JNTU
JNTU B.Tech II MATHEMATICS-II Previous Question Paper (April/May - 2018 - R13)
Part A is compulsory. Part B consists of 5 Units. Answer any one full question from each unit.
Part A
Find the integrand of conversion of the line integral of y i + 2x j over a closed curve ‘C’ by the application of Green’s theorem.
Answer
Green's theorem states that the line integral around a closed curve C is equal to the double integral over the region R enclosed by C: ∫C (M dx + N dy) = ∫∫R (∂N/∂x - ∂M/∂y) dA. Given F = y i + 2x j, we have M = y and N = 2x. The integrand is ∂N/∂x - ∂M/∂y = ∂(2x)/∂x - ∂(y)/∂y = 2 - 1 = 1.
Prove that f = (x^2 + xy^2)i + (y^2 + x^2y)j is conservative and find the scalar potential.
Answer
A vector field F is conservative if curl F = 0. Here, F = (x2 + xy2)i + (y2 + x2y)j. Calculating curl F = (∂/∂x(y2 + x2y) - ∂/∂y(x2 + xy2))k = (2xy - 2xy)k = 0. Since curl F = 0, F is conservative. To find the scalar potential ϕ, we solve ∂ϕ/∂x = x2 + xy2 and ∂ϕ/∂y = y2 + x2y. Integrating the first gives ϕ = (x3/3) + (x2y2/2) + g(y). Differentiating with respect to y gives x2y + g'(y) = y2 + x2y, so g'(y) = y2, which means g(y) = y3/3. Thus, ϕ = (x3/3) + (x2y2/2) + (y3/3) + C.
Define a periodic function and state the period of sin x + cos 2x + (1/3)sin 3x.
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Find the Fourier transform of f(x) = 1/(2a) for |x| ≤ α and 0 for |x| > α.
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Discuss about errors in polynomial interpolation.
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Derive the normal equation for fitting a straight line y = a + bx.
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Gauss-Seidal method is similar in principle to Jacobi method. Then what is difference between them?
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Evaluate 1/12 by the fixed point iteration method.
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State merits and demerits of Runge-Kutta method.
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How Euler’s method is modified?
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Part B
Verify divergence theorem for 2x^2y i - y^2 j + 4xz^2 k taken over the region of first octant of the cylinder y^2 + z^2 = 9 and x = 2.
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Evaluate ∫(3x + 4y) dx + (2x - 3y) dy, where C is the circle x^2 + y^2 = 4.
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Prove that ∫(r x dr) = 2 ∫ dS.
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Find the Fourier series for f(x) = -1 for -1 < x < 0 and 0 for 0 < x < 1. Also find the sum of the series at x = -1.
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If f(x) = x for 0 < x < π/2 and f(x) = π/2 - x for π/2 < x < π, then show that:
a) f(x) = 4/π [sin x - (sin 3x)/3^2 + (sin 5x)/5^2 - ... ]
b) f(x) = π/4 - 2/π [ (cos 2x)/1^2 + (cos 6x)/3^2 + (cos 10x)/5^2 + ... ]
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Find the missing term in the following data: x=0,1,2,3,4; y=1,3,9,-,81. Why this value is not equal to 3^3. Explain?
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Fit a straight line to the following data by the method of least square: x=0,1,2,3,4; y=1, 1.8, 3.3, 4.5, 6.5.
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Using Lagrange formula express the function (x^2+6x-1) / ((x^2-1)(x-4)(x-6)) as a sum of partial fractions.
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Solve the system of equations: 2x + y + z – 2w = -10; 4x + 2z + w = 8; 3x + 2y + 2z = 7; x + 3y + 2z – w = -5 using LU decomposition method.
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Find a root of sin x = (x+1)/(x-1) whose initial approximate root is -5.5 by using Iterative method.
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Using modified Euler’s method, find an approximate value of y when x = 1.3, given that dy/dx + y/x = 1/x^2, y(1) = 1.
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Determine the largest Eigen value and the corresponding Eigen vector of the matrix [4 1 0; 1 20 1; 0 1 4] to 3 correct decimal places using the power method.
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