Comprehensive Examples

Applied Mathematics & Experimental Design: Complete Question Bank

Questions organized by Syllabus Units for Semester End Examination preparation.

Unit-I: Equations (Linear, Quadratic, and Simultaneous)

1. Solve using factorization: x² − 7x + 10 = 0
2. Solve using factorization: x² − 9x + 20 = 0
3. Solve using the quadratic formula: 2x² + 5x − 3 = 0
4. Solve using the quadratic formula: 3x² − 5x − 2 = 0
5. Solve graphically: x + y = 4 and x − y = 2
6. Solve graphically: 2x + y = 5 and x − y = 1
7. Use Remainder Theorem to find the remainder when x³ + 2x² − 3x + 4 is divided by (x − 1).
8. Find the remainder when x³ − x² + 2x − 1 is divided by (x + 1).
9. Solve by cross-multiplication method: 3x − 2y = 4 and 2x + y = 5
10. Solve by elimination method: 4x + y = 9 and 2x − y = 3
11. Solve the system: x + 2y = 6 and 3x − y = 5
12. Solve the system: 3x + 2y = 11 and x + y = 5

Unit-II: Linear Differential Equations

13. Solve the differential equation: (D² − 5D + 6)y = 0
14. Solve the differential equation: (D² − 4D + 4)y = 0
15. Solve the differential equation: (D² + 4D + 4)y = e⁻²ˣ
16. Solve: d²y/dx² - 3 dy/dx + 2y = 0
17. Solve: d²y/dx² + 4 dy/dx + 4y = 0
18. Solve: d²y/dx² + 3 dy/dx = 0
19. Solve the variable coefficient equation: x² d²y/dx² - x dy/dx + y = 0
20. Solve the equation: (D² + 2D)y = 0

Unit-III: Laplace Transformations

21. Find the Laplace Transform of f(t) = t e^2t
22. Find the Laplace Transform of e^-2t cos 3t
23. Find the Laplace Transform of cos 2t
24. Find the inverse Laplace Transform of: 1 / [s(s + 2)]
25. Find the inverse Laplace Transform of: 1 / [(s + 1)(s + 2)]

Unit-IV: Frequency Distribution & Central Tendency

26. Calculate mean and standard deviation for: xᵢ (5, 10, 15, 20) with fᵢ (4, 6, 8, 2).
27. Find mean and variance for: xᵢ (4, 8, 12, 16) with fᵢ (5, 7, 6, 2).
28. Draw an ogive and find the Median wage and workers earning > ₹550 from a given wage distribution.
29. Draw an ogive and estimate the median wage from the distribution: 300–350 (4), 350–400 (10), 400–450 (12), 450–500 (8).
30. Find the mean using the short-cut method for Weight (60, 62, 64, 66) and Frequency (5, 9, 7, 4).
31. Find the mean using the short-cut method for Marks (40, 50, 60, 70) and Frequency (6, 10, 8, 6).
32. Draw a histogram for the class intervals 10–20, 20–30, 30–40, 40–50 with frequencies 6, 10, 8, 6.
33. Draw a histogram for the class intervals 0–10, 10–20, 20–30, 30–40 with frequencies 5, 9, 11, 5.
34. Find mean deviation about the mean for xᵢ (2, 4, 6, 8) and fᵢ (3, 7, 5, 5).
35. Find median and mode for values 12, 14, 16, 18 with frequencies 5, 9, 11, 5.
36. Find the missing frequencies f1 and f2 for a distribution with mean 1.46 and total frequency 200.
37. Find Q₁, Q₃ and inter-quartile range for the data: 8, 12, 15, 17, 19, 21, 25.
38. Find quartiles for the data set: 5, 9, 12, 15, 18, 21.
39. Construct a frequency distribution table with class size 5 for the data: 12, 15, 18, 22, 25, 27, 29, 31, 33, 35.
40. Construct a frequency table with class size 4 for the data: 6, 8, 10, 12, 14, 16, 18, 20.

Unit-V: Factorial Experiments & Regression

41. Explain the basic steps involved in regression analysis.
42. Explain regression analysis with a suitable example.
43. Explain YATE’S algorithm used in factorial experiments.
44. Explain factorial experiments involving two factors.