Engineering Mathematics-II PYQ 2022-23 AKTU Question Paper

AKTU · BTECH · Semester 2 · Engineering Mathematics-II · Session 2022-23 · PYQ

AKTU Engineering Mathematics-II previous year question paper 2022-23 for B.Tech Semester 2. Covers Ordinary Differential Equation of Higher Order, Laplace…

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Engineering Mathematics-II AKTU syllabus

  1. Unit 1: Ordinary Differential Equation of Higher Order
  2. Unit 2: Laplace Transform
  3. Unit 3: Sequence and Series
  4. Unit 4: Complex Variable – Differentiation
  5. Unit 5: Complex Variable – Integration

Questions in Engineering Mathematics-II AKTU PYQ 2022-23

  1. Q1a. Solve: (D³ + 2D² − 3D)y = eˣ, D = d/dx (2 marks, 2022-23)
  2. Q1b. Explain the first shifting property of the Laplace transform with example. (2 marks, 2022-23)
  3. Q1c. Discuss the convergence of sequence {uₙ}, where uₙ = sin(1/n). (2 marks, 2022-23)
  4. Q1d. Show that the function f(z) = |z|² is not analytic at origin. (2 marks, 2022-23)
  5. Q1e. Classify the singularity of f(z) = e^(1/z) / z. (2 marks, 2022-23)
  6. Q1f. Find the inverse Laplace transform of F(s) = 1 / (s² + 2s + 2). (2 marks, 2022-23)
  7. Q1g. Find the invariant points of the transformation w = (2z + 6) / (z + 7). (2 marks, 2022-23)
  8. Q2a. Solve the following differential equation: x² d²y/dx² + 2x dy/dx − 12y = x³ log x. (7 marks, 2022-23)
  9. Q2b. Find the Laplace transform of the function f(x) = x³ sin x. Hence, prove that ∫₀^∞ e^(−x) x³ sin x dx = 0. (7 marks, 2022-23)
  10. Q2c. Test the convergence of following series: 1/(1.2.3) + x/(4.5.6) + x²/(7.8.9) + ..., Where x is a real number. (7 marks, 2022-23)
  11. Q2d. Show that the function f(z) defined by f(z) = x³y⁵(x + iy) / (x⁶ + y¹⁰), z ≠ 0, f(0) = 0 is not analytic at the origin even though it satisfies Cauchy-Riemann equations at the origin. (7 marks, 2022-23)
  12. Q2e. Using Cauchy-integral formula, evaluate ∮_C sin 2z / [(z + 3)(z + 1)²] dz, where C is a rectangle with vertices at 3 ± i, −2 ± i. (7 marks, 2022-23)
  13. Q3a. Solve the following differential equation by the variation of parameters: d²y/dx² + y = cosec x. (7 marks, 2022-23)
  14. Q3b. Solve the differential equation by the changing the independent variable: x d²y/dx² − dy/dx − 4x³y = 8x³ sin x². (7 marks, 2022-23)
  15. Q4a. State convolution theorem of the Laplace transforms. Hence, find inverse Laplace transform of 1 / [s²(s + 1)²]. (7 marks, 2022-23)
  16. Q4b. Using Laplace transform, solve the following differential equation: d²y/dx² + y = 6 cos 2x, y(0) = 3 & y'(0) = 1. (7 marks, 2022-23)
  17. Q5a. Find a Fourier series to represent f(x) = x − x², −π ≤ x ≤ π. Hence, show that 1/1² − 1/2² + 1/3² − 1/4² + ... = π²/12. (7 marks, 2022-23)
  18. Q5b. Find the half range cosine series for the function f(x) = (x − 1)² in the interval (0, 1). Hence, prove that 1/1² + 1/3² + 1/5² + 1/7² + ... = π²/8. (7 marks, 2022-23)
  19. Q6a. Determine an analytic function f(z) = u + iv in terms of z whose real part u(x, y) is eˣ(x cos y − y sin y) and f(1) = e. (7 marks, 2022-23)
  20. Q6b. Find the bilinear transformation which maps the points z = 0, −1, i onto w = i, 0, ∞. Also, find the image of the unit circle |z| = 1. (7 marks, 2022-23)
  21. Q7a. Expand f(z) = (7z − 2) / (z³ − z² − 2z) in the following regions: (i) 0 < |z| < 1 (ii) 1 < |z| < 2 (iii) |z| > 2. (7 marks, 2022-23)
  22. Q7b. Using contour integration, evaluate the real integral ∫₀^π a dθ / (a² + sin²θ), a > 0. (7 marks, 2022-23)

AKTU paper codes: BAS203, KAS203

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