Download VTU B-Tech/B.E 2019 June-July 1st And 2nd Semester 18 Scheme 18MAT21 Advanced Calculus and Numerical Methods Question Paper

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Second Semester B.E. Degree Examination, June/July 2019

Advanced Calculus and Numerical Methods

Time: 3 hrs.

Max. Marks: 100

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

Module-1

1. 1. If F = ∇(x³ + y² + z³ – 3xyz), find div F and curl f. (06 Marks)

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   2. Find the angle between the surfaces x² + y² + z² = 9 and z = x² + y² – 3 at the point (2, -1, 2). (07 Marks)
   3. Find the value of a, b, c such that f= (axy + bz³ )i + (3x² - cz)j + (3)(z² - y)k is irrotational, also find the scalar potential (Φ) such that F =∇Φ). (07 Marks)

OR

1. 1. Find the total work done in moving a particle in the force field F = 3xyi – 5zj + 10xk along the curve x = t²+1, y = 2t², z = t³ from t = 1 to t = 2. (06 Marks)

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   2. Using Green's theorem, evaluate ∫(xy + y²)dx + x²dy, where C is bounded by y = x and y = x². (07 Marks)
   3. Using Divergence theorem, evaluate ∫∫ F ds, where F = (x² – yz)i + (y² – zx)j + (z² – xy)k taken over the rectangular parallelepiped 0 ≤ x ≤ a, 0 ≤ y ≤ b, 0 ≤ z ≤ c. (07 Marks)

Module-2

1. 1. Solve (D² - 3D + 2)y = 2x e^x sin 2x. (06 Marks)
   2. Solve (D²+1)y = sec x by the method of variation of parameter. (07 Marks)

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   3. Solve x²y" - 4xy' + y = cos(2 logx) (07 Marks)

OR

1. 1. Solve (D² – 4D + 4)y = e^2x + sin x. (06 Marks)
   2. Solve (x+1)²y" + (x+1)y' + y = 2sin[log (x+1)] (07 Marks)

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   3. The current i and the charge q in a series containing an inductance L, capacitance C, emf E, satisfy the differential equation L (d²q/dt²) + q/C = E, Express q and i interms of 't' given that L, C, E are constants and the value of i and q are both zero initially. (07 Marks)

Module-3

1. 1. Form the partial differential equation by elimination of arbitrary function from Φ(x + y + z, x² + y² + z²) = 0 (06 Marks)
   2. Solve ∂²z/∂x∂y = cos(2x + 3y) (07 Marks)
   3. Derive one dimensional heat equation in the standard form as ∂u/∂t = c² (∂²u/∂x²) (07 Marks)

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OR

1. 1. Solve (∂z/∂x) + z = 0 such that z = e^y where x = 0 and (∂z/∂x) + (nx - e^y) = e^y when x = 0. (06 Marks)
   2. Solve (mz— ny)(∂u/∂x) + (nx - lz)(∂u/∂y) + (ly - mx)(∂u/∂z) = 0 (07 Marks)
   3. Find all possible solutions of one dimensional wave equation (∂²u/∂t²) = c² (∂²u/∂x²) using the method of separation of variables. (07 Marks)

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Module-4

1. 1. Discuss the nature of the series Σ [n/(n+1)] * xⁿ from n=1 to infinity. (06 Marks)
   2. With usual notation prove that J_1/2(x) = √(2/πx) sin x (07 Marks)
   3. If x³ + 2x² - x + 1 = aP₃ + bP₂ + cP₁ + dP₀, find a, b, c and d using Legendre's polynomial. (07 Marks)

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OR

1. 1. Discuss the nature of the series x + (1/2²)*x² + (1.3)/(2².4²)*x³ + (1.3.5)/(2².4².6²)*x⁴ + ... (06 Marks)
   2. Obtain the series solution of Legendre's differential equation (1-x²)y" - 2xy' + n(n+1)y = 0 in terms of P_n(x) (07 Marks)
   3. Express x⁴ - 3x² + x interms of Legendre's polynomial. (07 Marks)

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Module-5

1. 1. Find the real root of the xsinx + cosx = 0 near x = π using Newton-Raphson method. Carry out 3 iterations. (06 Marks)
   2. From the following data, find the number of students who have obtained (i) less than 45 marks (ii) between 40 and 45 marks.
      

      Marks	30-40	40-50	50-60	60-70	70-80
      No. of Students	31	42	51	35	31

      (07 Marks)
   3. Evaluate ∫₀⁶ dx / (1 + x²) using Simpson's 1/3 rule by taking 7 ordinates. (07 Marks)

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OR

1. 1. Find the real root of the equation x log₁₀ x = 1.2 which lies between 2 and 3 using Regula-Falsi method. (06 Marks)
   2. Using Lagrange's interpolation formula, find y(2) from the given data:
      

      X	0	1	2	5
      y	2	3	12	147

      (07 Marks)
   3. Evaluate ∫₄^5.2 log_e x dx using Weddle's rule by taking six equal parts. (07 Marks)

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