Here are the answers to the questions based on the image:

27. (d) None of the above

28. (d) $A = \begin{bmatrix} 1 & 3 \\ 2 & 4 \end{bmatrix}$

29. (d) Both A and B are null sets

30. (c) $R = \{(1, 2), (2, 3)\}$

31. (a) $R = \{(3, 3), (4, 4), (6, 6)\}$

32. (b) 128

33. (d) $\{(1, 3), (1, 5), (3, 5)\}$

34. (d) None of the above

35. (b) Circular relation

36. (c) $K(T(x) + T(y)) = T(Kx + Ky)$ for all $k \in \mathbb{R}$

37. (c) $f = \{(7, 8), (8, 9), (9, 9)\}$

38. (c) $x = 3, y = 9$

39. (e) None of the above

40. (a) Linearly dependent

Would you like more details or have any questions?

Relative Questions:

1. What is the process for solving a system of linear equations?
2. How do you determine if matrices are orthogonal?
3. What properties define a null set in set theory?
4. How can you identify whether a relation is reflexive, symmetric, or transitive?
5. What is the significance of linear dependence and independence in vector spaces?

Tip: Always verify the properties of sets and relations when working with abstract algebra and linear algebra problems to avoid confusion.