Mathematics-IV PYQ 2022-23 AKTU Question Paper

AKTU · BTECH · Semester 3, 4 · Mathematics-IV · Session 2022-23 · PYQ

AKTU Mathematics-IV previous year question paper 2022-23 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 2022-23

  1. Q1a. Obtain a partial differential equation that governs the family of surfaces z = (x - α)² + (y - β)². (2 marks, 2022-23)
  2. Q1a. Find partial differential equation (PDE) by eliminating a and b from z = ax + by + a² + b². (2 marks, 2022-23)
  3. Q1b. Find the complete integral of the partial differential equation z = px + qy + p/(p + q), where p = ∂z/∂x and q = ∂z/∂y. (2 marks, 2022-23)
  4. Q1b. Solve the PDE: ∂²z/∂x² - ∂²z/∂y² = 0. (2 marks, 2022-23)
  5. Q1c. Classify the partial differential equation r + 2s + (sin²x)t + q = 0. (2 marks, 2022-23)
  6. Q1c. Classify the PDE: u_xx + u_yy - u_xy = 0. (2 marks, 2022-23)
  7. Q1d. Write down the two dimensional heat equation. (2 marks, 2022-23)
  8. Q1d. Write the wave equation in two dimensions. (2 marks, 2022-23)
  9. Q1e. What is the relation between the regression coefficients and the coefficient of correlation? (2 marks, 2022-23)
  10. Q1e. Find the arithmetic mean of the following frequency distribution: x: 1 2 3 4 5 6 7, f: 5 9 12 17 14 10 6. (2 marks, 2022-23)
  11. Q1f. The fourth central moment is 48. What must be its standard deviation in order that the distribution be mesokurtic? (2 marks, 2022-23)
  12. Q1f. Write the formula of Karl Pearson correlation coefficient and write the range of correlation coefficient. (2 marks, 2022-23)
  13. Q1g. A and B are any two independent events such that P(A) = 0.4, P(A∪Bᶜ) = 0.7. Find P(B), where Bᶜ is the complementary event of B. (2 marks, 2022-23)
  14. Q1g. If P(A) = 1/4, P(B) = 1/2, P(A∪B) = 5/8, then find the value of P(A∩B). (2 marks, 2022-23)
  15. Q1h. The random variable X follows the Normal distribution with mean 9 and standard deviation 3. Find x* such that P(X > x*) = 0.16. (2 marks, 2022-23)
  16. Q1h. Write probability mass function of binomial distribution with mean and variance of the distribution. (2 marks, 2022-23)
  17. Q1i. Write down the definition of the null hypothesis. (2 marks, 2022-23)
  18. Q1i. Define Null Hypothesis. (2 marks, 2022-23)
  19. Q1j. What is Statistical Quality Control (SQC)? Define in brief. (2 marks, 2022-23)
  20. Q1j. Discuss (in brief) Control Charts. (2 marks, 2022-23)
  21. Q2a. Find the general solution of the partial differential equation (y + z)p + (z + x)q = (x + y). (10 marks, 2022-23)
  22. Q2a. Solve (x²D² - y²D'²)z = xy, where D² = ∂²/∂x², D'² = ∂²/∂y². (10 marks, 2022-23)
  23. Q2b. Solve the partial differential equation by the method of separation of variables: 4(∂u/∂x) + (∂u/∂y) = 3u, given that u = 5e⁻ʸ - e⁻⁵ʸ when x = 0. (10 marks, 2022-23)
  24. Q2b. A string is stretched and fastened to two points l meter 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)… (10 marks, 2022-23)
  25. Q2c. Use the method of least squares to fit the curve y = abˣ for the following data: x: 2 3 4 5 6, y: 144 172.8 207.4 248.8 298.5. (10 marks, 2022-23)
  26. Q2c. Fit a parabolic curve of second degree to the following data: X: 0 1 2 3 4, Y: 1 1.8 1.3 2.5 6.3. (10 marks, 2022-23)
  27. Q2d. In a normal distribution, 12% of the items are under 30 and 85% items are under 60. Find the mean and standard deviation. (10 marks, 2022-23)
  28. Q2d. A bag contains 10 white and 15 black balls. If two balls are drawn in succession without replacement, find the probability that the first ball is white and the second ball is black. (10 marks, 2022-23)
  29. Q2e. The annual rainfall in Lucknow city is normally distributed with mean 45 cm. The rainfall during the last five years are 48, 42, 40, 44 and 43 cm respectively. Can we conclude that the average rainfall during the last five years is less than the normal rainfall? Test at 5% level of significance.… (10 marks, 2022-23)
  30. Q2e. The score of 10 candidates obtained in tests before and after attending coaching classes: Before: 54 76 92 65 75 78 66 82 80 78, After: 60 80 86 72 80 72 66 88 82 73. Is the coaching for the test effective? Test at 5% level of significance. (10 marks, 2022-23)
  31. Q3a. Find the solution of the partial differential equation [2D² + 5DD' + 3(D')²]z = eˣ. (10 marks, 2022-23)
  32. Q3a. Solve: (mz - ny)p + (nx - lz)q = ly - mx. (10 marks, 2022-23)
  33. Q3b. Find the complete integral of the partial differential equation (p² + q²)x = pz. (10 marks, 2022-23)
  34. Q3b. By Charpit's method, find the complete solution of PDE: px + qy - pq = 0. (10 marks, 2022-23)
  35. Q4a. A tightly stretched string with fixed end points x=0 and x=l is initially in a position given by y = a sin³(πx/l). If it is released from rest from this position, find the displacement. (10 marks, 2022-23)
  36. Q4a. Solve by the method of separation of variables, the heat equation u_t = u_xx, 0 < x < 1, t > 0 subject to conditions u(x,0) = x - x², u(0,t) = u(1,t) = 0. (10 marks, 2022-23)
  37. Q4b. An insulated rod of length l has its ends A and B maintained at 0°C and 100°C respectively until steady state conditions prevail. If B is suddenly reduced to 0°C and maintained at 0°C, find the temperature at a distance x from A at time t. (10 marks, 2022-23)
  38. Q4b. Solve the Laplace equation u_xx + u_yy = 0, x∈(0,1), y∈(0,1) with conditions u(x,0) = u(x,1) = 0, u(0,y) = 0, u(1,y) = f(y) by method of separation of variables. (10 marks, 2022-23)
  39. Q5a. If 4x - 5y + 33 = 0 and 20x - 9y = 107 are two lines of regression, find the mean values of x and y, the coefficient of correlation and the standard deviation of y if the variance of x is 9. (10 marks, 2022-23)
  40. Q5a. Calculate the correlation coefficient for the following heights (in inches) of fathers (X) and their sons (Y): X: 65 66 67 67 68 69 70 72, Y: 67 68 65 68 72 72 69 71. (10 marks, 2022-23)

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

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