JNTU
JNTU B.Tech I MATHEMATICS-II Previous Question Paper (December 2019/January 2020 - R16)
Note: This question paper contains two parts A and B. Part A is compulsory (25 marks). Part B consists of 5 Units. Answer any one full question from each unit.
Part A
Find L[t^2 e^t]
Answer
Using the First Shifting Theorem, which states that L[eatf(t)] = F(s-a) where F(s) = L[f(t)], we first find the Laplace transform of f(t) = t2.
For n = 2, L[t2] = 2! / s3 = 2 / s3. Applying the shifting theorem with a = 1:
Find inverse Laplace of 1 / (s^2 + 2s + 2)
Answer
To find the inverse Laplace transform of the given function, first complete the square in the denominator:
The expression becomes:
Using the shifting property L^-1 { F(s + a) } = e^(-at) f(t), where f(t) = L^-1 { 1 / (s^2 + 1^2) } = sin(t), we apply the shift with a = 1:
Evaluate the integral of sin^3(x) cos^4(x) from 0 to pi/2 using Beta-Gamma functions.
Answer
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Show that Gamma(1/2) = sqrt(pi), using the relation beta(m,n) = (Gamma(m)Gamma(n)) / Gamma(m+n).
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Evaluate the triple integral of dxdydz from 0 to 1.
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Evaluate the integral of dydx from 0 to x and 0 to x^2.
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Find the directional derivative of x^2 + y^2 + z^2 = r^2 at (0, r, 0) in the direction of j.
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Find Curl(grad f) where f = x^2 + y^2 + z^2.
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State Green’s theorem in a plane.
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Find the work done by F = x^2 i + y^2 j + z^3 k in moving a particle from (-1, 2, 1) to (2, 3, 4).
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Part B
Find inverse Laplace transform of L^-1[log((s+1)/(s-1))]
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Find L[t^2 e^t sin(3t)]
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Using Laplace transform solve y'' + 3y' + 2y = 3, y(0)=1, y'(0)=1.
Answer
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Prove the relation between Beta and Gamma functions.
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If beta(m, n) = beta(m+1, n) + beta(m, n+1), find k+p in terms of m and n.
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Prove that Gamma(m) Gamma(m + 1/2) = (sqrt(pi) / 2^(2m-1)) Gamma(2m).
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Show the integral relation involving Beta functions.
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Evaluate the integral of e^-(x^2+y^2) dx dy by changing to polar coordinates.
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Evaluate the integral of (xy + yz + zx) dxdydz for the given bounds.
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Evaluate the integral of xy dydx.
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Evaluate the triple integral for the given region.
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Prove the vector identity for Div(A x B).
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Evaluate the angle between the normals of the surface xy = z^2 at given points.
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Find the unit normal vector and directional derivative.
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Prove that Div(Curl V) = 0.
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Verify Divergence Theorem for F = x^2 i + y^2 j + z^2 k over the surface S of the solid cut by x+y+z=a in the 1st octant.
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Find the work done by the force along the given curve.
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Apply Stokes Theorem to evaluate F.dR.
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