Engineering Mathematics-II PYQ 2024-25 AKTU Question Paper

AKTU · BTECH · Semester 2 · Engineering Mathematics-II · Session 2024-25 · PYQ

AKTU Engineering Mathematics-II previous year question paper 2024-25 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 2024-25

  1. Q1a. Find the general solution of the following differential equation: d³y/dx³ + dy/dx = 0 (2 marks, 2024-25)
  2. Q1b. Find the Particular Integral for the following differential equation: y'' − 8y' + 16y = e^(4x) (2 marks, 2024-25)
  3. Q1c. Find Laplace Transform of f(t) = sin 2t cos 3t. (2 marks, 2024-25)
  4. Q1d. Find inverse Laplace Transform of F(s) = (s−1)/(s²+3s+2). (2 marks, 2024-25)
  5. Q1e. Test the convergence of the following sequence: a_n = { 1 if n = 2^p for some p ∈ N; 1/n otherwise } (2 marks, 2024-25)
  6. Q1f. Show that the following function is harmonic: h(x, y) = x² + xy − y² (2 marks, 2024-25)
  7. Q1g. Find the residue at the simple pole of the following function: f(z) = 8z³ / [(z−1)(z+1)³] (2 marks, 2024-25)
  8. Q2a. Find the general solution of the differential equation y'' − 2y' + 2y = x + e^x cos x (7 marks, 2024-25)
  9. Q2b. Solve the following differential equations using Laplace Transform: d³y/dt³ + 2d²y/dt² − dy/dt − 2y = 0; y(0) = 1, y'(0) = y''(0) = 2 (7 marks, 2024-25)
  10. Q2c. Test the convergence of the following series: ∑ [1·3·5…(2n−1) / 2·4·6…(2n)] x^(2n), n=1 to ∞ (7 marks, 2024-25)
  11. Q2d. If f(z) = u + iv is analytic, and u − v = (e^y − cos x + sin x) / (cosh y − cos x), find f(z) such that f(π/2) = (3−i)/2. (7 marks, 2024-25)
  12. Q2e. Evaluate the following integral using contour integration: ∫_c (12z−7) / [(z−1)²(2z+3)] dz where c is the circle |z| = 2. (7 marks, 2024-25)
  13. Q3a. Find the general solution of the differential equation: x² d²y/dx² − x dy/dx + 4y = x sin(log x). (7 marks, 2024-25)
  14. Q3b. Solve the following set of simultaneous linear differential equations: dx/dt = 3x + 8y; dy/dt = −x − 3y (7 marks, 2024-25)
  15. Q4a. Find the Laplace Transform of the following function: ∫₀ᵗ (eᵗ sin t / t) dt. (7 marks, 2024-25)
  16. Q4b. Use convolution theorem to evaluate L⁻¹[ p² / (p²+4)(p²+9) ] (7 marks, 2024-25)
  17. Q5a. Examine the convergence of the following series: 1 + (α+1)/(β+1) + (α+1)(2α+1)/[(β+1)(2β+1)] + (α+1)(2α+1)(3α+1)/[(β+1)(2β+1)(3β+1)] + … (7 marks, 2024-25)
  18. Q5b. Obtain the Fourier series for the function f(x) = x², −π ≤ x ≤ π. Hence, or otherwise show that 1/1² + 1/2² + 1/3² + 1/4² + … = ∑(1/n²) = π²/6 (7 marks, 2024-25)
  19. Q6a. If f(z) = { x³y⁵(x+iy) / (x⁶+y¹⁰) ; z ≠ 0 and 0 ; z = 0 }, show that f(z) is not analytic at z = 0 even if Cauchy-Riemann equations are satisfied at origin. (7 marks, 2024-25)
  20. Q6b. Show that f(z) = z|z| is nowhere analytic. (7 marks, 2024-25)
  21. Q7a. State Cauchy's Integral Theorem. Verify Cauchy's theorem for f(z) = e^(iz) integrated along the boundary of the rectangle 1−i, 1+i, −1+i, −1−i in counterclockwise direction. (7 marks, 2024-25)
  22. Q7b. Use contour integral to evaluate: ∫₀^(2π) cos 2θ / (5 + 4cosθ) dθ (7 marks, 2024-25)

AKTU paper codes: BAS203, KAS203

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