JNTU
JNTU B.Tech II MATHEMATICS - II Previous Question Paper (May/June - 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
Find ๐๐ข๐๐ ๐ where ๐ = ๐ฅ๐ฆ๐ + ๐ฆ๐ง๐ + ๐ง๐ฅ๐.
Answer
The curl of a vector field f = F1i + F2j + F3k is given by the determinant of the operator matrix. For f = xy i + yz j + zx k:
Find the circulation of an irrotational vector function over the rectangle ๐ฅ = ยฑ๐, ๐ฆ = ยฑ๐.
Answer
By definition, an irrotational vector function satisfies โ ร f = 0. The circulation of a vector field around a closed curve C is given by the line integral โฎC f ยท dr. By Stokes' theorem, this is equal to the surface integral of the curl over the enclosed area S:
Since the curl is zero, the circulation is 0.
Find the leading term in the Fourier series expansion of ๐(๐ฅ) = ๐^๐ฅ in (โ๐, ๐).
Answer
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State Fourier integral theorem.
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Write the normal equations for a quadratic curve.
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Prove that ฮ2๐ฅ(๐) = ๐(๐-1)๐ฅ(๐-2), ๐ is a positive integer and โ = 1.
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What is Interpolation?
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Under what conditions Gauss-Seidal method is applicable.
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Write the iteration formulae of modified Eulerโs method.
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If ๐๐ฆ/๐๐ฅ = ๐ฅ + ๐ฆ and ๐ฆ(0), โ = 0.1 then find ๐พ1 and ๐พ2 in Runge Kutta fourth order formula.
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Part B
If ๐ = 3๐ฅ2๐ฆ๐ง2๐ + ๐ฅ2๐ง2๐ + 2๐ฅ3๐ฆ๐ง๐. Show that โซ ๐ โ ๐๐ is independent of the path of integration. Hence evaluate the integral when ๐ถ any path joining is (0, 0, 0) to (1, 2, 3).
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Prove that if ๐ is the position vector of any point in space then ๐๐๐ is irrotational and is solenodial if ๐ = -3.
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State and prove shifting property of Fourier transform.
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Find the inverse Fourier sine transform ๐(๐ฅ) of ๐น๐ {๐} = (๐-๐๐)/๐ and hence deduce ๐น๐ -1{1/๐}.
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Find the Fourier series expansion of the function ๐(๐ฅ) = { sin ๐ฅ, -๐ โค ๐ฅ โค 0; 0, 0 โค ๐ฅ โค ๐ } and deduce 1/(1.3) - 1/(3.5) + 1/(5.7) + โฏ = (๐-2)/4.
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Fit a curve of the form ๐ฆ = ๐/๐ฅ + ๐๐ฅ to the data given below:
๐ฅ: 1.0, 1.2, 1.4, 1.6, 1.8
๐ฆ: 0, 0.1280, 0.5440, 1.2960, 2.4320
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Given ๐ข0 + ๐ข8 = 1.924; ๐ข1 + ๐ข7 = 1.959; ๐ข2 + ๐ข6 = 1.982 and ๐ข3 + ๐ข5 = 1.996 find ๐ข4, stating the assumption made.
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Given sin 45ยฐ = 0.7071, sin 50ยฐ = 0.7660, sin 55ยฐ = 0.8192 and sin 60ยฐ = 0.8660. Find sin 52ยฐ using Newtonโs interpolation formula.
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By using Regula Falsi method, find a positive root of the equation tan ๐ฅ + tanh ๐ฅ = 0.
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Solve the system of equations by Gauss seidal method:
4๐ฅ + 2๐ฆ + ๐ง = 10, ๐ฅ + 3๐ฆ + ๐ง = 7, ๐ฅ + 2๐ฆ + 5๐ง = 8.
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Find the successive approximate solution of the differential equation ๐ฆ' = ๐ฆ, ๐ฆ(0) = 1 by Picardโs method and compare it with exact solution.
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Using the shooting method, solve the first boundary value problem: ๐ข'' = ๐ข + 1, 0 < ๐ฅ < 1; ๐ข(0) = 0; ๐ข(1) = ๐ - 1.
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