JNTU
JNTU B.Tech I Year MATHEMATICS-II Previous Question Paper (December - 2017 - R16)
Note: This question paper contains two parts A and B. Part A is compulsory which carries 25 marks. Answer all questions in Part A. Part B consists of 5 Units. Answer any one full question from each unit. Each question carries 10 marks and may have a, b as sub questions.
Part A
Find the Laplace transform of cosh3(2t).
Answer
Using the identity cosh(3x) = 4cosh3(x) - 3cosh(x), we can express cosh3(2t) as:
Applying the linearity property of the Laplace transform and the standard formula L{cosh(at)} = s / (s2 - a2), we get:
Find the Laplace transform of e-t(2cos 5t - 3sin 5t).
Answer
Using the linearity property of the Laplace transform, we have:
Applying the first shifting theorem, which states that L{eatf(t)} = F(s - a), where F(s) = L{f(t)}:
For f(t) = cos 5t, F(s) = s / (s2 + 25). Thus, L{e-tcos 5t} = (s + 1) / ((s + 1)2 + 25).
For f(t) = sin 5t, F(s) = 5 / (s2 + 25). Thus, L{e-tsin 5t} = 5 / ((s + 1)2 + 25).
Substituting these back into the expression:
Evaluate the improper integral
Answer
Sign in to view the complete answer paper.
Evaluate the improper integral
Answer
Sign in to view the complete answer paper.
Find the area bounded by the curves y2 = 4ax and x2 = 4ay using double integration.
Answer
Sign in to view the complete answer paper.
Change the order of the integration
Answer
Sign in to view the complete answer paper.
Find ∇φ, when φ = 3x2y - y3z2 at the point (1, -2, -1).
Answer
Sign in to view the complete answer paper.
Find the directional derivative of the function f(x, y, z) = xy2 + yz2 at the point (1, -1, 3) in the direction of the vector i + 2j + 2k.
Answer
Sign in to view the complete answer paper.
If R = (t-1)i + t2j + (t-3)k and S = 2t2i + 6tk, evaluate
Answer
Sign in to view the complete answer paper.
Evaluate the line integral
Answer
Sign in to view the complete answer paper.
Part B
Find the Laplace transform of (sin t)/t. Hence find L(cos t)/t.
Answer
Sign in to view the complete answer paper.
Prove that
Answer
Sign in to view the complete answer paper.
Find the inverse Laplace transform of ln((s+1)/(s-1)).
Answer
Sign in to view the complete answer paper.
Find the inverse Laplace transform of 1/(s2(s2+a2)) using the convolution theorem.
Answer
Sign in to view the complete answer paper.
Prove that
Answer
Sign in to view the complete answer paper.
Evaluate
Answer
Sign in to view the complete answer paper.
Prove that
Answer
Sign in to view the complete answer paper.
Evaluate
Answer
Sign in to view the complete answer paper.
Evaluate the integral
Answer
Sign in to view the complete answer paper.
Find by triple integration, the volume of the paraboloid of revolution x2 + y2 = 4z cut off by the plane z = 4.
Answer
Sign in to view the complete answer paper.
Prove the following vector identities: a) ∇(φ1φ2) = φ1∇φ2 + φ2∇φ1 b) ∇×(∇φ) = 0.
Answer
Sign in to view the complete answer paper.
If R = xi + yj + zk, show that: a) ∇R = 3 b) ∇(1/r) = -R/r3 c) ∇(rn) = nrn-2R d) ∇×(a×R) = 2a, where a is a constant vector and r = |R|.
Answer
Sign in to view the complete answer paper.
State the Stokes' theorem. Verify it for the vector field F = (x2-y2)i - 2yzj + yzk over the upper half surface of x2 + y2 + z2 = 1, bounded by its projection on the xy-plane.
Answer
Sign in to view the complete answer paper.
State the Green's theorem in a plane. Verify it for ∮C (ex sin y) dx + (ex cos y) dy where C is the rectangle with the vertices (0,0), (π, 0), (π, π/2) and (0, π/2).
Answer
Sign in to view the complete answer paper.