Mathematics-IV PYQ 2021-22 AKTU Question Paper

AKTU · BTECH · Semester 3, 4 · Mathematics-IV · Session 2021-22 · PYQ

AKTU Mathematics-IV previous year question paper 2021-22 for B.Tech Semester 3/4. Covers Partial Differential Equations, Applications of PDE and Fourier…

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Mathematics-IV AKTU syllabus

  1. Unit 1: Partial Differential Equations
  2. Unit 2: Applications of PDE and Fourier Transform
  3. Unit 3: Statistical Techniques I
  4. Unit 4: Statistical Techniques II
  5. Unit 5: Statistical Techniques III

Questions in Mathematics-IV AKTU PYQ 2021-22

  1. Q1a. Solve the partial differential equation p + q = 1. (2 marks, 2021-22)
  2. Q1a. Solve the following partial differential equation (D² + DD')z = 0. (2 marks, 2021-22)
  3. Q1b. Calculate particular Integral (P.I.) of (D - 3D' + 2)z = e^(x+2y). (2 marks, 2021-22)
  4. Q1b. Derive a partial differential equation by eliminating the constants a and b from z = ax + a²y² + b. (2 marks, 2021-22)
  5. Q1c. Tell the classification of the following partial differential equation: 5(∂²u/∂x²) - 9(∂²u/∂x∂t) + 4(∂²u/∂t²) = 0. (2 marks, 2021-22)
  6. Q1c. Write radio wave equations. (2 marks, 2021-22)
  7. Q1d. Write down the two-dimensional wave equation. (2 marks, 2021-22)
  8. Q1d. Classify the partial differential equation u_xx + 3u_xy + u_yy = 0. (2 marks, 2021-22)
  9. Q1e. Calculate the moment generating function of the negative exponential function f(x) = λe^(-λx); x, λ > 0. (2 marks, 2021-22)
  10. Q1e. In an asymmetrical distribution mean is 16 and median is 20. Calculate the mode of the distribution. (2 marks, 2021-22)
  11. Q1f. If Regression Coefficients are 0.8 and 0.8, what would be the value of coefficient of correlation? (2 marks, 2021-22)
  12. Q1f. The lines of regression of y on x and x on y are respectively y = x + 5 and 16x - 9y = 94. Find the correlation coefficient. (2 marks, 2021-22)
  13. Q1g. A die is tossed twice. A success is getting 2 or 3 on a toss. Calculate mean. (2 marks, 2021-22)
  14. Q1g. Four persons are chosen at random from a group containing 3 men, 2 women and 4 children. Prove that the chance that exactly two of them will be children is 10/21. (2 marks, 2021-22)
  15. Q1h. Write Statement of Baye's theorem. (2 marks, 2021-22)
  16. Q1h. If the probability density function f(x) = kx³, 0 ≤ x ≤ 3; 0 elsewhere, find the value of k. Also find the probability between x = 1/2 and x = 3/2. (2 marks, 2021-22)
  17. Q1i. When we use F-test? (2 marks, 2021-22)
  18. Q1i. Explain t-test for small samples. (2 marks, 2021-22)
  19. Q1j. Explain one-way ANOVA classification. (2 marks, 2021-22)
  20. Q1j. What do you mean by statistical quality control (SQC)? (2 marks, 2021-22)
  21. Q2a. Solve the following partial differential equation by Charpit Method: px + qy = pq. (10 marks, 2021-22)
  22. Q2a. Solve the partial differential equation (D - D' - 1)(D - D' - 2)z = sin(2x + 3y). (10 marks, 2021-22)
  23. Q2b. Determine the solution of one dimensional heat equation ∂u/∂t = ∂²u/∂x² where the boundary conditions are u(0,t) = 0, u(l,t) = 0 (t > 0) and the initial condition u(x,0) = 3sin(πx/l), l being the length of the bar. (10 marks, 2021-22)
