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

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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. 1. Find the directional derivative of ∅= 4xz³ – 3x²y²z at (2, -1, 2) along 2i - 3j + 6k . (07 Marks)

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   2. If f=∇(x²y +7y³-3xyz) find div f and curl f (07 Marks)
   3. 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 ∅ if F = ∇∅. (07 Marks)

OR

2. 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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   2. 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)
   3. 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

3. 1. Solve (4D⁴ -8D³ -7D² +I1D + 6)y = 0 (06 Marks)

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   2. Solve (D² + 4D +3)y = e^-x (07 Marks)
   3. Using the method of variation of parameter solve y" + 4y = tan2x. (07 Marks)

OR

4. 1. Solve (D² - 1)y = 3 cos2x (06 Marks)

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   2. Solve x²y" + 5xy' + 8y= 2 logx (07 Marks)
   3. The differential equation of a simple pendulum is d²x/dt² + ω²x = F₀ Sinnt, where ω₀ and F₀ are constants. Also initially x = 0, dx/dt = 0 solve it. (07 Marks)

Module-3

5. 1. Find the PDE by eliminating the function from z = y f(x) + 2f(x) + log y (06 Marks)

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   2. 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 π/2 (07 Marks)
   3. Derive one-dimensional wave equation. (07 Marks)

OR

6. 1. Solve ∂²z/∂x² = a²z given that when x = 0, z = a sin y and ∂z/∂x = 0. (06 Marks)

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   2. Solve x(y – z) p + y (z – x) q = z (x−y). (07 Marks)
   3. Find all possible solution of U t = C² Uxx, one dimensional heat equation by variable separable method. (07 Marks)

Module-4

7. 1. Test for convergence for 2/2³ + 3/3² + 4/4³ + ... (06 Marks)

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   2. Find the series solution of Legendre differential equation (1 – x²)y" - 2xy' + n(n + 1) = 0 leading to Pₙ(x). (07 Marks)
   3. Prove the orthogonality property of Bessel's function as ∫₀¹ x Jₙ (αx) J ₙ (βx)dx = 0 , α ≠ β (07 Marks)

OR

8. 1. Test for convergence for ∫₀¹ x^-1 (1 - x²)^-1/2 dx (06 Marks)

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   2. Find the series solution of Bessel differential equation x ²y" + xy' + (x² – n²) y = 0 Leading to Jₙ(x) (07 Marks)
   3. Express the polynomial x³ + 2x² - 4x + 1 in terms of Legendre polynomials. (07 Marks)

Modue-5

9. 1. Using Newton's forward difference formula, find f(43) from the data:

      X	40	50	60	70	80	90
      f(x)	184	204	226	250	276	304

      (06 Marks)

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   2. Find the real root of the equation x log₁₀ x =1.2 by Regula falsi method between 2 and 3 (Three iterations). (07 Marks)
   3. Evaluate ∫₄^5.2 log x dx by Weddle's rule considering six intervals. (07 Marks)

OR

10. 1. Find f(9) from the data by Newton's divided difference formula:

       X	5	7	11	13	17
       f(x)	150	392	1452	2366	5202

       (06 Marks)

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    2. Using Newton – Raphson method, find the real root of the equation x sin x + cosx = 0 near X = π (07 Marks)
    3. By using Simpson's 1/3 rule, evaluate ∫₀⁶ dx/(1+x²) by considering seven ordinates. (07 Marks)

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