JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD
JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD B.Tech II MATHEMATICS-II Previous Question Paper (May - 2019 - R13)
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, c as sub questions.
PART-A
State Stokes theorem.
Answer
Stokes' theorem states that the surface integral of the curl of a vector field F over an open surface S is equal to the line integral of the vector field F around the closed boundary curve C of the surface S. Mathematically, it is expressed as:
where C is the boundary curve of the surface S, traversed in the positive direction relative to the surface normal.
Find the work done by a force yi + xj which displaces a particle from origin to a point (0, -1).
Answer
The work done W by a force field F = (y, x) along a path C is given by the line integral:
Since the differential form y dx + x dy is the exact differential of the function f(x, y) = xy, the force field is conservative. The work done is independent of the path and depends only on the endpoints (0, 0) and (0, -1):
Thus, the work done by the force is 0 units.
Write down the half range sine series for the function f(x) = e-x in (0, π).
Answer
Sign in to view the complete answer paper.
Find the finite Fourier cosine transform of f(x) = 2x, 0 < x < 4.
Answer
Sign in to view the complete answer paper.
Answer
Sign in to view the complete answer paper.
Find Δ2y4 given that y(2)=5, y(3)=8 and y(4)=12.
Answer
Sign in to view the complete answer paper.
State the theorem for the convergence of a root by the iteration method.
Answer
Sign in to view the complete answer paper.
Establish the iteration formula to find the square root of a number by Newton Raphson method.
Answer
Sign in to view the complete answer paper.
Write the Simpson's 1/3rd rule formula to evaluate a∫b y dx.
Answer
Sign in to view the complete answer paper.
Write finite difference formula for y''.
Answer
Sign in to view the complete answer paper.
PART-B
If f = (x2 + y2 + z2)-n, find div grad f and determine n if div grad f = 0.
Answer
Sign in to view the complete answer paper.
Verify Green’s theorem for (xy + y2) dx + x2dy where C is the curve made up of the line y = x and the parabola y = x2.
Answer
Sign in to view the complete answer paper.
Show that f = (6xy + z3)i + (3x2 - z)j + (3xz2 - y)k is irrotational. Find φ such that f = ∇φ.
Answer
Sign in to view the complete answer paper.
By using divergence theorem, evaluate ∫∫S f ⋅ n dS, where f = 4xi - 2y2j + z2k and S is the surface enclosing the region for which x2 + y2 ≤ 4, 0 ≤ z ≤ 3.
Answer
Sign in to view the complete answer paper.
Find the Fourier series for the function f(x) = ex, from x = 0 to x = 2π.
Answer
Sign in to view the complete answer paper.
Find the Fourier transform of f(x) = 0 for x < a, 1 for x ≥ a.
Answer
Sign in to view the complete answer paper.
Obtain the Fourier series for the function f(x) = x - x2 in the interval (-π, π). Hence show that ∑n=1∞ [(-1)n-1/n2] = π2/12.
Answer
Sign in to view the complete answer paper.
Find the Fourier cosine transform of eax.
Answer
Sign in to view the complete answer paper.
i) Prove Δ∇ = Δ - ∇ ii) Show that μδ = 1/2(Δ + ∇).
Answer
Sign in to view the complete answer paper.
Find the Lagrange’s interpolation polynomial for the following data and find the value of y(t) when t = 2.
Answer
Sign in to view the complete answer paper.
Find y(25), given that y(20)=24, y(24)=32, y(28)=35, y(32)=40, using Gauss forward difference formula.
Answer
Sign in to view the complete answer paper.
Fit a second degree curve to the following data by the method of least squares.
Answer
Sign in to view the complete answer paper.
Use bisection method to find a real root of the equation f(x) = x3 - 2log10x - 3 = 0.
Answer
Sign in to view the complete answer paper.
Using Crout’s method, solve the system of equations: 4x1 + x2 + x3 = 4; x1 + 4x2 - 2x3 = 4; 3x1 + 2x2 - 3x3 = 6.
Answer
Sign in to view the complete answer paper.
Use Newton-Raphson method to find a real root of the equation f(x) = x2 - 2 cos x - 1 = 0.
Answer
Sign in to view the complete answer paper.
Solve the system of equations using Jacobi’s method: 4x1 + x2 + x3 = 2; x1 + 5x2 + 2x3 = -6; x1 + 2x2 + 3x3 = -4, with the initial approximation x0 = (0.5, -0.5, -0.5)T.
Answer
Sign in to view the complete answer paper.
Find the largest eigen value and the corresponding eigen vector of the matrix A using the power method.
Answer
Sign in to view the complete answer paper.
Solve the differential equation dy/dx = 1 + xy2, subject to y(0) = 1 by Taylor series method and hence find y(0.2) and y(0.4).
Answer
Sign in to view the complete answer paper.