0, y = 0, x = a, y = b.
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Verify divergence theorem for 2x^2yi - y^2 j + 4xz^2k taken over the region of first octant of the cylinder y^2 + z^2 = 9 and x = 2.
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