JNTU
JNTU B.Tech II MATHEMATICS - II Previous Question Paper (May - 2019 - R15)
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
(25 Marks)
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 along the closed boundary curve C of that surface. Mathematically, it is expressed as:
where C is the boundary curve of the surface S, oriented counter-clockwise when viewed from the side of the surface normal.
Find the work done by a force y i + x j which displaces a particle from origin to a point (0, -1).
Answer
The work done W by a force field F = yi + xj 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 integral is independent of the path taken. By the Fundamental Theorem of Line Integrals:
Substituting the coordinates:
Therefore, 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
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Find the finite Fourier cosine transform of f(x) = 2x, 0 < x < 4.
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Evaluate Δ(e^ax log bx).
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Find Δ^2 y_4 given that y(2)=5, y(3)=8 and y(4)=12.
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State the theorem for the convergence of a root by the iteration method.
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Establish the iteration formula to find the square root of a number by Newton Raphson method.
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Write the simpson’s 1/3rd rule formula to evaluate the integral of y dx from a to b.
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Write finite difference formula for y''.
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PART-B
(50 Marks)
If f = (x^2 + y^2 + z^2)^-n, find div grad f and determine n if div grad f = 0.
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Verify Green’s theorem for (xy + y^2)dx + x^2 dy where C is the curve made up of the line y = x and the parabola y = x^2.
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Show that f = (6xy + z^3)i + (3x^2 - z)j + (3xz^2 - y)k is irrotational. Find φ such that f = ∇φ.
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By using divergence theorem, evaluate the surface integral of f.n dS, where f = 4xi - 2y^2 j + z^2 k and S is the surface enclosing the region x^2 + y^2 ≤ 4, 0 ≤ z ≤ 3.
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Find the Fourier series for the function f(x) = e^x, from x = 0 to x = 2π.
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Find the Fourier transform of f(x) = 1 for x ≥ a and 0 for x < a.
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Obtain the Fourier series for f(x) = x - x^2 in (-π, π). Hence show that summation n=1 to ∞ (-1)^(n-1)/n^2 = π^2/12.
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Find the Fourier cosine transform of e^-ax.
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i) Prove Δ∇ = Δ - ∇ ii) Show that μδ = 1/2(Δ + ∇).
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Find the Lagrange’s interpolation polynomial for the following data and find the value of y(t) when t = 2.
| t | 1.3 | 1.7 | 1.9 | 2.7 |
|---|---|---|---|---|
| y(t) | 1.44 | 1.65 | 2.56 | 3.76 |
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Find y(25), given y(20)=24, y(24)=32, y(28)=35, y(32)=40, using Gauss forward difference formula.
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Fit a second degree curve by the method of least squares: (1, 2.11), (2, 2.78), (4, 3.12), (6, 4.00), (8, 4.83), (9, 5.01).
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Use bisection method to find a real root of x^3 - 2 log10(x) - 3 = 0.
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Using Crout’s method, solve: 4x1 + x2 + x3 = 4; x1 + 4x2 - 2x3 = 4; 3x1 + 2x2 - 3x3 = 6.
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Use Newton-Raphson method to find a real root of x^2 - 2 cos(x) - 1 = 0.
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Solve using Jacobi’s method: 4x1 + x2 + x3 = 2; x1 + 5x2 + 2x3 = -6; x1 + 2x2 + 3x3 = -4, with initial x0 = (0.5, -0.5, -0.5)^T.
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Find the largest eigen value and the corresponding eigen vector of the matrix A = [[15, -4, 3], [-10, 12, 6], [-20, 4, 2]] using the power method.
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Solve dy/dx = 1 + xy^2, y(0) = 1 by Taylor series method and find y(0.2) and y(0.4).
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