  24. Q2b. A laterally insulated bar of length l has its ends A and B maintained at 0°C and 100°C respectively until steady state conditions prevail. If the temperature at B is suddenly reduced to 0°C and kept so while that of A is maintained at 0°C. Find the temperature at a distance x from A at any time t. (10 marks, 2021-22)
  25. Q2c. From the following data, determine the equations of line of regression of y on x and x on y. x: 6 2 10 4 8, y: 9 11 5 8 7. (10 marks, 2021-22)
  26. Q2c. Calculate the first four central moments about the mean of the following data: x: 0 1 2 3 4 5 6 7 8, f: 1 8 28 56 70 56 28 8 1. (10 marks, 2021-22)
  27. Q2d. In a test on 2000 electric bulbs, the life was normally distributed with average life of 2040 hours and S.D of 60 hours. Calculate the number of bulbs likely to burn for: (i) More than 2150 hours, (ii) less than 1950 hours, (iii) between 1920 hours and 2160 hours. (10 marks, 2021-22)
  28. Q2d. In a sample of 1000 cases, the mean of a certain test is 14 and S.D is 2.5. Assuming the distribution to be normal, find: (i) How many students score between 12 and 15? (ii) How many score above 18? (iii) How many score below 8? Given f(0.8)=0.2881, f(0.4)=0.1554, f(1.6)=0.4452, f(2.4)=0.4918. (10 marks, 2021-22)
  29. Q2e. The 9 items of a sample have the following values: 45, 47, 50, 52, 48, 47, 49, 53, 51. Does the mean of these values differ significantly from the assumed mean 47.5? [t₀.₀₅ = 2.31 for 8 d.f.]. (10 marks, 2021-22)
  30. Q2e. In an experiment on immunization of cattle from tuberculosis the following results were obtained: Inoculated: Affected 12, Unaffected 28; Not Inoculated: Affected 13, Unaffected 7. Examine the effect of vaccine in controlling the incidence of the disease. [χ²₀.₀₅,₁ = 3.84]. (10 marks, 2021-22)
  31. Q3a. Solve the partial differential equation x²(∂²z/∂x²) - y²(∂²z/∂y²) = xy. (10 marks, 2021-22)
  32. Q3a. Solve (y + zx)p - (x + yz)q = x² - y². (10 marks, 2021-22)
  33. Q3b. Use Cauchy's method of characteristics to solve the first order PDE: u_x + u_y = 1 + cos y, u(0,y) = sin y. (10 marks, 2021-22)
  34. Q3b. Solve (x²D² - 4xyDD' + 4y²D'²)z = x³y⁴. (10 marks, 2021-22)
  35. Q4a. Solve the following PDE by method of separation of variables: ∂u/∂t - ∂u/∂x + 2u = 0, u(x,0) = 10e^(-x) - 6e^(-4x). (10 marks, 2021-22)
  36. Q4a. Solve the following PDE by method of separation of variables: ∂z/∂x + ∂²z/∂y² = 0; z(x,0) = 0, z(x,π) = 0, z(0,y) = 4sin3y. (10 marks, 2021-22)
  37. Q4b. Determine the solution of Laplace equation ∂²u/∂x² + ∂²u/∂y² = 0 subject to the boundary conditions u(0,y) = u(l,y) = u(x,0) = 0 and u(x,a) = f(x). (10 marks, 2021-22)
  38. Q4b. A string is stretched and fastened to two points l m apart. Motion is started by displacing the string in the form u(x,0) = A sin(πx/l) from which it is released at time t=0. Show that the displacement of any point at a distance x from one end at time t is given by u(x,t) = A sin(πx/l) cos(πct/l). (10 marks, 2021-22)
  39. Q5a. Compute skewness and kurtosis, if the first four moments of a frequency distribution about the value 4 of the variable are 1, 4, 10 and 45. (10 marks, 2021-22)
  40. Q5a. Fit a parabolic curve of regression of y on x to the following data: x: 1.0 1.5 2.0 2.5 3.0 3.5 4.0, y: 1.1 1.3 1.6 2.0 2.7 3.4 4.1. (10 marks, 2021-22)

AKTU paper codes: BAS303, BAS303H, BAS403, BAS403H, KAS302, KAS402

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