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Download VTU BE 2020 Jan Question Paper 18 Scheme 18MAT21 Advanced Calculus and Numerical Methods First And Second Semester

Download Visvesvaraya Technological University (VTU) BE/B.Tech First And Second Semester (1st sem and 2nd sem) 2019-2020 Jan ( Bachelor of Engineering) 18 Scheme 18MAT21 Advanced Calculus and Numerical Methods Previous Question Paper

This post was last modified on 02 March 2020

VTU B.Tech 1st Year Last 10 Years 2011-2021 Question Papers


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Second Semester B.E. Degree Examination, Dec.2019/Jan.2020

Advanced Calculus and Numerical Methods

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Time: 3 hrs.

Max. Marks: 100

Note: Answer any FIVE full questions, choosing ONE full question from each module.

Module-1

    1. Find the directional derivative of Ø= 4xz³ – 3x²y²z at (2, -1, 2) along 2i - 3j + 6k . (07 Marks)
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    3. If f=?(x²y +7y³-3xyz) find div f and curl f (07 Marks)
    4. Find the constants a and b such that F = (axy + z ³ )i + (3x² — z)j+ (bx - y)k is irrotational. Also find a scalar potential Ø if F = ?Ø. (07 Marks)

OR

    1. If F = xyi + yz j + zxk evaluate ?c F.dr where C is the curve represented by x = t,. y = t ², z=t ³. (06 Marks)
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    3. Using Stoke's theorem Evaluate ?c F.dr if F = (x² +y² )i – 2xy j taken round the rectangle bounded by x = 0, x = a, y = 0, y = b (07 Marks)
    4. Using divergence theorem, evaluate ? F . n ds where F = 4xi – 2y²j + z²k taken around 0 < x < 1, 0 < y < 1, 0 < z < 1. (07 Marks)

Module-2

    1. Solve (4D4 -8D³ -7D² +I1D + 6)y = 0 (06 Marks)
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    3. Solve (D² + 4D +3)y = e-x (07 Marks)
    4. Using the method of variation of parameter solve y" + 4y = tan2x. (07 Marks)

OR

    1. Solve (D² - 1)y = 3 cos2x (06 Marks)
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    3. Solve x²y" + 5xy' + 8y= 2 logx (07 Marks)
    4. The differential equation of a simple pendulum is d²x/dt² + ?²x = F0 Sinnt, where ?0 and F0 are constants. Also initially x = 0, dx/dt = 0 solve it. (07 Marks)

Module-3

    1. Find the PDE by eliminating the function from z = y f(x) + 2f(x) + log y (06 Marks)
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    3. Solve ?²z/?x?y = sin x sin y given ?z/?x = -2 siny, when x = 0 and z = 0, when y is odd multiple of p/2 (07 Marks)
    4. Derive one-dimensional wave equation. (07 Marks)

OR

    1. Solve ?²z/?x² = a²z given that when x = 0, z = a sin y and ?z/?x = 0. (06 Marks)
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    3. Solve x(y – z) p + y (z – x) q = z (x-y). (07 Marks)
    4. Find all possible solution of U t = C² Uxx, one dimensional heat equation by variable separable method. (07 Marks)

Module-4

    1. Test for convergence for 2/2³ + 3/3² + 4/4³ + ... (06 Marks)
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    3. Find the series solution of Legendre differential equation (1 – x²)y" - 2xy' + n(n + 1) = 0 leading to P?(x). (07 Marks)
    4. Prove the orthogonality property of Bessel's function as ?0¹ x J? (ax) J ? (ßx)dx = 0 , a ? ß (07 Marks)

OR

    1. Test for convergence for ?0¹ x-1 (1 - x²)-1/2 dx (06 Marks)
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    3. Find the series solution of Bessel differential equation x ²y" + xy' + (x² – n²) y = 0 Leading to J?(x) (07 Marks)
    4. Express the polynomial x³ + 2x² - 4x + 1 in terms of Legendre polynomials. (07 Marks)

Modue-5

    1. Using Newton's forward difference formula, find f(43) from the data:
      X405060708090
      f(x)184204226250276304
      (06 Marks)
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    3. Find the real root of the equation x log10 x =1.2 by Regula falsi method between 2 and 3 (Three iterations). (07 Marks)
    4. Evaluate ?45.2 log x dx by Weddle's rule considering six intervals. (07 Marks)

OR

    1. Find f(9) from the data by Newton's divided difference formula:
      X57111317
      f(x)150392145223665202
      (06 Marks)
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    3. Using Newton – Raphson method, find the real root of the equation x sin x + cosx = 0 near X = p (07 Marks)
    4. By using Simpson's 1/3 rule, evaluate ?06 dx/(1+x²) by considering seven ordinates. (07 Marks)

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